Symplectic Geometric Algorithms for Hamiltonian Systems

Kang Feng Mengzhao Qin Symplectic Geometric Algorithms for Hamiltonian Systems

Kang Feng Mengzhao Qin

Symplectic Geometric Algorithms for Hamiltonian Systems With 62 Figures

ZHEJIANG PUBLISHING UNITED GROUP ZHEJIANG SCIENCE AND TECHNOLOGY PUBLISHING HOUSE

Authors Kang Feng (1920-1993) Institute of Computational Mathematics and Scientific/ Engineering Computing Beijing 100190, China

Mengzhao Qin Institute of Computational Mathematics and Scientific/ Engineering Computing Beijing 100190, China Email: [email protected]

ISBN 978-7-5341-3595-8 Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou ISBN 978-3-642-01776-6 e-ISBN 978-3-642-01777-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009930026 ¤ Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

“. . . In the late 1980s Feng Kang proposed and developed so-called symplectic algorithms for solving equations in Hamiltonian form. Combining theoretical analysis and computer experimentation, he showed that such methods, over long times, are much superior to standard methods. At the time of his death, he was at work on extensions of this idea to other structures . . . ”

Peter Lax

Cited from SIAM News November 1993

Kang Feng giving a talk at an international conference

“ A basic idea behind the design of numerical schemes is that they can preserve the properties of the original problems as much as possible . . . Different representations for the same physical law can lead to different computational techniques in solving the same problem, which can produce different numerical results . . .” Kang Feng (1920 – 1993) Cited from a paper entitled “How to compute property Newton’s equation of motion”

Prize certificate

Author’s photograph taken in Xi’an in 1989

Foreword

Kang Feng (1920–1993), Member of the Chinese Academy of Sciences, Professor and Honorary Director of the Computing Center of the Chinese Academy of Sciences, famous applied mathematician, founder and pioneer of computational mathematics and scientific computing in China. It has been 16 years since my brother Kang Feng passed away. His scientific achievements have been recognized more and more clearly over time, and his contributions to various fields have become increasingly outstanding. In the spring of 1997, Professor Shing-Tung Yau, a winner of the Fields Medal and a foreign member of the Chinese Academy of Sciences, mentioned in a presentation at Tsinghua University, entitled “The development of mathematics in China in my view”, that “there are three main reasons for Chinese modern mathematics to go beyond or hand in hand with the West. Of course, I am not saying that there are no other works, but I mainly talk about the mathematics that is well known historically: Professor Shiingshen Chern’s work on characteristic class, Luogeng Hua’s work on the theory of functions of several complex variables, and Kang Feng’s work on finite elements.” This high evaluation of Kang Feng as a mathematician (not just a computational mathematician) sounds so refreshing that many people talked about it and strongly agreed with it. At the end of 1997, the Chinese National Natural Science Foundation presented Kang Feng et al. with the first class prize for his other work on a symplectic algorithm for Hamiltonian systems, which is a further recognition of his scientific achievements (see the certificate on the previous page). As his brother, I am very pleased. Achieving a major scientific breakthrough is a rare event. It requires vision, ability and opportunity, all of which are indispensable. Kang Feng has achieved two major scientific breakthroughs in his life, both of which are very valuable and worthy of mention. Firstly, from 1964 to 1965, he proposed independently the finite element method and laid the foundation for the mathematical theory. Secondly, in 1984, he proposed a symplectic algorithm for Hamiltonian systems. At present, scientific innovation has become the focus of discussion. Kang Feng’s two scientific breakthroughs may be treated as case studies in scientific innovation. It is worth emphasizing that these breakthroughs were achieved in China by Chinese scientists. Careful study of these has yet to be carried out by experts. Here I just describe some of my personal feelings. It should be noted that these breakthroughs resulted not only from the profound mathematical knowledge of Kang Feng, but also from his expertise in classical physics and engineering technology that were closely related to the projects. Scientific breakthroughs are often cross-disciplinary. In addition, there is often a long period of time before a breakthrough is made-not unlike a long time it takes for a baby to be born, which requires the accumulation of results in small steps.

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The opportunity for inventing the finite element method came from a national research project, a computational problem in the design of the Liu Jia Xia dam. For such a concrete problem, Kang Feng found a basis for solving of the problem using his sharp insight. In his view, a discrete computing method for a mathematical and physical problem is usually carried out in four steps. Firstly, one needs to know and define the physical mechanism. Secondly, one writes the appropriate differential equations accordingly. In the third step, design a discrete model. Finally, one develops the numerical algorithm. However, due to the complexity of the geometry and physical conditions, conventional methods cannot always be effective. Nonetheless, starting from the physical law of conservation or variational principle of the matter, we can directly relate to the appropriate discrete model. Combining the variational principle with the spline approximation leads to the finite element method, which has a wide range of adaptability and is particularly suited to deal with the complex geometry of the physical conditions of computational engineering problems. In 1965, Kang Feng published his paper entitled “Difference schemes based on the variational principle”, which solved the basic theoretical issues of the finite element method, such as convergence, error estimation, and stability. It laid the mathematical foundation for the finite element method. This paper is the main evidence for recognition by the international academic community of our independent development of the finite element method. After the Chinese Cultural Revolution, he continued his research in finite element and related areas. During this period, he made several great achievements. I remember that he talked with me about other issues, such as Thom’s catastrophe theory, Prigogine’s theory of dissipative structures, solitons in water waves, the Radon transform, and so on. These problems are related to physics and engineering technology. Clearly he was exploring for new areas and seeking a breakthrough. In the 1970s, Arnold’s “Mathematical Method of Classical Mechanics” came out. It described the symplectic structure for Hamiltonian equations, which proved to be a great inspiration to him and led to a breakthrough. Through his long-term experience in mathematical computation, he fully realized that different mathematical expressions for the same physical law, which are physically equivalent, can perform different functions in scientific computing (his students later called this the “Feng’s major theorem”). In this way, for classical mechanics, Newton’s equations, Lagrangian equations and Hamiltonian equations will show a different pattern of calculations after discretization. Because the Hamiltonian formulation has a symplectic structure, he was keenly aware that, if the algorithm can maintain the geometric symmetry of symplecticity, it will be possible to avoid the flaw of artificial dissipation of this type of algorithm and design a high-fidelity algorithm. Thus, he opened up a broad way for the computational method of the Hamiltonian system. He called this way the “Hamiltonian way”. This computational method has been used in the calculation of the orbit in celestial mechanics, in calculations for the particle path in accelerator, as well as in molecular dynamics. Later, the scope of its application was expanded. For example, it has also been widely used in studies of the atmosphere and earth sciences and elsewhere. It

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has been effectively applied in solving the GPS observation operator, indicating that Global Positioning System data can be dealt with in a timely manner. This algorithm is 400 times more efficient than the traditional method. In addition, a symplectic algorithm has been successfully used in the oil and gas exploration fields. Under the influence of Kang Feng, international research on symplectic algorithm has become popular and flourishing, nearly 300 papers have been published in this field to date. Kang Feng’s research work on the symplectic algorithm has been well-known and recognized internationally for its unique, innovative, systematic and widespread properties, for its theoretical integrity and fruitful results. J. Lions, the former President of the International Mathematics Union, spoke at a workshop when celebrating his 60th birthday: “This is another major innovation made by Kang Feng, independent of the West, after the finite element method.” In 1993 one of the world’s leading mathematicians, P.D. Lax, a member of the American Academy of Sciences, wrote a memorial article dedicated to Kang Feng in SIAM News, stating that “In the late 1980s, Kang Feng proposed and developed so-called symplectic algorithms for solving evolution equations . . .. Such methods, over a long period, are much superior to standard methods.” E. J. Marsden, an internationlly wellknown applied mathematician, visited the computing institute in the late 1980s and had a long conversation with Kang Feng. Soon after the death of Kang Feng, he proposed the multi-symplectic algorithm and extended the characteristics of stability of the symplectic algorithm for long time calculation of Hamiltonian systems with infinite dimensions. On the occasion of the commemoration of the 16th anniversary of Kang Feng’s death and the 89th anniversary of his birth, I think it is especially worthwhile to praise and promote what was embodied in the lifetime’s work of Kang Feng — “ independence in spirit, freedom in thinking”. 1 Now everyone is talking about scientific innovation, which needs a talented person to accomplish. What type of person is needed most? A person who is just a parrot or who has an “independent spirit, freely thinking”? The conclusion is self-evident. Scientific innovation requires strong academic atmosphere. Is it determined by only one person or by all of the team members? This is also self-evident. From Kang Feng’s scientific career, we can easily find that the key to the problem of scientific innovation is “independence in spirit, freedom in thinking”, and that needs to be allowed to develop and expand. Kang Feng had planned to write a monograph about a symplectic algorithm for Hamiltonian systems. He had accumulated some manuscripts, but failed to complete it because he died too early due to sickness. Fortunately, his students and Professor Mengzhao Qin (see the photo on the previous page), one of the early collaborators, spent 15 years and finally completed this book based on Kang Feng’s plan, realizing his wish. It is not only an authoritative exposition of this research field, but also an 1

Yinke Chen engraved on a stele in 1929 in memory of Guowei Wang in campus of Tsinghua University.

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exposure of the academic thought of a master of science, which gives an example of how an original and innovative scientific discovery is initiated and developed from beginning to end in China. We would also like to thank Zhejiang Science and Technology Publishing House, which made a great contribution to the Chinese scientific cause through the publication of this manuscript. Although Kang Feng died 16 years ago, his scientific legacy has been inherited and developed by the younger generation of scientists. His scientific spirit and thought still elicit care, thinking and resonance in us. He is still living in the hearts of us.

Duan, Feng Member of Chinese Academy of Sciences Nanjing University Nanjing September 20, 2009

Preface It has been 16 years since Kang Feng passed away. It is our honor to publish the English version of Symplectic Algorithm for Hamiltonian Systems, so that more readers can see the history of the development of symplectic algorithms. In particular, after the death of Kang Feng, the development of symplectic algorithms became more sophisticated and there have been a series of monographs published in this area, e.g., Sanz-Serna & M.P. Calvo’s Numerical Hamiltonian Problems published in 1994 by Chapman and Hall Publishing House; E. Hairer, C. Lubich and G. Wanner’s Geometrical Numerical Integration published in 2001 by Springer Verlag; B. Leimkuhler and S. Reich’s Simulating Hamiltonian Dynamics published in 2004 by Cambridge University Press. The symplectic algorithm has been developed from ordinary differential equations to partial differential equations, from a symplectic structure to a multi-symplectic structure. This is largely due to the promotion of this work by J. Marsden of the USA and T. Bridge and others in Britain. Starting with a symplectic structure, J. Marsden first developed the Lagrange symplectic structure, and then to the multi-symplectic structure. He finally proposed a symplectic structure that meets the requirement of the Lagrangian form from the variational principle by giving up the boundary conditions. On the other hand, T. Bridge and others used the multisymplectic structure to derive directly the multi-symplectic Hamilton equations, and then constructed the difference schemes that preserve the symplectic structure in both time and space. Both methods can be regarded as equivalent in the algorithmic sense. Now, in this monograph, most of the content refers only to ordinary differential equations. Kang Feng and his algorithms research group working on the symplectic algorithm did some foundation work. In particular, I would like to point out three negative theorems: “ non-existence of energy preserving scheme”, “ non-existence of multistep linear symplectic scheme”, and “ non-existence of volume-preserving scheme form rational fraction expression”. In addition, generating function theory is not only rich in analytical mechanics and Hamilton–Jacobi equations. At the same time, the construction of symplectic schemes provides a tool for any order accuracy difference scheme. The formal power series proposed by Kang Feng had a profound impact on the later developed “ backward error series” work ,“ modified equation” and “ modified integrator”. The symplectic algorithm developed very quickly, soon to be extended to the geometric method. The structure preserving algorithm (not only preserving the geometrical structure, but also the physical structure, etc.) preserves the algebraic structure to present the Lie group algorithm, and preserves the differential complex algorithm. Many other prominent people have contributed to the symplectic method in addition to those mentioned above. There are various methods related to structure preserving algorithms and for important contributions the readers are referred to R. McLachlan & GRW Quispel “ Geometric integration for ODEs” and T. Bridges & S. Reich “ Numerical methods for Hamiltonian PDEs”. The book describes the symplectic geometric algorithms and theoretical basis for a number of related algorithms. Most of the contents are a collection of lectures given

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by Kang Feng at Beijing University. Most of other sections are a collection of papers which were written by group members. Compared to the previous Chinese version, the present English one has been improved in the following respects. First of all, to correct a number of errors and mistakes contained in the Chinese version. Besides, parts of Chapter 1 and Chapter 2 were removed, while some new content was added to Chapter 4, Chapter 7, Chapter 8, Chapter 9 and Chapter 10. More importantly, four new chapters — Chapter 13 to Chapter 16 were added. Chapter 13 is devoted to the KAM theorem for the symplectic algorithm. We invited Professor Zaijiu Shang , a former PhD student of Kang Feng to compose this chapter. Chapter 14 is called Variational Integrator. This chapter reflects the work of the Nobel Prize winner Professor Zhengdao Li who proposed in the 1980s to preserve the energy variational integrator, but had not explained at that time that it had a Lagrange symplectic type, which satisfied the Lagrange symplectic structure. Together with J. Marsden he proposed the variational integrator trail connection, which leads from the variational integrator. Just like J. Marsden, he hoped this can link up with the finite element method. Chapter 15 is about Birkhoffian Systems, describing a class of dissipative structures for Birkohoffian systems to preserve the dissipation of the Birkhoff structure. Chapter 16 is devoted to Multisymplectic and Variational Integrators, providing a summary of the widespread applications of multisymplectic integrators in the infinitely dimensional Hamiltonian systems. We would also like to thank every member of the Kang Feng’s research group for symplectic algorithms: Huamo Wu, Daoliu Wang, Zaijiu Shang, Yifa Tang, Jialin Hong, Wangyao Li, Min Chen, Shuanghu Wang, Pingfu Zhao, Jingbo Chen, Yushun Wang, Yajuan Sun, Hongwei Li, Jianqiang Sun, Tingting Liu, Hongling Su, Yimin Tian; and those who have been to the USA: Zhong Ge, Chunwang Li, Yuhua Wu, Meiqing Zhang, Wenjie Zhu, Shengtai Li, Lixin Jiang, and Haibin Shu. They made contributions to the symplectic algorithm over different periods of time. The authors would also like to thank the National Natural Science Foundation, the National Climbing Program projects, and the State’s Key Basic Research Projects for their financial support. Finally, the authors would also like to thank the Mathematics and Systems Science Research Institute of the Chinese Academy of Sciences, the Computational Mathematics and Computational Science and Engineering Institute, and the State Key Laboratory of Computational Science and Engineering for their support. The editors of this book have received help from E. Hairer, who provided a template from Springer publishing house. I would also like to thank F. Holzwarth at Springer publishing house and Linbo Zhang of our institute, and others who helped me successfully publish this book. For the English translation, I thank Dr. Shengtai Li for comprehensive proofreading and polishing, and the editing of Miss Yi Jin. For the English version of the publication I would also like to thank the help of the Chinese Academy of Sciences Institute of Mathematics. Because Kang Feng has passed away, it may not be possible to provide a comprehensive representation of his academic thought, and the book will inevitably contain some errors. I accept the responsibility for any errors and welcome criticism and corrections.

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We would also like to thank Springer Beijing Representation Office and Zhejiang Science and Technology Publishing House, which made a great contribution to the Chinese scientific cause through the publication of this manuscript. We are especially grateful to thank Lisa Fan, W. Y. Zhou, L. L. Liu and X. M. Lu for carefully reading and finding some misprints, wrong signs and other mistakes. This book is supported by National Natural Science Foundation of China under grant No.G10871099 ; supported by the Project of National 863 Plan of China (grant No.2006AA09A102-08); and supported by the National Basic Research Program of China (973 Program) (Grant No. 2007CB209603).

Mengzhao Qin Institute of Computational Mathematics and Scientific Engineering Computing Beijing September 20, 2009

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.

Preliminaries of Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.1.1 Differentiable Manifolds and Differentiable Mapping . . . . 40 1.1.2 Tangent Space and Differentials . . . . . . . . . . . . . . . . . . . . . . 43 1.1.3 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.1.4 Submersion and Transversal . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.2 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1.2.1 Tangent Bundle and Orientation . . . . . . . . . . . . . . . . . . . . . . 56 1.2.2 Vector Field and Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.3 Exterior Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.3.1 Exterior Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 1.3.2 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1.4 Foundation of Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1.4.1 Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 1.4.2 The Behavior of Differential Forms under Maps . . . . . . . . 80 1.4.3 Exterior Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 1.4.4 Poincar´e Lemma and Its Inverse Lemma . . . . . . . . . . . . . . . 84 1.4.5 Differential Form in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 1.4.6 Hodge Duality and Star Operators . . . . . . . . . . . . . . . . . . . . 88 1.4.7 Codifferential Operator δ . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 1.4.8 Laplace–Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 90 1.5 Integration on a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1.5.1 Geometrical Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1.5.2 Integration and Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . 93 1.5.3 Some Classical Theories on Vector Analysis . . . . . . . . . . . 96 1.6 Cohomology and Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 1.7 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 1.7.1 Vector Fields as Differential Operator . . . . . . . . . . . . . . . . . 99 1.7.2 Flows of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 1.7.3 Lie Derivative and Contraction . . . . . . . . . . . . . . . . . . . . . . . 103 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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Symplectic Algebra and Geometry Preliminaries . . . . . . . . . . . . . . . . . . 113 2.1 Symplectic Algebra and Orthogonal Algebra . . . . . . . . . . . . . . . . . . . 113 2.1.1 Bilinear Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.1.2 Sesquilinear Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.1.3 Scalar Product, Hermitian Product . . . . . . . . . . . . . . . . . . . . 117 2.1.4 Invariant Groups for Scalar Products . . . . . . . . . . . . . . . . . . 119 2.1.5 Real Representation of Complex Vector Space . . . . . . . . . 121 2.1.6 Complexification of Real Vector Space and Real Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.1.7 Lie Algebra for GL(n, F) . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.2 Canonical Reductions of Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . 128 2.2.1 Congruent Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.2.2 Congruence Canonical Forms of Conformally Symmetric and Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 130 2.2.3 Similar Reduction to Canonical Forms under Orthogonal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.3 Symplectic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 2.3.1 Symplectic Space and Its Subspace . . . . . . . . . . . . . . . . . . . 137 2.3.2 Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.3.3 Lagrangian Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 2.3.4 Special Types of Sp(2n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 2.3.5 Generators of Sp(2n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2.3.6 Eigenvalues of Symplectic and Infinitesimal Matrices . . . 158 2.3.7 Generating Functions for Lagrangian Subspaces . . . . . . . . 160 2.3.8 Generalized Lagrangian Subspaces . . . . . . . . . . . . . . . . . . . 162 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

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Hamiltonian Mechanics and Symplectic Geometry . . . . . . . . . . . . . . . . . 165 3.1 Symplectic Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3.1.1 Symplectic Structure on Manifolds . . . . . . . . . . . . . . . . . . . 165 3.1.2 Standard Symplectic Structure on Cotangent Bundles . . . . 166 3.1.3 Hamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 167 3.1.4 Darboux Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.2 Hamiltonian Mechanics on R2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.2.1 Phase Space on R2n and Canonical Systems . . . . . . . . . . . 169 3.2.2 Canonical Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.2.3 Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.2.4 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3.2.5 Hamilton–Jacobi Equations . . . . . . . . . . . . . . . . . . . . . . . . . 182 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

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Symplectic Difference Schemes for Hamiltonian Systems . . . . . . . . . . . . 187 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.1.1 Element and Notation for Hamiltonian Mechanics . . . . . . 187

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4.1.2

Geometrical Meaning of Preserving Symplectic Structure ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 4.1.3 Some Properties of a Symplectic Matrix . . . . . . . . . . . . . . . 190 4.2 Symplectic Schemes for Linear Hamiltonian Systems . . . . . . . . . . . 192 4.2.1 Some Symplectic Schemes for Linear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.2.2 Symplectic Schemes Based on Pad´e Approximation . . . . . 193 4.2.3 Generalized Cayley Transformation and Its Application . . 197 4.3 Symplectic Difference Schemes for a Nonlinear Hamiltonian System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.4 Explicit Symplectic Scheme for Hamiltonian System . . . . . . . . . . . . 203 4.4.1 Systems with Nilpotent of Degree 2 . . . . . . . . . . . . . . . . . . 204 4.4.2 Symplectically Separable Hamiltonian Systems . . . . . . . . . 205 4.4.3 Separability of All Polynomials in R2n . . . . . . . . . . . . . . . 207 4.5 Energy-conservative Schemes by Hamiltonian Difference . . . . . . . . 209 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.

The Generating Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.1 Linear Fractional Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.2 Symplectic, Gradient Mapping and Generating Function . . . . . . . . 215 5.3 Generating Functions for the Phase Flow . . . . . . . . . . . . . . . . . . . . . 221 5.4 Construction of Canonical Difference Schemes . . . . . . . . . . . . . . . . . 226 5.5 Further Remarks on Generating Function . . . . . . . . . . . . . . . . . . . . . . 231 5.6 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5.7 Convergence of Symplectic Difference Schemes . . . . . . . . . . . . . . . . 239 5.8 Symplectic Schemes for Nonautonomous System . . . . . . . . . . . . . . . 242 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

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The Calculus of Generating Functions and Formal Energy . . . . . . . . . . 249 6.1 Darboux Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6.2 Normalization of Darboux Transformation . . . . . . . . . . . . . . . . . . . . . 251 6.3 Transform Properties of Generator Maps and Generating Functions 255 6.4 Invariance of Generating Functions and Commutativity of Generator Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.5 Formal Energy for Hamiltonian Algorithm . . . . . . . . . . . . . . . . . . . . 264 6.6 Ge–Marsden Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

7.

Symplectic Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.1 Multistage Symplectic Runge–Kutta Method . . . . . . . . . . . . . . . . . . . 277 7.1.1 Definition and Properties of Symplectic R–K Method . . . . 277 7.1.2 Symplectic Conditions for R–K Method . . . . . . . . . . . . . . . 281 7.1.3 Diagonally Implicit Symplectic R–K Method . . . . . . . . . . 284 7.1.4 Rooted Tree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7.1.5 Simplified Conditions for Symplectic R–K Method . . . . . 297

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Symplectic P–R–K Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.2.1 P–R–K Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.2.2 Symplified Order Conditions of Explicit Symplectic R–K Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 7.3 Symplectic R–K–N Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 7.3.1 Order Conditions for Symplectic R–K–N Method . . . . . . . 319 7.3.2 The 3-Stage and 4-th order Symplectic R–K–N Method . 323 7.3.3 Symplified Order Conditions for Symplectic R–K–N Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 7.4 Formal Energy for Symplectic R–K Method . . . . . . . . . . . . . . . . . . . 333 7.4.1 Modified Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 7.4.2 Formal Energy for Symplectic R–K Method . . . . . . . . . . . 339 7.5 Definition of a(t) and b(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 7.5.1 Centered Euler Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 7.5.2 Gauss–Legendre Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 7.5.3 Diagonal Implicit R–K Method . . . . . . . . . . . . . . . . . . . . . . 347 7.6 Multistep Symplectic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.6.1 Linear Multistep Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.6.2 Symplectic LMM for Linear Hamiltonian Systems . . . . . . 348 7.6.3 Rational Approximations to Exp and Log Function . . . . . . 352 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

8.

Composition Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1 Construction of Fourth Order with 3-Stage Scheme . . . . . . . . . . . . . 365 8.1.1 For Single Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.1.2 For System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 8.2 Adjoint Method and Self-Adjoint Method . . . . . . . . . . . . . . . . . . . . . 372 8.3 Construction of Higher Order Schemes . . . . . . . . . . . . . . . . . . . . . . . 377 8.4 Stability Analysis for Composition Scheme . . . . . . . . . . . . . . . . . . . . 388 8.5 Application of Composition Schemes to PDE . . . . . . . . . . . . . . . . . . 396 8.6 H-Stability of Hamiltonian System . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

9.

Formal Power Series and B-Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 9.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 9.2 Near-0 and Near-1 Formal Power Series . . . . . . . . . . . . . . . . . . . . . . . 409 9.3 Algorithmic Approximations to Phase Flows . . . . . . . . . . . . . . . . . . . 414 9.3.1 Approximations of Phase Flows and Numerical Method . 414 9.3.2 Typical Algorithm and Step Transition Map . . . . . . . . . . . . 415 9.4 Related B-Series Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 9.4.1 The Composition Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 9.4.2 Substitution Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 9.4.3 The Logarithmic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

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10.

Volume-Preserving Methods for Source-Free Systems . . . . . . . . . . . . . . 443 10.1 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 10.2 Volume-Preserving Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.2.1 Conditions for Centered Euler Method to be Volume Preserving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.2.2 Separable Systems and Volume-Preserving Explicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 10.3 Source-Free System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 10.4 Obstruction to Analytic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 10.5 Decompositions of Source-Free Vector Fields . . . . . . . . . . . . . . . . . . 452 10.6 Construction of Volume-Preserving Schemes . . . . . . . . . . . . . . . . . . . 454 10.7 Some Special Discussions for Separable Source-Free Systems . . . . 458 10.8 Construction of Volume-Preserving Scheme via Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 10.8.1 Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 10.8.2 Construction of Volume-Preserving Schemes . . . . . . . . . . . 464 10.9 Some Volume-Preserving Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 467 10.9.1 Volume-Preserving R–K Methods . . . . . . . . . . . . . . . . . . . . 467 10.9.2 Volume-Preserving 2-Stage P–R–K Methods . . . . . . . . . . . 471 10.9.3 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 10.9.4 Some Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

11.

Contact Algorithms for Contact Dynamical Systems . . . . . . . . . . . . . . . 477 11.1 Contact Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 11.1.1 Basic Concepts of Contact Geometry . . . . . . . . . . . . . . . . . 477 11.1.2 Contact Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 11.2 Contactization and Symplectization . . . . . . . . . . . . . . . . . . . . . . . . . . 484 11.3 Contact Generating Functions for Contact Maps . . . . . . . . . . . . . . . . 488 11.4 Contact Algorithms for Contact Systems . . . . . . . . . . . . . . . . . . . . . . 492 11.4.1 Q Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.4.2 P Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.4.3 C Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.5 Hamilton–Jacobi Equations for Contact Systems . . . . . . . . . . . . . . . 494 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

12.

Poisson Bracket and Lie–Poisson Schemes . . . . . . . . . . . . . . . . . . . . . . . . 499 12.1 Poisson Bracket and Lie–Poisson Systems . . . . . . . . . . . . . . . . . . . . . 499 12.1.1 Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 12.1.2 Lie–Poisson Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 12.1.3 Introduction of the Generalized Rigid Body Motion . . . . . 505 12.2 Constructing Difference Schemes for Linear Poisson Systems . . . . 507 12.2.1 Constructing Difference Schemes for Linear Poisson Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

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12.2.2

Construction of Difference Schemes for General Poisson Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 12.2.3 Answers of Some Questions . . . . . . . . . . . . . . . . . . . . . . . . . 511 12.3 Generating Function and Lie–Poisson Scheme . . . . . . . . . . . . . . . . . 514 12.3.1 Lie–Poisson–Hamilton–Jacobi (LPHJ) Equation and Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 12.3.2 Construction of Lie–Poisson Schemes via Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 12.4 Construction of Structure Preserving Schemes for Rigid Body . . . . 523 12.4.1 Rigid Body in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . 523 12.4.2 Energy-Preserving and Angular Momentum-Preserving Schemes for Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 12.4.3 Orbit-Preserving and Angular-Momentum-Preserving Explicit Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 12.4.4 Lie–Poisson Schemes for Free Rigid Body . . . . . . . . . . . . 530 12.4.5 Lie–Poisson Scheme on Heavy Top . . . . . . . . . . . . . . . . . . . 535 12.4.6 Other Lie–Poisson Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 538 12.5 Relation Among Some Special Group and Its Lie Algebra . . . . . . . . 543 12.5.1 Relation Among SO(3), so(3) and SH1 , SU (2) . . . . . . . 543 12.5.2 Representations of Some Functions in SO(3) . . . . . . . . . . 545 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 13.

KAM Theorem of Symplectic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 549 13.1 Brief Introduction to Stability of Geometric Numerical Algorithms 549 13.2 Mapping Version of the KAM Theorem . . . . . . . . . . . . . . . . . . . . . . 551 13.2.1 Formulation of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 551 13.2.2 Outline of the Proof of the Theorems . . . . . . . . . . . . . . . . . 554 13.2.3 Application to Small Twist Mappings . . . . . . . . . . . . . . . . . 558 13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems559 13.3.1 Symplectic Algorithms as Small Twist Mappings . . . . . . . 560 13.3.2 Numerical Version of KAM Theorem . . . . . . . . . . . . . . . . . 564 13.4 Resonant and Diophantine Step Sizes . . . . . . . . . . . . . . . . . . . . . . . . . 568 13.4.1 Step Size Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 13.4.2 Diophantine Step Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 13.4.3 Invariant Tori and Further Remarks . . . . . . . . . . . . . . . . . . . 574 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

14.

Lee-Variational Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 14.1 Total Variation in Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . 581 14.1.1 Variational Principle in Lagrangian Mechanics . . . . . . . . . 581 14.1.2 Total Variation for Lagrangian Mechanics . . . . . . . . . . . . . 583 14.1.3 Discrete Mechanics and Variational Integrators . . . . . . . . . 586 14.1.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 14.2 Total Variation in Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . 591 14.2.1 Variational Principle in Hamiltonian Mechanics . . . . . . . . 591

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14.2.2 Total Variation in Hamiltonian Mechanics . . . . . . . . . . . . . 593 14.2.3 Symplectic-Energy Integrators . . . . . . . . . . . . . . . . . . . . . . . 596 14.2.4 High Order Symplectic-Energy Integrator . . . . . . . . . . . . . 600 14.2.5 An Example and an Optimization Method . . . . . . . . . . . . . 603 14.2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 14.3 Discrete Mechanics Based on Finite Element Methods . . . . . . . . . . . 606 14.3.1 Discrete Mechanics Based on Linear Finite Element . . . . . 606 14.3.2 Discrete Mechanics with Lagrangian of High Order . . . . . 608 14.3.3 Time Steps as Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 14.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 15.

Structure Preserving Schemes for Birkhoff Systems . . . . . . . . . . . . . . . . 617 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 15.2 Birkhoffian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 15.3 Generating Functions for K(z, t)-Symplectic Mappings . . . . . . . . . 621 15.4 Symplectic Difference Schemes for Birkhoffian Systems . . . . . . . . . 625 15.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 15.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

16.

Multisymplectic and Variational Integrators . . . . . . . . . . . . . . . . . . . . . . 641 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 16.2 Multisymplectic Geometry and Multisymplectic Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 16.3 Multisymplectic Integrators and Composition Methods . . . . . . . . . . 646 16.4 Variational Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 16.5 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

Introduction The main theme of modern scientific computing is the numerical solution of various differential equations of mathematical physics bearing the names, such as Newton, Euler, Lagrange, Laplace, Navier–Stokes, Maxwell, Boltzmann, Einstein, Schr¨odinger, Yang-Mills, etc. At the top of the list is the most celebrated Newton’s equation of motion. The historical, theoretical and practical importance of Newton’s equation hardly needs any comment, so is the importance of the numerical solution of such equations. On the other hand, starting from Euler, right down to the present computer age, a great wealth of scientific literature on numerical methods for differential equations has been accumulated, and a great variety of algorithms, software packages and even expert systems has been developed. With the development of the modern mechanics, physics, chemistry, and biology, it is undisputed that almost all physical processes, whether they are classical, quantum, or relativistic, can be represented by an Hamiltonian system. Thus, it is important to solve the Hamiltonian system correctly.

1. Numerical Method for the Newton Equation of Motion In the spring of 1991, the first author [Fen92b] presented a plenary talk on how to compute the numerical solution of Newton classical equation accurately at the Annual Physics Conference of China in Beijing. It is well known that numerically solving so-called mathematics-physics equations has become a main topic in modern scientific computation. The Newton equation of motion is one of the most popular equations among various mathematics-physics equations. It can be formulated as a group of second-order ordinary differential equations, f = ma = m¨ x. The computational methods of the differential equations advanced slowly in the past due to the restriction of the historical conditions. However, a great progress was made since Euler, due to contributions from Adams, Runge, Kutta, and St¨omer, etc.. This is especially true since the introduction of the modern computer for which many algorithms and software packages have been developed. It is said that the three-body problem is no longer a challenging problem and can be easily computed. Nevertheless, we propose the following two questions: 1◦ Are the current numerical algorithms suitable for solving the Newton equation of motion? 2◦ How can one calculate the Newton equation of motion more accurately? It seems that nobody has ever thought about the first issue seriously, which may be the reason why the second issue has never been studied systematically. In this book, we will study mainly the fundamental but more difficult Newton equation of motion that is in conservative form. First, the conservative Newton equation has two equivalent mathematical representations: a Lagrange variation form and a Hamiltonian form. The K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

2

Introduction

latter transforms the second-order differential equations in physical space into a group of the first-order canonical equations in phase space. Different representations for the same physical law can lead to different computational techniques in solving the same problem, which can produce different numerical results. Thus making a wise and reasonable choice among various equivalent mathematical representations is extremely important in solving the problem correctly. We choose the Hamiltonian formulation as our basic form in practice based on the fact that the Hamiltonian equations have symmetric and clean form, where the physical laws of the motion can be easily represented. Secondly, the Hamiltonian formulation is more general and universal than the Newton formulation. It can cover the classical, relativistic, quantum, finite or infinite dimensional real physical processes where dissipation effect can be neglected. Therefore, the success of the numerical methods for Hamiltonian equations has broader development and application perspectives. Thus, it is very surprising that the numerical algorithms for Hamiltonian equations are almost nonexistent after we have searched various publications. This motivates us to study the problem carefully to seek the answers to the previous two questions. Our approach is to use the symplectic geometry, which is the geometry in phase space. It is based on the anti-symmetric area metric, which is in contrast to the symmetric length metrics of Euclid and Riemann geometry. The basic theorem of the classic mechanics can be described as “the dynamic evolution of all Hamiltonian systems preserves the symplectic metrics, which means it is a symplectic (canonical) transformation”. Hence the correct discretization algorithms to all the Hamiltonian systems should be symplectic transformation. Such algorithms are called symplectic (canonical) algorithms or Hamiltonian algorithms. We have intentionally analyzed and evaluated the derivation of the Hamiltonian algorithm within the symplectic structures. The fact proved that this approach is correct and fruitful. We have derived a series of symplectic algorithms, found out their properties, laid out their theoretical foundation, and tested them with extremely difficult numerical experiments. In order to compare the symplectic and non-symplectic algorithm, we proposed eight numerical experiments: harmonic oscillator, nonlinear Duffing oscillator, Huygens oscillator, Cassini oscillator, two dimensional multi-crystal and semi-crystal lattice steady flow, Lissajous image, geodesic flow on ellipsoidal surface, and Kepler motion. The numerical experiments demonstrate the superiority of the symplectic algorithm. All traditional non-symplectic algorithms fail without exception, especially in preserving global property and structural property, and long-term tracking capability, regardless of their accuracy. However, all the symplectic algorithms passed the tests with long-term stable tracking capability. These tests clearly demonstrate the superiority of the symplectic algorithms. Almost all of the traditional algorithms are non-symplectic with few exceptions. They are designed for the asymptotic stable system which has dissipation mechanism to maintain stability, whereas the Hamiltonian system does not have the asymptotic stability. Hence all these algorithms inevitably contain artificial numerical dissipation, fake attractors, and other parasitics effects of non-Hamiltonian system. All these effects lead to seriously twist and serious distortion in numerical results. They can be used in short-term transient simulation, but are not suitable and can lead to wrong

Introduction

3

conclusions for long-term tracking and global structural property research. Since the Newton equation is equivalent to Hamiltonian equation, the answer to the first question is “No”, which is quite beyond expectation. The symplectic algorithm does not have any artificial dissipation so that it can congenitally avoid all non-symplectic pollution and become a “clean” algorithm. Hamiltonian system has two types of conservation laws: one is the area invariance in phase space, i.e., Liouville–Poincar´e conservation law; the other is the motion invariance which includes energy conservation, momentum and angular momentum conservation, etc. We have proved that all symplectic algorithms have their own invariance, which has the same convergence to the original theoretical invariance as the convergence order of the numerical algorithm. We have also proved that the majority of invariant tori of the near integrable system can be preserved, which is a new formulation of the famous KAM (Kolmogorov–Arnorld–Moser) theorem[Kol54b,Kol54a,Arn63,Mos62] . All of these results demonstrate that the structure of the discrete Hamiltonian algorithm is completely parallel to the conservation law, and is very close to the original form of the Hamiltonian system. Moreover, theoretically speaking, it has infinite longterm tracking capability. Hence, a correct numerical method to solve the Newton equation is to Hamiltonize the equation first and then use the Hamiltonian algorithm. This is the answer to the second question. We will describe in more detail the KAM theory of symplectic algorithms for Hamiltonian systems in Chapter 13. In the following we present some examples to compare the symplectic algorithm and other non-symplectic algorithms in solving Newton equation of motion.

(1)

Calculation of the Harmonic oscillator’s elliptic orbit

Calculation of the Harmonic oscillator’s elliptic orbit (Fig. 0.1(a)) uses Runge–Kutta method (R–K) with a step size 0.4. The output is at 3,000 steps. It shows artificial dissipation, shrinking of the orbit. Fig. 0.1(b) shows the results using Adams method with a step size 0.2. It is anti-dissipative and the orbit is scattered out. Fig. 0.1(c) shows the results of two-step central difference (leap-frog scheme). This scheme is symplectic to linear equations. The results are obtained with a step size 0.1. It shows that the results of three stages for 10,000,000 steps: the initial 1,000 steps, the middle 1,000 steps, and the final 1,000 steps. They are completely in agreement.

(2)

The elliptic orbit for the nonlinear oscillator

Fig. 0.2(a) shows the results of two-step central-difference. This scheme is nonsymplectic for nonlinear equations. The output is for step size 0.2 and 10,000 steps. Fig. 0.2(a) shows the initial 1,000 steps and Fig. 0.2(b) shows the results between 9,000 to 10,000 steps. Both of them show the distortion of the orbit. Fig. 0.2(c) is for the second-order symplectic algorithm with 0.1 step size, 1,000 steps.

4

Introduction

Fig. 0.1.

Calculation of the Harmonic oscillator’s elliptic orbit

Fig. 0.2.

Calculation of the nonlinear oscillator’s elliptic orbit

Introduction

Fig. 0.3.

(3)

5

Calculation of the nonlinear Huygens oscillator

The oval orbit of the Huygens oscillator

Using the R–K method, the two fixed points on the horizontal axes become two fake attractors. The probability of the phase point close to the two attractors is the same. The same initial point outside the separatrix is attracted randomly either to the left or to the right. Fig. 0.3(a) shows the results with a step size 0.10000005 and 900,000 steps, which approach the left attractor. Fig. 0.3(b) shows the results with a step size 0.10000004 and 900,000 steps, which approach the right attractor. Fig. 0.3(c) shows the results of the second-order symplectic algorithm with a step size 0.1. Four typical orbits are plotted and each contains 100,000,000 steps: for every orbit first 500 steps, the middle 500 steps, and the final 500 steps. They are in complete agreement.

(4)

The dense orbit of the geodesic for the ellipsoidal surface

The dense orbit of the geodesic for the ellipsoidal surface with irrational frequency ratio. The square of frequency ratio is 5/16, step size is 0.05658, 10,000 steps. Fig.0.4(a) is for the R–K method which does not tend to dense. Fig. 0.4(b) is for the symplectic algorithm which tends to dense.

6

Introduction

Fig. 0.4.

(5)

Geodesics on ellipsoid, frequency ratio

√

5 : 4, non dense (a), dense orbit (b)

The close orbit of the geodesic for the ellipsoidal surface

The close orbit of the geodesic for the ellipsoidal surface with rational frequency ratio. The frequency ratio is 11/16, step size is 0.033427, 100,000 steps and 25 cycles. Fig.0.5(a) is for the R–K method which does not lead to the close orbit. Fig. 0.5(b) is for the symplectic algorithm which leads to the close orbit.

Fig. 0.5.

(6)

Geodesics on ellipsoid, frequency ratio 11:16, non closed (a), closed orbit (b)

The close orbit of the Keplerian motion

The close orbit of the Keplerian motion with rational frequency ratio. The frequency ratio is 11/20, step size is 0.01605, 240,000 steps and 60 cycles. Fig. 0.6(a) is for the R–K method which does not lead to the close orbit. Fig. 0.6(b)is for the symplectic method which leads to the close orbit.

Introduction

Fig. 0.6.

2.

7

Geodesics on ellipsoid, frequency ratio 11:20, non closed (a), closed orbit (b)

History of the Hamiltonian Mechanics

We first consider the three formulations of the classical mechanics. Assume a motion has n degrees of freedom. The position is denoted as q = (q1 , · · · , qn ). The potential function is V = V (q). Then we have m

∂ d2 q = − V, d t2 ∂q

which is the standard formulation of the motion. It is a group of second-order differential equations in space Rn . It is usually called the standard formulation of the classical mechanics, or Newton formulation. Euler and Lagrange introduced an action on the difference between the kinetic energy and potential energy L(q, q) ˙ = T (q) ˙ − V (q) =

1 (q, ˙ M q) ˙ − V (q). 2

Using the variational principle the above equation can be written as d ∂L ∂L − = 0, d t ∂ q˙ ∂q which is called the variational form of the classical mechanics, i.e., the Lagrange form. In the 19th century, Hamilton proposed another formulation. He used the momentum p = M q˙ and the total energy H = T + V to formulate the equation of motion as ∂H ∂H p˙ = − , q˙ = , ∂q ∂p which is called Hamiltonian canonical equations. This is a group of the first-order differential equations in 2n phase space (p1 , · · · , pn , q1 , · · · , qn ). It has simple and symmetric form.

8

Introduction

The three basic formulations of the classical mechanics have been described in almost all text-books on theoretical physics or theoretical mechanics. These different mathematical formulations describe the same physics law but provide different approaches in problem solving. Thus equivalent mathematical formulation can have different effectiveness in computational methods. We have verified this in our own simulations. The first author did extensive research on Finite Element Method (FEM) in the 1960s [Fen65] which represents a systematic algorithm for solving equilibrium problem. Physical problems of this type have two equivalent formulations: Newtonian, i.e., solving the second-order elliptic equations, and variational formulation, i.e., minimization principle in energy functional. The key to the success of FEM in both theoretical and computational methods lies in using a reasonable variational formulation as the basic principle. After that, he had attempted to apply the FEM idea to the dynamic problem of continuum media mechanics, but not yet achieved the corresponding success, which appears to be difficult to accomplish even today. Therefore, the reasonable choice for computational method of dynamic problem might be the Hamiltonian formulation. Initially it is a conjecture and requires verification from the computational experiments. We have investigated how others evaluated the Hamiltonian system in history. First we should point out that Hamilton himself proposed his theory based on the geometric optics and then extended it to mechanics that appears to be a very different field. In 1834 Hamilton said, “This set of idea and method has been applied to optics and mechanics. It seems it can be applied to other areas and developed into an independent knowledge by the mathematicians”[Ham34] . This is just his expectation, and other peers in the same generation seemed indifferent to this set of theory, which was “beautiful but useless”[Syn44] to them. Klein, a famous mathematician, while giving a high appreciation to the mathematical elegance of the theory, suspected its applicability, and said: “. . . a physicist, for his problems, can extract from these theories only very little, and an engineer nothing”[Kle26] . This claim has been proved wrong at least in physics aspect in the later history. The quantum mechanics developed in the 1920s under the framework of the Hamiltonian formulation. One of the founders of the quantum mechanics, Schr¨odinger said, “Hamiltonian principle has been the foundation for modern physics . . . If you want to solve any physics problem using the modern theory, you must represent it using the Hamiltonian formulation”[Sch44] .

3.

The Importance of the Hamiltonian System

The Hamiltonian system is one of the most important systems among all the dynamics systems. All real physical processes where the dissipation can be neglected can be formulated as Hamiltonian system. Hamiltonian system has broad applications, which include but are not limited to the structural biology, pharmacology, semiconductivity, superconductivity, plasma physics, celestial mechanics, material mechanics, and partial differential equations. The first five topics have been listed as “Grand Challenges” in Research Project of American government.

Introduction

9

The development of the physics verifies the importance of the Hamiltonian systems. Up to date, it is undisputed that all real physical processes where the dissipation can be neglected can be written as Hamiltonian formulation, whether they have finite or infinite degrees of freedom. The problem with finite degrees of freedom includes celestial and man-made satellite mechanics, rigid body, and multi-body (including the robots), geometric optics, and geometric asymptotic method (including ray-tracing approximation method in wave-equation, and WKB equation of quantum mechanics), confinement of the plasma, the design of the high speed accelerator, automatic control, etc. The problem with infinite degrees of freedom includes ideal fluid dynamics, elastic mechanics, electrical mechanics, quantum mechanics and field theory, general relativistic theory, solitons and nonlinear waves, etc. All the above examples show the ubiquitous and nature of the Hamiltonian systems. It has the advantage that different physics laws can be represented by the same mathematical formulation. Thus we have confidence to say that successful development of the numerical methods for Hamiltonian system will have extremely broad applications. We now discuss the status of the numerical method for Hamiltonian systems. Hamiltonian systems, including finite and infinite dimensions, are Ordinary Differential Equations (ODE) or Partial Differential Equations (PDE) with special form. The research on the numerical method of the differential equations started in the 18th century and produced abundant publications. However, we find that few of them discuss the numerical method specifically for Hamiltonian systems. This status is in sharp contrast with the importance of the Hamiltonian system. Therefore, it is appealing and worthy to investigate and develop numerical methods for this virgin field.

4. Technical Approach — Symplectic Geometry Method The foundation for the Hamiltonian system is symplectic geometry, which is increasingly flourishing in both theory and practice. The history of symplectic geometry can be traced back to Astronomer Hamilton in the 19th century. In order to study the Newton mechanics, he introduced generalized coordinates and generalized momentums to represent the energy of the system, which is now called Hamiltonian function now. For a system with n degrees of freedom, the n generalized coordinates and momentums are spanned into a 2n phase space. Thus the Newton mechanics becomes the geometry in phase space. In terms of the modern concept, this is a kind of symplectic geometry. Later, Jacobi, Darboux, Poincar´e, Cartan, and Weyl did a lot of research on this topic from different points of view (algebra and geometry). However, the major development of the modern symplectic geometry started with the discovery of KAM theorem (1950s to the beginning of 1960s). In the 1970s, in order to research Fourier integral operator, quantum representation of the geometry, group representation theory, classification of the critical points, Lie Algebra, etc., people did a lot of work on symplectic geometry (e.g., Arnold[Arn89] , Guillemin[GS84] , Weinstein[Wei77] , Marsden[AM78] , etc.), which promoted the development in these areas. In the 1980s, the research on total

10

Introduction

symplectic geometry emerged subsequently, such as the research on “coarse” symplectic (e.g., Gromov et al.), fix point for symplectic map (e.g., Conley, Zehnder’s Arnold conjecture), the convexity of the matrix mapping (e.g., Atiyah, Guillemin, Sternberg et al.). The research on symplectic geometry is not only extremely enriched and vital, but its application is also widely applied to different areas, such as celestial mechanics, geometric optics, plasma, the design of high speed accelerators, fluid dynamics, elastic mechanics, optimal control, etc. Weyl[Wey39] said the following in his monograph on the history of the symplectic group: “I called it complex group initially. Because this name can be confused with the complex number, I suggest using symplectic, a Greek word with the same meaning.” An undocumented law for the modern numerical method is that the discretized problem should preserve the properties of the original problem as much as possible. To achieve this goal, the discretization should be performed in the same framework as the original problem. For example, the finite element method treats the discretized and original problem in the same framework of the Sobolev space so that the basic properties of the original problem, such as symmetry, positivity, and conservativity, etc., are all preserved. This not only ensures the effectiveness and reliability in practice, but also provides a theoretical foundation. Based on the above principle, the constructed numerical methods for the Hamiltonian system should preserve the Hamiltonian structure, which we call “Hamiltonian algorithm”. The Hamiltonian algorithm must be constructed in the same framework as the Hamiltonian system. In the following, we will describe the basic mathematical framework of the Hamiltonian system and derive the Hamiltonian algorithm from the same framework. This is our approach. We will use the Euclid geometry as an analogy to describe the symplectic geometry. The structure of an Euclid space Rn lies in the bilinear, symmetric, non-degenerate inner product, (x, y) = x, Iy, I = In . Since it is non-degenerate, (x, x) is alwayspositive when x = 0. Therefore we can define the length of the vector x as ||x|| = (x, x) > 0. All the linear operators that preserve the inner product, i.e., satisfy AT IA = I, form a group O(n), called the orthogonal group, which is a typical Lie group. The corresponding Lie algebra o(n) consists of all the transformation that satisfies AT + A = AI + IA = 0, which is infinitesimal orthogonal transformation. The symplectic geometry is the geometry on the phase space. The symplectic space, i.e, the symplectic structure in phase space, lies in a bilinear, anti-symmetric, and non-degenerate inner product, [x, y] = x, Jy,

J = J2n =

 O −In

which is called the symplectic inner product. When n = 1,   x [x, y] =  1 x2

 y1  , y2

In  , O

Introduction

11

which is the area of the parallel quadrilateral with vectors x and y as edges. Generally speaking, the symplectic inner product is an area metric. Due to the anti-symmetry of the inner product, [x, x] = 0 always holds for any vector x. Thus it is impossible to derive the concept of length of a vector from the symplectic inner product. This is the fundamental difference between the symplectic geometry and Euclid geometry. All transformations that preserve the symplectic inner product form a group, called a symplectic group, Sp(2n), which is also a typical Lie group. Its corresponding Lie algebra consists of all infinitesimal symplectic transformations B, which satisfy B T J + JB = 0. We denote it as sp(2n). Since the non-degenerate anti-symmetric matrix exists only for even dimensions, the symplectic space must be of even dimensions. The phase space exactly satisfies this condition. Overall the Euclid geometry is a geometry for studying the length, while the symplectic geometry is for studying the area. The one-to-one nonlinear transformation in the symplectic geometry is called symplectic transformation, or canonical transformation. The transformation whose Jacobian is always a symplectic matrix plays a major role in the symplectic geometry. For the Hamiltonian system, if we represent a pair of n-dim vectors with a 2n-dim vector z = (p, q), the Hamiltonian equation becomes ∂H dz = J −1 . dt ∂z Under the symplectic transformation, the canonical form of the Hamiltonian equation is invariant. The basic principle of the Hamiltonian mechanics is for any Hamiltonian system. There exists a group of symplectic transformation (i.e., the phase flow) GtH1 ,t0 that depends on H and time t0 , t1 , so that z(t1 ) = GtH1 ,t0 z(t0 ), which means that GtH1 ,t0 transforms the state at t = t0 to the state at t = t1 . Therefore, all evolutions of the Hamiltonian system are also evolutions of the symplectic transformation. This is a general mathematical principle for classical mechanics. When H is independent of t, GtH1 ,−t2 = GtH1 ,−t0 , i.e., the phase flow depends only on the difference in parameters t1 − t0 . We can let GtH = Gt,0 H . One of the most important issues for the Hamiltonian system is stability. The feature of this type of problems in geometry perspective is that its solution preserves the metrics. Thus the eigenvalue is always a purely imaginary number. Therefore, we cannot use the asymptotic stability theory of Poincar´e and Liapunov. The KAM theorem must be used. This is a theory about the total stability and is the most important breakthrough for Newton mechanics. The application of the symplectic geometry to the numerical analysis was first proposed by K. Feng [Fen85] in 1984 at the international conference on differential geometry and equations held in Beijing. It is based on a basic principle of the analytical mechanics: the solution of the system is a volume-preserved transformation (i.e., symplectic transformation) with one-parameter2 on symplectic 2

Before K.Feng’s work, there existed works of de Vorgelaere[Vog56] , Ruth[Rut83] and Menyuk[Men84] .

12

Introduction

integration. Since then, new computational methods for the Hamiltonian system have been developed and we have studied the numerical method of the Hamiltonian system from this perspective. The new methods make the discretized equations preserve the symplectic structure of the original system, i.e., to restore the original principle of the discretized Hamiltonian mechanics. Its discretized phase flow can be regarded as a series of discrete symplectic transformations, which preserve a series of phase area and phase volume. In 1988, K. Feng described his research work on the symplectic algorithm during his visit to Western Europe and gained the recognition from many prominent mathematicians. His presentation on “Symplectic Geometry and Computational Hamiltonian Mechanics” has obtained consistent high praise at the workshop to celebrate the 60th birthday of famous French mathematician Lions. Lions thought that K. Feng founded the symplectic algorithm for Hamiltonian system after he developed the finite element methods independent of the efforts in the West. The prominent German numerical mathematician Stoer said, “This is a new method that has been overlooked for a long time but should not be overlooked.” We know that we can not study the Hamiltonian mechanics without the symplectic geometry. In the meantime, the computational method of the Hamiltonian mechanics doesn’t work without the symplectic difference scheme. The classical R–K method is not suitable to solve this type of problems, because it cannot preserve the long-term stability. For example, the fourth-order R–K method obtains a completely distorted result after 200,000 steps with a step size 0.1, because it is not a symplectic algorithm, but a dissipative algorithm. We will describe in more detail the theory of symplectic geometry and symplectic algebra in Chapters 1, 2 and 3.

5.

The Symplectic Schemes

Every scheme, whether it is explicit or implicit, can be treated as a mapping from this time to the next time. If this mapping is symplectic, we call it a symplectic geometric scheme, or in short, symplectic scheme. We first search the classical difference schemes. The well-known Euler midpoint scheme is a symplectic scheme z n+1 = z n + J −1 Hz

 z n+1 + z n  2

.

The symplectic scheme is usually implicit. Only for a split Hamiltonian system, we can obtain an explicit scheme in practice by alternating the explicit and implicit stages. Its accuracy is only of first order. Symmetrizing this first-order scheme yields a second-order scheme (or so-called reversible scheme). There exist multi-stage R–K symplectic schemes among the series of R–K schemes. It is proved that the 2s-order Gauss multi-stage R–K scheme is symplectic. We will give more details on these topics in Chapters 4 , 7 and 8. The theoretical analysis and a priori error analysis will be described in Chapter 6 and 9.

Introduction

13

In addition, the first author and his group constructed various symplectic schemes with arbitrary order of accuracy using the generating function theory from the analytical mechanics perspective. In the meantime, he extended the generating function theory and Hamilton–Jacobi equations by constructing all types of generating function and the corresponding Hamilton–Jacobi equations. The generating function theory and the construction of the symplectic schemes will be introduced in Chapter 5.

6. The Volume-Preserving Scheme for Source-free System Among the various dynamical systems, one of them is called source-free dynamical system, where the divergence of the vector field is zero: dx = f (x), dt

div f (x) = 0.

  ∂xn+1 The phase flow to this system is volume-preserved, i.e., det = 1. Therefore, ∂xn the numerical solution should also be volume-preserved. We know that Hamiltonian system is of even dimensions. However, the source-free system can be of either even or odd dimensions. For the system of odd dimensions, the Euler midpoint scheme may not be volume-preserved. ABC (Arnold–Beltrami– Childress) flow is one of the examples. Its vector field has the following form: x˙ = A sin x + C cos y, y˙ = B sin x + A cos z, z˙ = C sin y + B cos x, which is a source-free system and the phase flow is volume-preserved. This is a split system and constructing the volume-preserving scheme is easy. Numerical experiments show that the volume-preserving scheme can calculate the topological structure accurately, whereas the traditional schemes can not[FS95,QZ93] . We will give more details in Chapter 10.

7.

The Contact Schemes for Contact System

There exists a special type of dynamical systems with odd dimensions. They have similar symplectic structure as the systems of even dimensions. We call them contact systems. The reader can find more details in Chapter 11. Consider the contact system in R2n+1 space

14

Introduction

⎤ ⎡ ⎤ ⎡ ⎤ y1 x1 x ⎥ ⎢ ⎥ ⎢ (2n + 1) − dim vector : ⎣ y ⎦ , where x = ⎣ ... ⎦ , y = ⎣ ... ⎦ , z = (z); z xn y ⎡ ⎤ ⎡ ⎤ ⎡n ⎤ a(x, y, z) a1 b1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (2n + 1) − dim v.f. : ⎣ b(x, y, z) ⎦ , where a = ⎣ ... ⎦ , b = ⎣ ... ⎦ , c = (c). ⎡

an

c(x, y, z)

bn

A contact system can be generated from a contact Hamiltonian function K(x, y, z): dx = −Ky + Kz x = a, dt dy = Kx = b, dt dz = Ke = c, dt

Ke (x, y, z) = K(x, y, z) − (x, Ky (x, y, z)). The contact structure in R2n+1 space is defined as ⎡

⎤ dx α = xd y + d z = [0, x, 1] ⎣ d y ⎦ . dz A transformation f is called the contact transformation if it could preserve the contact structure with a pre-factor μf . A scheme which can preserve the contact structure is called contact scheme[FW94,Shu93] . The contact schemes have potential applications in the propagation of the wave front[MF81,QC00] , the applications in thermal dynamics[MNSS91,EMvdS07] , and the characteristic method for the first-order differential equations[Arn88] . The symplectic algorithm, the volume-preserving algorithm, the contact algorithm, and the Lie–Poisson algorithm are all schemes that preserve the geometry structure of the phase space. We call these methods “geometric integration for dynamic system”[FW94,LQ95a] . The geometric integration was first introduced by the first author[FW94] and has been widely accepted and used by the international scientists. The 1996 workshop on the advance of the numerical method, held in England, mentioned the importance of the structure-preserving schemes for the dynamics system. In that workshop, a series of high-order structure preserving schemes has been proposed via the multiplicative extrapolation method[QZ92,QZ94] . We have extended the explicit schemes of Yoshida[Yos90] to all self-adjoint schemes. By using the product of the schemes and their adjoint, we have constructed very high order self-adjoint schemes. The details are described in Chapter 8. Concerning the Lie–Poisson algorithm we will describe more details in Chapter 12.

Introduction

15

8. Applications of the Symplectic Algorithms for Dynamics System (1) Applications of symplectic algorithms to large time scale system Nearly all systems of celestial mechanics and dynamic astrophysics are Hamiltonian or approximately Hamiltonian with few dissipations. Such systems can be described by canonical forms of Hamiltonian systems, which has now become one of the most important research areas of dynamical system. However, due to the complicated nonlinearity of those canonical Hamiltonian systems, few analytic solutions are available. Although sometimes approximate analytic solutions in form of power series can be obtained by the perturbation method, the long time dynamics, the quantity property, and the intrinsic nonlinearity are overlooked by such solutions. Thus, the numerical methods are required to study those problems to get more accurate and quantitative numerical solutions, which not only provide the information and images on the whole phase space of the given mechanical system for further qualitative analysis, but also lead to some important results for the system. There are two ways to analyze Hamiltonian system qualitatively. One way is to get the numerical solution of the canonical Hamiltonian system directly by the numerical methods, and the other is simpler discretization process to the equation of motion, which becomes a simple mapping question which making computing easier. The later method reduces the computational effort so that it can be performed by normal computers to study the large time scale evolution of dynamical systems. Traditional numerical methods for dynamics system can be categorized into singlestep methods, e.g. the R–K method, and multi-step methods, e.g. the widely used Adams method for the first order differential equations, and Cowell methods for the second order differential equations. However, all the methods have the artificial numerical dissipations so that the corresponding total energy of the Hamiltonian system will change linearly. This will distort basic property of Hamiltonian system and lead to wrong results for a long time computation. By quantitative analysis, we know that the dissipation of the total energy will accumulate errors of the numerical trajectories of the celestial bodies. The errors will increase at least squarely with respect to the integration time step. In the 1980s, the first author and his group established the theory of the symplectic algorithms for Hamiltonian system. The significance of this theory is not only to present a new kind of algorithms, but also to elucidate the reason for the false dissipation of the traditional methods, i.e.,that the main truncation error terms of those non-symplectic methods are dissipative terms, whereas the main truncation error terms of symplectic algorithms are not dissipative terms. Thus the numerical energy of the system will not decrease linearly, but change periodically. Due to the conservation of the symplectic structure of the system, which is the basic property, the symplectic algorithms have the long time capacity to simulate the evolution of the celestial bodies. As the energy is a very important parameter of such a system, the numerical results of symplectic algorithms, which can preserve the energy approximatively, are more

16

Introduction

reasonable. Furthermore, because the errors of the energy are controlled, the errors of numerical trajectories of celestial bodies are no longer along the track by (t − t0 )2 laws of the fast-growing, and with only a t − t0 linear growth, this to the long arc computation is extremely advantageous. For the advantages of the symplectic algorithms, nowadays they have been widely used in the study of dynamical astronomy, especially in the qualitative analysis of the evolution of solar system, e.g. to analyze the stable motion area, space distributions and trajectory resonance of little planets, long time evolution of large planets and extra-planets, and other hot topics in the dynamical astronomy.

(2)

Applications of symplectic algorithms to qualitative analysis

We first use two simple examples to illustrate the special affects of symplectic schemes on the qualitative analysis in dynamics astronomy[LZL93,JLL02,LL95,LL94,Lia97,LLZW94] . Example I. The Keplerian motions. It is the elliptical motions of two-body problem. The corresponding Hamiltonian function is: H(p, q) = T (p) + V (q), where p and q are the generalized coordinates and generalize momentum, T and V are the kinetic and potential energies. The analytic solution is a fixed ellipse. When we simulate this problem by the R–K methods and symplectic algorithms, the former ones shrink the ellipse gradually, whereas the later ones preserve the shape and size of the ellipse (see the numerical trajectories after 150 and 1000 steps respectively in Fig. 0.7(a), e=0.7 and Fig. 0.7(b), e=0.9 where e is the eccentricity of the ellipse). This means the non-symplectic R–K methods have the false energy dissipation and the symplectic algorithms preserve the main character of the Kepler problem because of the conservation of the symplectic structure. Example II. The axial symmetry galaxy’s stellar motion question. Its simplified dynamic model corresponding to the Hamiltonian function is: H(p, q) =

1 2 1 2 (p1 + p22 ) + (q12 + q22 ) + (2q12 q2 − p32 ). 2 2 3

To obtain the basic character of the dynamics of this system, we compute it with order 7 and order 8 Runge–Kutta–Fehlberg methods (denoted as RKF(7) and RKF(8) resp.), as well as the order 6 explicit symplectic algorithm (SY6). The numerical results are listed in Fig. 0.8 to Fig. 0.10. In these figures (Fig. 0.8 to Fig. 0.9), we see that the symplectic algorithm preserves the energy H very well in both of the two cases (ordered LCN= 0 and disorder region LCN> 0), while the RKF methods increase the energy with the evolution of time ΔH. In Fig. 0.10 (a) and Fig. 0.10 (b), the symplectic algorithms present numerically the basic characters of the system: the fixed curve in case of LCN= 0 and the chaos property in case of LCN> 0.

Introduction

17

Fig. 0.7.

Comparison of calculation of Keplerian motion by R–K and symplectic methods.

Fig. 0.8. LCN=0.

Curves of ΔH obtained by RKF(8)[left] and SY6 [right] both with H0 = 0.553,

Fig. 0.9. Curves of ΔH obtained by RKF(8)[left] and SY6 [right] both with H0 = 0.0148, LCN > 0

The symplectic algorithms can preserve the symplectic structure of Hamiltonian systems and the basic evolutionary property of such dynamical systems. Therefore,

18

Introduction

Fig. 0.10. Poincar´e section obtained by RKF(8)[left] with H0 = 0.553,LCN=0 and SY6 [right] with H0 = 0.0148,LCN>0

the symplectic algorithms were widely used to study the dynamical astronomy. Currently, it is a hot topic to study the dynamical evolution of the solar system, such as the long-term trajectory evolution of large planet and extra-planet, the space distribution of little planets in main zone (Kirkwood interstice phenomenon), trajectory resonance, the evolution of satellite system of a large planet, the birth and evolution of planet loops and the trajectory evolution of a little planet near Earth. All these problems require numerical simulation for a very long time, e.g. 109 years or more for the solar system. Thus, the time steps for the numerical methods shall be large enough due to limitations of our computers, while the basic property of the system should be preserved. This excludes all the non-symplectic methods, whilst just lower order symplectic algorithms are valid for the task. In recent years, many astronomers in Japan and America, e.g. Kinoshita[KYN91] , Bretit[GDC91] and Wisdom[WHT96,WH91] , have done a large amount of research on the evolution of the solar system. The following contribution of Wisdom has been widely cited. He derived the Hamiltonian function in Jacobin coordinates of the solar system as

H(p, q) =

n−1 

Hi (p, q) + εΔH(p, q),

i=1

where Hi (p, q) is corresponding Hamiltonian function for a two-body system, ε  1 is a small parameter. By splitting the Hamiltonian function, explicit symplectic algorithms with different orders can be constructed. The advantage of those symplectic algorithms is that the truncation errors are as small as order of ε than those of algorithms constructed by the ordinary splitting for the Hamiltonian function (i.e., H(p, q) = T (p) + V (q)). Even the lower order symplectic algorithms obtained by this splitting method are very effective in a study of the evolution of the solar system. Since the 1980s , Chinese astronomers have also made some progress in the applications of symplectic algorithms to the research of dynamical astronomy, such as[WH91]

Introduction

19

1◦ For the restrictive three-body system constituted by solar, the major planet and the planetoid, some new results have been obtained after studying its corresponding resonance of 1:1 orbit and the triangle libration point. These results can successfully explain the distribution of stability region [ZL94,ZLL92] of Trojan planetoid, as well the actual size of the stable region of distributed triangle libration points corresponding to several relate major planet. 2◦ Adopting the splitting method of Wisdom for the Hamiltonian function to study the long-term trajectories evolution of some little planets. H(p, q) = H0 (p, q) + εH1 (q), where H0 (p, q) is the Hamiltonian function for an integrable system, ε  1 is a little parameter. The numerical results obtained by using this splitting method are very reasonable because the energy is preserved in a controlled range and no false dissipation occurs. 3◦ Application of symplectic algorithms to galaxy system. The bar phenomenon and the evolution of stars in NGC4736 Galaxy were simulated successfully by the symplectic algorithms. 4◦ Some useful results on how to describe the evolutionary features of celestial dynamical system were obtained by further study on the symplectic integrators and the existence of their formal integrations, as well as the changes of all kinds of conservation laws. Besides the research on Hamilton systems in dynamics astronomy mentioned above, the small diffusion situation were also discussed and applied. In view of the fact that the diffusion factor is relatively weak, a mixed symplectic algorithm constituted by the explicit scheme and the centered Euler scheme is applied for the conservative part (the main part corresponding to mechanic system) and the dissipative part, which is remarkably effective, because it could maintain the features of Hamilton system in the main part of this system.

(3) Applications of symplectic algorithms to quantitative computations Because the structure could be preserved by the symplectic algorithms, the errors of their numerical energy don’t accumulate linearly. When the celestial systems are integrated by symplectic algorithms, errors of trajectories will increase linearly as t − t0 , whereas errors of the non-symplectic methods increase rapidly as (t − t0 )2 . We show some examples next. Taking the trajectory of Lageos satellite as background, we consider two mechanics system of the Earth perturbation problems. The first one just takes into account the nonspherical perturbation of the Earth and the second one takes into account the nonspherical perturbation of the Earth and the perturbation of atmospheric resistance. The former problem corresponds to a Hamiltonian system, while the later one corresponds to a quasi-Hamiltonian system because of very small dissipation. We use the RKF7(8) methods and revised order 6 symplectic algorithm (denoted as SY6) to compute the

20

Introduction

two problems, respectively. The numerical results of the errors Δ(M + ω) of main 1000 cycles trajectory are listed in Table 0.1 and Table 0.2. From the two tables, we can clearly see that the errors of the non-symplectic methods, though very small at the beginning, increase rapidly as (t − t0 )2 ; whereas the errors of symplectic algorithm increase linearly as t − t0 . The results of symplectic algorithms are much better. This indicates that though the accuracy order of symplectic algorithms is the same as for other methods, they have more application value in the quantitative computations. We also improve the RKF7(8) for energy conserving methods by compensating the energy at every time step. We denote such method as the RKH method whose numerical results are also listed in the two tables. From the results, we can see that we have made much improvement of the schemes. The results of the energy error by the RKH are almost same with those by the symplectic algorithm. Thus the RKH methods not only have high order accuracy, but also can preserve the energy approximately as the symplectic algorithms. Table 0.1.

Errors of trajectories with nonspherical perturbation of the EarthΔ(M + ω)

method

N of steps / circle

100 circles

1000 circles

10000 circles

FKF7(8)

100

1.5 E − 10

1.4 E − 08

1.3 E − 06

SY6

50

0.5 E − 09

0.6 E − 08

1.0 E − 07

RKH

100

0.9 E − 11

0.9 E − 10

0.9 E − 09

Table 0.2.

Errors of trajectories with perturbation of atmospheric resistanceΔ(M + ω)

method

N of steps / circle

100 circles

1000 circles

10000 circles

FKF7(8)

100

1.4 E − 410

1.3 E − 08

1.3 E − 06

SY6

50

0.6 E − 09

0.7 E − 08

1.0 E − 07

RKH

100

2.1 E − 11

3.5 E − 10

6.2 E − 09

(4)

Applications of symplectic algorithms to quantum systems

The governing equation of the time evolution of quantum system is the Schr¨odinger equation ∂ψ ˆ ˆ =H ˆ 0 (r) + Vˆ (t, r), i = Hψ, H (0.1) ∂t ˆ is Hermitian. where the operator H According to the basic theory of quantum mechanics, the initial state of a quantum system uniquely determines all the states after initial moment of time. That is to say, if the state function ψ(t1 , r) is given at time t1 , then the solution (so-called wave function) of Equation (0.1) is determined as

Introduction

21

ψ(t, r) = a(t, r) + ib(t, r), where functions a and b are real. Such a solution can be generated by a group of time evolutionary operators t1 ,t2 }, i.e., {UH ˆ t1 ,t2 ψ(t1 , r). ψ(t2 , r) = UH ˆ ˆ They are independent of the Every operator is unitary and depends on t1 , t2 and H. state ψ(t1 , r) at time t1 . Therefore, the time evolutions of the quantum system are evolutions of unit transformation in this sense. Every operator can induce an operator, which acts on the real function vector. The two components of the real functions vector are the real part and the image part of the wave function, i.e.,  b(t , r)   b(t , r)  2 1 t1 ,t2 = SH . ˆ a(t2 , r) a(t1 , r) t1 ,t2 The operator SH preserves the inner product and symplectic wedge product for ˆ any two real function vectors. It is simply called norm-preserving symplectic evolution. The quantum system is a Hamiltonian system (with infinite dimensions) and the time evolution of the Schr¨odinger equation can be rewritten as a canonical Hamiltonian system for the two real functions of the wave function as the generalized momentum and generalized coordinates. The norm of wave function is the conservation law of the canonical system. Thus it is reasonable to integrate such a system by the norm-preserving symplectic numerical methods. To apply such a method to the infinite dimensional system, we should first space discretize the system into a finite dimensional canonical Hamiltonian system, which also preserves the norm of wave ˆ 0 (r) for the evolutionfunction. Suppose the characteristic functions of the operator H ary Schr¨odinger equation with some given boundary conditions contain the discrete states and continuous states. ˆ is independent on time explicitly, the energy of the quanWhen the Hamiltonian H ˆ tum system ψ|, H|ψ = Z T HZ is a conservation law both for the canonical system and norm-preserving symplectic algorithm. Such a norm-preserving symplectic algorithm with the fourth order accuracy can be constructed by the order 4 diagonal Pad´e approximations to the exponential function eλ . In the following, we take an example to introduce the method to discretize the time involved Schr¨odinger equation to a canonical system[LQHD07,QZ90a] . Consider the time evolution of an atom moving in one dimensional space by the action of some strong field V (t, x) is

i

∂ψ ˆ = Hψ, ∂t

ˆ =H ˆ 0 (r) + Vˆ (t, r), H 2

ˆ 0 = − 1 ∂ + V0 (x), H 2 ∂ x2  0, 0 < x < 1, V0 (x) = ∞, x ≤ 0 or x ≥ 1.

22

Introduction

In contrast to the characteristic function expanding method, we don’t make any truncation for the wave function when discretizing the Schr¨odinger equation. Therefore, ˆ0. the resulting canonical system contains all the characteristic states of H The numerical conservation laws of explicit symplectic algorithms will converge to the corresponding conservation laws of the system as the time step tends to zero. Thus, although numerical energy and norm of the wave function presented by explicit symplectic algorithms will not be preserved exactly, they will converge to the true energy and norm of the wave function of the system as the time step reduces. The time dependent Schr¨odinger equation (TDSE) in one dimensional space by the action of some strong field V (t, x) is i

∂ψ ˆ = Hψ, ∂t

ˆ =H ˆ 0 (x) + εVˆ (t, x), H 2

ˆ 0 = − 1 ∂ + V0 (x), H 2 ∂x2  0, 0 < x < 1, V0 (x) = ∞, x ≤ 0 or x ≥ 1. ⎧ ⎪ 0 < x < 0.5, ⎨ 2x, V (x) = 2x − 2x, 0.5 ≤ x ≤ 1, ⎪ ⎩ 0, x ≤ 0 or x ≥ 1. Using the similar method √ as before, we expand the wave function as the characterˆ 0 to discretize the TDSE. istic functions {Xn (x) = 2 sin nπx, n = 1, 2, · · ·} of H Because the Hamiltonian operator is real, the discrete TDSE is a separable linear canonical Hamiltonian system with the parameters as follows. S = (Smn ),

⎧ 1 1 − (−1)n ⎪ ⎪ + , ⎪ ⎪ 2 n2 π 2 ⎪ ⎪ ⎪ ⎪ ⎨ 0, vmn =

Smn =

n2 π 2 δmn + εvmn , 2

m = n, |m − n| = 1, 3, 5, · · · , m−n 2

−16mn(1 − (−1) (m2 − n2 )2 π2

)

, |m − n| = 2, 4, 6, · · · , n = 2, 4, 6, · · · , ⎪ ⎪ ⎪ ⎪ m−n ⎪ ⎪ ⎪ −8|2mn − (−1) 2 (m2 − n2 )| ⎪ ⎩ , |m − n| = 2, 4, 6, · · · , n = 1, 3, 5, · · · . 2 2 2 2 (m − n ) π

The initial state is taken as ψ(0, x) =

1+i |X1 (x) + X2 (x)|, 2

ε = 5π 2 .

The energy of the system is conserved because the Hamiltonian does not depend on the time explicitly. E(b, a) = e0 = 42.0110165. The norm of wave function keeps unitary, i.e., N (b, a) = n0 = 1. We take the Euler midpoint rule, order 2 explicit symplectic algorithm and the order 2 R–K method to compute the problem with the same time step h = 10−3 . The numerical results are as follows:

Introduction

23

1◦ The R–K method can not preserve the energy and the norm of wave function, as evident by ER–K in Fig. 0.11(left) and NR–K in Fig. 0.11(right). 2◦ The Euler midpoint rule can preserve the energy and norm, as evident by EE in Fig. 0.11(left) and NE in Fig. 0.11(right). Note that for EE in Fig. 0.11(left), there is a very small increase at some time because of the implicity of the Euler scheme.

Fig. 0.11.

Energy [left] and norm [right] comparison among the 3 difference schemes

˜ k , ak ; h) 3◦ The explicit symplectic algorithms can preserve exactly the energy E(b k k   ˜ and norm N (b , a ; h), as evident by ES in Fig. 0.11(left) and NS in Fig. 0.11(right). If we want to get further insight into these conservation laws within smaller scales, we find that as the time steps get smaller, the numerical energy of symplectic algorithm converges to the true energy of the system e0 = 42.0110165 and the numerical norm converges to unit n0 = 1. See Table 0.3 showing the numerical energy and norm as well as their errors. The errors are defined as CE (h) = max |ESk − e0 |, k

CN (h) = max |NSk − n0 |. k

Actually, the numerical energy and norm obtained by symplectic algorithm oscillate slightly, as shown by ES and NS in Fig. 0.12. However, the amplitude of their oscillations will converge to zero, if the time step tends to zero. As the time step tends to zero, we have e(h) −→ e0 ,

CE (h) = maxk |ESk − e0 | −→ 0,

n(h) −→ n0 ,

CN (h) = maxk |NSk − n0 | −→ 0.

24

Introduction

Table 0.3.

The change of energy and norm of the wave function with the step size h

e(h)

CS (h)

n(h)

CN (h)

42.0169964

0.0445060

0.9996509

0.0003106

42.0110763

0.0004195

0.9999965

0.0000030

42.0110171

0.0000018

0.9999990

0.0000000

10

42.0110165

0.0000000

1.0000000

0.0000000

10−7

42.0110165

0.0000000

1.0000000

0.0000000

exact value

42.0110165

0.0000000

1.0000000

0.0000000

−3

10

−4

10

−5

10

−6

Fig. 0.12.

Energy E and norm N obtained from explicit symplectic scheme

In all, for a quantum system with real Hamiltonian function independent of time explicitly, the explicit symplectic algorithms can preserve the energy and norm of the wave function to any given accuracy. They overcome the main disadvantages of the traditional numerical methods. Next, we look at the quantum system with real Hamiltonian function, which is dependent on time explicitly. In this case, the resulting system after semi-discretization is an m-dimensional, separable, linear, Hamiltonian canonical system. The energy of the system is not conserved any more, but the norm of the wave function is still a quadratic conservation law. The TDSE for an atom in one dimensional space with the action of some strong field V (t, x) = εx sin(ωt) is i

∂ψ ˆ = Hψ, ∂t

ˆ =H ˆ 0 (x) + εVˆ (t, x), H

2 ˆ 0 = − 1 ∂ + V0 (x). H 2

2∂x

By the similar method as before, we expand the wave function as the characteristic √ ˆ 0 to discretize the TDSE. Because functions Xn x = 2 sin nπx(n = 1, 2, · · ·) of H

Introduction

Fig. 0.13.

25

ω = 3π 2 /2, ε = π 2 /2: Graph of norm[left]; graph of probability[right]

the Hamiltonian operator is real, the discrete TDSE is a separable linear canonical Hamiltonian system with the parameters as follows. n2 π 2

S(t) = (s(t)mn ), s(t)mn = δm,n + εv(t)mn ; 2 ⎧ sin(ωt), m = n, ⎪ ⎪ ⎪ ⎨ 0, |m − n| = 2, 4, 6, · · · , v(t)mn = ⎪ ⎪ ⎪ ⎩ 8mn sin (ω t) , |m − n| = 1, 3, 5, · · · . (m2 − n2 )2 π 2

√ The initial state is taken as ψ(0, x) = X1 (x) = 2 sin(πx). The energy of the system is not conserved in this case because the Hamiltonian depends on the time explicitly. The norm of wave function remains unitary, i.e., N (b, a) = n0 = 1. We take the Euler midpoint rule scheme, order 2 explicit symplectic algorithm and the order 2 R–K method to compute the problem with the same time step h = 4 × 10−3 . The numerical results are as follows: 1◦ The R–K method increases the norm of wave function rapidly, see NR–K in Fig. 0.13(left). It leads to unreasonable results, see in Fig. 0.13(right). 2◦ The Euler midpoint rule scheme can preserve the norm, see NE in Fig.0.13(left). These results are in good agreement with the theoretical results. See Fig. 0.13(right) π for the results for weak fields ε = . When ω = ΔE1n , i.e., resonance occurs, the 2 basic state and the first inspired state will intermix and the variation period of the energy is identical to the period of intermixing. See the corresponding results in Fig. 0.14(left) and Fig. 0.14(right). When ω = ΔE1n there will not be intermixing. See the corresponding numerical results in Fig. 0.15(left) and Fig. 0.15(right), where O is the basic state. When the field is strong, the selection rule is untenable, and no resonance occurs, but the basic state will intermix with the first, second, . . . inspired states. See the results for ω =

5π 2 3π 2 in Fig. 0.16(left) and Fig. 0.16(right) and ω = = ΔE12 4 2

in Fig. 0.17(left) and Fig. 0.17(right).

26

Introduction

Fig. 0.14.

ω = 3π 2 /2, ε = π 2 /2: Graph of probability[left]; graph of norm[right]

Fig. 0.15.

ω = 5π 2 /4, ε = 3π 2 /2: Graph of probability[left]; graph of norm[right]

3◦ The order 2 explicit symplectic algorithms can not preserve the norm exactly. The numerical norms oscillate near the unit. See NS in Fig. 0.13, where changes of numerical energy and states of intermixing obtained by symplectic algorithms are similar to the results of Euler midpoint rule scheme. We can conclude that for this system the R–K method can not preserve the norm of wave function and its results are unreasonable; the Euler scheme can preserve the norm and its results are in agreement with the theoretical results; the second order scheme obtains the numerical norm which oscillates near the unit and its energy and states of intermixing are the same as for the results of Euler scheme. Thus, the Euler scheme (an implicit symplectic scheme) and the second order explicit symplectic algorithm are good choices for studying the quantum system with the Hamiltonian dependent on time explicitly. They overcome the drawbacks of the traditional R–K methods.

(5)

Applications to computation of classical trajectories

Applications of symplectic algorithms to computation of classical trajectories of A2 B molecular reacting system [LDJW00] . To study the classical or semi-classical trajectories of the dynamical system, microscopic chemistry is an effective theory method.

Introduction

Fig. 0.16.

ω = 5π 2 /4, ε = 50π 2 : Graph of probability[left]; graph of norm[right]

Fig. 0.17.

ω = 3π 2 /2, ε = 50π 2 : Graph of probability[left]; graph of norm[right]

27

The classical trajectory method regards the atom approximatively as a point and the system as a system of some points, and advances the process of action as the classical motions of point system in potential energy plane of the electrons. It was Bunker who first applied the R–K method to computations of classical trajectory of molecular reacting system. Karplus et al. did a large number of computations by all kinds of numerical methods and screened out the R–K–G (Runge–Kutta–Gear) method to prolong the computation time from 10−15 s to 10−12 s. The R–K–G method made rapid progress in the theoretical study of reacting dynamics of microscopic chemistry and was widely used for computation of classical trajectory. However, its valid computation time is much less than 10−8 s which is necessary time for study of chemical reactions. Moreover, there were many differences between the numerical quantities and theoretical quantities of some parameters. The classical trajectory method describes the microscopic reaction system approximately as a Hamiltonian system which naturally has symplectic structure. Thus, it is expected that the symplectic algorithms will overcome the shortages of the R–K–G method and improve the numerical results. Here we take the mass of the proton as the unit mass and 4.45 × 10−14 s as unit time. Consider the classical motions of the A2 B type molecules like H2 O and SO2 moving in the electron potential energy plane of the reaction system and preserving the

28

Introduction

symmetry of C2v . Set the masses of A and B to be mA = 1 and mB = 2 resp., the center of mass of the molecule be the origin of some coordinate, the C2 axes be z axes, and the coordinates of two atoms A and the atom B be (y1 , z1 ), (y2 , z2 ) and (y3 , z3 ) reps. in the fixed coordinate system. By Banerjee’s coordinates separating method, we can get the generalized coordinates of the A2 B molecule as q1 = z1 + z2 − 2z3 ,

q2 = y2 − y1 ,

and the generalized mass as M1 = 0.25, M2 = 0.5, further the generalized momentum as d q1 d q2 p1 = 0.25 , p2 = 0.5 , dt dt and the kinetic energy of system as K(p) = 2p21 + p22 . The potential energy suggested by Banerjee, who introduced the symmetry C2v  and notation D = q12 + q22 , was V (q) = 5π 2 (D2 − 5D + 6.5) + 4D−1 +0.5π 2 (|q2 | − 1.5)2 + |q2 |−1 . The Hamiltonian function for the A2 B molecular system is H(p, q) = K(p) + V (q), and the canonical equations for the classical trajectories are d p1 ∂V =− = −f1 (q), dt ∂t d p2 ∂V =− = −f2 (q), dt ∂ q2

d q1 ∂K = = g1 (p), dt ∂ p1 d q2 ∂K = = g2 (p). dt ∂ p2

It is a separated Hamiltonian system, which can be integrated by explicit symplectic algorithms. We can obtain its numerical solutions of some initial values as tk = kh,

pk1 = p1 (tk ),

q1k = q1 (tk ),

pk2 = p2 (tk ),

q2k = q2 (tk ),

and further its classical trajectories of A2 B system and the changes of kinetic energy, potential energy and total energy with time by following relations: y3 = 0,

z3 = −

q1 ; 4

y2 = −y1 =

q2 , 2

z2 = z 1 =

The initial values are taken as q1 (0) = 3,

3 2

q2 (0) = ;

p1 = 0,

p2 = 0.

q1 . 4

Introduction

Fig. 0.18.

29

The potential energy curve of the electronic potential function in phase space

We compute this system with order 4 explicit symplectic algorithm and R–K method. The time step is taken as h = 0.01 for both. The numerical classical trajectories, kinetic energy, potential energy and total energy are recorded. Fig. 0.18 shows the potential energy curve of the electronic potential function in phase space. If |q1 | → +∞, then V (q) → +∞; if |q2 | → 0 or |q2 | → +∞, then V (q) → +∞. By the theoretical analysis, we know that the total energy of the system will be conserved all the time, the three atoms will oscillate nearly periodically, and the whole geometry structure of the system may be reversed but kept periodic. The changes of the total energy with time are shown in Fig.0.19, where we can see that the total energy obtained by symplectic algorithms are preserved up to 6.23 × 10−9 s, whereas the R–K method reduces them rapidly with time. The motion trajectories of the system in the plane by the symplectic algorithms and R–K method are shown in Fig. 0.20 (a), (c), (e) and (b), (d), (f) resp., where we can see that the numerical results of symplectic algorithms are coincident with the theoretical results but the results of R–K method are not. We also applied the order 1 and 2 symplectic algorithms, the Euler method and the revised Euler method to compute the same problem. The conclusions are almost the same. Because all the traditional methods such as R–K methods, Adams methods and Euler methods can not preserve the symplectic structure of this microscopic system, they will bring false dissipations inevitably, which will make their numerical results meaningless after long-term computations. On the contrary, symplectic algorithms can preserve the structure and do not bring any false dissipations. Therefore, they are suitable for long-term computations and greatly improve the classical trajectory methods for studying the microscopic dynamical reactions of chemical systems.

30

Introduction

Fig. 0.19.

The changes of the total energy with the time

(6) Applications to computation of classical trajectories of diatomic system [Dea94,DLea96] Consider the classical motion of AB diatomic molecule system in electron potential energy plane. Set the masses of A and B to be m1 and m2 resp., the center of mass to be the origin of some coordinate with fixed axes Ox, the coordinates of two atoms A and B to be −x1 and x2 resp. Then the generalized coordinate is q = x2 + x1 and the m1 m2 dq . Further, the generalized momentum is p = M m1 + m2 dt p2 and the generalized kinetic energy is U (p) = . Take the potential function as the 2M

generalize mass is M =

Morse potential

V (q) = D{e−2a(q−qe ) − 2e−a(q−qe ) }, where the parameters D, a, qe were derived by E. Ley and Koo recently. Thus, the total energy for such system is H(p, q) = U (p) + V (q), and the canonical Hamiltonian system for the classical trajectory is dp d V (q) =− = −f (q), dt dt dq d U (p) = = g(p). dt dt

It is a separable system. By explicit symplectic algorithms, we can get its numerical solutions as tk = kh, pk = p(tk ), q k = q(tk ), and advance its classical trajectories of AB two-atom system as x1 =

m2 q , m 1 + m2

x2 =

m1 q , m 1 + m2

Introduction

31

Fig. 0.20. The motion trajectories of the system in the plane,(a) and (b) period range from 4.45 × 10−10 s to (4.45 × 10−10 + 4.45 × 10−13 )s. (c) and (d) period range from 6.23 × 10−9 s to (6.23 × 10−9 + 4.45 × 10−13 )s. (e) and (f) period range from 6.23 × 10−9 s to (6.23 × 10−9 + 4.45 × 10−13 )s. (a), (c), (e) is the symplectic algorithm path, (b), (d), (f) is the R–K method path

as well as the changes of kinetic energy, potential energy and total energy with the variation of time. We compute some states of two homonuclear molecules Li2 and N2 and two heteronuclear molecules CO and CN by using the order 1, 2 and 4 explicit symplectic algorithms and compare the numerical results of total energy and classical trajectories with the Euler method and order 2 and 4 order R–K methods. In Fig. 0.21, Fig. 0.22 and Fig. 0.23, we show the numerical results of the classical trajectories, total energy and the trajectories in p − q phase space obtained by order 4 explicit symplectic algorithm and order 4 R–K method respectively. The parame-

32

Introduction

Fig. 0.21.

Classical orbit of two homonuclear molecules Li2

Fig. 0.22.

Comparison of energy of two homonuclear molecules Li2

ters in those computations are taken as the time step h = 0.005, the initial values √ ˚ a= q(0) = qe , p(0) = 2M D − 0.0001, and D = 8541cm−1 , qe = 2.67328A, −1 ˚ ,A ˚ = 0.1nm. The results show that the symplectic algorithms can preserve 0.867A the energy after 106 time steps and the facts that the two Li atoms oscillate periodically and their trajectories in phase remain invariant are simulated by the symplectic algorithm. The results are opposite for the R–K method. The numerical total energy, and the oscillation period and amplitude of the two atoms were reduced, after 3000 time steps. Furthermore, the trajectories in the phase space became flat to q axis after 50000 time steps and lost entirely their shape as manifested in the theory analysis and experiments (Fig. 0.21, Fig. 0.22, Fig. 0.23). The results of the other molecules N2 , CO and CN are similar. Thus, we can draw the conclusion that the symplectic algorithms can preserve the symplectic structure and the basic properties of the microscopic system. Therefore they are capable of long time computations for such systems.

Introduction

Fig. 0.23.

(7)

33

The trajectories in p − q phase space

Applications to atmospheric and geophysical science

Recently, the symplectic algorithms have been applied to study the observation operator of the global positioning system (GPS) by Institute of Atmospheric Physics of the Chinese Academy of Science[WZJ95,WJX01] . Numerical weather forecasting needs very large amount of atmospheric information from GPS. One of the key problems in this field is how to reduce largely the computational costs and to compute it accurately for a long time. The symplectic algorithms provide rapid and accurate numerical algorithms for them to deal with the information of GPS efficiently. The computational costs of the symplectic algorithms are one four hundredth of the costs of traditional algorithms. For the complicated nonlinear system of atmosphere and ocean, symplectic algorithms can preserve its total energy, total mass, total potential so well that the relative errors of potential height is below 0.0006 (see Fig. 0.24). Another application of symplectic algorithms to geophysics is carried out by Institute of Geophysics to prospect for the oil and natural gas[GLCY00,LLL01a,LLL01b,LLL99] , which has obtained several great achievements. For example, the spread waves of earthquake under the framework of Hamiltonian system and the corresponding symplectic algorithms have been investigated. Moreover, “the information of oil reserves and geophysics and its process system ” has been produced, and the task of prospecting for 1010 m3 of natural gas, which has obtained. Fig. 0.25 shows the numerical results of prestack depth migration in the area of Daqing Xujiaweizi by applying symplectic algorithms to Marmousi model. Recently, Liuhong et.al. proposed a new method[LYC06] to calculate the depth extrapolation operator via exponential of pseudo-differential operator in lateral varied medium. The method offers the phase of depth extrapolation operator by introducing lateral differential to velocity, which in fact is an application of Lie group method.

34

Introduction

Fig. 0.24.

The relative errors of potential height is below 0.0006 after 66.5 days

Fig. 0.25. Numerical results of prestack depth migration in the area of Daqing Xujiaweizi obtained by applying symplectic algorithms to Marmousi model

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Chapter 1. Preliminaries of Differentiable Manifolds

Before introducing the concept of differentiable manifold, we first explain what mapping is. Given two sets X, Y, and a corresponding principle, if for any x ∈ X, there exists y = f (x) ∈ Y to be its correspondence, then f is a mapping of the set X into the set Y , which is denoted as f : X → Y. X is said to be the domain of definition of f , and f (x) = {f (x) | x ∈ X} ⊂ Y is said to be the image of f . If f (X) = Y , then f is said to be surjective or onto; if f (x) = f (x ) ⇒ x = x , then f is said to be injective (one-to-one); if f is both surjective and injective (i.e., X and Y have a one-to-one correspondence under f ), f is said to be bijective. For a bijective mapping f , if we define x = f −1 (y), then f −1 : Y → X is said to be the inverse mapping of f . In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). For example, for two groups G and G and a mapping f : G → G , a → f (a), if f (a, b) = f (a) · f (b), ∀a, b ∈ G, then f is said to be a homomorphism from G to G . A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structures, i.e., properties such as identity element, inverse element, and binary operations. An isomorphism is a bijective homomorphism. If f is a G → G homomorphic mapping, and also a one-to-one mapping from G to G , then f is said to be a G → G isomorphic mapping. An epimorphism is a surjective homomorphism. Given two topological spaces (x, τ ) and (y, τ ), if the mapping f : X → Y is one-to-one, and both f and its inverse mapping f −1 : Y → X are continuous, then f is said to be a homeomorphism. If f and f −1 are also differentiable, then the mapping is said to be diffeomorphism. A monomorphism (sometimes called an extension) is an injective homomorphism. A homomorphism from an object to itself is said to be an endomorphism. An endomorphism that is also an isomorphism is said to be an automorphism. Given two manifolds M and N , a bijective mapping f from M to N is called a diffeomorphism if both f : M → N and its inverse f −1 : N → M are differentiable (if these functions are r times continuously differentiable, f is said to be a C r -diffeomorphism). Many differential mathematical methods and concepts are used in classical mechanics and modern physics: differential equations, phase flow, smooth mapping, manifold, Lie group and Lie algebra, and symplectic geometry. If one would like to construct a new numerical method, one needs to understand these basic theories and concepts. In this book, we briefly explain manifold, symplectic algebra, and symplectic geometry. In a series of books[AM78,Che53,Arn89,LM87,Ber00,Wes81] can be found these materials. K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

40

1. Preliminaries of Differentiable Manifolds

1.1 Differentiable Manifolds The concept of manifold is an extension of Euclidean space. Roughly speaking, a manifold is an abstract mathematical space where every point has a neighborhood that resembles Euclidean space (homeomorphism). Differentiable manifold is one of the manifolds that can have differentiable structures.

1.1.1 Differentiable Manifolds and Differentiable Mapping Definition 1.1. A Hausdorff space M with countable bases is called an n-dimensional topological manifold, if for any point in M there exists an open neighborhood homeomorphic to an open subset of Rn . Remark 1.2. Let (U, ϕ), (V, ψ) be two local coordinate systems (usually called chart) on the topological manifold M . (U, ϕ), (V, ψ) are said to be compatible, if U ∩ V = Ø, or the change of coordinates ϕ ◦ ψ −1 and ψ ◦ ϕ−1 are smooth when U ∩ V = Ø. Definition 1.3. A chart is a domain U ⊂ Rn together with a 1 to 1 mapping ϕ : W → U of a subset W of the manifold M onto U . ϕ(x) is said to be the image of the point x ∈ W ⊂ M on the chart U . Definition 1.4. A collection of charts ϕi : Wi → Ui is an atlas on M if 1◦ Any two charts are compatible. 2◦ Any point x ∈ M has an image on at least one chart. Remark 1.5. If a smooth atlas on a topological manifold M possesses with its all compatible local coordinate systems (chart), then this smooth atlas is called the maximum atlas. Definition 1.6. If an n-dimensional topological manifold M is equipped with the maximal smooth atlas A, then (M, A) is called the n-dimensional differentiable manifold, and A is called the differentiable structure on M . Definition 1.7. Two atlases on M are equivalent if their union is also an atlas (i.e., if any chart of the first atlas is compatible with any chart of the second). Remark 1.8. Suppose M is the n-dimensional topological manifold, A = {(Uλ , ϕλ )} is a smooth atlas on M . Then there exists a unique differentiable structure A∗ , which contains A. Hence, a smooth atlas determines a unique differentiable structure on M . The local coordinate system will be called (coordinate) chart subsequently. Definition 1.9. A differentiable manifold structure on M is a class of equivalent atlases. Definition 1.10. A differentiable manifold M is a set M together with a differentiable manifold structure on it. A differentiable manifold structure is induced on set M if an atlas consisting of compatible charts is prescribed.

1.1 Differentiable Manifolds

41

Below are examples of differentiable manifold. Example 1.11. Rn is an n-dimensional differentiable manifold. Let A ={(Rn , I)}, where I is the identity mapping. Example 1.12. S n is an n-dimensional differentiable manifold. We only discuss the n = 1 case. Let U1 = {(u1 , u2 ) ∈ S 1 |u1 > 0},

U2 = {(u1 , u2 ) ∈ S 1 |u1 < 0},

U3 = {(u1 , u2 ) ∈ S 1 |u2 > 0},

U4 = {(u1 , u2 ) ∈ S 1 |u2 < 0}.

Define ϕi : Ui → (−1, 1), such that (s.t.) ϕi (u1 , u2 ) = u2 ,

i = 1, 2;

ϕi (u1 , u2 ) = u1 ,

i = 3, 4.

Note that on ϕ1 (U1 ∩ U3 )   2 2 2 2 1 − (u2 )2 ϕ3 ◦ ϕ−1 1 : u −→ ( 1 − (u ) , u ) −→ is smooth, then A ={(Uk , ϕk )} is a smooth atlas on S 1 . Example 1.13. RP n is an n-dimensional differentiable manifold. Let Uk = {[(u1 , · · · , un+1 )] | (u1 , · · · , un+1 ) ∈ S n , uk = 0},

k = 1, · · · , n + 1

defines ϕk : Uk → Int B n (1), s.t. ϕk ([(u1 , · · · , un+1 )]) = uk |uk |−1 (u1 , · · · , uk−1 , uk+1 , · · · , un+1 ), n     where B n (1) = (u1 , · · · , un ) ∈ Rn  (ui )2 ≤ 1 . It is easy to prove that A

={(Uk , ϕk )} is a smooth atlas on RP n .

i=1

Example 1.14. Let M, N be m- and n-dimensional differentiable manifolds, respectively, then M ×N is a m+n dimensional differentiable manifold (product manifold). Suppose A = {(Uα , ϕα )}, B = {(Vα , ψα )} are smooth atlases on M, N respectively. Denote A × B={(Uα × Vλ, ϕα × ψλ )}, where ϕα × ψλ : Uα × Vλ → ϕα (Uα ) × ψλ (Vλ ),(ϕα × ψλ )(p, q) = ϕα (p), ψλ (q) , (p, q) ∈ Uα × Vλ , then A × B is a smooth atlas on M × N . Definition 1.15. Let M, N be m- and n-dimensional differentiable manifolds, respectively. A continuous mapping f : M → N is called C k differentiable at p ∈ M , if the local representation f = ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) is C k differentiable for the charts (U, ϕ), (V, ψ) corresponding to points p and f (p), and f (U ) ⊂ V . If f is C k differentiable in each p ∈ M , then f is called C k differentiable, or called C k mapping. See Fig. 1.1.

42

1. Preliminaries of Differentiable Manifolds

Fig. 1.1.

A differentiable mapping

Example 1.16. Let M1 , M2 be m- and n-dimensional differentiable manifolds, respectively. Define θ1 : M1 × M2 → M1 , θ2 : M1 × M2 → M2 , such that θ1 (p, q) = p,

θ2 (p, q) = q,

∀ (p, q) ∈ M1 × M2 ,

then θ1 , θ2 are all smooth mappings. If the charts on M1 , M2 are denoted by (U, ϕ), (V, ψ), then it is easy to show that (U × V, ϕ × ψ) is the chart on M1 × M2 . Thus, the local coordinate expression of θ1 , θ1 = ϕ ◦ θ1 ◦ (ϕ × ψ)−1 : (ϕ × ψ)(U × V ) −→ ϕ(U ),

θ1 (u, v) = u

is a smooth mapping. Therefore, θ1 is a smooth mapping. Likewise, θ2 is also a smooth mapping. Example 1.17. Let M, N1 , N2 be differentiable manifolds. f1 : M −→ N1 ,

f2 : M −→ N2

k

are C -mapping. Define f : M −→ N1 × N2 ,

f (p) = (f1 (p), f2 (p)),

∀ p ∈ M,

then f is a C k -mapping. ∀ p0 ∈ M , let N1 contain the chart (V, ψ) of f1 (p0 ), and let N2 contain the chart (W, χ) of f2 (p0 ), and M contain the chart (U, ϕ) of p0 . Assume f1 (U ) ⊂ V, f2 (U ) ⊂ W , and f1 : ψ ◦ f1 ◦ ϕ−1 : ϕ(U ) −→ ψ(V ), f2 : χ ◦ f2 ◦ ϕ−1 : ϕ(U ) −→ χ(W ) are all C k -mapping, and (V × W, ϕ × χ) is a chart that contains (f1 (p0 ), f2 (p0 )) = f (p0 ) on the product manifold N1 × N2 , which satisfies f (U ) ⊂ V × W . Then we have f = (ψ × χ) ◦ f ◦ ϕ−1 : ϕ(U ) −→ (ψ × χ)(U × W ), i.e., f is C k -mapping.

f = (f1 , f2 ),

1.1 Differentiable Manifolds

43

Remark 1.18. According to the definition, if f : M → N, g : N → L are C k mappings, then g ◦ f : M → L is also a C k -mapping. Definition 1.19. Let M, N be differentiable manifolds, f : M → N is a homeomorphism. If f, f −1 are smooth, then f is called diffeomorphism from M to N . If there exists a diffeomorphism between differentiable manifolds M and N , then M and N are called differentiable manifolds under diffeomorphism, denoted by M  N . If we define two smooth atlases (R, I), (R, ϕ) on R, and ϕ : R → R, ϕ(u) = u3 , √ −1 because the change of coordinates I ◦ ϕ (u) = 3 u in u = 0 is not differentiable, then (R, I) and (R, ϕ) determine two different differentiable structures A, A on R. However, if we define f : (R, A) → (R, A ), {f (u)} = u3 , then (R, A)  (R, A ). In fact, there exist examples that are not homeomorphism in a differentiable manifold, like the famous Milnor exotic sphere.

1.1.2 Tangent Space and Differentials In order to establish the differential concept for differentiable mapping on a differentiable manifold, we first need to extend the concept of tangent of curve and tangent plane of surface in Euclidean space. If we take the tangent vector in Euclidean space not simply as a vector with size and direction, but as a linear mapping, which satisfies the Leibniz rule, from the differentiable functional space to R, then the definition of tangent vector can be given similarly for a manifold. Let M be the m-dimensional differentiable manifold, p ∈ M be a fixed point. Let C ∞ (p) be the set of all smooth functions that are defined in some neighborhood of p. Define operations on M that have the following properties: (f + g)(p) = f (p) + g(p), (αf )(p) = αf (p), (f g)(p) = f (p)g(p). Definition 1.20. A tangent vector Xp at the point p ∈ M is a mapping Xp : C ∞ −→ R, that has the following properties: 1◦ Xp (f ) = Xp (g), if f, g ∈ C ∞ (p) are consistent in some neighborhood of the point p. 2◦ Xp (αf + βg) = αXp (f ) + βXp (g), ∀ f, g ∈ C ∞ (p), ∀ α, β ∈ R. 3◦ Xp (f g) = f (p)Xp (g) + g(p)Xp (f ), ∀ f, g ∈ C ∞ (p) (which is equivalent to the derivative operation in Leibniz rule). Denote Tp M ={All tangent vectors at the point p ∈ M } and define operation: (Xp + Yp )(f ) = Xp (f ) + Yp (f ), (kXp )(f ) = kXp (f ), ∀ f ∈ C ∞ (p).

44

1. Preliminaries of Differentiable Manifolds

It is easy to verify that Tp M becomes the vector space that contains the above operation, which is called the tangent space at the point p of the differential manifold M. Remark 1.21. By definition of the tangent vector, it is easy to know that if f is the constant function, Xp (f ) = 0 for Xp ∈ Tp M . Lemma 1.22. Let (U, ϕ) be the chart that contains p ∈ M , and let x1 , · · · , xm , ϕ(p) = (a1 , · · · , am ) be the coordinate functions. If f ∈ C ∞ (p), then there exists a function gi in some neighborhood W of p ∈ M , such that f (q) = f (p) +

m  (xi (q) − ai )gi (q),

∀ q ∈ W,

i=1

 

and gi (p) =









∂f  ∂f  ∂  ∂f ◦ ϕ−1  .  where i  = i  (f ) =  i ∂x p ∂x p ∂x p ∂ui ϕ(p)

Proof. Assume ϕ(p) = O ∈ Rm , and f is well defined in some neighborhood of p. Let W = ϕ−1 (B m ). Then ∀ q ∈ W and we have f (q) − f (p) = f ◦ ϕ−1 (u) − f ◦ ϕ−1 (O). After calculation, we obtain f (q) − f (p) =

m 

ui g i (u),

i=1

where g i (u) =

 1 ∂f ◦ ϕ−1 0

∂ui

(su1 , · · · , sum ) d s (i = 1, · · · , m). Let g i (ϕ(q)) =

gi (q), then gi is smooth on W , and satisfies m 

xi (q)gi (q), i=1   ∂f ◦ ϕ−1  ∂f  gi (p) = g i (O) =  = i . i

f (q) = f (p) +

∂u

O

∂x

p



Hence lemma is proved.

   ∂  ∂  ∂f ◦ ϕ−1  ∞ Theorem 1.23. Define : C (p) → R, (f ) = , ∀f ∈    ∂xi p ∂xi p ∂ui ϕ(p)  ∂  C ∞ (p), then i  (i = 1, · · · , m) is a group of bases for Tp M . Therefore, dim Tp M ∂x

p

= m, and for Xp ∈ Tp M, we have Xp =

m  i=1

Xp (xi )



∂   . ∂xi p

1.1 Differentiable Manifolds

45

Proof. ∀ Xp ∈ Tp M, as f ∈ C ∞ (p). By Lemma 1.22, we know f = f (p) +

m 

(xi − ai )gi ,

i=1

then  ∂f  ∂  Xp [(x − a )gi ] = Xp (x ) i  = Xp (xi ) i  (f ). Xp (f ) = ∂x p i=1 ∂x p i=1 i=1 m 

i

m 

i

m

i

i The  decomposed coefficients, {Xp (x )}, of Xp with respect to (w.r.t.) the bases

∂   (i = 1, · · · , m) are called coordinates of the tangent vector Xp w.r.t. the ∂xi p



chart(U, ϕ).

Remark 1.24. By Theorem 1.23 we know: if the coordinates of Xp w.r.t. chart (U, ϕ) are defined as (Xp (x1 ), · · ·, Xp (xm )), then Tp M and Rm are isomorphisms,  and the  ∂ m basis for Tp M corresponds exactly to the standard basis for R , i.e.,  → ei = ∂xi p (0, · · · , 1, 0, · · · , 0). 1. Definition and properties of differentials of mappings The definition of differentials of a mapping is as follows: Definition 1.25. Let f : M → N be a smooth mapping. ∀ p ∈ M, Xp ∈ Tp M, we define f∗p : Tp M → Tf (p) N that satisfies: f∗p (Xp )(g) = Xp (g ◦ f ),

∀ g ∈ C ∞ (f (p)).

This linear mapping f∗p is called the differential of f at the p ∈ M . Definition 1.26. The differential of the identity mapping I is an identity mapping, i.e., I∗p : Tp M → Tp M . Remark 1.27. Let M, N, L be differentiable manifolds, p ∈ M , and f : M → N, g : N → L are smooth mappings, then (g ◦ f )∗p = g∗f (p) ◦ f∗p . Remark 1.28. If f : M → N is a diffeomorphism, then f∗p : Tp M → Tf (p) N is a isomorphism. Proposition 1.29. Let x1 , · · · , xm , y 1 , · · · , y n be the coordinate functions of (U, ϕ), (V, ψ) respectively, then  f∗p where fj = y j ◦ f .

 

∂   ∂xi p

=

n   ∂fj   i j=1



∂  ,  ∂x p ∂y j f (p)

46

1. Preliminaries of Differentiable Manifolds

Proof. Since

 f∗p

 





∂  ∂  ∂fk   (y k ) = i  (y k ◦ f ) = i  , ∂xi p ∂x p ∂x p

therefore, by Theorem 1.23 we have  f∗p

 

∂   ∂xi p

n    ∂fj  ∂y k    i j

=

i,j=1

∂x

p ∂y

n   ∂fj  i

=

i,j=1

f (p)



∂  (y k ).  ∂x p ∂y j f (p)



Therefore the proposition is completed. Let Xp =

n 

αi

i=1

have

n    ∂  j ∂  , f (X ) = β , by Proposition 1.29, we   ∗p p ∂xi p ∂y j f (p) j=1

⎛

⎛

∂f1 ⎞ β1 ⎜ ∂x1 ⎜ .. ⎟ ⎜ .. ⎝ . ⎠=⎜ . ⎝ ∂fn βn

 This matrix

∂fi ∂xj

 n×m

∂x1

···

⎞

∂f1 ⎛ 1 ⎞ α ∂xm ⎟

.. .

···

∂fn ∂xm

⎟⎜ ⎟⎝ ⎠

.. ⎟ . . ⎠ αm

is the Jacobian matrix of f at p w.r.t. charts (U, ϕ), (V, ψ).

Its rank rkp f is called the rank of f : M → N at the p. From the above equations, we can easily observe that f∗p is equivalent to Df(ϕ(p)) under the assumption of isomorphism, where Df(ϕ(p)) is the differential at ϕ(p) of the local representation of f , f = ψ ◦ f ◦ ϕ−1 . 2. Geometrical meaning of differential of mappings A smooth on M is a smooth mapping c : (a, b) → M . The tangent vector,   curve  d  on Tc(t0 ) M is called the velocity vector of c at t0 . Let f : M → N be a c∗t0  dt

t0

smooth mapping. Then, f ◦ c is a smooth curve on N that passes f (p). By composite differentiation, we have d     d   = f∗p0 ◦ c∗t0 , (f ◦ c)∗t0   d t t=t0 d t t=t0 i.e., f∗p0 transforms the velocity vector of c at t0 to the velocity vector of f ◦ c at t0 .

1.1.3 Submanifolds The extension of the curve and surface on Euclidean space to the differentiable manifold is the submanifold. In the following section, we focus on the definitions of three submanifolds and their relationship. First, we describe a theorem.

1.1 Differentiable Manifolds

1.

47

Inverse function theorem

Theorem 1.30. Let M, N be m-dimensional differentiable manifolds, f : M → N is a smooth mapping, p ∈ M . If f∗p : Tp M → Tf (p) N is an isomorphism, then there exists a neighborhood, W of p ∈ M , such that 1◦ f (W ) is a neighborhood of f (p) in N. 2◦ f |W : W → f (W ) is a diffeomorphism (this theorem is an extension of the inverse function theorem for a manifold). Proof. Consider charts (U, ϕ) on M about p ∈ M and (V, ψ) on N about f (p) ∈ N , so that f (U ) ⊂ V . Then, the local representation f = ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) is a smooth mapping. Since f∗p : Tp M → Tf (p) N is an isomorphism, Dfˆ(ϕ(p)) : Rm → Rm is also an isomorphism. By the inverse function theorem, there exists a neighborhood O of ϕ(p) ∈ Rm such that f(O) is a neighborhood of ψ(f (p)) in Rm , and f : O → f(O) is a diffeomorphism. O has to be chosen appropriately. Let O ⊂ ϕ(U ), and f(O) ⊂ ψ(V ). Let W = ϕ−1 (O). Then, W is the neighborhood of p, which meets our requirement.  Remark 1.31. Given a chart (V, ψ) on N , f (p) ⊂ V , choose ϕ = ψ ◦ f and some neighborhood U of p, such that ϕ(U ) ⊂ V . By 2◦ of Theorem 1.30, f |W : W → f (W ) is a diffeomorphism and (U, ϕ) is a chart on M . Hence f = ψ ◦ f ◦ ϕ−1 = I is an identity mapping from ϕ(U ) to ψ(V ). Example 1.32. Suppose f : R → S 1 , defined by f (t) = (cos t, sin t). Using the chart of Example 1.11, we obtain    π π cos t, t ∈ kπ − , , kπ +  f (t) = 2 2 − sin t, t ∈ (kπ, (k + 1)π). Obviously, f (t) = 0, ∀ t ∈ R. However, f : R → S 1 is not injective. Thus, f is not a diffeomorphism. This example shows that f∗p : Tp M → Tf (p) N isomorphism and f : M → N homeomorphism at some neighborhood of p are only local properties. We have discussed the case where f∗p : Tp M → Tf (p) N is an isomorphism. In the following section we turn to the case when f∗p is injective. 2. Immersion Definition 1.33. Let M, N be differentiable manifolds, and f : M → N a smooth mapping, and p ∈ M . If f∗p : Tp M → Tf (p) N is injective (i.e., rkp f = m), then f is said to immerse at p. If f immerses at every p ∈ M , then f is called an immersion. Below are some examples of immersion. Example 1.34. Let U ∈ Rm be an open subset, α : U → Rn , α(u1 , · · · , um ) = (u1 , · · · , um , 0, · · · , 0). By definition, α is obviously an immersion, and is often called a model immersion.

48

1. Preliminaries of Differentiable Manifolds

Proposition 1.35. Let M, N be m- and n-dimensional differentiable manifolds respectively; f : M → N is a smooth mapping, p ∈ M . If f immerses at p, then there exist charts (U, ϕ) on M about p ∈ M and (V, ψ) on N about f (p) ∈ N in which the coordinate description f = ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) has the form f(u1 , · · · , um ) = (u1 , · · · , um , 0, · · · , 0). Proof. Choose charts (U1 , ϕ1 ) and (V1 , ψ1 ) appropriately so that p ∈ U1 , f (p) ∈ V1 , ϕ1 (p) = 0 ∈ Rm , ψ1 (f (p)) = 0 ∈ Rn and f (U1 ) ⊂ V1 . Since f immerses     at p, the rank of Jacobian Jf(0) = ∂ fij  is m, where f = (f1 , · · · , fn ). We can ∂u 0 assume that the first m rows in the Jacobian matrix Jf(0) are linearly independent. Then, define a mapping for ϕ1 (U1 ) × Rn−m → Rn = Rm × Rn−m by g(u, v) = f(u) + (0, v). It is easy to prove that g(u,!0) = f(u) maps origin 0 to itself in Rn and the rank of 0 Jg (O ) = Jf(O) is n, where 0 denotes a m × (n − m) zero matrix, and by In−m the inverse function theorem, g is a diffeomorphism from a neighborhood of origin of Rm to a neighborhood of origin of Rn . Shrink U1 , V1 so that they become U, V , and −1  let ϕ = ϕ1 |U, ψ = g −1 ◦ (ψ|V ). Since ψ ◦ f ◦ ϕ−1 = g −1 ◦ ψ1 ◦ f ◦ ϕ−1 ◦f = 1 =g g(u, 0), the proposition is proved.  Remark 1.36. By definition of immersion, if f : M → N immerses at p ∈ M , then f immerses in some neighborhood of p. Remark 1.37. By Proposition 1.35, f limited in some neighborhood of p has a local injective expression f = (u1 , · · · , um , 0, · · · , 0). Then, f limited in some neighborhood of p is injective. Note that this is only a local injection, not total injective. Definition 1.38. Let N, N  be differentiable manifolds, N  ⊂ N . If the inclusion map i : N  → N is an immersion, then N  is said to be an immersed submanifold of N. Remark 1.39. Suppose f is an immersion and injective (such f would henceforth be called injective immersion), M is a smooth atlas A = {(Uα , ϕα )}. Denote f A = {(f (Uα ), ϕα ◦ f −1 )}. Then, it is easy to prove that {f (M ), f A} is a differentiable manifold. Since f has a local expression f = ϕα ◦ f −1 ◦ f ◦ ϕ−1 α = I : ϕα (Uα ) → ϕα (Uα ), f : M → f (M ) is a diffeomorphism, i.e., f∗p : Tp M → Tf (p) f (M ) is an isomorphism. Since f is an immersion, the inclusion map i : f (M ) → N is also an immersion. Hence, f (M ) is an immersed submanifold of N . From the following example, we can see that the manifold topology of an immersed submanifold may be inconsistent with its subspace topology and can be very complex.

1.1 Differentiable Manifolds

49

Example 1.40. T 2 = S 1 × S 1 = {(z1 , z2 ) ∈ C × C | |z1 | = |z2 | = 1}. Define f : R → T 2 , s.t. f (t) = (e2πit , e2πiα ), where α is an irrational number. We can prove that f (R), which is a differentiable manifold derived from f , is an immersed submanifold of T 2 , and f (R) is dense in T 2 . We may regard T 2 as a unit square on the plane of R2 , which is also a 2D manifold that has equal length on opposite sides. It can be represented by a pair of ordered real numbers (x, y), where x, y are mod Z real numbers. Define ϕ : R2 −→ S 1 × S 2 ,

1

2

ϕ(u1 , u2 ) = (e2πiu , e2πiu )

and define“ ∼ ” : (u1 , u2 )∼ (v1 , v 2 ) ⇔ u1 = v1 (mod Z), u2 = v 2 (mod Z), and 1 1 1 1 let W = u10 − , u10 + × u20 − , u20 + . Then, (ϕ(W ), ϕ−1 ) is a chart 2

2

2

2

of T 2 = S 1 × S 1 that contains f (t0 ). Choose a neighborhood U of t0 ∈ R so that f (U ) ⊂ ϕ(W ). Then, the local expression of f , f = (t, αt) is an immersion at t0 . It is easy to prove that if ϕ−1 f (R) is dense in R2 , then f (R) is dense in T 2 = S 1 × S 1 . By definition, f is injective. It is concluded that the topology of T 2 is different from the topology of f (U ), which derives from f (R). 3. Regular submanifolds The type of submanifold given below has a special relationship to its parent differential manifold, which is similar to that of Euclidean space and its subspace. Definition 1.41. Let M  ⊂ M have the subspace topology, and k be some nonnegative integer, 0 ≤ k ≤ m. If there exists a chart (U, ϕ) of M that contains p in every p ∈ M  , so that 1◦ ϕ(p) = O ∈ Rm . 2◦ ϕ(U ∩ M  ) = {(u1 , · · · , um ) ∈ ϕ(U ) | uk+1 = · · · = um = 0}. Then M  is said to be a k-dimensional regular submanifold of M , and the chart is called submanifold chart. Let A = {(Uα , ϕα )} be a set that contains all submanifold charts on M . Denote "α = Uα ∩ M  , ϕ "α ), π : Rk × Rm−k → Rk . " "α , ϕ "α )}, where U "α = π ◦ (ϕα |U A={(U  "α is an open set of M  , and ϕ "α → ϕ "α ) Since M has the subspace topology, U " :U "α (U #"α  "α to ϕ "α ) ⊂ Rk . Moreover, Uα U is a homeomorphism for U "α (U = M , and hence α "α , ϕ "β , ϕ "α ∩ U "β = Ø, we have A" is an atlas of M  . ∀ (U "α ), (U "β ) ∈ A" and U 1 k −1 1 k "−1 ϕ $β ◦ ϕ α (u , · · · , u ) = π ◦ ϕβ ◦ ϕα (u , · · · , u , 0, · · · , 0).

Obviously, A" is a smooth atlas of M  , which determines a differentiable structure of M  . Thus, M  is a k-dimensional differentiable manifold. Below is an example of regular submanifold. Example 1.42. Let M, N be m- and n-dimensional differentiable manifolds respectively, and f : M → N be a smooth mapping. Then, the graph of f gr(f ) = {(p, f (p)) ∈ M × N | p ∈ M } is an m-dimensional closed submanifold of M × N ( closed regular submanifold ).

50

1. Preliminaries of Differentiable Manifolds

Proof. Consider charts (U, ϕ), (V, ψ), p0 ∈ U, f (p0 ) ∈ V, ϕ(p0 ) = O ∈ Rm , ψ(f (p0 )) = O  ∈ Rn , f (U ) ⊂ V , and define G : ϕ(U ) × ψ(V ) → Rn+m = Rm × Rn , so that G(u, v) = (u, v − f(u)). It is easy to prove that G(gr(f)) = {(u, O ) | u ∈ ϕ(U )}, and the rank of 

%

JG (O, O ) =

Im −Df(O)

O In

&

is n + m. Since G(O, O  ) = (O, O  ), G homeomorphically maps some neighborhood " of (O, O ) on ϕ(U ) × ψ(V ) to some neighborhood V" of (O, O ) on Rn+m . Denote U " ), W = (ϕ × ψ)−1 (U

χ = G ◦ (ϕ × ψ)|W.

Then, (W, χ) is a chart of M × N that contains (p0 , f (p0 )), and χ(p0 , f (p0 )) = (O, O ) ∈ Rn+m , χ(W ∩ gr(f )) = {(u, v) ∈ χ(W )|v = 0}. The proof can be obtained.



Remark 1.43. If N  is a regular submanifold N , f : M → N is a smooth mapping, f (M ) ⊂ N  , then f : M → N  is also a smooth mapping. Let (U, ϕ), (V, ψ) be " is a induced chart of N  from N . Then, by the fact that N  is charts of N , then (V" , ψ) a regular submanifold of N , we know ψ ◦ f ◦ ϕ−1 (u) = (ψ" ◦ f ◦ ϕ−1 (u), 0). Then, the smoothness of f : M → N leads to the smoothness of f : M → N  . Remark 1.44. Let M  be a k-dimensional regular submanifold of M , then i : M  → M is the inclusion mapping. Take a submanifold chart (U, ϕ) of M that induces the " ) → ϕ(U ) has the form " , ϕ) "−1 : ϕ( "U chart (U " of M  . Then, i = ϕ ◦ i ◦ ϕ i (u1 , · · · , uk ) = (u1 , · · · , uk , 0, · · · , 0). Thus, i∗p : Tp M  → Tp M is injective, which means that the regular submanifold is definitely an immersed submanifold. 4. Embedded submanifolds Definition 1.45. Let f : M → N be an injective immersion. If f : M → f (M ) is a homeomorphism, where f (M ) has the subspace topology of N , then f (M ) is an embedded submanifold of N . Proposition 1.46. Suppose f : M → N is an embedding, then f (M ) is a regular submanifold of N , and f : M → f (M ) is a diffeomorphism.

1.1 Differentiable Manifolds

51

Proof. Since f is an embedding, f is an immersion ,∀ q ∈ f (M ), ∃ p ∈ M so that f (p) = q. Let charts (U, ϕ), (V, ψ), p ∈ U, f (p) ∈ V so that ϕ(p) = O ∈ Rm , ψ(q) = O ∈ Rn , f (U ) ⊂ V , and f(u1 , · · · , um ) = (u1 , · · · , um , · · · , 0). Since f : M → f (M ) is a homeomorphism, if U is an open subset of M , then f (U ) is an open subset of f (M ), and there exists an open subset W1 ⊂ N so that f (U ) = W1 ∩ f (M ). Denote W = V ∩ W1 , χ = ψ|W . Then, χ(q) = O ∈ Rn , and χ(W ∩ f (M )) = {(u1 , · · · , un ) ∈ χ(W ) | um+1 = · · · = un = 0}, i.e., (W, χ) is a submanifold chart of N that contains q, which also means that f (M ) $, χ is a regular submanifold of N . Let (W ") be a chart of f (M ) induced from (W, χ). −1 Then from χ ◦ f ◦ ϕ (u) = (" χ ◦ f ◦ ϕ−1 (u), 0), we conclude that f : M → f (M ) is a diffeomorphism.  Remark 1.47. If f is an immersion, then we can appropriately choose the charts of M, N , such that f has the local expression f(u1 , · · ·, um ) = (u1 , · · · , um , · · · , 0). Therefore, it is easy to see that f can be an injective immersion in the neighborhood U of p, and f : U → f (U ) is a homeomorphism. Obviously, f (U ) has the induced subspace topology from N . Therefore, f | U : U → N is an embedding. Definition 1.48. Let X, Y be two topological spaces, and f : X → Y be continuous. If for every compact subset K in Y , we have f −1 (K) to be a compact subset in X, then f is said to be a proper mapping. Proposition 1.49. Let f : M → N be an injective immersion. If f is a proper mapping, then f is an embedding. Proof. It would be sufficient to prove f −1 : f (M ) → M is continuous. Assume there exist an open set W of M and a sequence of points {qi } of f (M ) s.t. qi ∈ / f (W ),but {qi } converges to some point qi of f (W ). Denote pi = f −1 (qi ), p0 = f −1 (q0 ), p0 ∈ W . Since {q0 , qi }(i = 1, 2, · · ·) is compact, and f is a proper mapping, {p0 , pi }(i = 1, 2, · · ·) is a compact set of M . Let p1 ∈ M be the convergence of {pi }. Since f is continuous, {f (pi )} converges to f (p1 ), i.e., f (p1 ) = f (p0 ). Thus, p0 = p1 . Therefore, when i is large enough, there exists pi ∈ W , and qi = f (pi ) ∈ f (W ). This / f (W ).  is in contradiction with qi ∈ Remark 1.50. Let f be an injective immersion. If M is compact, then f is a proper mapping. By Proposition 1.49, f is an embedding.

1.1.4 Submersion and Transversal Below we will discuss the local property of f when f∗p : Tp M → Tf (p) N is surjective. Definition 1.51. f is smooth, p ∈ M . If f∗p : Tp M → Tf (p) N is surjective, then f is a submersion at p; if f is a submersion at every p ∈ M , then f is said be a submersion. Similar to the proposition for f that immerses at p, we have the following proposition.

52

1. Preliminaries of Differentiable Manifolds

Proposition 1.52. Given a smooth f and p ∈ M , if f submerses at p, then there exists chart (U, ϕ) on M about p and (V, ψ) on N about f (p) ∈ N in which f = ψ ◦ f ◦ ϕ−1 : ϕ(U ) → ψ(V ) has the form f(u1 , · · · , um ) = (u1 , · · · , un ). Proof. Take charts (U1 , ϕ1 ), (V, ψ1 ), p ∈ U1 , f (p) ∈ V, ϕ1 (p) = O ∈ Rm , ψ1 (f (p))    ∂ fi  = O ∈ Rn and f (U1 ) ⊂ V . Since f is a submersion, Jf(O) =  has rank ∂uj O n, where f = (f1 , · · · , fn ). We assume that the first n rows of Jf(O) are linearly independent. Let g : ϕ1 (U1 ) → ψ(V ) × Rm−n satisfy g(u1 , · · · , um ) = (f(u1 , · · · , um ), un+1 , · · · , um ). ! Jf(O) Then, g(O) = O, and Jg (O) = has rank n. By the inverse function O Im−n theorem, g(O) maps a neighborhood W at O diffeomorphically to a neighborhood of −1 g(W ) ⊂ ψ(V ) × Rm−n . Let U = W ∩ U1 , ϕ = g ◦ (ϕ1 |U ), then ψ ◦ f ◦ ϕ−1 = 1 ◦g −1 m n 1 m 1 n β ◦ g ◦ g = β, and β : R → R , β(u , · · · , u ) = (u , · · · , u ) is a projection  from Rm → Rn . Remark 1.53. By Definition 1.51, if f : M → N is a submersion at p ∈ M , then f is a submersion in some neighborhood of p. Remark 1.54. If f : M → N is a submersion, then f is an open mapping (i.e., open set mapping to an open set). Furthermore, f (M ) is an open subset of N . Let G be an open subset of M , ∀ q ∈ f (G). There exists a p ∈ G, s.t. f (p) = q. Since f is a submersion, there exist charts (U, ϕ), (V, ψ), p ∈ U, q ∈ V , s.t. U ⊂ G, and f : ϕ(U ) → ψ(V ), f(u1 , · · · , um ) = (u1 , · · · , un ). Let H = β(ϕ(U )), where β(u1 , · · · , um ) = (u1 , · · · , un ), s.t. H ⊂ ψ(V ). Thus, ψ −1 (H) is a neighborhood of q ∈ N , ψ −1 (H) ⊂ f (G), i.e., f (G) is an open subset of N . Next, we consider under what condition would f −1 (q0 ) be a regular submanifold of M , and ∀ q0 ∈ N be fixed. Definition 1.55. Given f : M → N is smooth, p ∈ M , if f∗p : Tp M → Tf (p) N is surjective, then p is said to be a regular point of f (i.e., f submerses at p), otherwise p is said to be a critical point of f , and q ∈ N is called a regular value of f , if q ∈ / f (M ) or q ∈ f (M ), but each p ∈ f −1 (q) is a regular point of f ; otherwise, q is called a critical value of f . Remark 1.56. When dim M < dim N , as a result of dim Tp M = dim M < dim N = dim Tf (p) N , for q ∈ f (M ), p ∈ f −1 (q), p cannot be a regular point of f . Hence, q ∈ N is a regular value of f ⇔ q ∈ / f (M ). Theorem 1.57. Let f : M → N be smooth, q ∈ N ; if q is a regular value of f , and f −1 (q) = Ø, then f −1 (q) is an (m − n)-dimensional regular submanifold of M . Moreover, ∀p ∈ f −1 (q), Tp {f −1 (q)} = ker f∗p .

1.1 Differentiable Manifolds

53

Proof. Since q is a regular value, ∀ p ∈ f −1 (q), f submerses at p by definition. By the Proposition 1.52, there exist charts (U, ϕ), (V, ψ), p ∈ U, f (p) = q ∈ V, ϕ(p0 ) = O ∈ Rm , ψ(q) = O ∈ Rn , and ψ ◦ f ◦ ϕ−1 (u1 , · · · , um ) = f(u1 , · · · , um ) = (u1 , · · · , un ), ϕ{U ∩ f −1 (q)} = {(u1 , · · · , um ) ∈ ϕ(U ) | u1 = · · · = un = 0}, i.e., (U, ϕ) is a submanifold chart of M that contains p. Therefore, f −1 (q) is a regular submanifold of M , and dim f −1 (q) = m − n. Note that f |f −1 (q) : f −1 (q) → M, f |f −1 (q) = f ◦ i, i : f −1 (q) → M is an inclusion mapping. Since f |f −1 (q) = q is a constant mapping, f∗p ◦i∗p = (f |f −1 (q) )∗p = 0, i.e., i∗p (Tp {f −1 (q)}) ⊂ ker f∗p . Furthermore, because q is a regular value of f , f∗p (Tp M ) = Tq N , and dim ker f∗p = dim Tp M − dim f∗p (Tp M ) = m − n =  dim f −1 (q). Therefore, we have Tp {f −1 (q)} = ker f∗p . Remark 1.58. Given f : M → N is smooth, dim M = dim N, M is compact. If q ∈ N is a regular value of f , then f −1 (q) = Ø or f −1 (q) consists of finite points. By Theorem 1.57, if f −1 (q) = Ø, then f −1 (q) is a 0-dimensional regular submanifold of M . By definition, we have ϕ(U ∩ f −1 (q)) = O ∈ Rm , i.e., every point in f −1 (q) is an isolated point. Moreover due to the compactness of f −1 (q), f −1 (q) must consist of finite points. Below, we give some applications of Theorem 1.57. Example 1.59. Let f : Rn+1 → R, and f (u1 , · · · , un+1 ) =

n+1 

(ui )2 .

i=1

From the Jacobian matrix of f at (u1 , · · · , un+1 ), we know f is not a submersion at (u1 , · · · , un+1 ) ⇔ u1 = · · · = un+1 = 0. Therefore, any non-zero real number is a regular value of f . According to the Theorem 1.57, the n-dimensional unit sphere S n = f −1 (1) is an n-dimensional regular submanifold on Rn+1 .  Example 1.60. Let f : R3 → R, and f (u1 , u2 , u3 ) = (a − (u1 )2 + (u2 )2 )2 + (u3 )2 , a > 0. The assumption tells us that any non-zero real number is a regular point of f . Then, 0 < b2 < a2 is a regular value of f . Therefore, by Theorem 1.57, T 2 = f −1 (b2 ) is a 2-dimensional regular submanifold on R2 . If M  is a regular submanifold of M , then dim M − dim M  =codim M  is called the M -codimension of M  . Denote M  = {p ∈ M | fi (p) = 0 (i = 1, · · · , k)} and consider the mapping F : M −→ Rk ,

F (p) = (f1 (p), · · · , fk (p)).

If fi : M → R is smooth, then F is smooth too, and M  = F −1 (O). Proposition 1.61. Suppose M  is a subset of M . Then, M  is a k-codimensional regular submanifold of M if and only if for all q ∈ M  , there exists a neighborhood U of q ∈ M and a smooth mapping F : U → Rk , s.t. 1◦ U ∩ M  = F −1 (O). 2◦ F : U → Rk is a submersion.

54

1. Preliminaries of Differentiable Manifolds

Proof. Necessity. By the definition of the regular submanifold, if M  is a k-codimensional regular submanifold of M , then ∀p ∈ M  , there exists a submanifold chart (U, ϕ) of M that contains p s.t. ϕ(p) = O ∈ Rm , and ϕ(U ∩ M  ) = {(u1 , · · · , um ) ∈ ϕ(U ) | um−k+1 = · · · = um = 0}. Let us denotes the projection by π : Rm = Rm−k × Rk → Rk , let F = π ◦ ϕ : U → Rk . Then, F is smooth, and F −1 (O) = (π ◦ ϕ)−1 (O) = U ∩ M  , F∗q = π∗ϕ(q) ◦ ϕ∗q . Since ϕ∗q is an isomorphism and π∗ϕ(q) is surjective, F submerses at q. Sufficiency. If ∀ q ∈ U ∩ M  , F submerses at q, then O ∈ Rk is a regular value of F . By Theorem 1.57, F −1 (O) is a k-codimensional regular submanifold of U , i.e.,  M  is a k-codimensional regular submanifold of M . We know that if q ∈ N is a regular value of f : M → N , and f −1 (q) = Ø, then f −1 (q) is a regular submanifold of M . Assume that Z is a regular submanifold of N . Then, under what condition would f −1 (Z) be a regular submanifold of M ? For this question, we have the following definition. Definition 1.62. Suppose Z is a regular submanifold of N , f : M → N is smooth, p ∈ M . Then, we say f is transversal to Z at p, if f (p) ∈ / Z or when f (p) ∈ Z has f∗p Tp M + Tf (p) Z = Tf (p) N, denoted by f p Z. If ∀p ∈ M , f p Z, then f is transversal to Z, denoted by f Z. Remark 1.63. If dim M + dim Z < dim N , then f Z ⇔ f (M ) ∩ Z = Ø; if q ∈ N is a regular value of f , then ∀p ∈ f −1 (q), f p Z; if f : M → N is a submersion, then for any regular submanifold Z of N , f Z. For transversality, we focus on its geometric property. Example 1.64. M = R, N = R2 , Z is x-axis in R2 , f : M → N, f (t) = (t, t2 ). When t = 0, as a result of f (t) ∈ / Z, f t Z;      d  ∂ ∂  d  , 2 = , When t = 0, Jf (0) = (1, 0) , note that f∗0  =(1, 0) 1 dt 0 ∂u ∂u d u1 f∗0 T0 M = T(0,0) Z. Therefore, f∗0 T0 M + T(0,0) Z = T(0,0) N is impossible to establish. Thus, f is not transversal to Z at 0. However, if we change f to f (t) = (t, t2 − 1), we obtain: /   t = ±1, f (t) ∈  when d ∂   = Z, so f t Z; when t = 1, Jf (1) = (1, 2) , therefore f∗1 +   dt 1 ∂u1 (1,0)  ∂  , i.e., f∗1 T1 M + T(1,0) Z = T(1,0) N , and hence f 1 Z. Similarly, we have 2 2 ∂u

(1,0)

f −1 Z. Thus, f Z. Submanifold transverse: Let Z, Z  be two regular submanifolds of N , i : Z  → N is an inclusion mapping. If iZ, then submanifold Z  is transversal to Z, denoted as Z  Z. If Z  Z, ∀p ∈ Z ∩ Z  , by definition, we have i∗p (Tp Z  ) + Tp Z = Tp N, i.e.,

1.1 Differentiable Manifolds

55

Tp Z  + Tp Z = Tp N. We assume that f : M → N is smooth, and Z is a k-codimensional regular submanifold of N , p ∈ M, f (p) = q ∈ Z. According to the Proposition 1.61, there exists a submanifold chart (V, ψ) of N that contains q, s.t. π ◦ ψ : V → Rk is a submersion, and Z ∩ V = (π ◦ ψ)−1 (O). Now, take a neighborhood of p in M , s.t., f (U ) ⊂ V , then π ◦ ψ ◦ f : U → Rk . Proposition 1.65. f p Z ⇔ π ◦ ψ ◦ f : U → Rk submerses at p. Proof. Since π ◦ ψ submerses at f (p), O ∈ Rk is a regular value of π ◦ ψ. From Z ∩ V = (π ◦ ψ)−1 (O), we know for every q ∈ Z ∩ V , there exists a (π ◦ ψ)∗q Tq N = To Rk . By Theorem 1.57, ker(π ◦ ψ)∗q = Tq Z. Therefore, f∗p Tp M + Tq Z = Tq N ↔ (π◦ψ)∗q (f∗p Tp M ) = To Rk ↔ (π◦ψ◦f )∗p (Tp M ) = To Rk , i.e., π◦ψ◦f submerses at p.  Remark 1.66. Extending from the conclusion of Proposition 1.65, we have f Z ↔ O ∈ Rk are regular values of π ◦ ψ ◦ f : U → Rk . Remark 1.67. Since f p Z, i.e., π ◦ ψ ◦ f : U → Rk submerses at p. By Proposition 1.52, we can choose a coordinate chart s.t. π ◦ ψ ◦ f ◦ ϕ−1 : ϕ(U ) → Rk has the form (π ◦ ψ ◦ f ◦ ϕ−1 )(u1 , · · · , um ) = (um−k+1 , · · · , um ). Then, f = ψ ◦ f ◦ ϕ−1 can be represented by f = (η1 (u1 , · · · , um ), · · · , ηn−k (u1 , · · · , um ), um−k+1 , · · · , um ). Theorem 1.68 (Extension of Theorem 1.57). Suppose f : M → N is smooth, Z is a k-codimensional regular submanifold of N . If f Z and f −1 (Z) = Ø, then f −1 (Z) is a k-codimensional regular submanifold of M , and ∀p ∈ f −1 (Z), −1 Tp {f −1 (Z)} = f∗p {Tf (p) Z}.

Proof. ∀ p ∈ f −1 (Z), there exists q ∈ Z, denoted by q = f (p). Since Z is a kcodimensional regular submanifold of N , there exists a submanifold chart (V, ψ) of N that contains p. Let U = f −1 (V ). From f Z, we know that O ∈ Rk is a regular value of π ◦ ψ ◦ f , and U ∩ f −1 (Z) = (π ◦ ψ ◦ f )−1 (O). By Theorem 1.57, U ∩ f −1 (Z) is a k-codimensional regular submanifold of U , and Tp {f −1 (Z)}

= ker(π ◦ ψ ◦ f )∗p −1 = f∗p {(π ◦ ψ)−1 ∗q (O)} −1 {Tf (p) Z}. = f∗p

The theorem is proved.



56

1. Preliminaries of Differentiable Manifolds

1.2 Tangent Bundle The tangent bundle of a differentiable manifold M is the disjoint union of the tangent spaces of M . It is useful, in distinguishing between the tangent space and bundle, to consider their dimensions, n and 2n respectively. In other words, the tangent bundle accounts for dimensions in the positions in the manifold as well as directions tangent to it. Since we can define a projection map, for each element of the tangent bundle giving the element in the manifold whose tangent space the first element lies in, tangent bundles are also fiber bundles.

1.2.1 Tangent Bundle and Orientation In this section, we will discuss two invariable properties under diffeomorphism– tangent bundle and orientation. 1. Tangent Bundle Definition 2.1. The Triple (T M, M, π) is called tangent bundle # of differentiable manifold M (sometimes simply called T M ), where T M = Tp M , projection map p∈M

π : T M → M satisfies π(Xp ) = p, ∀ Xp ∈ T M . For every p ∈ M, π −1 (p) = Tp M is called fiber at p of tangent bundle T M . Proposition 2.2. Let M be an m-dimensional differentiable manifold, then T M is a 2m-dimensional differentiable manifold, and π : T M → M is a submersion. Proof. Let (U, ϕ) be a chart on M , and its coordinate function be x1 , · · · , xm . Then,  ∂  ai i  . Define ϕU : π −1 (U ) → ϕ(U ) × Rm , s.t. ∀ Xp ∈ π −1 (U ), Xp = ∂x p i ϕU (Xp ) = (ϕ(p); a1 , · · · , am ), obviously ϕU is a 1 to 1 mapping. Note that as (U, ϕ) takes all the charts on M , all the corresponding π−1 (U ) constitutes a covering of T M . Hence, if the topology of π −1 (U ) is given, the subset of π −1 (U ) is open, iff the image of ϕU is an open set of ϕ(U ) × Rm . It is easy to prove that by the 1 to 1 correspondence of ϕU , the topology of ϕU on the Rm ×Rm = R2m subspaces can be lifted on π −1 (U ). The topology on T M can be defined as follows: W is called an open subset of T M , iff W ∩ π −1 (U ) is an open subset of π −1 (U ). It is easy to deduce that T M constitutes a topological space that satisfies the following conditions: 1◦ T M is a Hausdorff space that has countable bases. 2◦ π −1 (U ) is an open subset of T M , and ϕU is a homeomorphism from π −1 (U ) to an open subset of R2m . Furthermore, it can be proved that the manifold structure on T M can be naturally induced from the manifold structure on M . We say that {(π −1 (U ), ϕU )} = A is a smooth atlas of T M . For any chart (π−1 (U ), ϕU ), there exists a (π −1 (V ), ψV ) ∈

1.2 Tangent Bundle

57

A, and π −1 (U ) ∩ π −1 (V ) = Ø. Let x1 , · · · , xm and y 1 , · · · , y m be the coordinate functions of the charts (U, ϕ), (V, ψ). Then,  & ∂  ∂xi ϕ−1 (u) i   %% & & ∂y j  ∂  ai i  = ψV ∂x ϕ−1 (u) ∂y j ψ◦ϕ−1 (u) j i

1 m ψV ◦ ϕ−1 U (u; a , · · · , a ) = ψV

%

ai

% =

−1

ψ◦ϕ

(u);

 i

i ∂y

1 

 a ∂xi 

ϕ−1 (u)

,···,



a

i

m 

&  . ∂x ϕ−1 (u)

i ∂y

i

It is easy to conclude that T M is a 2m-dimensional manifold, A is a differentiable structure on T M . From the charts (U, ϕ) of M and (π −1 (U ), ϕU ) of T M , we know m π  = ϕ ◦ π ◦ ϕ−1 U : ϕ(U ) × R → ϕ(U ) has the form: π (u; a1 , · · · , am ) = u. 

By the definition of submersion, π is a submersion.

Given below are examples of two trivial tangent bundles (if there exists a diffeomorphism from its tangent bundle T M to M × Rm , and this diffeomorphism limited on each fiber of T M (Tp M ) is a linear isomorphism from Tp M to {p} × Rm ). Example 2.3. Let U be an open subset of Rm and T U  U × Rm .     ∂  ∂  (i = 1, · · · , m) is the basis of ∀ Xu ∈ T U, Xu = ai i  , where ∂ui u ∂u u i Tu U . Then, it is easy to prove that Xu −→ (u; a1 , · · · , am ) is a diffeomorphism from T U to U × Rm . Moreover, since each fiber Tu U of T U is a linear space, maps limited on Tu U is a linear isomorphism from Tu U to {u} × Rm , i.e., T U is a trivial tangent bundle. Example 2.4. T S 1 is a trivial tangent bundle, i.e., T S 1  S 1 × R. Let A={(U, ϕ), (V, ψ)} be a smooth atlas on S 1 , where U = {(cos θ, sin θ)|0 < θ < 2π}, ϕ(cos θ, sin θ) = θ, V = {(cos θ, sin θ)| − π < θ < π}, ψ(cos θ, sin θ) = θ, ' θ, 0 < θ < π, −1 ψ ◦ ϕ (θ) = θ − 2π, π < θ < 2π. Define f : T S 1 → S 1 × R, s.t. ⎧ ⎪ ⎪ ⎨ (p; a), p ∈ U, f (Xp ) = ⎪ ⎪ ⎩ (p; b), p ∈ V,



Xp = a

∂   , ∂x p

Xp = b

∂   , ∂y p



58

1. Preliminaries of Differentiable Manifolds

where x, y are the coordinate functions on (U, ϕ), (V, ψ) respectively. When p ∈ U ∩ V , we have ∂y  ∂  ∂  ∂   =   =  . ∂x p ∂x p ∂y p ∂y p Therefore, f has the definition and is a 1 to 1 correspondence. Moreover, f and f −1 are smooth. Hence, T S 1 is a trivial tangent bundle. Apart from trivial tangent bundles, there exists a broad class of nontrivial tangent bundles. For an example, T S 2 is a nontrivial tangent bundle. Definition 2.5. Let f : M → N be smooth. Define a mapping T f : T M → T N , s.t. T f |Tp M = f∗p ,

∀ p ∈ M,

then T f is called the tangent mapping of f . Remark 2.6. ∀ Xp ∈ Tp M , there exist charts (U, ϕ) on M about p and (V, ψ) on N about f (p), s.t. f (U ) ⊂ V . By π1 : T M → M, π2 : T N → N , it is naturally derived that (π1−1 (U ), ϕU ), (π2−1 (V ), ψV ) are charts on T M, T N , and T f (π1−1 (U )) ⊂ π2−1 (V ). Note that 1 m ψV ◦ T f ◦ ϕ−1 U (u; a , · · · , a )    ∂f1   ∂fn  , = ψ ◦ f ◦ ϕ−1 ai i  −1 , · · · , ai i  −1 U ; ∂x ϕ (u) ∂x ϕ (u) i i

which may be simplified as    ˆ ψV ◦ T f ◦ ϕ−1 U (u; α) = f (u); Df (u)α , where α = (a1 , · · · , am ). Therefore, T f is a smooth mapping. Remark 2.7. Let M, N, L be the differentiable manifolds. By the definition of tangent mapping, if f : M → N and g : N → L are smooth, then T (g ◦ f ) = T g ◦ T f. Remark 2.8. If f : M → N is a diffeomorphism, then T f : T M → T N is also a diffeomorphism. 2. Orientation Next, we introduce the concept of orientation for differentiable manifolds. Given V as a m-dimensional vector space, {e1 , · · · , em }, {e1 , · · · , em } as V ’s two m  aij ei (j = 1, · · · , m), then ordered bases, if ej = i=1

(e1 , · · · , em ) = (e1 , · · · , em )A,

1.2 Tangent Bundle

59

where A = (aij )m×m . If det A > 0, we call {ei } and {ej } concurrent; otherwise, if det A < 0, we call {ei } and {ej } reverse. Then, a direction μ of V can be expressed by a concurrent class [{ej }] equivalent to {ej }. The other direction −μ can be expressed by an equivalent class to the reverse direction of {ej }. (V, μ) is called an orientable vector space. Let (V, μ), (W, ν) be two orientable vector spaces. A : V → W is a linear isomorphism from V to W . If the orientation of W , which is induced by A, is consistent with ν, i.e., Aμ = ν, then A preserves orientations. Otherwise, A reverses orientations. In the below section, we extend the orientation concept to differentiable manifolds. Definition 2.9. Let M be an m-dimensional differentiable manifold, ∀p ∈ M, μp is the orientation of Tp M , s.t. ϕ∗q : (Tq M, μq ) −→ (Tϕ(q) Rm , νϕ(q) ),

∀q ∈ U

are all linear isomorphisms that preserves orientations, where (U, ϕ) is a chart that contains p, and  ∂   ∂   νϕ(q) = , · · · , .   ∂u1 ϕ(q) ∂um ϕ(q) Then, μ = {μp | p ∈ M } is the orientation on M , and (M, μ) is called an orientable differentiable manifold. Remark 2.10. The Definition 2.9 shows that if (M, μ) is an orientable differentiable manifold, W is an open subset of M , then ∀p ∈ M and there exists an orientation μp of Tp M . This gives an orientation on W , denoted by μ|W . Then, (W, μ|W ) is also an orientable differentiable manifold. Specifically, if (U, ϕ) is a chart on M , then (U, μp ) is an orientable differentiable manifold. Remark 2.10 shows that M may be locally orientable. Next, we discuss how to construct a global orientation. Proposition 2.11. Let M be an m-dimensional differentiable manifold, then M is orientable, iff there exists a smooth atlas, A = {(Uα , ϕα )}, on M , s.t. ∀ (Uα , ϕα ), (Uβ , ϕβ ) ∈ A, if Uα ∩ Uβ = Ø, then det Jϕβ ◦ϕ−1 (ϕα (q)) > 0, α

∀ q ∈ Uα ∩ Uβ ,

where Jϕβ ◦ϕ−1 (ϕα (q)) is the Jacobian matrix of ϕβ ◦ ϕ−1 α at ϕα (q). α Proof. Necessity. Since M is orientable, select one of the orientations of M , μ = {μp | p ∈ M }. According to Definition 2.9, ∀p ∈ M , there exists a chart (U, ϕ) on M about p, s.t. ∀q ∈ U ,  ∂   ∂   , · · · , . ϕ∗q μq =   ∂u1 ϕ(q) ∂um ϕ(q) Denote a set consisting of all such charts by A. Then, A is a smooth atlas of M , and the properties of A described in the proposition are easy to prove.

60

1. Preliminaries of Differentiable Manifolds

Sufficiency. Let A be an atlas that satisfies all the properties of the proposition. Choose (Uα , ϕα ), (Uβ , ϕβ ) ∈ A and Uα ∩ Uβ = Ø, and use x1 , · · · , xm and y 1 , · · · , y m to represent the coordinate functions of (Uα , ϕα ), (Uβ , ϕβ ) respectively. Note that  ∂  ∂    ∂  ∂    −1 (ϕα (q)), = , · · · , , · · · ,     J ∂x1 p ∂xm p ∂y 1 p ∂y m p ϕβ ◦ϕα (ϕα (q)) > 0, we have and by supposition Jϕβ ◦ϕ−1 α  ∂  ∂    ∂  ∂    , · · · , , · · · , =     , ∂x1 p ∂xm p ∂y 1 p ∂ym p 

i.e., M is orientable.

Remark 2.12. If f : M → N is a diffeomorphism, f A = {f (Uα ), ϕα ◦ f −1 } is a smooth atlas. Pick two charts on N , (f (ϕα ), ϕα ◦ f −1 ), (f (ϕβ ), ϕβ ◦ f −1 ), we have det Jϕβ ◦f −1 ◦f ◦ϕ−1 (ϕα (q)) = det Jϕβ ◦ϕ−1 (ϕα (q)), ∀ q ∈ Uα ∩ Uβ . If M is α α orientable, then N is possible, which means orientation is an invariable property under diffeomorphism. Proposition 2.13. Let M be a connected differentiable manifold; if M is orientable, then M has only two orientations. Proof. If μ = {μp | p ∈ M } is an orientation of M , then −μ is also an orientation. Therefore, M has at least two orientations. Assume there exists another orientation, denoted as ν = {νp | p ∈ M }. Let S = {p ∈ M | μp = νp }. ∀p ∈ S, take charts (U, ϕ), (V, ψ) of M about p, s.t. μ, ν satisfy all the requirements of Definition 2.9. As a result of μp = νp , we have det Jψ◦ϕ−1 (ϕ(p)) > 0. By continuity, there exists a neighborhood of ϕ(p), W ⊂ ϕ(U ∩ V ), s.t. det Jψ◦ϕ−1 (ϕ(u)) > 0,

∀ u ∈ W.

Denote O = ϕ−1 (W ). Then, O is a neighborhood of p in M , and O ⊂ S, i.e., S is an open subset of M . Similarly, M \S is also an open subset of M . Since M is connected, we have either S = Ø or S = M . If S = Ø, then μ = −ν; if S = M , then μ = ν.  Remark 2.14. By the Proposition 2.13, any connected open set on an orientable differentiable manifold M has two and only two orientations. Remark 2.15. Let (U, ϕ), (V, ψ) be two charts on M , and U and V be connected. If U ∩ V = Ø, then det Jψ◦ϕ−1 preserves the orientation on ϕ(U ∩ V ). Example 2.16. S 1 is an orientable differential manifold. Let the smooth atlas of S 1 be A = {(U+ , ϕ+ ), (U− , ϕ− )}, where

1.2 Tangent Bundle

U+ = S 1 \{(0, −1)},

61

U− = S 1 \{(0, 1)},

ϕ± : U± → R, s.t. ϕ+ (u1 , u2 ) = Since

u1 , 1 + u2

1 ϕ+ ◦ ϕ−1 − (u) = − , u

we have det Jϕ+ ◦ϕ−1 (u) = −

ϕ− (u1 , u2 ) =

−u1 . u2 − 1

∀ u ∈ ϕ− (U+ ∩ U− ),

1 > 0, u2

∀ u ∈ ϕ− (U+ ∩ U− ).

Similarly det Jϕ− ◦ϕ−1 (u) > 0, +

∀ u ∈ ϕ+ (U+ ∩ U− ),

i.e., S 1 is orientable. Example 2.17. M¨obius strip is a non-orientable surface. Define equivalent relation“∼” on [0, 1] × (0, 1): (u, v) ∼ (u, v),

0 < u < 1,

0 < v < 1,

(0, v) ∼ (1, 1 − v), 0 < v < 1, [0, 1] × (0, 1)\ ∼ is a M¨obius strip, A = {(U, ϕ), (V, ψ)} is its smooth atlas   1 U = M \{0} × (0, 1), V = M\ × (0, 1), 2   1 1 × (0, 1), ϕ : U −→ (0, 1) × (0, 1), ψ : V −→ − , 2 2

which satisfies: ϕ(u, v) = (u, v), ⎧ ⎪ ⎨ (u, v), ψ(u, v) = ⎪ ⎩ (u − 1, 1 − v),

1 2

0≤ , 1 < u ≤ 1, 2

  1 × (0, 1), (u, v) ∈ 0, 2 −1 ψ ◦ ϕ (u, v) =   ⎪ ⎩ (u − 1, 1 − v), (u, v) ∈ 1 , 1 × (0, 1), ⎧ ⎪ ⎨ (u, v),

2

i.e.,

  1 × (0, 1), (u, v) ∈ 0, 2 det Jψ◦ϕ−1 (u, v) =   ⎪ 1 ⎩ −1, (u, v) ∈ , 1 × (0, 1). ⎧ ⎪ ⎨ 1,

2

By the Remark 2.15, M¨obius strip is a nonorientable surface.

62

1. Preliminaries of Differentiable Manifolds

Definition 2.18. Let M, N be two orientable differential manifolds, and f : M → N be a local diffeomorphism ( diffeomorphism in the neighborhood of any p ∈ M ). If for every p ∈ M , there exists a f∗p : Tp M → Tf (p) N that preserves (or reverses) the orientation, then f is said to preserve the orientation (or reverse the orientation). Proposition 2.19. f : M → N is a diffeomorphism; if M is a connection, then f preserves the orientation or reverses the orientation. Proof. Let S = {p ∈ M | f∗p : Tp M → Tf (p) N preserves the orientation}. ∀ p ∈ S, because f∗p preserves orientation, det Jf (p) > 0. From the continuity, there exists U , s.t. det Jf (q) > 0, ∀ q ∈ U , i.e., U ⊂ S. Similarly, M \S is also an open subset of M . Since M is connected, S = Ø or S = M . When S = Ø, then f preserves the inverse orientation, otherwise (S = M ) f preserves the orientation. 

1.2.2 Vector Field and Flow Similar to Euclidean space, differentiable manifold also has the concept of vector field and curve of solution. Definition 2.20. Let M be a differentiable manifold. If map X : M → T M has the property π ◦ X = I : M → M , then X is said to be a vector field of M , and is also called a section in the tangent bundle T M , where π : T M → M is a projection. If the map X is smooth, then X is called a smooth vector field. Proposition 2.21. X is a smooth vector field on M , iff for every f ∈ C ∞ (M ) there exists a Xf ∈ C ∞ (M ), and C ∞ (M ) = {all smooth f unctions on M }, Xf : M → R, Xf (p) = Xp (f ), ∀ p ∈ M, f ∈ C ∞ (M ). Proof. Necessity. Suppose (U, ϕ) is a chart of M , and (π −1 (U ), ϕU ) is an induced natural chart of T M . Suppose X can be expressed as  ∂  Xp = ai (p) i  , ∀ p ∈ U, ∂x p i by

 = ϕU ◦ X ◦ ϕ−1 : ϕ(U ) −→ ϕ(U ) × Rm , X  X(u) = (u; a1 ◦ ϕ−1 (u), · · · , am ◦ ϕ−1 (u)),

we know, if X is smooth, then a1 , · · · , am are smooth too. Since  ∂f  (Xf )(p) = Xp f = ai (p) i  , ∀ p ∈ U, ∂x p i (Xf )|U is smooth. Sufficiency. ∀ p ∈ M , let (U, ϕ) be a chart on M about p, where its coordinate "i on the entire M , and satisfy x "i = function x1 , · · · , xm may be expanded to smooth x i x on some neighborhood of p on V ⊂ U . Then,

1.2 Tangent Bundle

Xq =



Xq (" xi )

i

∂   , ∂xi q

63

∀ q ∈ U, 

xi ) is smooth, i.e., X is also smooth. by the supposition Xq ("

Definition 2.22. Let X be a smooth vector field on the differentiable manifold M . The solution curve of X through p refers to a smooth mapping c : J → M s.t. c(0) = p, and   d c∗t  = Xc(t) , ∀ t ∈ J, dt

t

i.e., the velocity vector at t of a smooth curve c is exactly the value of the vector field at p on M . Proposition 2.23. Let f : M → N be a diffeomorphism, X be a smooth vector field on M . If we denote f∗ X = T f ◦ X ◦ f −1 : N → T N , then f∗ X is a smooth vector field on N , and c is a solution curve of X through p ∈ M , iff f ◦ c is a solution curve of f∗ X through f (p). Proof. By the definition of tangent mapping, we have π2 ◦ (f∗ X) = π2 ◦ (T f ◦ X ◦ f −1 ) = f ◦ (π1 ◦ X) ◦ f −1 = I. Since f −1 , X, T f are smooth, f∗ X is a smooth vector field on N . If c : J → M is a solution curve of X through p, then f ◦ c(0) = f (0) = f (p), and     d  d  = f∗c(t) ◦ c∗t (f ◦ c)∗t   dt

dt

t

t

= f∗c(t) (Xc (t)) = (f∗ X)f ◦c(t) ,

∀ t ∈ J. 

Therefore the proposition is completed.

Remark 2.24. Let X be a smooth vector field on the differentiable manifold M , and (U, ϕ) be a chart on M . By Proposition 2.23, we have ϕ∗ (X | U ) to be a smooth vector field of ϕ(U ). Remark 2.25. If ϕ∗ (X | U ) has an expression {ϕ∗ (X | U )}u =

m  i=1

then

ai (u)

∂   , ∂ui u

∀ u ∈ ϕ(U ),

m ∂  ∂(ϕ ◦ c)i  ∂   (ϕ ◦ c)∗t ,   =  ∂t t ∂t t ∂ui ϕ◦c(t) i=1

∀ t ∈ J,

where (ϕ ◦ c)i is the i-th component of ϕ ◦ c. Therefore, according to Proposition 2.23, there exists a solution curve, c : J → U , of X through p, iff ϕ ◦ c is a solution of

64

1. Preliminaries of Differentiable Manifolds

⎧ i ⎨ d u = ai (u1 , · · · , um ), ⎩

dt

i = 1, · · · , m,

u(0) = ϕ(p).

Strictly speaking, a vector field on Rn is a mapping A : Rn → T (Rn ),i.e., ∀ x ∈ Rn ,

A(x) ∈ Tx Rn .

Since (e1 )x , · · · , (en )x form a basis on Tx Rn , we can write fore n  A(x) = Ai (x)(ei )x .



 ∂ , · · · , ∂ , there∂x1 ∂xn

i=1

If Ai (x) ∈ C ∞ , then A(x) is called a smooth vector field on Rn . The set of all smooth vector fields on Rn is denoted by X (Rn ). For any vector A(x), B(x) ∈ X (Rn ), define: (αA + βB)(x) = αA(x) + βB(x), α, β ∈ R, (f A)(x) = f (x)A(x),

f (x) ∈ C ∞ (Rn ).

Then, X (Rn ) is a C ∞ module of C ∞ -vector on Rn . If we denote A(x) ∈ X (Rn ) by (A1 (x), · · · , An (x)) , i.e., A(x) =

n 

Ai (x)(ei )x = (A1 (x), · · · , An (x)) ,

i=1

then A(x) is a C ∞ n-value function.

1.3 Exterior Product Exterior product is one of the algebraic operations. It has quite an interesting geometric background. In this section, we would like to construct a new linear space from an original linear space so that the new space has not only the linear space algebraic structure, but also a new algebraic operation — exterior product. This forms a basis for the differential form introduced later. These materials can be found in a series of books[AM78,Che53,Arn89,Ede85,Fla] . In R3 , let the vectors be a1 = a11 i + a12 j + a13 k, a2 = a21 i + a22 j + a23 k, a3 = a31 i + a32 j + a33 k, where a1 , a2 , a3 are linearly independent. Then,

1.3 Exterior Product

65

( ) V = x ∈ R3 | x = α1 a1 + α2 a2 + α3 a3 , 0 ≤ α1 , α2 , α3 ≤ 1 a spanned parallelepiped by vectors a1 , a2 , a3 . tween a1 , a2 , a3 as follows:   a11  a1 ∧ a2 ∧ a3 =  a21  a31

We introduce a new operation, ∧ bea12 a22 a32

a13 a23 a33

   .  

The geometric meaning of a1 ∧a2 ∧a3 is the orientable volume of V , where orientation means the sign of the volume is positive or negative. If the right hand law is followed, the volume has the plus sign, otherwise it has the minus sign . It is easy to see that operation ∧ satisfies the following laws: 1◦ Multilinear. Let a2 = βb + γc, b, c be vectors, β, γ be real numbers. Then, a1 ∧ (βb + γc) ∧ a3 = β(a1 ∧ b ∧ a3 ) + γ(a1 ∧ c ∧ a3 ). 2◦

Anti-commute a1 ∧ a2 ∧ a3 = −a2 ∧ a1 ∧ a3 , a1 ∧ a2 ∧ a3 = −a3 ∧ a2 ∧ a1 , a1 ∧ a2 ∧ a3 = −a1 ∧ a3 ∧ a2 .

From 2◦ we know that if a1 , a2 , a3 has two identical vectors, then a1 ∧a2 ∧a3 = 0. Example 3.1. Let e1 , e2 , e3 be a basis in R3 , which are not necessarily orthogonal, and let a1 , a2 , a3 be three vectors in R3 , which can be represented by a1 = a11 e1 + a12 e2 + a13 e3 , a2 = a21 e1 + a22 e2 + a23 e3 , a3 = a31 e1 + a32 e2 + a33 e3 . By multilinearity and anti-commutativity of ∧, after the computation, we have    a11 a12 a13    a1 ∧ a2 ∧ a3 =  a21 a22 a23  e1 ∧ e2 ∧ e3 .  a31 a32 a33  Example 3.2. Let e1 , e2 , e3 be a basis in R3 , and a1 = a11 e1 + a12 e2 + a13 e3 , a2 = a21 e1 + a22 e2 + a23 e3 . Then,

  a a1 ∧ a2 =  11 a21

   a a12  e ∧ e2 +  12 a22  1 a22

   a a13  e ∧ e3 +  13 a23  2 a23

 a11  e ∧ e1 . a21  3

The geometric significance of this formula is that the projection of the parallelepiped spanned by the pair of vectors a1 and a2 onto the coordinate plane e1 e2 , e2 e3 , e3 e1 is equal to A12 , A23 , and A31 respectively. Abstracting from the multilinearity and the anti-commutativity, which is satisfied by the operation wedge, we can obtain the concept of exterior product.

66

1. Preliminaries of Differentiable Manifolds

1.3.1 Exterior Form 1. 1- Form In this section, Rn is an n-dimensional real vector space, where the vectors are denoted by ξ, η, · · · ∈ Rn . Definition 3.3. A form of degree 1 (or a 1-form) on Rn is a linear function ω : Rn → R, i.e., ω(λ1 ξ1 + λ2 ξ2 ) = λ1 ω(ξ1 ) + λ2 ω(ξ2 ),

λ1 , λ2 ∈ R,

ξ1 , ξ2 ∈ Rn .

The set of all 1-forms on Rn is denoted by Λ1 (Rn ). For ω1 , ω2 ∈ Λ1 (Rn ), define (λ1 ω1 + λ2 ω2 )(ξ) = λ1 ω1 (ξ) + λ2 ω2 (ξ),

λ1 , λ2 ∈ R.

Then, Λ1 Rn becomes a vector space, i.e., the dual space (Rn )∗ of Rn . Let ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 1 0 0 ⎢ 0 ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ e1 = ⎢ . ⎥ , e2 = ⎢ . ⎥ , · · · , en = ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ 0 0 1 be the standard basis on Rn . x1 , x2 , · · · , xn forms the coordinate system on Rn , i.e., if ξ = a1 e1 + a2 e2 + · · · + an en , then xi (ξ) = ai , especially xi (ej ) = δij . Obviously, xi ∈ Λ1 (Rn ). For any ω ∈ Λ1 (Rn ), ω(ξ) = ω

n 

 ai ei =

i=1

n 

ai ω(ei ) =

i=1

i=1 

ω(ei )xi (ξ),

n

and so ω = ω(e1 )x1 + ω(e2 )x2 + · · · + ω(en )xn . Thus, x1 , · · · , xn is a basis on Λ1 (Rn ), Λ1 (Rn ) = {xi }i=1,···,n . Example 3.4. If F is a uniform force field on a Euclidean space R3 , then its work A on a displacement ξ is a 1-form acting on ξ, ωF (ξ) = (F, ξ) = F1 a1 + F2 a2 + F3 a3 ,

ξ = a1 e1 + a2 e2 + a3 e3

or ωF = F1 x1 + F2 x2 + F3 x3 . 2. 2-Forms Definition 3.5. An exterior form of degree 2 (or a 2-form) is a bilinear, skewsymmetric function ω2 : Rn × Rn → R, i.e., ω 2 (λ1 ξ1 + λ2 ξ2 , ξ3 ) = λ1 ω 2 (ξ1 , ξ3 ) + λ2 ω 2 (ξ2 , ξ3 ), ω 2 (ξ1 , ξ2 ) = −ω 2 (ξ2 , ξ1 ),

ξ1 , ξ2 , ξ3 ∈ Rn ,

λ1 , λ2 ∈ R.

1.3 Exterior Product

67

The set of all 2-forms on Rn is denoted by Λ2 (Rn ) = Λ2 . Similarly, if we define the sum of two 2-forms ω12 , ω22 and scalar multiplication as follows (λ1 ω12 + λ2 ω22 )(ξ1 , ξ2 ) = λ1 ω12 (ξ1 , ξ2 ) + λ2 ω22 (ξ1 , ξ2 ), ω12 , ω22 ∈ Λ2 (Rn ),

λ1 , λ2 ∈ R,

then Λ2 (Rn ) becomes a vector space on R. Property 3.6. The skew-symmetric condition ω 2 (ξ1 , ξ2 ) − ω 2 (ξ2 , ξ1 ) is equivalent to ω 2 (ξ, ξ) = 0, ∀ ξ ∈ Rn since from the latter it follows that 0

= ω 2 (ξ1 + ξ2 , ξ1 + ξ2 ) = −ω 2 (ξ1 , ξ1 ) + ω 2 (ξ1 , ξ2 ) + ω2 (ξ2 , ξ1 ) + ω 2 (ξ2 , ξ2 ) = ω 2 (ξ1 , ξ2 ) + ω 2 (ξ2 , ξ1 ).

i.e., ω 2 (ξ1 , ξ2 ) = ω 2 (ξ2 , ξ1 ). Example 3.7. Let S(ξ1 , ξ2 ) be the oriented area of the parallelogram constructed on the vector ξ1 and ξ2 of the oriented Euclidean plane R2 , i.e.,    ξ11 ξ12   , S(ξ1 , ξ2 ) =  ξ21 ξ22  where ξ1 = ξ11 e1 + ξ12 e2 ,

ξ2 = ξ21 e1 + ξ22 e2 .

Example 3.8. Let v be a given vector on the oriented Euclidean space R3 . The triple scalar product on other two vectors ξ1 and ξ2 is a 2-form:    v1 v2 v3    ω(ξ1 , ξ2 ) = (v, [ξ1 , ξ2 ]) =  ξ11 ξ12 ξ13  ,  ξ21 ξ22 ξ23  where v =

3  i=1

vi eji , ξj =

3 

ξji ei (j = 1, 2).

i=1

3. k-Forms We denote the set of all permutations of the set {1, 2, · · · , k} by Sk and its element by σ = {σ(1), σ(2), · · · , σ(k)} = {i1 , i2 , · · · , ik } ∈ νk ,  1, if σ ∈ νk is even, ε(σ) = −1, if σ ∈ νk is odd.

68

1. Preliminaries of Differentiable Manifolds

Definition 3.9. An exterior form of degree k (or a k-form) is a function of k vectors that is k-linear and skew-symmetric: ω(λ1 ξ1 + λ2 ξ1 , ξ2 , · · · , ξk ) = λ1 ω(ξ1 , ξ2 , · · · , ξk ) + λ2 ω(ξ1 , ξ2 , · · · , ξk ), ω(ξi1 , ξi2 , · · · , ξik ) = ε(σ)ω(ξ1 , ξ2 , · · · , ξk ), ξ1 , ξ1 , ξ2 , · · · , ξk ∈ Rn ,

λ1 , λ2 ∈ R,

σ ∈ νk ,

where σ = (i1 , i2 , · · · , ik ) ∈ Sk . Example 3.10. The oriented volume of the parallelepiped with edges ξ1 , ξ2 , · · · , ξn in the oriented Euclidean space Rn is an n-form.    ξ11 · · · ξ1n     ξ21 · · · ξ2n    V (ξ1 , ξ2 , · · · , ξn ) =  . ..  ,  .. .    ξn1 · · · ξnn  where ξi = ξi1 e1 + · · · + ξin en . The set of all k-forms in Rn is denoted by Λk (Rn ). It forms a real vector space if we introduce operations of addition. (λ1 ω1k + λ2 ω2k )(ξ1 , ξ2 , · · · , ξk ) = λ1 ω1k (ξ1 , ξ2 , · · · , ξk ) + λ2 ω2k (ξ1 , ξ2 , · · · , ξk ), ω1k , ω2k ∈ Λk (Rn ),

λ1 , λ2 ∈ R.

it Question 3.11. Show that if ηj =

k 

aji ξi (j = 1, · · · , k), then

i=1

ω k (η1 , η2 , · · · , ηk ) = det (aji )ω k (ξ1 , ξ2 , · · · , ξk ).

1.3.2 Exterior Algebra 1. The exterior product of two 1-forms In the previous section, we have defined various exterior forms. We now introduce one more operation: exterior multiplication of forms. As a matter of fact, these forms can be generated from the 1-forms by an operation called exterior product. Definition 3.12. For ω1 and ω2 ∈ Λ1 (Rn ), the exterior product of ω1 and ω2 denoted by ω1 ∧ ω2 is defined by the formula    ω1 (ξ1 ) ω2 (ξ1 )    , ξ 1 , ξ2 ∈ Rn , (ω1 ∧ ω2 )(ξ1 , ξ2 ) =  ω1 (ξ2 ) ω2 (ξ2 ) 

1.3 Exterior Product

69

which denotes the oriented area of the image of the parallelogram with sides ω(ξ1 ) and ω(ξ2 ) on the ω1 , ω2 plane. It is not hard to verify that ω1 ∧ ω2 really is a 2-form and has properties ω1 ∧ ω2 = −ω2 ∧ ω1 , (λ1 ω1 + λ2 ω1 ) ∧ ω2 = λ1 ω1 ∧ ω2 + λ2 ω1 ∧ ω2 . Now suppose we have chosen a system of linear coordinates on Rn , i.e., we are given n independent 1-forms, x1 , x2 , · · · , xn . We will call these forms basic. The exterior products of the basic forms are the 2-forms xi ∧ xj . By skew-symmetry, xi ∈ Λ1 (Rn ), xi ∧ xi = 0, xi ∧ xj = −xj ∧ xi ,   xi (ξ1 ) xj (ξ1 )    (xi ∧ xj )(ξ1 , ξ2 ) =  xi (ξ2 ) xj (ξ2 )     ai aj   = ai bj − aj bi ,  = bi bj    where ξ1 = ai ei , ξ2 = bi ei . It is the oriented area of the parallelogram with i

i

sides (xi (ξ1 ), xi (ξ2 )) and (xj (ξ1 ), xj (ξ2 )) in the (xi , xj )-plane. For any ω ∈ Λ2 (Rn ), ω(ξ1 , ξ2 ) = =

n 

ω(ai ei , bj ej ) =

i,j=1 

n 

(ai bj − aj bi )ω(ei , ej ) =

i<j

where ξ1 =

 i

ai bj ω(ei , ej )

i,j=1



ω(ei , ej )(xi ∧ xj )(ξ1 , ξ2 ),

i<j

ai ei , ξ2 =



bi ei . Thus,

i

ω=



ω(ei , ej )xi ∧ xj ,

i<j

i.e., {xi ∧ xj }i<j generate Λ2 (Rn ). In addition, if  aij xi ∧ xj = 0, i<j

acting on el , ek (l < k), we get akl =



aij (xi ∧ xj )(el , ek ) = 0.

i<j

Thus, {xi ∧xj }i<j are linearly independent and they form a base of Λ2 (Rn ), which  n  . implies that the dimension of Λ2 (Rn ) is 2

70

1. Preliminaries of Differentiable Manifolds

In the oriented Euclidean space R3 , the base of Λ2 (R3 ) is x1 ∧ x2 , x2 ∧ x3 , and x3 ∧ x1 = −x1 ∧ x3 . Any 2-form ω ∈ Λ2 (R3 ) can be represented as ω = P x2 ∧ x3 + Qx3 ∧ x1 + Rx1 ∧ x2 . 2.

Exterior monomials

Definition 3.13. For ω1 , · · · , ωk ∈ Λ1 (Rn ), we define their exterior product ω1 ∧ · · · ∧ ωk as follows:    ω1 (ξ1 ) · · · ω1 (ξk )      .. .. (ω1 ∧ · · · ∧ ωk )(ξ1 , · · · , ξk ) =  . . .    ωk (ξ1 ) · · · ωk (ξk )  In other words, the value of a product of 1-forms on the parallelepiped ξ1 , · · · , ξk is equal to the oriented volume of the image of the parallelepiped in the oriented Euclidean coordinate space Rn under the mapping ξ → (ω1 (ξ), · · · , ωk (ξ)). Question 3.14. Prove that ω1 ∧ · · · ∧ ωk really is a k-form. Property 3.15. We have the following properties: 1◦ (λ ω1 + λ ω1 ) ∧ ω2 ∧ · · · ∧ ωk = λ ω1 ∧ · · · ∧ ωk + λ ω1 ∧ · · · ∧ ωk . 2◦ ωσ(1) ∧ ωσ(2) ∧ · · · ∧ ωσ(k) = ε(σ)ω1 ∧ ω2 ∧ · · · ∧ ωk . 3◦ If i = j, ωi = ωj , then ω1 ∧ · · · ∧ ωk = 0. 4◦ If ω1 , · · · , ωk are linearly dependent, then ω1 ∧ · · · ∧ ωk = 0. k  aij ωj (i = 1, · · · , k), then 5◦ If βi = j=1

β1 ∧ · · · ∧ βk = det (aij )ω1 ∧ · · · ∧ ωk . Proof. Here we only prove 5◦ , the others are easy. By the linearity of the exterior product, * k + * k +   β1 ∧ · · · ∧ βk = a1i1 ωi1 ∧ · · · ∧ akik ωik i1 =1

=

k 

ik =1

a1i1 · · · akik ωi1 ∧ · · · ∧ ωik

i1 ,···,ik =1

=



a1i1 · · · akik ε(i1 , · · · , ik )ω1 ∧ · · · ∧ ωk

(by 2◦ )

i1 ,···,ik ∈νk

= det (aij )ω1 ∧ · · · ∧ ωk . The proof can be obtained.



1.3 Exterior Product

71

Theorem 3.16. {xi1 ∧ · · · ∧ xik }i1 <···
n 

ξij ej (i = 1, · · · , k), ξi ∈ Rn , then

j=1

ω(ξ1 , · · · , ξk ) = ω

n 

ξi,j1 ej1 , · · · ,

j1 =1

n 

=

 ξkjk ejk

jk =1

ξij1 · · · ξkjk ω(ej1 , · · · , ejk )

j1 ,···,jk =1



=

n 



ε(j1 , · · · , jk )

i1 <···
· ξ1j1 · · · ξkjk ω(ξi1 , · · · , ξik )  ω(ei1 , · · · , eik )xi1 ∧ · · · ∧ xik (ξ1 , · · · , ξk ). = i1 <···
So ω=



ω(ei1 , · · · , eik )xi1 ∧ · · · ∧ xik .

i1 <···
Thus, {xi1 ∧ · · · ∧ xik }i1 <··· n, dim (Λk (Rn )) = 0, Λk (Rn ) = {0}. Therefore, the theorem is completed.



Definition 3.17. Let ω k = ω1 ∧· · ·∧ωk , ω l = ωk+1 ∧· · ·∧ωk+l , ωi ∈ Λ1 (Rn ) (i = 1, · · · , k + l). Define their product ω k ∧ ω l to be the monomial ωk ∧ ωl

= (ω1 ∧ · · · ∧ ωk ) ∧ (ωk+1 ∧ · · · ∧ ωk+l ) =

ω1 ∧ · · · ∧ ωk ∧ ωk+1 ∧ · · · ∧ ωk+l .

Property 3.18. If ω1 , ω2 , and ω3 are monomials, then 1◦ (λ1 ω1 + λ2 ω2 ) ∧ ω3 = λ1 ω1 ∧ ω3 + λ2 ω2 ∧ ω3 .

72

1. Preliminaries of Differentiable Manifolds

2◦

ω1 ∧ ω2 = (−1)kl ω2 ∧ ω1 , ω1 ∈ Λk , ω2 ∈ Λl .

3◦

(ω1k ∧ ω2l ) ∧ ω3m = ω1k ∧ (ω2l ∧ ω3m ), ω1 ∈ Λk , ω2 ∈ Λl , ω3 ∈ Λm .

3. Exterior product of forms We now turn to define the exterior product of an arbitrary k-form ω k with an arbitrary l-form ω l . Definition 3.19. The exterior product ω k ∧ ω l of a k-form ω k on Rn with an l-form ω l on Rn is the (k + l)-form on Rn , defined by the formula 

(ω k ∧ ω l )(ξ1 , · · · , ξk+l ) =

ε(σ)ω k (ξi1 , · · · , ξik )ω l (ξj1 , · · · , ξjl ),

i1 <···
(3.1) where σ = (i1 , i2 , · · · , ik , j1 , · · · , jl ) is a permutation of the numbers (1, · · · , k + l). Example 3.20. For k = l = 1, (ω1 ∧ ω2 )(ξ1 , ξ2 ) = ω1 (ξ1 )ω2 (ξ2 ) − ω1 (ξ2 )ω2 (ξ1 )    ω (ξ ) ω2 (ξ1 )  , =  1 1 ω1 (ξ2 ) ω2 (ξ2 )  which agrees with the definition of multiplication of 1-forms. Example 3.21. For ω1 ∈ Λ2 and ω2 ∈ Λ1 , (ω1 ∧ ω2 )(ξ1 , ξ2 , ξ3 ) = ω1 (ξ1 , ξ2 )ω2 (ξ3 ) − ω1 (ξ1 , ξ3 )ω2 (ξ2 ) + ω1 (ξ2 , ξ3 )ω2 (ξ1 ). Proposition 3.22. ωk ∧ ω l defined above is actually a (k + l)-form. The linearity of ω k ∧ω l is based on the linearity of ω k and ω l . The skew-symmetry is based on the following. Lemma 3.23. If ω is a k-linear function on R, then the following conditions are equivalent. 1◦ ω(ξσ(1) , · · · , ξσ(k) ) = ε(σ) ω(ξ1 , · · · , ξk ), ∀ σ ∈ νk . 2◦ ω(ξ1 , · · · , ξi , · · · , ξj , · · · , ξk ) = −ω(ξ1 , · · · , ξj , · · · , ξi , · · · , ξk ), ∀ i = j. 3◦ ω(ξ1 , · · · , ξk ) = 0, if ξi = ξj , ∀ i = j. 4◦ ω(ξ1 , · · · , ξk ) = 0, where ξi = ξi+1 (1 ≤ i ≤ n − 1). 5◦ ω(ξ1 , · · · , ξk ) = 0, ξ1 , · · · , ξk are linearly dependent. The proof is left to the reader. Now we turn to prove Proposition 3.22. For this we only need to prove 4◦ of Lemma 3.23.

1.3 Exterior Product

73

Proof. By 4◦ in Lemma 3.23, we only need to prove that if ξi = ξi +1 , then (ω k ∧ ω l )(ξ1 , · · ·, ξi , ξi +1 , · · · , ξk+l ) = 0. Consider the terms of the right hand side in (1.2). If i , i + 1 ∈ (i1 , · · · , ik ), then ω k (ξi1 , · · · , ξik ) = 0. Therefore, ω k (ξi1 , · · · , ξik )ω l (ξj+1 , · · · , ξj+l ) = 0. When i , i + 1 ∈ (j1 , · · · , jl ), the case is similar. If i ∈ (i1 , · · · , ik ) and i + 1 ∈ (j1 , · · · , jl ), i.e., i1 < · · · < ih < i < ih+1 < · · · < ik and j1 < · · · < jn < i + 1 < jn+1 < · · · < jl , then there is another term such that i1 < · · · < ih < i + 1 < ih+1 < · · · < ik , j1 < · · · < jn < i < jn+1 < · · · < jl . The condition ξi +1 = ξi implies that ω k (ξ1 , · · · , ξi , · · · , ξik )ω l (ξj1 , · · · , ξi +1 , · · · , ξjl ) = ω k (ξ1 , · · · , ξi +1 , · · · , ξik )ω l (ξj1 , · · · , ξi , · · · , ξjl ). However, the sign ε(σ) = −ε(σ  ) where σ = (i1 , · · · , i , · · · , ik , j1 , · · · , i + 1, · · · , jl ), σ  = (i1 , · · · , i + 1, · · · , ik , j1 , · · · , i , · · · , jl ). Thus, the right side in 1◦ is equal to 0, i.e., (ω k ∧ ω l )(ξ1 , · · · , ξk+l ) = 0. 

Hence lemma is proved.

Theorem 3.24. The exterior product of forms defined above is skew-commutative, distributive, and associative, i.e., 1◦ skew-commutative: ω k ∧ ω l = (−1)kl ω l ∧ ω k . 2◦ distributive : (λ1 ω1k + λ2 ω2k ) ∧ ω l = λ1 ω1k ∧ ω l + λ2 ω2k ∧ ω l . 3◦ associative : (ω k ∧ ω l ) ∧ ω m = ω k ∧ (ω l ∧ ω m ). For monomials, it coincides with the exterior product defined above. Distributivity follows from the fact that every term in Equation (1.2) is linear with respect to ω k and ω l . Since (ω k ∧ ω l )(ξ1 , · · · , ξk+l )  ε(i1 , · · · , ik , j1 , · · · , jl )ω k (ξi1 , · · · , ξik )ω l (ξj1 , · · · , ξjl ) = i1 <···
=



(−1)kl ε(j1 , · · · , jl , i1 , · · · , ik ) ω l (ξj1 , · · · , ξjl )ω k (ξi1 , · · · , ξik )

i1 <···
= (−1)kl (ω l ∧ ω k )(ξ1 , · · · , ξk+l ),

74

1. Preliminaries of Differentiable Manifolds

we get skew-commutativity. In order to prove associativity, we first prove that for monomials the exterior product defined by Definition 3.19 coincides with the exterior product in Definition 3.17. Since we have not get proved the equivalence of the Definition 3.17 of exterior product of k 1-forms with the Definition 3.19 , we will temporarily denote the exterior product of k 1-forms by the symbol ∧, so that our monomials have the form ω k = ω1 ∧ · · · ∧ωk

and ω l = ωk+1 ∧ · · · ∧ωk+l .

where ω1 , · · · , ωk+l are 1-forms. Lemma 3.25. The exterior product of two monomials is a monomial. (ω1 ∧ · · · ∧ωk ) ∧ (ωk+1 ∧ · · · ∧ωk+l ) = ω1 ∧ · · · ∧ωk ∧ωk+1 ∧ · · · ∧ωk+l . Proof. ((ω1 ∧ · · · ∧ωk ) ∧ (ωk+1 ∧ · · · ∧ωk+l ))(ξ1 , · · · , ξk+l )  ε(i1 , · · · , ik , j1 , · · · , jl )(ω1 ∧ · · · ∧ωk )(ξi1 , · · · , ξik ) = i1 <···
·(ωk+1 ∧ · · · ∧ωk+l ) (ξj1 , · · · , ξjl )  = ε(i1 , · · · , ik , j1 , · · · , jl ) i1 <···
· =

det

k+1≤j≤k+l; 1≤m≤k

det

1≤i≤k+l; 1≤j≤k+l

det

1≤i≤k; 1≤m≤k

ωi (ξim )

ωj (ξjm )

ωi (ξj ) = ω1 ∧ · · · ∧ωk+l (ξ1 , · · · , ξk+l ).

Therefore, the lemma is completed. Thus,



ω1 ∧ω2 = ω1 ∧ ω2 , ω1 ∧ω2 ∧ω3 = (ω1 ∧ω2 ) ∧ ω3 = (ω1 ∧ ω2 ) ∧ ω3 ,   ω1 ∧ω2 ∧ω3 = ω1 ∧ ω2 ∧ω3 = ω1 ∧ (ω2 ∧ ω3 ).

It follows that (ω1 ∧ ω2 ) ∧ ω3 = ω1 ∧ (ω2 ∧ ω3 ), and denoted by ω1 ∧ ω2 ∧ ω3 . Thus, ω1 ∧ω2 ∧ω3 = ω1 ∧ ω2 ∧ ω3 . In general, we have ω1 ∧ω2 ∧ · · · ∧ωk = (· · · ((ω1 ∧ ω2 ) ∧ ω3 ) ∧ · · · ∧ ωk ). We now prove associativity. By Theorem 3.16, ω k ∈ Λk , ω l ∈ Λl and ω m ∈ Λm can be represented by the following formulae resp.

1.4 Foundation of Differential Form

ωk =

ai1 ···ik xi1 ∧ · · · ∧ xik ,



bj1 ···jl xj1 ∧ · · · ∧ xjl ,  = ch1 ···hm xh1 ∧ · · · ∧ xhm .

ωl = ωm



75

By distributivity and associativity for monomials, (ω k ∧ ω l ) ∧ ω m

=



abc ((xi1 ∧ · · · ∧ xik ) ∧ (xj1 ∧ · · · ∧ xjl )) ∧ (xh1 ∧ · · · ∧ xhm )

k,l,m

=



abc (xi1 ∧ · · · ∧ xik ) ∧ ((xj1 ∧ · · · ∧ xjl ) ∧ (xh1 ∧ · · · ∧ xhm ))

k,l,m

= ω k ∧ (ω l ∧ ω m ).

Based on linear space Λk (k = 0, 1, 2, · · · , n), we may construct a bigger linear space Λ, which is a direct sum of Λ1 , Λ1 , · · · , Λn , i.e., ˙ 1+ ˙ · · · +Λ ˙ n. Λ = Λ0 +Λ Each element ω may be represented as ω = ω0 + ω1 + · · · + ωn ,

ω i ∈ Λi ,

and this kind of expression is unique. In Λ there is not only algebraic structure of the linear space, but also the definition of the exterior product. Direct sum Λ is the Grassmann algebra produced by the linear space which contained the entire real number field and the linear space. 1, x1 , · · · , xn ,

xi ∧ xj (i < j), · · · , x1 ∧ x2 ∧ · · · ∧ xn

form the basis of Λ, whose dimension is dim(Λ) =

n   n  = 2n . i i=0

1.4 Foundation of Differential Form There is no strict definition on how to define df for a smooth function and dx for dx1 , · · · , dxn , in classical mathematical analysis. The differential of the independent variable is equal to its increment in classical mathematical analysis, which is improper in a general sense. Here, we always regard dx1 , · · · , dxn as some basis of a linear space, which is called the differential space.

76

1. Preliminaries of Differentiable Manifolds

1.4.1 Differential Form A vector ξ on Rn is a vector from 0 into ξ. A tangent vector (x, ξ) ∈ Tx Rn on Rn at x is a vector from x to x + ξ i.e., a fixed vector starting from x to x + ξ. Tangent vector (x, ξ) is usually denoted by ξx . For (x, ξ), (x, η) ∈ Tx Rn define: α(x, ξ) + β(x, η) = (x, αξ + βη),

α, β ∈ R

or αξx + βηx = (αξ + βη)x .Then, the tangent space Tx Rn to Rn at x forms a vector space, and (e1 )x , · · · , (en )x is its standard basis. The set , T (Rn ) = Tx Rn x∈Rn

is called the tangent bundle on Rn , see Section 1.2. Notice that the tangent bundle T (Rn ) consists of all fixed vectors on Rn . The mapping π : T Rn → Rn defined by the following formula: π(ξx ) = x,

∀ξx ∈ Tx Rn

is called the tangent bundle projection. π −1 (x) = Tx Rn is called the fiber of the tangent bundle over the point. The dual space of Tx Rn denoted by Tx∗ Rn is called the cotangent vector space to Rn at x consisting of all linear functions from Tx Rn into Rn . Its element is called a covector (covariant vector) or a cotangent vector to Rn at x, and , T ∗ (Rn ) = Tx∗ Rn x∈Rn

is called the cotangent bundle. The cotangent bundle projection: π ∗ : T ∗ (Rn ) −→ Rn is similarly defined :

π ∗ (ωx ) = x,

∀ ωx ∈ Tx∗ Rn .

We now introduce a natural topological structure into T ∗ Rn . The element in T ∗ Rn can be represented as a vector consisting of 2n components (q1 , · · · , qn , p1 , · · · , pn ), where (q1 , · · · , qn ) ∈ Rn , (p1 , · · · , pn ) ∈ Tq∗ Rn , viewed as Rn . Thus, topology in T ∗ Rn is the product topology in Rn × Rn . T ∗ Rn equipped with such a topology forms a 2n-dimensional manifold. For any tangent space Tx Rn to Rn at x denoted by Tx Rn = {(x, ξ)|ξ ∈ Rn }, n k n the set of all k-forms # onkTx R nis denoted by Λ (Tx R ). k n Λ (Tx R ) is called the exterior k-forms bundle on Rn . The Λ (R ) = x∈Rn

direct sum Λn (Rn ) of k-form bundles Λk (Rn )(k = 0, 1, · · · , n), ˙ 1 (Rn )+ ˙ · · · +Λ ˙ n (Rn ) Λ(Rn ) = Λ0 (Rn )+Λ is called the bundle of exterior forms on Rn ( where Λ0 (Rn ) = Rn ).

1.4 Foundation of Differential Form

77

Definition 4.1. A differential k-form is a mapping ω : Rn → Λk (Rn ), such that ω(x) ∈ Λk (Tx Rn ). If ϕ1 (x), · · · , ϕn (x) is the dual basis to (e1 )x , · · · , (en )x on Rn , then  ω(x) = ai1 ···ik (x)ϕi1 (x) ∧ · · · ∧ ϕik (x), i1 <···
where ai1 ···ik (x) ∈ F(Rk ), the totality of functions on Rn . ω is called continuous, differentiable, etc., if ai1 ···ik is continuous, differentiable etc., respectively. From now on, we shall assume that forms that are differentiable will always mean (C ∞ ). The set of all differentiable k-differential forms on Rn is denoted by Ωk (Rn ), in particular Ω0 (Rn ) = C ∞ (Rn ). ∀ ω1 , ω2 ∈ Ωk (Rn ), f ∈ C ∞ (Rn ), define: (ω1 + ω2 )(x) = ω1 (x) + ω2 (x), (f ω)(x) = f (x) · ω(x). Then, Ω∞ (Rn ) forms a C ∞ (Rn )-module, i.e., vector space over a ring. ∀ ω k ∈ Ωk (Rn ), ω l ∈ Ωl (Rn ), define their exterior product ω k ∧ ωl ∈ Ωk+l (Rn ) as (ωk ∧ ω l )(x) = ω k (x) ∧ ω l (x). By this formula, we have: f · ω = f ∧ ω,

f ∈ C ∞ (Rn ) (= Ω0 (Rn )).

By the Theorem 3.24, the exterior product of differential forms defined above is distributive, skew-symmetric, and associative. If f : Rn → R is differentiable, then Df (x) ∈ Λ1 (Rn ), where Df is the derivative of f (x) at x. Thus, we get a differential 1-form df ∈ Ω1 (Rn ), defined as df (ξx ) = Df (x)ξ =

n 

Di f (x)ξi ,

ξ=



ξi ei .

i=1

Replacing f with xi for any x = (x1 , · · · , xi , · · · , xn ) yields: dxi (ξx ) = Dxi (ξ) = ξi , or dxi ((ej )x ) = Dxi (ej ) = δij . Thus, dx1 , · · · , dxn form the dual basis to (e1 )x , · · · , (en )x . ∀ ω ∈ Ωk (Rn ), ω can be written as  ω(x) = ai1 ···ik (x)dxi1 ∧ · · · ∧ dxik , i1 <···
where ai1 ···ik (x) ∈ C (R ). Represented by dx1 , · · · , dxn , the differential df 1-form is n

78

1. Preliminaries of Differentiable Manifolds

df = D1 f dx1 + · · · + Dn f dxn or in a classical notation ∂f ∂f dx1 + · · · + dxn , ∂xn ∂x1

df = since df (ξx ) = D f (x)(ξ) =

n 

ξ i Di f =

i=1

n 

Di f dxi (ξx ),

∀ ξ x ∈ Tx R n .

i=1

Theorem 4.2. Every differential k-form on the space Rn with a given coordinate system x1 , · · · , xn can be represented uniquely in the form  ωk = ai1 ···ik (x)dxi1 ∧ · · · ∧ dxik , i1 <···
where the ai1 ···ik (x) are smooth functions on Rn . As a particular case of Theorem 4.2, let k = 1. Thus, we have: Theorem 4.3. Every differential 1-form on the space Rn with a given coordinate system x1 , · · · , xn can be represented uniquely with smooth function ai (x) as follows: ω = a1 (x)dx1 + · · · + an (x)dxn . Example 4.4. Calculate the value of the forms ω = dr2 (r2 = (x1 )2 + (x2 )2 ) on the vectors ξ1 , ξ2 , ξ3 . (Fig. 4.1), the results in full in the Table 4.1. x2

6 2 1 6 ξ1 Fig. 4.1.

Table 4.1.

ξ2

ξ3

=

~

0 1 2 Example 4.4 graphical representations

3

- x1

Example 4.4 table representations ω1 ω2 ω3

ξ1 0 0 0

ξ2 −1 −2 −8

ξ3 1 −2 0

For example, calculate the value of ω3 in vectors ξ1 , ξ2 , ξ3 as follows:

1.4 Foundation of Differential Form

79

ω3 (x)(ξi ) = dr2 = 2x1 dx1 + 2x2 dx2 , ω3 (x)(ξ1 ) = 2 · 0 · dx1 (ξ1 ) + 2 · 0 · dx2 (ξ1 ) = 0, ω3 (x)(ξ2 ) = 2 · 2 · dx1 (ξ2 ) + 2 · 2 · dx2 (ξ2 ) = 4 × (−1) + 4 × (−1) = −8, ω3 (x)(ξ3 ) = 2 · 2 · dx1 (ξ3 ) + 2 · 2 · dx2 (ξ3 ) = 4 × 1 + 4 × (−1) = 0. Example 4.5. Calculate ω1 = dx1 ∧dx2 , ω2 = x1 dx3 ∧dx2 −x2 dx2 ∧dx1 , and ω3 = rdr ∧ dϕ(x1 = r cos ϕ, x2 = r sin ϕ) in vectors (ξ1 , η1 ), (ξ2 , η2 ), and (ξ3 , η3 )(see Fig. 4.2), result as follows: Table 4.2.

Example 4.5 table representations ω1 ω2 ω3

(ξ1 , η1 ) 1 2 1

(ξ2 , η2 ) 1 1 1

(ξ3 , η3 ) −1 −3 −1

x2

6 3 ξ3 2 6 η2 1 ξ2 Fig. 4.2.

6 η3

R

0 1 Example 4.5 graphical representations

η1

 - ξ1 2

- x1

Example 4.6. Calculate the value of the forms ω1 = dx2 ∧ dx3 , ω2 = x1 dx3 ∧ dx2 , ω3 = dx3 ∧ dr 2 on the vectors ξ, η at the point x, where r2 = (x1 )2 + (x2 )2 + (x3 )2 , ξ = (1, 1, 1) , η = (1, 2, 3) , x = (2, 0, 0). The detailed calculation is shown bellow as follows:      dx2 (ξ) dx3 (ξ)   1 1  2 3    = 1,  ω1 (ξ, η) = dx ∧ dx (ξ, η) =  = dx2 (η) dx3 (η)   2 3  ω2 (ξ, η) = 2 · dx3 ∧ dx2 (ξ, η) = −2ω1 (ξ, η) = −2, ω3 (ξ, η) = dx3 ∧ dr2 (ξ, η) = dx3 ∧ (2x1 dx1 + 2x2 dx2 + 2x3 dx3 )(ξ, η) = 2x1 dx3 ∧ dx1 (ξ, η) − 2x2 dx2 ∧ dx3 (ξ, η) = 2 · 2 · dx3 ∧ dx1 (ξ, η) − 2 · 0 · dx2 ∧ dx3 (ξ, η)      1 1   dx3 (ξ) dx1 (ξ)     = −8.  = 4 = 4 dx3 (η) dx1 (η)  3 1 

80

1. Preliminaries of Differentiable Manifolds

1.4.2 The Behavior of Differential Forms under Maps First, we consider the behavior of differential forms under maps. Let f : Rn → Rm be a differential mapping. Df (x) is the linear transformation from Rn to Rm , which is the derivative of f , and DF (x) =

∂ fi the Jacobian of f at x. It induces a linear ∂ xj

transformation f∗ from the tangent space Tx Rn to Rn at x into the tangent space Tf (x) Rm to Rm at f (x), i.e., f∗ (ξx ) = (Df (x)(ξ))f (x) ,

∀ ξx ∈ Tx Rn .

Definition 4.7. Let f ∗ : Ωk (Tf (x) Rm ) → Ωk (Tx Rn ) be a linear map: (f ∗ ω(x))(ξ1 , · · · , ξk ) = ω(f (x))(f∗ ξ1 , · · · , f∗ ξk ),

ξi ∈ T x R n .

f ∗ can be expanded on Ωk (Rm ): (f ∗ ω)(x) = f ∗ ω(x). f ∗ ω is called the pull-back of ω under f , which is the dual transformation of f∗ . Theorem 4.8. Let f : Rn → Rm , h : Rm → Rl , f, h ∈ C ∞ . Then, n  ∂ fi d xj . 1◦ f ∗ (d y i ) = j 2◦ 3◦ 4◦ 5◦

j=1

∂x

f ∗ (ω1 + ω2 ) = f ∗ ω1 + f ∗ ω2 . f ∗ (ω ∧ η) = f ∗ ω ∧ f ∗ η. f ∗ (g · ω) = (g ◦ f ) · f ∗ ω, ∀ g ∈ C ∞ (Rm ). (h ◦ f )∗ = f ∗ ◦ h∗ .

Proof. The proof is not difficult. We only prove 1◦ , 4◦ and 5◦ . ∀ ξx ∈ Tx Rn , f ∗ (d y i )(ξx ) = d xi (f∗ ξx ) = (f∗ ξ)i =

n n   ∂f i ∂f i j ξ = d x (ξx ). j ∂xj ∂xj j=1 j=1

Thus, we have f ∗ (dy i ) =

n  ∂f i j dx . ∂xj j=1



The theorem is proved. Example 4.9. Let y = f (x1 , x2 ) = (x1 )2 + (x2 )2 , ω = dy, f ∗ ω = d x(f∗ ξx ) =

∂f ∂f d x1 (ξx ) + 2 d x2 (ξx ) ∂x1 ∂x

= 2x1 d x1 + 2x2 d x2 .

1.4 Foundation of Differential Form

81

Before proving 4◦ , let us understand the following theorem. Theorem 4.10. Let f = (f 1 , · · · , f m ) = (y 1 , · · · , y m ), f : Rn → Rm be a differ i  i ∂f ∂y = . For ω ∈ Ωk (Rm ), entiable mapping with Jacobian j j ∂x



ω=

∂x

m×n

m×n

ai1 ,···,ik (y)dy i1 ∧ · · · ∧ dy ik ,

i1 <···
we have 

f ∗ω =

i1 <···
where Δ

· · · ik · · · jk

i1 j1

!

is the

· · · ik · · · jk

i1 j1

· · · ik · · · jk

i1 j1

ai1 ,···,ik (f (x))Δ

! dxj1 ∧ · · · ∧ dxjk ,

!

 -minor of matrix

∂y i ∂xj

 i = 1, · · · , m j = 1, · · · , n

Proof. * ∗

f ω =f

∗



+ ai1 ,···,ik (y)dy ∧ · · · ∧ dy i1

ik

i1 <···
=



ai1 ,···,ik (f (x))f ∗ (dyi1 ) ∧ · · · ∧ f ∗ (dy ik )

i1 <···
=

 i1 <···


⎞ ⎛ ⎞ n n i1 ik   ∂y ∂y ai1 ,···,ik (f (x)) ⎝ dxj1 ⎠ ∧ · · · ∧ ⎝ dxjk ⎠ j1 jk ∂x ∂x j =1 j =1 ⎛

1

n 

k

i1

ik

∂y ∂y · · · jk dxj1 ∧ · · · ∧ dxjk j1 ∂x ∂x i1 <···
ai1 ,···,ik (f (x))

i1 <···
··· =

j1 <···<jk (j1 ,···,jk )∈νk

ik

∂y j1 ∧ · · · ∧ dxjk  dx ∂xjk %  i1 ai1 ,···,ik (f (x))Δ j1

i1 <···
· · · ik · · · jk

& dxj1 ∧ · · · ∧ dxjk .

The proof can be obtained.



The following deduction gives the proof for 4◦ . Proof. If f : Rn → Rn is differentiable, then f ∗ (gdx1 ∧ · · · ∧ dxn ) = g ◦ f det (f  )dx1 ∧ · · · ∧ dxn ,  i ∂f where f  = Df = . ∂xj



.

82

1. Preliminaries of Differentiable Manifolds

For proof of 5◦ : Proof. [f ∗ (h∗ ω)](x)(ξ1 , · · · , ξp ) =

(h∗ ω)(f (x))(f∗ ξ1 , · · · , f∗ ξp )

= ω(h(f (x)))(h∗ f∗ ξ1 , · · · , h∗ f∗ ξp ) =

ω(h ◦ f )(x)((hf )∗ ξ1 , · · · , (hf )∗ ξp )

= [(h ◦ f )∗ ω](x)(ξ1 , · · · , ξp ). Therefore, (h ◦ f )∗ = f ∗ ◦ h∗ .



1.4.3 Exterior Differential We now define an operator similar to differentiation in classical mathematical analysis. We have introduced the function differential on the manifold, namely if f ∈ C ∞ (Rn ), and df |x ∈ Tx Rn , then df is a 1-form on Rn . Therefore, we may say operator d : Ω0 (Rn ) → Ω1 (Rn ) maps the 0-form defined on Rn onto the 1-form on Rn . We need to extend this operator to the exterior algebra Ω(Rn ) on M . Definition 4.11. The exterior differential operator d on an exterior algebra ω(Rn ) of manifold M is a mapping d : Ωk (Rn ) −→ Ωk+1 (Rn ), where k = 0, 1, · · · , n. Exterior algebra may be represented by the local coordinate system of M as  ω= ai1 ,···,ik dxi1 ∧ · · · ∧ dxik , 1≤i1 <···
which is a k-form. Then, dω

=



dai1 ,···,ik ∧ dxi1 ∧ · · · ∧ dxik

1≤i1 <···
=



n  ∂ai

1 ,···,ik

1≤i1 <···
∂xj

dxj ∧ dxi1 ∧ · · · ∧ dxik .

Here d is called an exterior differential operator. In particular, if ω = f ∈ Ω0 (Rn ) = C ∞ (Rn ), then dω =

n  ∂f j dx . ∂xj j=1

From this, we can see that when ω ∈ Ωn (Rn ), dω = 0.

1.4 Foundation of Differential Form

83

Theorem 4.12. Exterior differential operator d has the following properties: 1◦ d(ω + η) = dω + dη. 2◦ ∀ ω k ∈ Ωk , ω l ∈ Ωl , has d(ω k ∧ ω l ) = dω k ∧ ω l + (−1)k ω k ∧ dω l . 3◦ d(dω) = 0 or in simple form d2 ω = 0. 4◦ f ∗ d ω = df ∗ ω. Proof. 1◦ The proof is obvious. 2◦ Let  ω k (x) =

ai1 ,···,ik dxi1 ∧ · · · ∧ dxik ,

i1 <···


ωl (x) =

bj1 ,···,jl dxj1 ∧ · · · ∧ dxjl .

j1 <···<jl

Then, 

ωk ∧ ωl =

ai1 ,···,ik (x)bj1 ,···,jl (x)dxi1 ∧ · · · ∧ dxik ∧ dxj1 ∧ · · · ∧ dxjl .

i1 <···
By definition, 

d(ω k ∧ ω l ) =

i1 <···
*



=

i1 <···
n 

n  ∂(ab) i dx ∧ dxi1 ∧ · · · ∧ dxik ∧ dxj1 ∧ · · · ∧ dxjl i ∂x i=1

∂ai1 ···ik ∂bj1 ···jl bj1 ···jl + ai1 ···ik i ∂x ∂xi i=1

+ dxi ∧ dxi1 ∧ · · · ∧ dxik

∧dxj1 ∧ · · · ∧ dxjl 

=

n 

bj1 ···jl

i1 <···


+

n 

∂ai1 ···ik i dx ∧ dxi1 ∧ · · · ∧ dxik ∧ dxj1 ∧ · · · ∧ dxjl ∂xi

ai1 ···ik

i1 <···
∂bj1 ···jl (−1)k dxi ∧ dxi1 ∧ · · · ∧ dxik ∧ dxj1 ∧ · · · ∧ dxjl ∂xi

= dω k ∧ ω l + (−1)k ω k ∧ dω l .

3◦

The proof is as follows:

d(dω)



=

i1 <···


=

i1 <···
= 0, since

n 

∂ 2a dxi ∧ dxj ∧ dxi1 ∧ · · · ∧ dxik i j ∂x ∂x i,j=1 &  % ∂2a ∂2a − dxi ∧ dxj ∧ dxi1 ∧ · · · ∧ dxik i j j i ∂x ∂x ∂x ∂x i<j

84

1. Preliminaries of Differentiable Manifolds

4◦

∂2a ∂2a = . ∂xi ∂xj ∂xj ∂xi By Theorem 4.8, 1 ◦ ∀ ω = g ∈ C ∞ (Rn ), we have: *m + m  ∂g  ∂g i ∗ ∗ dy ◦ f · f ∗ (dy i ) = f dg = f i i ∂y ∂y i=1 i=1 m n   ∂g(f (x)) ∂f i j · dx ∂y i ∂xj j=1 i=1

=

n  ∂g(f (x))

=

j=1

∂xj

Furthermore, ∀ ω ∈ Ωk (Rm ), *  ∗ ∗ f (dω) = f * =f

i1 <···


∗

dxj = d(g ◦ f ) = df ∗ g.

m  ∂a i dy ∧ dy i1 ∧ · · · ∧ dy ik i ∂y i=1

+

+

da ∧ dy ∧ · · · ∧ dy i1

ik

i1 <···


=

  f ∗ (da) ∧ f ∗ dyi1 ∧ · · · ∧ dy ik

i1 <···


=

  d(f ∗ a) ∧ f ∗ dyi1 ∧ · · · ∧ dy ik

i1 <···
*

=d



∗

f a∧f

∗

+



dy ∧ · · · ∧ dy i1

ik



(by 3◦ )

i1 <···
= df ∗ ω. Therefore, the theorem is completed.



1.4.4 Poincar´e Lemma and Its Inverse Lemma Definition 4.13. A differential form ω is closed if dω = 0, and is exact if there exists a differential form η such that ω = dη. Clearly if ω is exact, then it is closed by the formula d2 ω = 0. However, the converse is not always true. The following Poincar´e lemma asserts that in the neighbourhood of each point, the closed is equivalent to the exact. Before stating the Poincar´e lemma, we introduce a notion. Definition 4.14. An open set A ⊂ Rn is star-shaped with respect to 0, if for any x ∈ A the set {αx | α ∈ [0, 1]} ⊂ A.

1.4 Foundation of Differential Form

85

Evidently, Rn is a star-shaped open set, and every convex set containing 0 is a star-shaped open set with respect to 0. Theorem 4.15 (Poincar´e lemma). Let A ⊂ Rn be an open star-shaped set with respect to 0. Then, every closed form on A is exact. Proof. We will construct an R-linear mapping H : Ωk (A) → Ωk−1 (A), such that d ◦ H + H ◦ d = id: Ωk (A) → Ωk (A), i.e., ω = d ◦ H(ω) + Hdω. Then, from dω = 0, if follows that ω = d(H(ω)). Taking η = H(ω), we get ω = dη. Let



ω=

ai1 ···ik (x)dxi1 ∧ · · · ∧ dxik .

i1 <···
Define 

H(ω)(x) =

k 

%j−1

(−1)

0

i1 <···
1

tk−1 ai1 ···ik (tx)dtxij

& ik . i j , · dx ∧ · · · ∧ dx ∧ · · · ∧ dx i1

where the symbol “  ” over dxij indicates that it is omitted. 

dH(ω) =

k 

%0

i1 <···
+



k 

* j−1

(−1)

i1 <···


= k

i1 <···
+



1 0

k 

& tk−1 a(tx)dt dxij ∧ dxi1 ∧ · · · ∧ dxik

n 

1 0

i=1

%-

+

∂a(tx) i t dx dt ∂xi k

xij ∧ dxi1 ∧ · · · ∧ dxik

& tk−1 a(tx)dt dxi1 ∧ · · · ∧ dxij ∧ · · · ∧ dxik

(−1)j−1

i1 <···
. ij ∧ · · · ∧ dxik . ∧dx

On the other hand,

1

(−1)j−1

n  i=1

1 0

tk

∂a(tx) dtxij dxi ∧ dxi1 ∧ · · · ∂xi (4.1)

86

1. Preliminaries of Differentiable Manifolds n  ∂a i dx ∧ dxi1 ∧ · · · ∧ d xik , i ∂x i1 <···


dω =

1

k

n  k  j + (−1)

1

tk

0

i=1 j=1

∂a(tx) dtxij dxi ∧ dxi1 ∧ · · · ∂xi

/ ik . i j . ∧ dx ∧ · · · ∧ dx The second term of the right hand side in this equality coincides with the second term in Equation (1.3) except for the sign. Adding them together, we get -



dH(ω) + Hdω = k

1 0

i1 <···
tk−1 a(tx)dt dxi1 ∧ · · · ∧ dxik

n 



1

∂a(tx) dtxi dxi1 ∧ · · · ∧ dxik i ∂x i1 <···
tk

0

Notice that ktk−1 a(tx) +

n 

tk xi

i=1

Thus,

-

1

 dk ∂a(tx) t a(tx) . = i dt ∂x

* ktk−1 a(tx) +

0

-

n 

∂a(tx) i tk x i

i=1 1

= 0

Then, we have dH(ω) + Hdω =

∂x

+ dt

 d k t a(t, x) dt = a(x). dt 

ai1 ···ik (x)dxi1 ∧ · · · ∧ dxik = ω,

i1 <···
i.e., dH + Hd = id. Therefore, the theorem is completed.



1.4.5 Differential Form in R3 We now assume that R3 is a three-dimensional oriented Euclidean space. The square of the length element in R3 has the form

1.4 Foundation of Differential Form

87

ds2 = (dx1 )2 + (dx2 )2 + (dx3 )2 . 1 2 and a 2-form ωA by For any vector A ∈ R3 , we define a corresponding 1-form ωA the formula 1 ωA (ξ) = (A, ξ),

2 ωA (ξ, η) = (A, [ξ, η]),

∀ ξ, η ∈ R3 ,

where ( , ) stands for usual inner product and ( , [ ]) for triple scalar product. 1 Let A = A1 e1 + A2 e2 + A3 e3 and ωA = a1 dx1 + a2 dx2 + a3 dx3 . Then by 3  ai dx1 (ej ) = Aj ; on the other hand, ωA (ej ) = definition, on one hand, ωA (ej ) = i=1

1 = A1 dx1 + A2 dx2 + A3 dx3 . Similarly, we (A, ej ) = Aj . Thus, aj = Aj , i.e., ωA can get 2 ωA = A1 dx2 ∧ dx3 + A2 dx3 ∧ dx1 + A3 dx1 ∧ dx2 .

It is easy to observe that 2 1 ωA =∗ (ωA ),

1 ωA = b (A).

Here, the top left hand corner “*” (“b”) represents the Hodge (sharp) operator respectively, namely ∗ : ∧k (Rm ) → ∧n−k (Rm ); b : R → R∗ . We now introduce three operators that play an important role in classical vector analysis, i.e., gradient, curl, and divergence. Definition 4.16. Let f ∈ C ∞ (R3 ) and A ∈ X (R3 ). The grad f and curl A ∈ X (R3 ) and div (A) ∈ C ∞ (R3 ) defined as follows 1 = df, ωgrad f

2 1 ωcurl = dωA , A

2 3 dωA = div A = ωA ,

and

where ω 3 = dx1 ∧ dx2 ∧ dx3 is the volume element in R3 . By this definition, ∂f ∂f ∂f 1 = df = dx1 + 2 dx2 + dx3 . ωgrad 1 f ∂x ∂x ∂x3 Thus, grad f =

∂f ∂f ∂f e1 + 2 e2 + 3 e3 ∂x1 ∂x ∂x

 = and so

∂f ∂f ∂f , , ∂x1 ∂x2 ∂x3



=

∂f , ∂x

88

1. Preliminaries of Differentiable Manifolds

  1 dωA = d A1 (x)dx1 + A2 (x)dx2 + A3 (x)dx3   ∂A1 1 ∂A ∂A = dx + 21 dx2 + 31 dx3 ∧ dx1 1

2 ωcurl = A

∂x

∂x



∂x



∂A2 1 ∂A ∂A + dx + 22 dx2 + 32 dx3 ∧ dx2 ∂x1 ∂x ∂x



+  =



∂A3 1 ∂A ∂A dx + 23 dx2 + 33 dx3 ∧ dx3 ∂x1 ∂x ∂x

∂A3 ∂A − 32 ∂x2 ∂x



+



dx2 ∧ dx3 +

∂A2 ∂A − 21 ∂x1 ∂x





∂A1 ∂A − 13 ∂x3 ∂x



dx3 ∧ dx1

dx1 ∧ dx2 ,

where  curl A =

∂A3 ∂A − 32 ∂x2 ∂x

  e1   ∂ =  1  ∂x  A1



 e1 +

e2

e3

∂ ∂x2

∂ ∂x3

A2

A3

∂A1 ∂A − 13 ∂x3 ∂x

    ,   



 e2 +

∂A2 ∂A − 21 ∂x1 ∂x



3 2 ωdiv = dωA = d(A1 dx2 ∧ dx3 + A2 dx3 ∧ dx1 + A3 dx1 ∧ dx2 ) A   ∂A1 ∂A ∂A + 22 + 33 dx1 ∧ dx2 ∧ dx3 . = 1

∂x

∂x

∂x

Therefore, div A = Since

∂A1 ∂A ∂A + 22 + 33 . ∂x1 ∂x ∂x

2 1 ωcurl grad f = d ωgrad f = d (d f ) = 0,   2 1 = d (d ωA ) = 0, div curl (A)ω 3 = d ωcurl A

we easily get two equalities in classical vector analysis: curl grad = 0,

div curl = 0.

1.4.6 Hodge Duality and Star Operators Let us introduce the Hodge operator as a linear transformation: ∗ : Λp −→ Λn−p . Definition 4.17. Denoting ∗υ as a element in Λn−p , ∀ u ∈ Λp , we have

e3

1.4 Foundation of Differential Form

89

uΛ ∗ υ = (u, υ)en . For brevity, we write uΛ ∗ υ as u ∗ υ. If υ is a scalar, u must also be a scalar. By the above formula, we get ∗υ = υen . Example 4.18. If α=



ai1 ···ip dxi1 ∧ · · · ∧ dxip ,

i1 <···
then ∗α =



bj1 ···jn−p dxj1 ∧ · · · ∧ dxjn−p ,

i1 <···
where bj1 ···jn−p =



εi1 ···ip j1 ···jn−p ai1 ···ip ,

i1 <···
εi1 ···ip j1 ···jn−p is the generalized Kronecker symbol. Star operators in 3-dimensional space have the following properties: Property 4.19. 1◦ ∗ d x = d y ∧ d z. 2◦ ∗ d y = d z ∧ d x. 3◦ ∗ d z = d x ∧ d y. 4◦ ∗ (d x ∧ d y ∧ d z) = 1. 5◦ Let ω = a1 d x1 + a2 d x2 + a3 d x3 , then % % % & & & ∂ a3 ∂ a2 ∂ a3 ∂ a1 ∂ a1 ∂ a2 1 2 ∗dω = − + − + − d x d x d x3 . ∂ x2 ∂ x3 ∂ x3 ∂ x1 ∂ x1 ∂ x2 ∂ a1 ∂ a2 ∂ a3 + + . ∂ x1 ∂ x2 ∂ x3

6◦

∗d∗ω =

7◦ 8◦ 9◦

grad = d (operation on Λ0 (R3 )). rot = ∗d (operation on Λ1 (R3 )). div = ∗d∗ (operation on Λ2 (R3 )).

1.4.7 Codifferential Operator δ We know that the exterior differential operator d : Λk (M ) → Λk+1 (M ) is a linear differential operator of first order, which increases one order in the form of differential form. However, Hodge star operator ∗ : Λk (M ) → Λn−k (M ) essentially is a dual operator. From this, one may ask whether it is possible to define a kind of −1 linear differential operator δ : Λk (M ) → Λk−1 (M ). The answer is yes. Definition 4.20. The codifferential operator δ is a kind of linear differential operator of −1 order. δ: Λk (M ) → Λk−1 (M ). It can be represented by δ = −(−1)g (−1)n(k+1) ∗ d ∗ . If our manifold is an oriented Riemann manifold, then g = 1.

90

1. Preliminaries of Differentiable Manifolds

Definition 4.21. The k-form ω is called coclosed, if δω = 0; it is called coexact, if there exists a θ ∈ Λk+1 (M ) s.t. ω = δθ. The relation between operators δ, d, and ∗ leads to the following theorem. Theorem 4.22. A codifferential operator δ has the following properties: 1◦ δ 2 = 0. 2◦ ∗ δd = dδ∗, ∗dδ = δd∗. 3◦ d ∗ δ = δ ∗ d = 0. 4◦ ∗ (δω) = (−1)k d(∗ω), ω ∈ Λk (M ). 5◦ δ(∗ω) = (−1)n−k+1 (dω), ω ∈ Λk (M ).

1.4.8 Laplace–Beltrami Operator Definition 4.23. Linear mapping Δ = d δ + δ d,

Λk (M ) −→ Λk (M )

is called Laplace–Beltrami operator on Riemann manifold. If k = 0, for f ∈ Λ0 (M ), we have δf = 0, and then Δf = δdf. Theorem 4.24. The Laplace–Beltrami operator obeys the following rule: 1◦ Δ = (d + δ)2 . 2◦ d · Δ = Δ · d = d · δ · d. 3◦ δ · Δ = Δ · δ = δ · d · δ. 4◦ ∗ Δ = Δ ∗ . Example 4.25. Let f ∈ Λ0 (R3 ). For a rectangular coordinate system δ, we have Δf = δdf = δ

∂f i dx . ∂xi

Since δ = −(−1)g (−1)n(k+1) ∗ d∗ = ∗d∗, % &  2 ∂f i ∂ f dx . Δf = ∗d ∗ = i ∂x (∂xi )2 Obviously, for R3 , the Laplace–Beltrami operator Δ : Λ0 (R3 ) → Λ0 (R3 ) is the 3  ∂2 . usual Laplace operator Δ = i 2 i=1

(∂x )

The corresponding relationship in action on the form operators d and δ and action on the coefficient of vector analysis can be summarized as follows:

1.5 Integration on a Manifold

91

d d d  Λ1  Λ2  Λ3 δ δ δ grad curl div coef.: scalar  vector  vector  scalar −div curl −grad We can easily obtain two equations in classical vector analysis: Λ0

form:

d d = 0 : rot grad = 0, div rot = 0. −rot grad = 0,

δδ = 0 :

−div rot = 0.

1.5 Integration on a Manifold The integral of an n-form on an n-manifold is defined in terms of integral over sets in Rn by means of partition of unity subordinate of an atlas.

1.5.1 Geometrical Preliminary Chains. A singular k-cube in M ⊂ Rn is a continuous function c : [0, 1]k → M , where [0, 1]n = [0, 1] × · · · × [0, 1], R0 = [0, 1]0 = {0}. 12 3 0 n

A singular 0-cube in M is a function c : [0, 1]0 = {0} → M , i.e., a point in M . A singular 1-cube in M is a usual curve. The standard n-cube in Rn is the identity mapping I n : [0, 1]n → Rn , I n (x) = x, ∀ x ∈ [0, 1]n . Definition 5.1. A k-chain C in M is a linear combination of finite singular k-cubes ci in M , i.e., c = α1 c1 + · · · + αr cr ,

αi ∈ R,

i = 1, · · · , r.

The set of all k-chains in M is denoted by C k (M ). C k (M ) forms a vector space on R if we introduce in C k (M ) the addition and multiplication by scalar by the following formulae: c1 + c2 =

r 

αi1 ci +

i=1

αc1 = α

r  i=1

αi1 ci =

r 

αi2 ci =

i=1 r  i=1

(ααi1 )ci ,

r  i=1

(αi1 + αi2 )ci ,

92

1. Preliminaries of Differentiable Manifolds

where cj =

r 

αij ci (j = 1, 2) are two k-chains in M . Without loss of generality,

i=1

we assume that different chains c1 and c2 are generated by the same set of k-cubes {c1 , · · · , cr }. For example, let c1 = c1 + 2c2 , c2 = c1 + c3 , where c3 = c2 . We only need to rewrite c1 and c2 as c1 = c1 + 2c2 + 0 · c3 , c2 = c1 + 0 · c2 + c3 . Boundary of Chains Corresponding to the exterior operator d: Ωk → Ωk+1 , there is a boundary operator ∂ : C k (M ) → C k−1 (M ), as defined below: ∂ [0, 1] = {1} − {0},   ∂ [0, 1]2 = ∂ [0, 1] × [0, 1] = {1} × [0, 1] − {0} × [0, 1] −[0, 1] × {1} + [0, 1] × {0}. For general [0, 1]k , ∀ x ∈ [0, 1]k−1 , denote k (x) = I k (x1 , · · · , xi−1 , 0, xi , · · · , xk−1 ), I(i,0) k (x) = I k (x1 , · · · , xi−1 , 1, xi , · · · , xk−1 ). I(i,1) k k We call I(i,0) and I(i,1) as (i, 0)- and (i, 1)-surface respectively, and

∂I k = ∂[0, 1]k =

k  

n (−1)i+α I(i,α) .

i=1 α=0,1

For any k-cube c : [0, 1]k → M , the (i, α)-surface is defined as k . c(i,α) = c ◦ I(i,α)

The boundary ∂c of the k-cube c is ∂c =

k  

k (−1)i+α c ◦ I(i,α) =

i=1 α=0,1

The boundary of any k-chain c =

k  

(−1)i+α c(i,α) .

i=1 α=0,1



αj cj is

j

∂c =



αj ∂cj .

j

Theorem 5.2. For any k-chain c, ∂(∂c) = 0, or briefly ∂ 2 = 0. k )(j,β) . For x ∈ [0, 1]k−2 , we have Proof. Firstly, assume i ≤ j and consider (I(i,α) k k )(j,β) (x) = I(i,α) (x1 , · · · , xj−1 , β, xj , · · · , xk−2 ) (I(i,α)

= I k (x1 , · · · , xi−1 , α, xi , · · · , xj−1 , β, xj , · · · , xk−2 ).

1.5 Integration on a Manifold

93

Similarly, we have k k (I(j+1,β) )(i,α) (x) = I(j+1,β) (x1 , · · · , xi−1 , α, xi , · · · , xk−2 )

= I k (x1 , · · · , xi−1 , α, xi , · · · , xj−1 , β, xj , · · · , xk−2 ). k k )(j,β) (x) = (I(j+1,β) )(i,α) (x), it is easy to see that for any Thus, if i ≤ j, (I(i,α) k-cube c, (c(i,α) )(j,β) = (c(j+1,β) )(i,α) as i ≤ j. Now, + * k   i+α ∂(∂c) = ∂ (−1) c(i,α) i=1 α=0,1

=

k   k−1  

(−1)i+α+j+β (c(i,α) )(j,β) .

i=1 α=0,1 j=1 β=0,1

In this sum, (c(i,α) )(j,β) and (c(j+1,β) )(i,β) occur simultaneously with the opposite sign. Then, all terms disappear in pairs and ∂(∂c) = 0. Consequently, for any k-chain r  αi ci , where ci (i = 1, · · · , r) are the k-cubes, c= i=1

∂(∂c) = ∂

* r 

+ ai ∂ci

=

i=1

r 

ai ∂(∂ci ) = 0.

i=1



The theorem is proved.

Definition 5.3. A k-chain c is called a cycle if ∂c = 0. A k-chain c is called a boundary if there is a k + 1-chain c1 such that c = ∂c1 . Obviously, boundaries imply cycles. However, the converse is not always true.

1.5.2 Integration and Stokes Theorem For any ω ∈ Ωk ([0, 1]k ), there is a f ∈ C ∞ ((0, 1)k ) such that ω = f dx1 ∧ · · · ∧ dxk . Define the integral of ω on [0, 1]k as ω= f, [0,1]k

i.e.,

-

[0,1]k

ω=

[0,1]k

[0,1]k

f dx1 · · · dxk ,

where the right hand side is a Riemannian integral of f on [0, 1]k . If ω ∈ C k (M ) and c is a k-cube in M , we define the integral of ω on c as ω= c∗ ω. c

[0,1]k

94

1. Preliminaries of Differentiable Manifolds

In particular, Ik

f dx1 ∧ · · · ∧ dxk =

[0,1]k

(I k )∗ (f dx1 ∧ · · · ∧ dxk )

-

f (x1 , · · · , xk ) dx1 · · · dxk .

= [0,1]k

If c is a 0-cube, we define

ω = ω(c(0)). c

The integral of ω on a k-chain c =



ai ci is

i

ω=



c

ai

ω, ci

i

where ci are k-cubes in M . Example 5.4. For k = 1, c : [0, 1] → R2 is a curve defined by x = cos 2πθ, y = sin 2πθ (1 ≤ θ ≤ 1), i.e., a circle on (x, y)-plane. Let ω = P (x, y)dx + Q(x, y)dy, then, the integral of ω on c is ω = P (x, y)dx + Q(x, y)dy c

c

-

c∗ (P (x, y)dx + Q(x, y)dy)

= -

[0,1] 1

= 0



− P (cos 2πθ, sin 2πθ)2π sin 2πθ

 +Q(cos 2πθ, sin 2πθ)2π cos 2πθ d θ, which is the usual integral along a curve. Theorem 5.5 (Stokes theorem). If ω is a (k − 1)-form on an open set M ⊂ Rn and c is a k-chain in M , then dω = c

ω. ∂c

Proof. 1◦ Firstly, we assume c = I k and ω is a (k − 1)-form on [0, 1]k . Then, ω is a sum of (k − 1)-forms of the type 4i ∧ · · · ∧ dxk . f dx1 ∧ · · · ∧ dx

1.5 Integration on a Manifold

95

Thus, it suffices to prove the theorem for each of these. 4i ∧ · · · ∧ dxk ) dω = d(f dx1 ∧ · · · ∧ dx Ik

Ik

= -

 ∂f 4i ∧ · · · ∧ dxk dxi ∧ dx1 ∧ · · · ∧ dx ∂xi [0,1]k (−1)i−1

= [0,1]k

∂f 1 4i ∧ · · · ∧ dxk dx ∧ · · · ∧ dx ∂xi

-

∂f 1 (x , · · · , xk )dx1 · · · dxk ∂xi

= (−1)i−1 [0,1]k

-



= (−1)i−1 [0,1]k−1

f (x1 , · · · , 1, · · · , xk )

 4i ∧ · · · ∧ dxk . −f (x1 , · · · , 0, · · · , xk ) dx1 ∧ · · · ∧ dx Notice that -

[0,1]k−1

k∗ 4i ∧ · · · ∧ dxk ) I(j,α) (f dx1 ∧ · · · ∧ dx

-

= [0,1]k−1

 ∗  ∗   k k f (x1 , · · · , xj−1 , α, xj , · · · , xk ) I(j,α) dx1 ∧ · · · ∧ dxi ∧ · · · ∧ I(j,α) dxk

⎧ 0, ⎪ ⎨ = ⎪ ⎩

i = j,

[0,1]k−1

4i · · · dxk , f (x1 , · · · , α, · · · , xk ) dx1 · · · dx

i = j.

Thus, ∂I k

=

4i ∧ · · · ∧ dxk f dx1 ∧ · · · ∧ dx n  

(−1)j+α [0,1]k−1

j=1 α=0,1

-

k∗ 4i ∧ · · · ∧ dxk ) I(j,α) (f dx1 ∧ · · · ∧ dx



= (−1)i+1 [0,1]k−1

f (x1 , · · · , xi−1 , 1, · · · , xk )

 xi ∧ · · · ∧ d xk . −f (x1 , · · · , 0, · · · , xk ) d x1 ∧ · · · ∧ d. In other words,

-

Ik

2◦

dω =

For a singular k-cube c, since

ω. ∂I k

96

1. Preliminaries of Differentiable Manifolds

∂c =

k  

(−1)i+α c(i,α)

i=1 α=0,1

=

k  

k (−1)i+α c ◦ I(i,α) ,

i=1 α=0,1

by definition of integration, -

-

k  

ω= ∂c

(−1)

i+α

ω k c◦I(i,α)

i=1 α=0,1

=

k  

(−1)

i=1 α=0,1

-

-

dω =

∗

c

∂I k

-

∗

c dω = Ik

c∗ ω,

c ω= ∗ I(i,α)

-

-

∗

i+α

c∗ ω,

dc ω = Ik

∂I k

-

dω =

and so c

ω, for any singular k-cube c. ∂c

Finally, if c is a k-chain, i.e., c =

k 

αi ci , where ci are singular k-cubes, then

i=1

dω = c

k 

ai

dω = ci

i=1

k 

-

ai

ω= ∂ci

i=1

ω. ∂c



Therefore, the theorem is completed. Example 5.6. Consider the 1-form ω 1 = p1 dq1 + · · · + pn dqn ,

on R2n with the coordinates p1 , · · · , pn , q1 , · · · , qn ; dω 1 = dp1 ∧ dq1 + · · · + dpn ∧ dqn = dp ∧ dq. Thus, - dp ∧ dq = ci

p dq. ∂ci

- dp ∧ dq = 0.

In particular, if ci is a cycle , i.e., ∂ci = 0, then ci

1.5.3 Some Classical Theories on Vector Analysis Here, we assume R3 is the oriented 3-dim Euclidian space. By Subsection 1.4.5, every 1 2 vector field A on R3 corresponds to a 1-form ωA and a 2-form ωA :

1.5 Integration on a Manifold

97

A = (A1 (x), A2 (x), A3 (x)), 1 ωA = A1 (x)dx1 + A2 (x)dx2 + A3 (x)dx3 , 2 = A1 (x)dx2 ∧ dx3 + A2 (x)dx3 ∧ dx1 + A3 (x)dx1 ∧ dx2 . ωA

Suppose a 1-chain c1 represents a curve l (with the same orientation). Then, 1 ωA = A1 dx1 + A2 dx2 + A3 dx3 = A · dl, c1

c1

l 1 ωA

on a l-chain c1 representing a curve l is the circuwhich shows that the integral of 1 on a lation of the field A over the curve l. If A is a force field, then the integral of ωA 1-chain c1 is the work done by A along the curve l. Suppose a 2-chain c2 represents an oriented surface S. Then, 2 ωA = A1 dx2 ∧ dx3 + A2 dx3 ∧ dx1 + A3 dx1 ∧ dx2 c2

c2

-

A1 dx2 dx3 + A2 dx1 dx3 + A3 dx1 dx2

= S

=

Adn. S

2 on a 2-chain c2 representing an oriented surface In other words, the integral of ωA S is the flux of the field A though the surface S. Applying the Stokes’ theorem to different cases, we can get three important theorems in classical calculus, Green theorem, Gauss theorem, and Stokes theorem.

Theorem 5.7 (Green theorem). Let c2 represent a 2-dim domain D, ∂c2 the boundary l of D, and ω = P (x, y)dx + Q(x, y)dy ∈ C 1 (R2 ). Then, dω = d(P dx + Qdy), -

c2

-

c2

ωdl = ∂c2

P dx + Qdy l

-  -  ∂Q ∂P  ∂Q ∂P  = − dx ∧ dy = − dxdy. ∂x ∂y ∂x ∂y c2 D

Proof. Since

 ∂Q ∂P  − dx ∧ dy, d (P dx + Qdy) = ∂x ∂y the Stokes theorem ωdl = dω implies ∂c

c

-  ∂Q ∂P  − dxdy = P dx + Qdy, ∂x ∂y D l

which is the classical Green theorem.



98

1. Preliminaries of Differentiable Manifolds

Theorem 5.8 (Gauss theorem). If a 3-chain c3 represents a domain in R3 , ∂c3 2 represents the boundary S = ∂D of D, and ωA = Ax dy∧dz+Ay dz∧dx+Az dx∧dy is a 2-form on R3 , then -

2 dωA c3

-  ∂Ax ∂Ay ∂Az  = + + dx ∧ dy ∧ dz ∂x ∂y ∂z c3 = div Adxdydz, -

-

D

ω = ∂c3

Ax dydz + Ay dxdz + Az dxdy. S

Thus, by Stokes’ theorem, we obtain Gauss theorem. Theorem 5.9 (Classical Stokes theorem). div Adxdydz = Ax dydz + Ay dxdz + Az dxdy. D

S

If a 2-chain C in R3 represents an oriented surface S, its boundary is l = ∂S = 1 = Ax dx + Ay dy + Az dz is a 1-form on R3 . Then, the Stokes’ theorem ∂c2 and ωA shows Ax dx + Ay dy + Az dz = curl A ds. l

Here we use the equality:

S 1 dωA

1

= ωcurl A .

1.6 Cohomology and Homology The set of all closed k-forms, denoted by Z k , forms a subspace of Ωk and the set of all exact k-forms, denoted by B k , also forms a subspace of Ωk . The quotient space H k (M, R) =

Zk ker dk ≡ , k B imdk−1

is called the k-th cohomology space. An element in H k is an equivalent class of closed forms, in which two closed forms ω1 and ω2 are equivalent if there is a (k − 1)-form θ such that ω1 − ω2 = dθ, where d is called the exterior differential operator. It is a mapping d : Ωk (M ) −→ Ωk+1 (M ). The kernel of d is the subspace Z k of the closed k-form, the image of d is the space B k of exact differential form. For every k-form, we have dω k = 0. Then Bk ⊆ Z k.

1.7 Lie Derivative

99

Similarly, the set of all k-cycles, denoted by Zk , and the set of all k-boundaries, denoted by Bk , form a subspace of the vector space C k . The corresponding quotient space Hk (M, R) =

Zk ker ∂k = Bk im∂k+1

is called the k-th homology space. An element in Hk is a class of cycles differing from one another only by a boundary. Definition 6.1. The dimension of H k (Hk ), denoted by bk (bk ), is called the k-th Betti number. Theorem 6.2 (De Rham theorem). The two Betti numbers are the same, i.e., bk = bk . Example 6.3. M ⊂ Rn is an open set. We consider H 0 (M, R) and H0 (M, R), B 0 = {0}, for there is no such form smaller than 0-form. If ω ∈ Ω0 (M ), then ω = f ∈ C ∞ (M ), df = 0 means that f is equal to a constant (local), and: Z 0 (M ) = R × · · · × R . 12 3 0 m

If M has components connected by m paths. Since H 0 (M, R) = Z 0 (M ), and b = dim (H 0 (M, R)) = m, it is easy to see that , Z0 (M ) is generated by all points in M and m1 , m2 ∈ M ⊂ Z0 (M ) is equivalent, iff they have components connected × · · · × R and b0 = dim (H0 (M, R)) = m. by the same path. Thus, H0 (M, R) = R 12 3 0 0

m

We have b0 = b0 . The De Rham theorem holds good for k = 0 in this case.

1.7 Lie Derivative The Lie derivative may be defined in several equivalent ways. In this section, to keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields. The Lie derivative can also be defined to act on general tensors, as discussed later in the article.

1.7.1 Vector Fields as Differential Operator Let X(x) = n  i=0

n 

Xi (x)ei = (X1 (x), · · · , Xn (x)) be a vector field on Rn , and ω1 =

i=0

ai (x) dx be a 1-form on Rn . ω 1 , X(x) = ω 1 (x), X(x) is the function on i

100

1. Preliminaries of Differentiable Manifolds

Rn , where  ,  is the dual bracket between Tx Rn and Tx∗ Rn . This defines the natural bilinear mapping of Λ1 (Rn ) × X (Rn ) −→ C ∞ (Rn ). If ω 1 = df =

 ∂f dxi , then ∂xi df, X =

n n   ∂f ∂f X = Xi (x) i . i i ∂x ∂x i=1 i=1 n 

∂f , ∀ f ∈ C ∞ (Rn ). Thus, any smooth i ∂x i=1 vector field may be viewed as a linear partial differential operator on Rn of order 1, without zero terms and smooth coefficient, i.e., there is a correspondence between X(x) ∈ X (Rn ) and LX :

Denote df, X = LX f , i.e., LX f =

X(x) =

n 

Xi

Xi (x)ei −→ LX =

i=1

n 

Xi (x)

i=1

It is one to one, and so hereafter, we can also write X(x) =

∂ . ∂xi

n 

Xi (x)ei as

i=1

X(x) =

n 

Xi (x)

i=1

∂ . ∂xi

Definition 7.1. For any two vector fields X, Y ∈ X (Rn ), define [X, Y ] = XY − Y X, i.e., ∀ f ∈ C ∞ (R),

[X, Y ]f = X(Y f ) − Y (Xf ),

where Xf is viewed as LX f , and [X, Y ] is called the commutator or the Poisson bracket of X, Y . Proposition 7.2. Let X =

n  i=1

[X, Y ] =

 ∂ ∂ ,Y = Yi (x) i , then i ∂x ∂x i=1 n

Xi (x)

n  n  

Xk

i=1 k=1

Proof. ∀ f ∈ C ∞ (Rn ),

∂Yi ∂Xi  ∂ − Y . k ∂xk ∂xk ∂xi

1.7 Lie Derivative

101

[X, Y ]f = X(Y f ) − Y (Xf ) =X

n  i=1

=

n 

 ∂f  ∂f  X − Y i ∂xi ∂xi i=1 n

Yi

n  

Xk

k=1 i=1

=

n  n  

Xk

i=1 k=1

∂Yi ∂f ∂ 2f ∂Xi ∂f ∂f 2  + Xk Yi i k − Yk k − Yk Xi i k i i k ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂Yi ∂Xi  ∂f − Y . k ∂xk ∂xk ∂xi

Thus, [X, Y ] =

n  n  

Xk

i=1 k=1

∂Yi ∂Xi  ∂ − Y . k ∂xk ∂xk ∂xi

The i-th component of [X, Y ] is [X, Y ]i =

n  

Xk

k=1

∂Yi ∂Xi  . − Y k ∂xk ∂xk

In this manner, [X, Y ] may be represented by ∂Y X − ∂X Y , where ∂Y is the Jaco∂x ∂x ∂x bian of Y .  Theorem 7.3. Let X1 , X2 , X3 ∈ X (Rn ) and f, g ∈ C ∞ (Rn ). Then, 1◦ ◦

[X1 , X2 ] = −[X2 , X1 ].

2

[α1 X1 + α2 X2 , X3 ] = α1 [X1 , X3 ] + α2 [X2 , X3 ].

3◦

[f X1 , gX2 ] = f (X1 g)X2 − g(X2 f )X1 + f g[X1 , X2 ].

4◦

[X1 , [X2 , X3 ]] + [X2 , [X3 , X1 ]] + [X3 , [X1 , X2 ]] = 0.

The equality 4◦ is called Jacobi identity. Proof. 1◦ and 2◦ are evident. 4◦ follows immediately from the expansion 3◦ : [f X1 , gX2 ] = (f X1 )(gX2 ) − (gX2 )(f X1 ) = f (X1 g)X2 + f gX1 X2 − g(X2 f )X1 − f gX2 X1 = f (X1 g)X2 − g(X2 f )X1 + f g[X1 , X2 ]. The proof can be obtained.



1.7.2 Flows of Vector Fields In Subsection 1.2.2, we have already discussed vector field and the flow of a general differentiable manifold. Now, we focus on applications to the dynamic system.

102

1. Preliminaries of Differentiable Manifolds

(1) Phase space: The set of all possible states of a process is called phase space. For example, consider the motion of system in classical mechanics, whose future and past are uniquely determined by the initial position and initial velocities of all points in the system. The phase space of a mechanical system is the set whose elements are the sets of positions and velocities of all points of the given system. This is exactly the tangent space we discussed earlier. (2) Differentiable process: A process is called differentiable if its phase space has the structure of a differentiable manifold, and the change of state with time is described by differentiable functions. Let M be the phase space. A point of this space is a defined state of the process. Assume that at instant t = 0 the process was in state x. Then at another moment, the process t will be in another state. Denote this new state of the process by g t x. We have defined for each t a mapping g t g t : M −→ M. This is called the transformation in time t, which takes the state at instance 0 to the state at the instant t g 0 = id, g t+s = g t · g s . Let y = g s x be the state after time s, and the state z = g t y again after time t. The effect is the same as advancing x to time t + s, i.e., z = gt+s x. Thus, we can define a one parameter transformation group. Definition 7.4. The mapping family {g t }of set M to itself is called a one parameter transformation group on filed M , if ∀ s, t ∈ R satisfies: g t+s = g t · g s ,

g 0 = I.

Definition 7.5. The set M and the corresponding one parameter transformation group {g t } that maps M to itself compose (M, {g t }), which is called the phase flow, where set M is called the phase space, and its element is called the phase point. (3) Diffeomorphism: If there exists a 1−1 map f : U → V so that f and f −1 : V → U are both differentiable, then f is said to be a diffeomorphism. (4) One parameter diffeomorphism groups: The one parameter differmorphism group {gt } on manifold M is a collection of mappings from the direct product R × M to M : g : R × M −→ M, g(t, x) = gt · x, t ∈ R, x ∈ M which satisfies: 1◦ g is a differentiable mapping. 2◦ ∀ t ∈ R, g t : M → M is a differomorphism. 3◦ Family {g t , t ∈ R} is one parameter transformation group for M . Dynamical system: the autonomous differential equation defined by a field X is the equation

1.7 Lie Derivative

d x(t) = X(x(t)), dt

x|t=0 = x0

103

(initial value).

The image of the mapping x(t) is called a phase space, and the graph of the mapping x(t) is called an integral curve. The integral curve lies in the direct product of the time axis and the phase space, which is called M × R extended space. Such an equation is called a dynamic system, whose phase flow g t , M → M, {g t , t ∈ R} composes a group: g t · g s = g t+s ,

g 0 = id,

g −t = {gt }−1 .

Let X be a smooth vector field on Rn . The solution curve through x0 is a differentiable mapping t → x(t): I → Rn , where I is an interval in R and 0 ∈ I, such that d x(t) = x(t) ˙ = X(x(t)), t ∈ I, x(0) = x0 . (7.1) dt

It can be represented by its components as d xi (t) = Xi (x1 (t), · · · , xn (t)), dt

t ∈ I;

i = 1, · · · , n,

xi (0) = xi0 ,

where x(t) = (x1 (t), · · · , xn (t)). It is well known that there is a unique differentiable solution x(t, x0 ) for the above system, which depends differentiably on the initial value x0 in some neighborhood. The mapping φtX from (some neighborhood of) Rn to itself is defined by φtX (x0 ) = x(t, x0 ). Suppose t is small enough, and is a diffeomorphism in (some neighborhood of) Rn . It has the following properties: 1◦ φtX1 +t2 = φtX1 · φtX2 . 2◦ φ0X = id. t −1 3◦ φ−t . X = (φX ) Thus, such a class {φtX } of the mappings φtX forms a group, called as a local 1parameter transformations group in Rn , or a local dynamics system in Rn , or a flow in Rn .

1.7.3 Lie Derivative and Contraction Let ϕ : Rn → Rn be a diffeomorphism and Y ∈ X (Rm ) be a smooth vector field on Rn . The pullback ϕ∗ Y of Y is a smooth vector field on Rn , defined by the formula (ϕ∗ Y )(x) = (Dϕ−1 )(y)Y (y) = ϕ−1 ∗ (y)Y (y),

y = ϕ(x).

Definition 7.6. Let X and Y be two vector fields. The Lie derivative LX Y of Y with respect to X is defined by LX Y = where φtX is the flow of X.

d  t∗  φ Y t=0 , dt X

104

1. Preliminaries of Differentiable Manifolds

Theorem 7.7. Let X and Y be∗ two vector fields. Then we have ∗ ∀ f ∈ C ∞ (Rn ). 1◦ (φtX Y )f = φtX (Y φ−t X f ), 2◦ 3◦

∗ d t∗ (φ f ) = φtX (Xf ). dt X

LX Y = [X, Y ].

Proof. 1◦

By definition,

∗

∗

(φtX Y )f (x) = df (x), (φtX · Y )(x) = df (x), Dφ−t X (y)Y (y)



 y = φtX (x)

 = (Dφ−t X (y)) df (x), Y (y)  = d(f ◦ φ−1 X (y)) , Y (y)   = df"(y), Y (y) f" = f ◦ φtX

= (Y f")(y) = (Y f")(φtX (x))  ∗ ∗ = φtX (Y f")(x) = φtX Y f (φ−t X (y)) (x)  ∗ ∗ = φtX Y (φ−t X f ) (x). 2◦

The proof is as follows: n     d  t∗  d  ∂f φX f = f φtX (x) = Xi φtX (x) dt dt ∂xi i=1

  ∗ = (Xf ) φtX (x) = φtX (Xf ). 3◦

The proof is as follows:  ∗ d  t∗   d  t∗  (LX Y )f = φX Y f t=0 = φX (Y · φ−t X f ) t=0 dt



dt

∗

= φtX XY ·

∗ φ−t X f

 ∗ ∗  − φtX Y φ−t X Xf t=0

= XY f − Y Xf = [X, Y ]f,

∀ f.

Thus, we get LX Y = [X, Y ]. 

Therefore, the theorem is completed.

By the equality 3◦ , the Jacobi identity about the Poisson bracket { , } shows that the operator LX is a { , }-derivative on the algebra X (Rn ) with binary operator [ , ], i.e., LX {X1 , X2 } = {LX X1 , X2 } + {X1 , LX X2 }. Definition 7.8. ∀ω ∈ Ωk (Rn ), the Lie derivative LX ω of ω with respect to a vector field X ∈ X (Rn ) is defined by   d ∗  1 ∗ LX ω = φtX ω  = lim φtX ω − ω . dt

t=0

t→0

t

1.7 Lie Derivative

105

Theorem 7.9. The Lie derivative LX with respect to X ∈ X (Rn ) has the following properties: n  ∂f Xi i , f ∈ C ∞ (Rn ), i.e., the Lie derivatives of a 1◦ LX f = Xf = ∂x i=1 function f with respect to X is the directional derivative in direction X(x). 2◦ LX is a Λ-derivative, i.e., LX is R-linear, LX (αω1 + βω2 ) = αLX ω1 + βLX ω2 , LX (ω1 ∧ ω2 ) = LX ω1 ∧ ω2 + ω1 ∧ LX ω2 . 3◦

LX d = d LX .

Proof. 1◦ We have

∗ d t∗ φ f = φtX (Xf ), and so dt X



LX f = 2◦

 ∗ ∗ d   (φt f ) = (φtX Xf )t=0 = Xf. d t t=0 X

It is obvious that LX is R-linear,  ∗ d  LX (ω1 ∧ ω2 ) =  φtX (ω1 ∧ ω2 ) dt

=

t=0

 d  

dt

 =



∗

∗

φtX ω1 ∧ φtX ω2

t=0



 



∗ ∗ d  d   φt ω1 ∧ ω2 + ω1 ∧  φtX ω2 d t t=0 X d t t=0

= LX ω1 ∧ ω2 + ω1 ∧ LX ω2 . 3◦

The proof is as follows: 

LX dω =



∗ ∗ d  d   φt dω =  dφtX ω d t t=0 X d t t=0



= d

∗ d  φt ω = d LX ω. d t t=0 X



Therefore, the theorem is completed.

Definition 7.10. Let X ∈ X (Rn ) and ω ∈ Ωk (Rn ). The contraction iX ω of X and ω is defined by iX ω(ξ1 , · · · , ξk−1 ) = ω(X(x), ξ1 , · · · , ξk−1 ),

ξi ∈ Tk Rn ,

i = 1, · · · , k − 1.

iX f = 0, for f ∈ C ∞ (Rn ) = Ω0 (Rn ). iX maps k-forms into (k − 1)-forms, i.e., Ωk (Rn ) −→ Ωk−1 (Rn ).

106

1. Preliminaries of Differentiable Manifolds

Theorem 7.11. Let ω1 , ω2 ∈ Ωk (Rn ), ω3 ∈ Ωl (Rn ). Then, 1◦

iX is a Λ-antiderivative, i.e., R-linear. R-linear : iX (α1 ω1 + α2 ω2 ) = α1 iX ω1 + α2 iX ω2 ,

α1 , α2 ∈ R.

anti-derivative : iX (ω1 ∧ ω3 ) = iX ω1 ∧ ω3 + (−1)k ω1 ∧ iX ω3 . 2◦

if X+gY = f iX + giY , ∀ f, g ∈ C ∞ (Rn ), X, Y ∈ X (Rn ).

3◦

iX df = LX f, f ∈ C ∞ (Rn ).

4◦

LX = iX d + diX , (Cartan’s Magic formula).

5◦

Lf X = f LX + df ∧ iX .

Proof.

1◦

R-linearity of ix is evident.

iX1 (ω1 ∧ ω3 )(ξ2 , · · · , ξk+l−1 ) = (ω1 ∧ ω3 )(X1 (x), ξ2 , · · · , ξk+l ) = (ω1 ∧ ω3 )(ξ1 , ξ2 , · · · , ξk+l )  =

(denote ξ1 = X1 (x)) ε(σ)ω1 (ξσ(1) , · · · , ξσ(k) )ω3 (ξσ(k+1) , · · · , ξσ(k+l) )

σ(1)<···<σ(k); σ(k+1)<···<σ(k+l)  

=

+

i∈{σ(1)···σ(k)}

 

.

i∈{σ(k+1)···σ(k+l)}

In the first part, it must be σ(1) = 1 since σ(1) < · · · < σ(k). Similarly, in the second part, σ(k + 1) = 1. Set σ  = {σ(2), · · · , σ(k), σ(k + 1), · · · , σ(k + l)}, σ  = {σ(1), · · · , σ(k), σ(k + 2), · · · , σ(k + l)}.

Then, ε(σ) = ε(σ ),

if

ε(σ) = (−1)k ε(σ  ), if

Thus,

i ∈ {σ(1), · · · , σ(k)}, i ∈ {σ(k + 1), · · · , σ(k + l)}.

1.7 Lie Derivative

107

iX (ω1 ∧ ω3 )(ξ2 , · · · , ξk+2 )  

=

ε(σ  )ω1 (ξ1 , ξσ(2) , · · · , ξσ(k) )

σ(2)<···<σ(k), σ(k+1)<···<σ(k+l)

· ω3 (ξσ(k+1) , · · · , ξσ(k+l) )  

+

(−1)k ε(σ  )ω1 (ξσ(1) , · · · , ξσ(k) )

σ(1)<···<σ(k), σ(k+1)<···<σ(k+l)

· ω3 (ξ1 , ξσ(k+2) , · · · , ξσ(k+l) )  

=

ε(σ  )(iX ω1 )(ξσ (1) , · · · , ξσ (k−1) )

σ  (1)<···<σ  (k), σ  (k−1)<···<σ  (k+l−1)

· ω3 (ξσ (k) , · · · , ξσ (k+l−1) )  

+

(−1)k ε(σ  )ω1 (ξσ (1) , · · · , ξσ (k) )(iX ω3 )

σ  (1)<···<σ  (k), σ  (k+1)<···<σ  (k+l−1)

· (ξσ (k+1) , · · · , ξσ (k+l−1) ) =

(iX ω1 ∧ ω3 )(ξ2 , · · · , ξk+l ) + (−1)k (ω1 ∧ iX ω3 )(ξ2 , · · · , ξk+l ).

Thus, we get the equality iX (ω1 ∧ ω3 ) = iX ω1 ∧ ω3 + (−1)k ω1 ∧ iX ω3 . 2◦

The proof is as follows: (if X+gY ω)(ξ1 , · · · , ξk−1 ) = ω(f X + gY, ξ1 , · · · , ξk−1 ) = f (x)ω(X(x), ξ1 , · · · , ξk−1 ) + g(x)ω(Y (x), ξ1 , · · · , ξk−1 ) = f (x)iX ω(ξ1 , · · · , ξk−1 ) + g(x)iY ω(ξ1 , · · · , ξk−1 ) = (f (x)iX + g(x)iY )(ξ1 , · · · , ξk−1 ).

This is the equation 2◦ . 3◦ iX df = df (X) = Xf = LX f . 4◦ By induction with respect to k. k = 0. i.e., 3◦ . Suppose that 4◦ holds good for k. Then for k + 1, ω can be written as a sum of the forms like ω 1 ∧ df where ω 1 ∈ Ωk (Rn ) and f ∈ C ∞ (Rn ). By linearity of LX , without loss of generality we may assume ω = ω 1 ∧ df . Then, LX (ω ∧ df ) = LX ω ∧ df + ω ∧ LX df = (iX dω + diX ω) ∧ df + ω ∧ dLX f. On the other hand,

108

1. Preliminaries of Differentiable Manifolds

(iX d + diX )(ω ∧ df ) = iX d(ω ∧ df ) + (diX )(ω ∧ df ) = iX (dω ∧ df ) + d(iX ω ∧ df + (−1)k ω ∧ iX df ) = iX dω ∧ df + (−1)k+1 dω ∧ iX df + diX ω ∧ df +(−1)k dω ∧ iX f + (−1)2k ω ∧ dLX f = (iX d + diX )ω ∧ df + ω ∧ dLX f, thus, LX (ω ∧ df ) = (iX d + diX )(ω ∧ df ), i.e., LX = iX d + diX . 5◦

The proof is as follows: Lf X = (dif X + if X d)ω = d(f iX ω) + f iX dω = df ∧ iX ω + f diX ω + f iX dω = (f LX + df ∧ iX )ω,

so Lf X = f LX + df ∧ iX . 

Therefore, the theorem is completed.

Every k-form ω on Rn can also be considered as a function on X k (Rn ): X k (Rn ) → C ∞ (Rn ), i.e., ω(X1 , · · · , Xk )(x) = ω(x)(X1 (x), · · · , Xk (x)),

Xi ∈ X (Rn ),

i = 1, · · · , k.

It is linear, skew-symmetric. Theorem 7.12. Let ω ∈ Ωk (Rn ), Xi (i = 1, · · · , k) be vector fields on Rn . Then, k  1◦ (LX ω)(X1 , · · · , Xk ) = LX (ω(X1 , · · · , Xk ))− ω(X1 , · · · , LX Xi , · · · , Xk ). i=1

2◦ dω(X0 , · · · , Xk ) =

k 

i , · · · , Xj , · · · , Xk ) (−1)i LXi (ω(X0 , · · · , X

i=0

+



i , · · · , X j , · · · , Xk ). (−1)i+j ω(LXi Xj , X0 , · · · , X

i<j

Proof. 1◦ By the definition of LX and the Theorem 7.7, since

1.7 Lie Derivative

109

∗

(φtX ω)(X1 , · · · , Xk ) = ω(φtX ω)(φtX∗ X1 , · · · , φtX∗ Xk ) ∗

∗

−t t = ω(φ−t X X1 , · · · , φX Xk )(φX X) ∗

∗

∗

−t = φtX (ω(φ−t X X1 , · · · , φX Xk )),

we have (LX ω)(X1 , · · · , Xk ) =



∗ d  (φt ω)(X1 , · · · , Xk ) d t t=0 X



=

  t∗ ∗ d −t∗ φ ω(φ−t  X X1 , · · · , φX Xk ) d t t=0 X

k    = LX ω(X1 , · · · , Xk ) − ω(X1 , · · · , LX Xi , · · · , Xk ). i=1

2◦

By induction with respect to k, k = 0 is evident: df (X) = LX f,

f ∈ C ∞ (Rn ) = Ω0 (Rn ).

Suppose 2◦ holds for k − 1. Then for k, by 1◦ , dω(X0 , X1 , · · · , Xk ) = (iX0 dω)(X1 , · · · , Xk ) = (LX0 ω)(X1 , · · · , Xk ) − (diX0 ω)(X1 , · · · , Xk ) k    ω(X1 , · · · , LX0 Xi , · · · , Xk ) = LX0 ω(X1 , · · · , Xk ) − i=1

−(diX0 ω)(X1 , · · · , Xk ), where iX0 ω ∈ Ωk−1 (Rn ). By inductive hypothesis, (diX0 ω)(X1 , · · · , Xk ) k    i , · · · , Xk ) (−1)i−1 LXi iX0 ω(X1 , · · · , X

=

i=1

+



i , · · · , X  j , · · · , Xk ) (−1)i+j iX0 ω(LXi Xj , X1 , · · · , X

1≤i≤j≤k k    i , · · · , Xk ) (−1)i−1 LXi ω(X0 , X1 , · · · , X

=

i=1

+



1≤i≤j≤k

Thus, we get

i , · · · , X j , · · · , Xk ). (−1)i+j−1 ω(LXi Xj , X0 , · · · , X

110

1. Preliminaries of Differentiable Manifolds

d ω(X0 , · · · , Xk ) = LX0

k  ω(X1 , · · · , Xk ) + (−1)j ω(X1 , · · · , LX0 Xj , · · · , Xk )





j=1

+

k 

   i , · · · , Xk ) (−1)i LXi ω(X0 , · · · , X

i=1



+

i , · · · , X  j , · · · , Xk ) (−1)i+j ω(LXi Xj , X0 , · · · , X

1≤i<j
=

k 

  i , · · · , Xk ) (−1)i LXi ω(X0 , · · · , X

i=0

+



i , · · · , X j , · · · , Xk ). (−1)i+j ω(LXi Xj , X0 , · · · , X

i<j

Finally , the theorem is completed.



Bibliography

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Chapter 2. Symplectic Algebra and Geometry Preliminaries

In order to deeply understand Hamiltonian mechanics, it is necessary to know basic concepts of symplectic algebra and geometry.

2.1 Symplectic Algebra and Orthogonal Algebra Symplectic algebra and orthogonal algebra have several similar concepts. First, we start with the bilinear form.

2.1.1 Bilinear Form 1. Bilinear form Definition 1.1 (Bilinear Form). Let Fn be an n-dimensional linear space. A bilinear form on Fn is a mapping ϕ : Fn × Fn → Fn that satisfies: 1◦ ϕ(αu + βv, y) = αϕ(u, y) + βϕ(v, y). 2◦ ϕ(x, αu + βv) = αϕ(x, u) + βϕ(x, v), ∀α, β ∈ F, u, v, x, y ∈ Fn . It is obvious that there exists a 1-1 correspondence between the matrix space M (n, F) and the space of the bilinear form on Fn . As a matter of fact, given a matrix A ∈ M (n, F), there is a bilinear form ϕA on Fn corresponding to ϕA (x, y) = x Ay =

n 

aij xi yj .

i,j=1

Conversely, given a bilinear form on Fn , there is also a matrix A ∈ M (n, F) corresponding to A = Aϕ = [aij ] = [ϕ(ei , ej )] ∈ M (n, F), such that ϕ(x, y) = x Ay, where e1 , e2 , · · · is a set of basis of Fn . Definition 1.2 (Symmetric or Antisymmetric). A bilinear form ϕ = ϕA is called symmetric or antisymmetric if ϕ(x, y) = ϕ(y, x) or ϕ(x, y) = −ϕ(y, x),

∀ x, y ∈ Fn ,

i.e., A = A, A = −A, respectively. K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

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Definition 1.3 (Conformally Symmetric). A bilinear form A is called conformally symmetric if ϕ(x, y) = 0 ⇐⇒ ϕ(y, x) = 0, i.e., {(x, y) ∈ Fn × Fn | ϕ(x, y) = 0} = {(x, y) ∈ Fn × Fn | ϕ(y, x) = 0}. A matrix A is called conformally symmetric if x Ay = 0 ⇔ x A y = 0, which is equivalent to saying that ϕA is conformally symmetric. ϕ is a non-degenerate map (or non-singular) if ∃x = 0 s.t. ϕ(x, y) = 0, ∀y ∈ Fn . Proposition 1.4. It is evident that the following claims are equivalent: 1◦ ϕA is non-degenerate. 2◦ If ∀ y ∈ Fn , ϕA (x, y) = 0, then x = 0. 3◦ A is non-degenerate. Definition 1.5 (Conformally Identical). A bilinear form ϕA is called conformally identical to ϕB if {(x, y) ∈ Fn × Fn | ϕA (x, y) = 0} = {(x, y) ∈ Fn × Fn | ϕB (y, x) = 0}. Proposition 1.6. The following claims are equivalent: 1◦ ϕA is conformally identical to ϕB . 2◦ ϕA (x, y) = 0, iff ϕB (x, y) = 0. 3◦ x Ay = 0, iff x By = 0. 4◦ ∃μ ∈ F, μ = 0, such that A = μB. The equivalence of 1◦ , 2◦ and 3◦ is trivial. Next, we prove the equivalence between 1◦ and 4◦ . Theorem 1.7. {(x, y) ∈ Fn × Fn | ϕA (x, y) = 0} = {(x, y) ∈ Fn × Fn | ϕB (y, x) = 0}, iff ∃ μ ∈ F, μ = 0, such that A = μB. Proof. The sufficiency is trivial. We only need to prove the necessity. Without loss of generality, we can assume F = R. Then, we have ker(A) = {y ∈ Rn | Ay = 0} = {y ∈ Rn | x Ay = 0, ∀ x ∈ Rn }, ker(B) = {y ∈ Rn | By = 0} = {y ∈ Rn | x By = 0, ∀ x ∈ Rn }. By our assumption, x Ay = 0 ⇔ x By = 0. Hence, ker(A) = ker(B), denoted as V , i.e., V = ker(A) = ker(B). Then ∀ v ∈ Rn , Av = 0 ⇐⇒ Bv = 0. Since x Ay = 0 ⇔ x By = 0, {Av}⊥ = {Bv}⊥ , and so

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{Av} = {Bv}. This shows that there exists μ(v) ∈ R, μ = 0 such that Av = μ(v)Bv. Next, we show μ(v) is a non-zero constant. Take a basis of Rn = {v1 , · · · , vr , vr+1 , · · · , vn }, such that {vr+1 , · · · , vn } = V = ker(A) = ker(B). Thus, Avi = 0 (i = 1, · · · , r), A(v1 + v2 + · · · + vr ) = 0. The above shows that there exist μ1 , · · · , μr , μ (all of which are no-zero), such that Avi = μi Bvi , i = 1, · · · , r, A(v1 + v2 + · · · + vr ) = μB(v1 + v2 + · · · + vr ). Then,

μ1 Bv1 + · · · + μr Bvr = μB(v1 + v2 + · · · + vr ) = μBv1 + · · · + μBvr .

After manipulation, we get (μ1 − μ)Bv1 + · · · + (μr − μ)Bvr = 0, i.e., B ((μ1 − μ)v1 + · · · + (μr − μ)vr ) = 0. Since (μ1 − μ)v1 + · · · + (μr − μ)vr ∈ {v1 , · · · , vr }, it must be (μ1 − μ)v1 + · · · + (μr − μ)vr = 0. Then by the linear independence of v1 , · · · , vr , we have μ = μ1 = · · · = μr . Therefore, Avi = μBvi ,

i = 1, · · · , r.

Similarly for i = r + 1, · · · , n, Avi = 0 = μBvi . Thus, we have obtained A = μB.  From Theorem 1.7, we can easily derive the following theorems. Theorem 1.8. ϕA is conformally symmetric, iff ∃ μ ∈ F, μ = 0, such that A = μA. Theorem 1.9. A ∈ M (n, R) is conformally symmetric, iff A = ±A, i.e., A is symmetric or antisymmetric. 2.

Quadratic forms induced by bilinear form n  Given a bilinear form ϕA (x, y) = aij xi yj , we can get a quadratic bilinear form ϕA (x, x) =

n  i,j

i,j

aij xi xj . Obviously, we have the following propositions:

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Proposition 1.10. ϕA (x + y, x + y) − ϕA (x, x) − ϕA (y, y) = ϕA (x, y) + ϕA (y, x). Proposition 1.11. ϕA is antisymmetric, iff ϕA (x, x) = 0, ∀x ∈ Fn . Proposition 1.12. ∀ x ∈ Fn , ϕA (x, x) = ϕB (x, x), iff A + A = B + B  . Proposition 1.13. If A = A , B = B  , then ∀ x ∈ Fn , ϕA (x, x) = ϕB (x, x) ⇔ A = B. Proposition 1.14. The following assertions are equivalent : 1◦ ϕ ) ( ) ( A = ϕB . n 2◦ (x, y) ∈ F × Fn | ϕA (x, y) = 1 = (x, y) ∈ Fn × Fn | ϕB (x, y) = 1 . 3◦ ϕA (x, y) = 1 ⇔ ϕB (x, y) = 1. 4◦ x Ay = 1 ⇔ x By = 1.

2.1.2 Sesquilinear Form 1. Sesquilinear form In a complex field C, there is an automorphism C  z = x + i y → z = x − i y ∈ C, such that z1 · z2 = z1 · z2 ,

z1 + z2 = z1 + z2 .

This leads to a new kind of binary forms. Definition 1.15 (Sesquilinear). A sesquilinear form on Cn is a mapping φ : Cn × Cn → C, such that for all u, v, x ∈ Cn , α, β ∈ C, we have 1◦ φ(αu + βv, x) = α φ(u, x) + β φ(v, x). 2◦ φ(x, αu + βv) = α φ(x, u) + β φ(x, v). Similarly, there exists a 1-1 correspondence between the complex matrix space M (n, C) and the space of sesquilinear forms on Cn . In fact, a complex matrix A ∈ M (n, C) has a natural correspondence to a sesquilinear form φA , which satisfies φA (x, y) = x∗ Ay =

n 

aij xi yj .

i,j

Conversely, a sequilinear form φ has a natural correspondence to a complex matrix A = Aφ = [aij ] = [φ(ei , ej )] ∈ M (n, C), which satisfies φ(x, y) = x∗ Aφ y. Definition 1.16 (Hermitian). φ = φA is Hermitian or anti-Hermitian, if ∀x, y ∈ Cn , φ(x, y) =φ(y, x) or φ(x, y) = −φ(y, x), i.e., A∗ = A or A∗ = −A. Such a complex matrix A is called Hermitian or anti-Hermitian.

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Definition 1.17 (Conformally Hermitian). φ = φA is called conformally Hermitian, if φ(x, y) = 0 ⇔ φ(y, x) = 0, then {(x, y) ∈ Cn × Cn | φ(x, y) = 0} = {(x, y) ∈ Cn × Cn | φ(y, x) = 0}, or

x∗ Ay = 0 ⇐⇒ x∗ A∗ y = 0.

The matrix A satisfying the above condition is called conformally Hermitian. φ = φA is non-degenerate iff x ∈ Cn , x = 0, such that φA (x, y) = 0, ∀y ∈ Cn ; or iff ∀y ∈ Cn , φ(x, y) = 0, then x = 0; or iff A is non-degenerate. 2. Hermitian forms induced by sesquilinear forms n  aij xi yj , we can induce a From a sesquilinear form on Cn , φA (x, y) = x∗ Ay = i,j

correspondent Hermitian form on Cn , φA (x, x) = x∗ Ax =

n 

aij xi xj .

i,j

If A is Hermitian, then ∀x ∈ Cn , φA (x, x) ∈ R. Remark 1.18. Hermitian forms have properties similar to Propositions 1.10 – 1.14, and Theorems 1.7 – 1.8. Remark 1.19. Hermitian analogue of Theorem 1.9 as follows. Theorem 1.20. A ∈ M (n, C) is conformally Hermitian, iff ∃μ ∈ C, |μ| = 1, such that A∗ = μA; or iff ∃ θ ∈ R, such that A∗ =eiθ A. Proof. A = A∗ ∗ = μA∗ = μμA = |μ|2 A. Thus, |μ|2 = 1.



2.1.3 Scalar Product, Hermitian Product A scalar product on Fn is a non-degenerate conformally symmetric bilinear form φG (x, y), where G = ±G. Symmetric products in Rn : (x, y)S = ϕS (x, y) = x Sy, S  = S, |S| = 0. Anti-symmetric product in Rn :[x, y]K = ϕK (x, y) = x Ky, K  = −K, |K| = 0. Remark 1.21. There does not exist any anti-symmetric scalar product in F2n+1 since |K| = 0 if K  = −K. Hermitian products in Cn are non-degenerate Hermitian forms in Cn , i.e., x, y = x∗ Gy = φG (x, y), Typical Examples are given below:

G∗ = G,

|G| = 0.

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Example 1.22 (Symmetric case). Euclidean scalar product (Euclidean form) in Rn : (x, y) = (x, y)I = x y =

n 

I  = I.

xi yi ,

i=1

This induces a Euclidean length measure |x|2 = (x, x) =

n 

x2i .

i=1

Example 1.23 (Anti-symmetrical case). Standard symplectic scalar product (symplectic form) in R2n : [x, y] = [x, y]J = x Jy =

n 

(xi yn+i − xn+i yi ),

i=1

where 0 −In

J= As n = 1, we have

In 0

!

J  = J −1 = −J.

= J2n ,

  x [x, y] = x1 y2 − x2 y1 =  1 x2

 y1  , y2 

which represents the oriented area of the parallelogram formed by the vector x, y in R2 (see the image below). (x2 , y2 )

6 y

For general n, we get n    xi  [x, y] = x Jy =  xn+i 

i=1

yi yn+i

   

 :

O

x

-

(x1 , y1 )

which represents a sum of oriented areas of the parallelograms formed by projecting vectors x, y ∈ R2n to the (xi , xn+i ) coordinate planes. Remark 1.24 (Pfaffian theorem). For any n, there exists a polynomial Pn (xij ) with integer coefficients in variables xij (i < j) such that 2

det K = det [kij ] = [Pn (kij )] ,

∀ anti-symmetric matrix K.

Example 1.25 (Hermitian case). Unitary product in Cn : w, z = w, zI = w ∗ Iz = w ∗ z,

w, z ∈ Cn , I ∗ = I,

w, z induces z = z ∗ z, which is the Euclidean length measure in Cn .

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2.1.4 Invariant Groups for Scalar Products This topic is discussed in detail in books[Wey39,Wey40,Art57] . General linear group: GL(n, F)={A ∈ M (n, F) | detA = 0}. Special linear group: SL(n, F)={A ∈ GL(n, F) | detA=1}. Orthogonal group: invariant group for the Euclidean scalar product (x, y)=ϕI (x, y) = x y, O(n, F) = {A ∈ GL(n, F) | (Ax, Ay) = (x, y), ∀x, y ∈ Fn }. From definition, we have A ∈ O(n, F) ⇐⇒ A IA = A A = I. In particular, we denote O(n, R) as O(n), i.e., O(n, R) ≡ O(n). Symplectic group: invariant group for the anti-symmetric scalar product [x, y] = ϕJ (x, y) = x Jy, ( ) Sp(2n, F) = A ∈ GL(2n, F) | [Ax, Ay] = [x, y], ∀x, y ∈ F2n . From definition, we have A ∈ Sp(n, F) ⇐⇒ A¯ JA = J. We denote Sp(n, F) ≡ Sp(n). Unitary group: invariant group for the Hermitian scalar product x, y=x∗ y. ) ( U (n, C) = A ∈ GL(n, C) | Ax, Ay = x, y, ∀x, y ∈ Cn . From definition, we have A ∈ U (n, C) ⇐⇒ A∗ IA = I. Similarly, we denote U (n, C) ≡ U (n). Invariant Group for Scalar Product ϕG , φG . Here: ϕG (x, y) = x Gy, G = ±G, |G| = 0, φG (x, y) = x∗ Gy,

G∗ = G, |G| = 0.

The invariant group for ϕG is ) ( G(G, n, F) = A ∈ GL(n, F) | ϕG (Ax, Ay) = ϕG (x, y), ∀ x, y ∈ Fn ( = A ∈ GL(n, C) | A GA = G} ) ( = A ∈ GL(n, C) | A−1 = G−1 A G . The symmetric case G = S, S  = S, ϕS (x, y) = (x, y)S = x Sy, where G is called an S-orthogonal group.

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O(S, n, F) = G(S, n, F) ( ) = A ∈ GL(n, F) | (Ax, Ay)S = (x, y)S , ∀ x, y ∈ Fn ( = A ∈ GL(n, F) | A SA = S} ) ( = A ∈ GL(n, F) | A−1 = S −1 A S . ¯ n, F) ≡ O(n, F). Special case: O(I, Anti-symmetrical case: G = K, K  = −K, ϕK (x, y) = [x, y]K = x Ky, where G is called a K-symplectic group. Sp(K, n, F) = G(K, 2n, F) ) ( = A ∈ GL(2n, F) | [Ax, Ay]K = [x, y]K , ∀ x, y ∈ F2n ( ) = A ∈ GL(2n, F) | A KA = K ) ( = A ∈ GL(2n, F) | A−1 = K −1 A K . Special case: Sp(J, 2n, F) ≡ Sp(2n, F). Hermitian case: G = H ∈ M (n, C), H ∗ = H, |H| = 0, φH (x, y) = x, yH = x∗ Hy, ) ( U (H, n, C) = A ∈ GL(n, C) | Ax, AyH = x, yH , ∀ x, y ∈ Cn ) ( = A ∈ GL(n, C) | A∗ HA = H ( ) = A ∈ GL(n, F) | A−1 = H −1 A∗ H . Special case: U (I, n, C) ≡ U (n, C) ≡ U (n). Conformally Invariant Group for scalar Product ϕG . CG(G, n, F) =

( (

) A ∈ GL(n, F) | ϕG (Ax, Ay) = 0 ⇐⇒ ϕG (x, y) = 0

) A ∈ GL(n, F) | ∃ μ ∈ F, μ = 0, such that A GA = μG ) ( = A ∈ GL(n, F) | ∃ μ ∈ F, μ−1 = 0, A−1 = μ−1 G−1 A G .

=

When G = S, S  = S, we denote CG(S, n, F) as CO(S, n, F). When G = K, K  = −K, we denote CG(K, n, F) as CSp(K, n, F). When G = H, H ∗ = H, we have ) A ∈ GL(n, C) | Ax, AyH = 0 ⇐⇒ x, yH = 0 ( ) = A ∈ GL(n, C) | ∃ μ ∈ C, μ = 0, such that A∗ HA = μH ( ) = A ∈ GL(n, C) | ∃ μ ∈ C, μ = 0, A−1 = μ−1 H −1 A∗ H .

CU (H, n, C) =

(

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2.1.5 Real Representation of Complex Vector Space Consider a mapping from Cn to R2n ρ : z = x + i y → ρ(z) = [x, y] , z ∈ Cn , x, y ∈ Rn . Evidently ρ : Cn → R2n is injective, and it satisfies the following properties: Property 1.26. 1◦ ρ(z + w) = ρ(z) + ρ(w), ∀z, w ∈ Cn . 2◦ ρ(αz) = αρ(z), ∀α ∈ R. ! ! −y 0 −I 3◦ ρ(iz) = ρ(−y + ix) = = x I 0 4◦ 5◦

ρ((α + iβ)z) = (αI − βJ)ρ(z), ρ(0) = 0 ∈ R2n , 0 ∈ Cn .

x y

! = −Jρ(z).

α + iβ ∈ C.

A −B For C = A + iB ∈ M (n, C), set R(C) = B A R : C → R(C), M (n, C) → M (2n, R) is injective. Assume C = A + iB ∈ M (n, C), w = Cz. Then,

! ∈ M (2n, R).Similarly,

w = u + iv = (A + iB)(x + iy) = (Ax − By) + i(Bx + Ay), i.e., u v

! =

A B

−B A

!

x y

! ,

or ρ(w) = R(C)ρ(z) = ρ(Cz). Analogously, R satisfies the following properties: Property 1.27. 1◦ R(On ) = O2n , On ∈ M (n, C). 2◦ R(In ) = I2n , In ∈ M (n, C). 3◦ R(αC) = αR(C), ∀ α ∈ R. −B A

−A −B

!

4◦

R( i C) = R( iA − B) =

5◦ 6◦

R(C1 + C2 ) = R(C1 ) + R(C2 ), ∀ C1 , C2 ∈ M (n, C). R(C1 · C2 ) = R(C1 )R(C2 ).

= −JR(C).

C invertible ⇐⇒ R(C) invertible. The last assertion follows from the theorem below. Theorem 1.28.

  A det(A + iB) = 0 ⇐⇒  B

 −B   0. = A 

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Real Representation of Unitary Group: If det(H) = 0, H = P + iQ ∈ M (n, C) is Hermitian, iff P  = P,

Q = −Q.

For w = u + iv, z = x + iy ∈ Cn , definew, zH = w ∗ Hz. w ∗ Hz = (u + iv)∗ H(x + iy) = (u + iv)∗ (P + iQ)(x + iy) ! ! P −Q x   = (u , v ) + i(u , v ) Q P y

Q −P

P Q

!

x y

! .

The above equation shows that the Hermitian scalar product of z and w, w, zH consists of two parts: its real part is a Euclidean scalar product in R2n (whose measure is a symmetric matrix S), denoted by the round bracket and its imaginary part can be taken as a new scalar product in R2n (whose measure is an anti-symmetric matrix K), denoted by the square bracket. Therefore, we have w, zH =(W, Z)S + i [W, Z]K , where ! u W = ρ(w) = , v ! x Z = ρ(z) = ∈ R2n , y H = P + i Q,

P  = P,

P −Q Q = −Q, S = S = Q P ! Q P  . K = −K = −P Q 



! ,

Let T = R(C) = R(A + iB). Then, w, zH = Cw, CzH ⇐⇒ (W, Z)S + i [W, Z]K = (T W, T Z)S + i [T W, T Z]K . From this, we can derive the following equivalent conditions: Proposition 1.29. 1◦ U((H, n, C)  C = A + i B ) = C ∈ GL(n, C) | Cw, CzH = w, zH , ∀ w, z ∈ Cn . ! A −B ◦ 2 T = , det T = 0, B A (W, Z)S = (T W, T Z)S , 3◦ 4◦

[W, Z]K = [T W, T Z]K .

T ∈ GL(n, C), T ∈ O(S, 2n, R) ∩ Sp(K, 2n, R), where K = SJ. T J = JT, det T = 0, and T  ST = S, T  SJT = SJ.

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Hence, GL(n, C) is identified with its real image GL(n, R) = {T ∈ GL(2n, R) | T J = JT }, and U (H, n, C) is identified with its real image in GL(2n, R). Since H = P + i Q is non-degenerate, and S = R(H), K = SJ are also nondegenerate, we have: if T J = JT, T  ST = S, then T  SJT = T  ST J = SJ; if T J = JT and T  SJT = SJ, then T  ST = T  SJ(−J)T = −T  SJT J = −SJ 2 = −1 S; if T  ST = S and T  SJT = SJ, then |T |2 = 1, T invertible, ST = T  S. −1 Hence, if SJT = T  SJ = ST J, then JT = T J. Therefore, we have U (H, n, C) = GL(n, C) ∩ O(S, 2n, R) ∩ Sp(K, 2n, R) = GL(n, C) ∩ O(S, 2n, R) = GL(n, C) ∩ Sp(K, 2n, R) = O(S, 2n, R) ∩ Sp(K, 2n, R), where H = P + iQ, P  = P, Q = −Q, S = R(H), K = SJ, |H| = 0. In particular, if H = I, S = I2n , K = J2n , then U (n, C) = GL(n, C) ∩ O(2n, R) ∩ Sp(2n, R) = GL(n, C) ∩ O(2n, R) = GL(n, C) ∩ Sp(2n, R) = O(2n, R) ∩ Sp(2n, R).

2.1.6 Complexification of Real Vector Space and Real Linear Transformation In a complex vector space V , we not only have the additive operator, but also the scalar multiplication by complex numbers: for u ∈ V, α + iβ ∈ C, then (α + iβ)u ∈ V . This can be realized by a scalar multiplication by real numbers u ∈ V, α ∈ R ⇔ αu ∈ V plus a single scalar multiplication by an imaginary unit i, i: u ∈ V ⇔ iu ∈ V , which can be seen as an operator l : V −→ V,

u −→ i u = l(u).

l is a real linear transformation of V , i.e., l(αu + βv) = αl(u) + βl(v),

u, v ∈ V,

α, β ∈ R.

Moreover, a complex linear transformation T of V satisfies: the additive property: T (u + v) = T (u) + T (v), the multiplication property: T ((α + i β)u) = (α + i β)T (u). The latter is simply the commutativity of complex linear transformation and complex scalar multiplication, which can be realized by

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T (i u) = i T (u) ∼ T l = lT. Let L(V, R) = {T | T (u + v) = T (u) + T (v), T (αu) = αT (u), ∀ u, v ∈ V, α ∈ R}, L(V, C) = {T | T (u + v) = T (u) + T (v), T ((α + i β)u) = (α + iβ)T (u),

∀ u, v ∈ V, α, β ∈ R}.

Evidently, T ∈ L(V, C) ⇔ iff T ∈ L(V, R) and T (i u) = i T (u). Thus, the operator l satisfies l ∈ L(V, R), i2 = −1, and T ∈ L(V, C) iff T ∈ L(V, R),

and

T l = lT.

These observations form the basis for the following definitions and the method of complexification of real vector space. Definition 1.30. A real vector space V = V (R) is complexifiable if there exists an operator l in L(V, R) such that l2 = −I. We can easily see that V is complexifiable iff dimV (R) = 2n. This is because the operator equation l2 = −I in L(V, R) corresponds to the matrix equation X 2 = −I in M (m, R), m =dimR (V ), which has no real solution for m = 2n + 1. (Since from X 2 = −I it follows that all eigenvalues of X are ±i, while for m = 2n + 1, X ∈ M (m, R) has at least one real eigenvalue.) When m = 2n, there is a special solution X = ±J, (±J)2 = −I. If we introduce an isomorphism J on R2n , which satisfies J 2 = −I, then we say that R2n is equipped with the complex structure. Hence, we can define operation (a + ib)u = au + bJu, and R2n becomes the complex n-dimensional space. Cn is called R2n complexifiable.

2.1.7 Lie Algebra for GL(n, F) 1. Lie algebra Definition 1.31. For B1 , B2 ∈ M (n, F), we define a commutator of B1 , B2 as follows: {B1 , B2 } = B1 B2 − B2 B1 , which satisfies the following properties: 1◦ {B1 , B2 } = −{B2 , B1 }. 2◦ {B1 , {B2 , B3 }} + {B2 , {B3 , B1 }} + {B3 , {B1 , B2 }} = 0. The equation 2◦ is called Jacobi identity.

2.1 Symplectic Algebra and Orthogonal Algebra

125

Definition 1.32. A Lie algebra is a vector space equipped with a binary operation L × L → L, which satisfies the Jacobi identity. Hence, M (n, F), equipped with the above commutator, becomes a Lie algebra, denoted as gl(n, F). Since gl(n, F) is the tangent vector space to GL(n, F) at I, gl(n, F) is called Lie algebra of the Lie group GL(n, F). Definition 1.33. The Lie algebra of the Lie group SL(n, F) is defined as follows: sl(n, F) = {B ∈ gl(n, F) | trB = 0}. Remark 1.34. If trB1 = trB2 = 0, then tr{B1 , B2 } = tr(B1 B2 − B2 B1 ) = 0. Therefore, sl(n, F) is closed under { , }. As the matter of fact, for any A, B, tr{A, B} is always equal to 0. Definition 1.35. The Lie algebra of the Lie group G(G, n, F) is defined as follows: g(G, n, F) = {B ∈ gl(n, F)|B  G + GB = 0}. Remark 1.36. g(G, n, F) is closed under { , }, i.e., {B1 , B2 } ∈ g(G, n, F),

∀ B1 , B2 ∈ g(G, n, F).

If Bi G = −GBi (i = 1, 2), then {B1 , B2 } G = (B1 B2 − B2 B1 ) G = (B2 B1 − B1 B2 )G = B2 B1 G − B1 B2 G = B2 (−GB1 ) − B1 (−GB2 ) = G(B2 B1 − B1 B2 ) = −G{B1 , B2 }. 2. Exponential matrix transform Definition 1.37. For B ∈ M (n, C), define exp(B) =

∞ n   1 k 1 k B = lim B . n→∞ k! k! k=0

k=0

Take the Chebyshev matrix norm ||B|| = max i

m 

|bij |.

j=1

* 2

Then, |bij | ≤ ||B||, ||B || ≤ ||B|| . Hence, the n series k

k

∞  1 k B k!

+

k=0

uniformly convergent, and convergent uniformly if ||B|| ≤ β, ∀ β > 0.

are always ij

126

2. Symplectic Algebra and Geometry Preliminaries

Proposition 1.38. We have the following results: 1◦ exp (T −1 BT ) = T −1 exp (B)T, ∀ T ∈ GL(n, C). 2◦ If B has eigenvalues λ1 , · · · , λn (with multiplicities), then exp B has the eigenvalues exp λ1 , · · · , exp λn . n  bii . 3◦ det (exp B) = exp (tr B), tr B = 4◦ 5◦

i=1

exp B ∈ GL(n, C), ∀ B ∈ M (n, C). If B1 , B2 ∈ M (n, C), B1 B2 = B2 B1 , then exp (B1 + B2 ) = exp B1 · exp B2 = exp B2 · exp B1 .

6◦ 7◦

exp B  = (exp B) , exp B = exp B. ∀ t ∈ R, if exp (tB) ∈ GL(n, C), then exp ((t1 + t2 )B) = exp (t1 B) · exp (t2 B), exp (0 · B) = I, exp (−tB) = (exp (tB))−1 .

Therefore, the mapping t ∈ R → exp (tB) ∈ GL(n, C) for a given B is a group homomorphism of the additive group R into the multiplicative group GL(n, C). Theorem 1.39. There exists a neighborhood W of On in M (n, C) such that the exponential mapping exp : M (n, C) → GL(n, C) is a diffeomorphism on W . ∞  Bk

is uniformly convergent in ||B|| ≤ ρ (ρ > 0). Therefore, k! Fij = (exp B)ij are the entire analytic functions of the complex variables bkl (k, l = 1, · · · , n), Fij (bkl ) = δij + bij + · · · + (terms of degree) ≥ 2.

Proof.

The series

k=0

Therefore, the Jacobian matrix is

'

∂Fij (bkl ) ∂(expB)ij kl = = δij , ¯ ∂bkl ∂ bkl

1, k = i, l = j, 0, otherwise. kl ) = 1 = 0, by implicit function theorem, there exists a neighborhood Since det (δij W ⊂ {||B|| < ρ}, such that B → exp B is a diffeomorphism on W . 

kl where δij =

Remark 1.40. The inverse function to B → exp B = A is A −→ log A = log (I + (A − I)) =

∞  (−1)k k=1

which is uniformly convergent for ||A − I|| ≤ ρ < 1.

k

(A − I)k ,

2.1 Symplectic Algebra and Orthogonal Algebra

127

Proposition 1.41. The function A(t) = exp (tB), ∀ t ∈ R satisfies the following properties: A(0) = I, A(t1 + t2 ) = A(t1 )A(t2 ) = A(t2 )A(t1 ), A(−t) = (A(t))−1 , d A(t) = BA(t), dt

d A(t)|t=0 = B. dt

Proposition 1.42. For A1 , A2 ∈ GL(n, C), its commutator is defined by −1 {A1 , A2 }G = A1 A2 A−1 1 A2

(commutator in Lie group).

Then for Ai (t) = exp tBi (i = 1, 2), ∀ t ∈ R, we have {A1 (t), A2 (t)}G = I + t2 {B1 , B2 }g + o(t4 ), {A1 (t), A2 (t)}G |t=0 = In , d {A1 (t), A2 (t)}G |t=0 = 0, dt

) 1 d2 ( A1 (t), A2 (t) G |t=0 = {B1 , B2 }g , 2 d t2 where {B1 , B2 }g = B1 B2 − B2 B1 is the commutator in Lie algebra. Proposition 1.43. If B ∈ g(G, n, F) and f (λ) = G(−B)k , (f (B)) G = Gf (−B).

n 

αk λk , αk ∈ F, then (B k ) G =

k=0

Theorem 1.44. A(t) = exp (tB) ∈ G(G, n, F), ∀ t ∈ R iff B ∈ g(G, n, F). Proof. Let C(t) = A (t)GA(t), then C(0) = G, d d A (t) d A(t) C(t) = GA(t) + A (t)G dt dt dt

= BA (t)GA(t) + A (t)GBA(t)

(1.1)

= A (t)(B  G + GB)A(t). d

C(t) = 0. Then, in order Thus, A(t) ∈ G(G, n, F), ∀t ∈ R iff C(t) ≡ G, i.e., dt to prove the theorem, we need only to show that the latter condition is equivalent to B ∈ g(G, n, F), i.e., B  G + GB = 0.  d d If C(t) = 0, then C(t)t=0 = I  (B  G + GB)I = B  G + GB = 0, and so dt dt B ∈ g(G, n, F). Conversely, if B  G + GB = 0, by (1.1), d C(t) = 0, dt then C(t) = C(0) = G, i.e., A (t)GA(t) = G, ∀ t ∈ R.



128

3.

2. Symplectic Algebra and Geometry Preliminaries

Lie algebra of conformally invariant groups CG(G, n, F)

Definition 1.45. Define ( cg(G, n, F) = B ∈ gl(n, F) | B  G + GB = βG, for some β ∈ F}. Since B1 , B2 ∈ cg(G, n, F), then {B1 , B2 } G = (B1 B2 − B2 B1 ) G = B2 B1 G − B1 B2 G = B2 (β1 G − GB1 ) − B1 (β2 G − GB2 ) = β1 β2 G − β1 GB2 − (β2 G − GB2 )B1 −β1 β2 G + β2 GB1 + (β1 G − GB1 )B2 = G(B2 B1 − B1 B2 ) = −G{B1 , B2 }. {B1 , B2 } ∈ cg(G, n, F). This shows that cg(G, n, F) is closed under { , }. Therefore, cg(G, n, F) is a subalgebra of gl(n, F) equipped with the induced binary operation { , }, called as the Lie algebra of the conformally invariant group CG(G, n, F). Theorem 1.46. Let A(t)= exp(tB), then A(t) ∈ CG(G, n, F) iff B ∈ cg(G, n, F).

2.2 Canonical Reductions of Bilinear Forms In Section 2.1, we have seen that for a given bilinear form ϕ on Fn , there is a matrix G in M (n, F) such that under natural base, ϕ can be represented by G: ϕ(x, y) = x Gy,

Gij = ϕ(ei , ej ).

The representative matrix G will change as the base changes. In this section, we want to make sure how the matrix G changes. Let Fn = {e1 , · · · , en } = {f1 , · · · , fn }, where e1 , · · · , en is the standard base and f1 , · · · , fn is a new base. Then, fi =

n 

T = [tij ] ∈ GL(n, F),

tji ej ,

j=1

⎤ ⎡ x1 ⎥ ⎢ ⎢ x = ⎣ ... ⎦ = T u = T ⎣ xn ⎡ ⎤ ⎡ y1 ⎢ ⎥ ⎢ y = ⎣ ... ⎦ = T v = T ⎣ ⎡

x=

y=

n 

xi ei =

n 

i=1

i=1

n 

n 

i=1

yi ei =

i=1

ui f¯i ,

vi f¯i ,

yn

⎤ u1 .. ⎥ , . ⎦ un ⎤ v1 .. ⎥ . . ⎦ vn

2.2 Canonical Reductions of Bilinear Forms

129

Assume that under the new base f¯1 , · · · , f¯n , the representative matrix of ϕ is G. Then, Gij = ϕ(f¯i , f¯j ) = ϕ

% n

tki ek ,

k=1

=

n 

n 

& tlj el

l=1

tki ϕ(ek , el )tlj = (T  GT )ij ,

k,l=1

i.e., G = T  GT, ϕ(x, y) = ϕ

n 

xi ei ,

i=1

n 

n n    yj ej = xi ϕ(ei , ej )yj = xi Gij yj = x Gy

j=1

i,j=1

i,j=1

n n n     ui f¯i , vj f¯j = ui ϕ(f¯i , f¯j )vj = ϕ i=1

=

n 

j=1

i,j=1

ui Gij vj = u Gv.

i,j=1

2.2.1 Congruent Reductions Definition 2.1. Let G1 and G2 ∈ M (n, F). G1 is congruent to G2 , if there exists a non-singular matrix T ∈ GL(n, F), such that T  G1 T = G2 , denoted by G1 ∼ G2 . Thus, the representative matrices of a bilinear form ϕ on Rn under different bases are congruent to one another. If G1 is congruent to G2 , then the equality G1 = εG1 implies the equality G2 = εG2 with the same parity ε and rank G1 = rank G2 . Let φ be a conjugate bilinear form on Cn . G is the representative matrix of φ under the standard base, i.e., ⎤ ⎡ ⎤ ⎡ x1 y1 ⎥ ⎢ ⎥ ⎢ φ(x, y) = x∗ Gy, y = ⎣ ... ⎦ , x = ⎣ ... ⎦ ∈ Cn . yn If f¯1 , · · · , f¯n is another base on Cn and f¯j =

xn n 

tij ei , T = [tij ] ∈ GL(n, C),

i=1

then similarly we can get G = T ∗ GT, where G is the representative matrix of φ under the base f¯1 , · · · , f¯n .

130

2. Symplectic Algebra and Geometry Preliminaries

Definition 2.2. Let G1 and G2 ∈ M (n, C). G1 is congruent to G2 if there exists a c matrix T ∈ GL(n, C), such that T ∗ G1 T = G2 , denoted by G1 ∼ G2 . c

If G1 ∼ G2 , then rank G1 = rank G2 and the equality G∗1 = εG1 implies the equality G∗2 = εG2 with the same factor ε. 

Remark 2.3. G is a conformal Hermitian matrix, i.e., G = εG with G, ε ∈ C, and iθ |ε| = 1, if and only if ε = eiθ and G∗ = eiθ G, if and only if e 2 G is a Hermitian matrix. Definition 2.4. Let ϕ(x, y) = ϕG (x, y) = x Gy be a bilinear form induced by G. For a subspace U ⊂ Rn /Cn , the subspace U ϕ ⊂ Rn /Cn defined by U ϕ = {x ∈ Rn /Cn | ϕ(x, y) = x Gy = 0, ∀y ∈ U } is called the G-orthogonal complement of U .

2.2.2 Congruence Canonical Forms of Conformally Symmetric and Hermitian Matrices We list congruence canonical forms of conformally symmetric and Hermitian matrices in Table 2.1 as a comparison. F = R/C. 1.

Alternative canonical forms ! I I 1 Let T = √ , then T  2 I −I

O I

I O

! T =

I O

O −I

! . where Fn = V +

U, U = {x ∈ Fn | ϕ(x, y) = 0, ∀ y ∈ Fn } = (Fn )ϕ , dim V = r, dim U =  0 1  n − r, ϕ is non-singular, and J canonical symplectic quadratic form . −1 0 ' i = 2k − 1, δk,j , i, j = 1, · · · , 2n. Then, Let T = [tij ], tij = δk+n,j , i = 2k, ⎡

0 1 ⎢ ±1 0 ⎢ ⎢ T ⎢ ⎢ ⎣

⎤ ..

. 0 1 ±1 0

⎥ ⎥ ⎥ ⎥T = ⎥ ⎦

O ±In

In O

! . 2n×2n

Thus, the canonical forms listed in Table 2.1 have the following alternative forms:

2.2 Canonical Reductions of Bilinear Forms

Table 2.1.

Canonical form of conformal matrix and Hermitian matrices Matrix

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Canonical form

G = G, in C or R, J J ..

.

s-block

J

r

⎤

∃ u, v ∈ R , s.t.

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

ϕ(u, v) = 0. Let a1 = u, a2 =

ϕ(a1 , a2 )

On−r

⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎣

G = G, in C, . . Ir . ... ... . . . On−r

∃ u ∈ Cr , s.t.

⎤

−iθ 2

⎥ ⎥ ⎥ ⎥ ⎦

⎤

Let a1 =

|ϕ(u, u)|

∃ u ∈ Cr , s.t. ϕ(u, u) = 0. Let a1 = ⎤

Iq

{a1 } ∩ {a1 }ϕ = {0},

u  |ϕ(u, u)|

{a1 }ϕ = {a2 , · · · , ar }, ⎤ ⎡ ±1 0 ⎦ ⎣ G∼ 0 G1

=⇒ ϕ(a1 , a1 )

⎥ ⎥ ⎥ ⎦

= signϕ(u, u)

On−r

= ±1

p+q =r

⎤

Ir 0

Ir On−r 0 ⎢ Is ⎣



{a1 }ϕ = {a2 , · · · , ar }, ⎤ ⎡ ±1 0 ⎦ ⎣ G∼ 0 G1

u

=⇒ ϕ(a1 , a1 ) = ±1

Ip

⎡

{a1 } ∩ {a1 }ϕ = {0},

ϕ(u, u) = 0.

⎥ ⎦

−Iq

0 ⎣ −Ir

{a1 }ϕ = {a2 , · · · , ar }, 5 6 1 0 G∼ 0 G1

u ϕ(u, u)

=⇒ ϕ(a1 , a1 ) = 1

Ip

⎡



Let a1 =

∃ u ∈ Rr , s.t.

G = G, in R,

−iθ −e 2

⎦ ∼ G = −G, On−r ! ∼ G = G,

in R

or

in C,

or

(G∗ = G, in C ),

in C,

⎤

Is 0

⎥  ⎦ ∼ G = G,

σId

in R

On−r ⎡

0 ⎢ e−iθ I s ⎢ ⎢ ⎣

6

{a1 } ∩ {a1 }ϕ = {0},

ϕ(u, u) = 0.

G∗ = G, G∗ = eiθ G ⎡ ⎤ Ip ⎢ ⎥ −Iq ⎣ ⎦, On−r ⎢ e ⎢ ⎢ ⎣

O G1

= −ϕ(a2 , a1 ) = 1

On−r p+q =r

⎡

= {0}, 5 J G∼ O

= ϕ(a2 , a2 ) = 0,

2s = r

⎡

{a1 , a2 } ∩ {a1 , a2 }ϕ

v =⇒ ϕ(a1 , a1 ) ϕ(u, v)

⎤

Is 0 σe

− iθ 2

⎥ ⎥ ∗ ⎥ G = eiθ G, ⎦

Id On−r

in C,

131

132

2. Symplectic Algebra and Geometry Preliminaries

where s = min(p, q), d = |p − q|, σ = sign(p − q), p + q = r = 2s + d, p − q = 1 2

1 2

1 2

1 2

σd, p = (r + σd) = s + (1 + σ)d, q = (r − σd) = s + (1 − σ)d. 2. Invariants under congruences Theorem 2.5. Let G be a conformally symmetric matrix in Fn (= Rn or Cn ), i.e., G = εG. Then, the quantities ε(G), r(G) and s(G) are the invariants under congruences. Moreover, if ε = −1, then r = 2s, if ε = 1, then p(G), q(G), d(G) and σ(G) are invariants under congruences. If G is conformal Hermitian, i.e., G∗ = εG with ε = eiθ , then the quantities ε(G), r(G), s(G), p(G), q(G) and σ(G) are invariants under congruences. Theorem 2.6 (Sylvester’s law of inertia). Let ϕ(x) be a quadratic form in Rn and x = T y, det(T ) = 0. If ϕ(x) = x21 + x22 + · · · + x2p − x2p+1 − · · · − x2n 2 = y12 (x) + y22 (x) + · · · + yq2 (x) − yq+1 (x) − · · · − yn2 (x),

(2.1)

then p = q. Similarly, let φ(x) be a quadratic form in Cn and x = T y, det T = 0. If φ(x) = |x1 |2 + |x2 |2 + · · · + |xp |2 − |xp+1 |2 − · · · − |xn |2 = |y1 |2 + |y2 |2 + · · · + |yq |2 − |yq+1 |2 − · · · − |yn |2 , then p = q. Proof. If p > q, then p + (n − q) > n. Thus, the equations x1 = 0, · · · , xp = 0, yq+1 (x) = 0, · · · , yn (x) = 0 has a non-zero solution ξ = 0. By (2.1), 2 2 y12 (ξ) + · · · + yq2 (ξ) + ξp+1 + · · · + ξn2 = ξ12 + · · · + ξp2 + yq+1 (ξ) + · · · + yn2 (ξ) = 0,

and thus ξp+1 = 0, . . . , ξn = 0. Then, we have ξ = 0, which is a contradiction. This shows that p ≤ q. Similarly, q ≤ p, then p = q.  ! G0 0 is conjugate Theorem 2.7 (Witt). If a non-singular Hermitian matrix 0 G1 ! G0 0 congruent to , then G1 is conjugate congruent to G2 . 0 G2 ! ! 1 0 1 0 ∼ . Proof. We first prove the case G0 = 1, i.e., 0 G1 0 G2 ! ! ! 1 0 1 0 a ¯b T = . Then, Let T = and T ∗ 0 G1 0 G2 c d ! ! ! ! 5 aa + c∗ G c a¯b + c∗ G d 6 1 1 a c∗ 1 0 1 0 a ¯b = ¯ = ∗ ∗  ∗ 0 G2 0 G c d ¯ba + d G1 c ¯b¯b + d G1 d b d 1 ,i.e.,

2.2 Canonical Reductions of Bilinear Forms



aa + c∗ G1 c = 1,

a¯b + c∗ G1 d = 0,

¯ba + d∗ G1 c = 0,

¯b¯b + d∗ G1 d = G2 .

133

Let dλ = d + λcb , d∗λ G1 dλ = (d∗ + λ¯bc∗ )G1 (d + λc¯b ) = d∗ G1 d + λ¯bc∗ G1 d + λd∗ G1 c¯b + |λ|2¯bc∗ G1 c¯b = G2 − ¯b¯b − λ¯ba¯b − λ¯ba¯b + |λ|2 (¯b¯b − aa¯b¯b ) = G2 − (1 + λa + aλ − (1 − |a|2 )|λ|2 )¯b¯b . If λ satisfies the equation 1 + λa + aλ − (1 − |a|2 )|λ|2 = 0, then d∗λ G1 dλ = G2 . It only needs to take ⎧ 1 ⎨ − , if a = 1, 2 λ= ⎩ 1 , if a = 1. 1−a

If the order r of G0 is larger than 1, ⎡ then ±1 c ⎢ .. G0 ∼ ⎣ . 0 and thus ⎡ ⎡ ⎤ . ⎛ ⎞ .. ⎢ ⎢ ⎥ ±1 ⎢ ⎢ ⎥ .. ⎟ ⎢ ⎢ ⎜ ⎥ .. ⎢ ⎢ ⎝ ⎠ . 0 ⎥ . c ⎢ ⎢ ⎥ .. ∼ ⎢ ⎢ ⎥ ±1 r . ⎢ ⎢ ⎥ · · · ⎢ ⎢ ⎥ . ··· ··· ··· .. ⎢ ⎢ ⎥ G ⎣ ⎣ ⎦ 1 0 .. . We denote

⎡

⎛

⎤

0

⎥ ⎦

±1 ⎛ ⎜ ⎝

,

r×r

⎞

±1 .. ···

⎞

⎢ ±1 ⎢ ⎟ ⎢ ⎜ .. ⎠ ⎢ ⎝ . ⎢ $1 = ⎢ G ±1 ⎢ r−1 ⎢ ··· ··· ··· ⎢ ⎣ 0

.

··· 0

.. . .. . .. . .. . .. .

⎟ ⎠ ±1 ···

⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥, ⎥ ··· ⎥ ⎥ G1 ⎦

r

.. . .. . .. . .. . .. .

⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥. ⎥ ··· ⎥ ⎥ G2 ⎦

134

2. Symplectic Algebra and Geometry Preliminaries

⎡

⎛

⎞

⎢ ±1 ⎢ ⎟ ⎢ ⎜ .. ⎠ ⎢ ⎝ . ⎢ $2 = ⎢ G ±1 r−1 ⎢ ⎢ · · · · · · ··· ⎢ ⎣ 0

.. . .. . .. . .. . .. .

⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥, ⎥ ··· ⎥ ⎥ G2 ⎦

c $ $1 ∼ then by the result just proved above, G G2 . By recursions, we can finally get c

G 1 ∼ G2 . The Witt theorem gives another proof of the invariance of index p. If Ip 0

0 −In−p

!

c

∼

Ip 0

0 −In−p

! ,

and p > p , then Ip−p 0

   c  ∗ 0 ∼ −In−p  

! −In−p

,

i.e., there exists a matrix T ∈ GL(n − p , C), such that Ip−p 0

0 −In−p

!

= −T ∗ T.

The (1,1) element of the given matrix is 1 = −(|ti1 |2 + · · · + |tn−p |2 ), which is a contradiction.



2.2.3 Similar Reduction to Canonical Forms under Orthogonal Transformation For comparison, we list the canonical forms of Hermitian, conformal Hermitian matrices, real symmetric, and anti-symmetric matrices under unitary or orthogonal transformations in the Table 2.2. The content can be found in any standard textbook.

2.2 Canonical Reductions of Bilinear Forms

Table 2.2.

H, S and K under unitary or orthogonal transformations Matrices

Canonical form If λ1 is an eigenvalue of H, then :

H ∗ = H, ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∃ w1 = 0, s.t.Hw1 = λ1 w1 ,

λk ∈ R

w1 , w1  = 1.

⎤

λ1 ..

λr

0 ..

w1 , Hw1  = w1 , λ1 w1  = λ1 .

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

.

.

w1 , Hw1  ∈ R ⇒ λ1 ∈ R. z ∈ {w1 }⊥ ⇒ Hz1 ∈ {w1 }⊥ So, ∃ T0 ∈ U (n, C), s.t. T0−1 HT0 =

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

H ∗ = ei θ H, ..

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 ..

. 0

S  = S, ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

! ,

S is a real matrix, S  = S ⇒ S ⎤

. e−iθ/2 λr

0 H1

where H1∗ = H1

λk ∈ R

e−i θ/2 λ1

λ1 0

λk ∈ R

is a Hermitian matrix. If λ1 is an eigenvalue, then: ∃ w1 ∈ R, s.t. Sw1 = λ1 w1 , (w1 , w1 ) = 1.Similarly, x ∈ {w1 }⊥ ⇒ Sx ∈ {w1 }⊥ . Analogous, ∃ T0 ∈ O(n, R), ! s.t. 0 λ1 T0−1 ST0 = , 0 S1 where S1 = S1

⎤

λ1 ..

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

. λr

0 ..

. 0

K anti-symmetrical, iK is a Hermitian matrix, its eigenvalues are real. If λ1 = 0 is an eigenvalue of iK,

⎡

0 ⎢ −λ1 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

then ∃ w1 = u − i v = 0,

K  = −K, λk > 0, 2s = r λ1 0 .. . 0 λs −λs 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ On−r

⎡

0 ⎣ −A

or A 0

⎤ ⎦

On−r where A = diag{λ1 , λ2 , · · · , λs }

s.t. i K(u − i v) = λ1 (u − i v). (1) Kv = λ1 u, ⇒ Therefore : (2) Ku = −λ1 v. λ1 (u, v) = (v, Kv) = 0. Sinceλ1 = 0, we have(u, v) = 0. From (1) and (2), we get u = 0, v = 0. Suppose : (u, u) = (v, v) = 1. Then : (u, Ku) = 0, (u, Kv) = λ1 , (v, Kv) = 0, (v, Ku) = −λ1 . If x ∈ {u, v}⊥ , then : Kx ∈ {u, v}⊥ . Thus, ∃ T0 ∈ O(n, R), s.t. ⎡ 0 T0−1 KT0 = ⎣ −λ1 where K1

= K1

⎤

λ1 0

⎦, K1

135

136

2. Symplectic Algebra and Geometry Preliminaries

Next, we consider Jordan canonical forms. Let us first recall the Jordan canonical form for a general real matrix A ∈ M (n, R). A Jordan canonical form viewed in real space is different from the Jordan canonical form viewed in complex space. 1. Elementary divisors in complex space In complex space, the elementary divisor corresponding to a paired-complex conjugate eigenvalue α ± iβ, β = 0 is of the form [λ − (α + iβ)]p ,

[λ − (α − iβ)]p .

The corresponding Jordan blocks are ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

α + iβ

⎤

1 α + iβ

.. ..

. .

1 α + iβ

⎡

⎥ ⎥ ⎥ ⎥ ⎦

α − iβ

⎢ ⎢ ⎢ ⎢ ⎣

,

⎤

1 α − iβ

..

.

..

.

p×p

1 α − iβ

⎥ ⎥ ⎥ ⎥ ⎦

. p×p

The elementary divisor corresponding to a real eigenvalue γ is of the form (λ − γ)q . Its Jordan block is

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

γ

⎤

1 γ

..

.

..

.

⎥ ⎥ ⎥ ∼ (λ − γ)q . ⎥ 1 ⎦ γ q×q

2. Elementary divisor in real space In real space, the elementary divisor corresponding to a paired complex conjugate eigenvalues α± iβ, (β = 0) is of the form [λ2 − 2αλ + (α2 + β 2 )]p . Its Jordan block is ⎡ 0 1 ⎢ −(α2 + β 2 ) 2α ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤ 1 0 1 −(α2 + β 2 ) 2α

1 .. ..

.

. 1 0 −(α2 + β 2 )

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ ⎥ ⎥ 1 ⎦ 2α 2p×2p

The elementary divisor and the Jordan block corresponding to a real eigenvalue γ is the same as in complex space, i.e.,

2.3 Symplectic Space

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

γ

⎤

1 γ

137

..

.

..

.

⎥ ⎥ ⎥ ∼ (λ − γ)q . ⎥ 1 ⎦ γ q×q

2.3 Symplectic Space A symplectic vector space is a vector space V equipped with a nondegenerate, skewsymmetric, bilinear form ω called the symplectic form. Explicitly, a symplectic form is a bilinear form ω : V × V → R that is 1◦ Skew-symmetric: ω(u, v) = −(v, u) ∀u, v ∈ V . 2◦ Nondegenerate: if ω(u, v) = 0 ∀v ∈ V , then u = 0. Working in a fixed basis, ω can be represented by a matrix. The two conditions above imply that this matrix must be skew-symmetric and nons-singular. This is not the same as a symplectic matrix, which represents a symplectic transformation of the space.

2.3.1 Symplectic Space and Its Subspace 1. Comparison between symplectic and Euclidian space[Tre75,LM87,FQ91a,Wei77] In this section, we restrict ourselves to R2n . The symbol ⇔, which stands for “if and only if”, will be widely adopted under J orthogonality and I Orthogonality. Sympl. Structure J-Sympl. Matrix Euclidian Structure I-Unit Matrix [x, y] = x Jy,

(x, y) = x Iy;

[y, x] = −[x, y],

(y, x) = (x, y);

[x, x] = 0, ∀ x,

(x, x) > 0, ∀ x = 0;

[x, y] = (x, Jy).

(x, y) = [x, J −1 y].

J-Orthogonality

I-Orthogonality

xy ⇐⇒ [x, y] = 0 ⇐⇒ yx,

x⊥y ⇐⇒ (x, y) = 0 ⇐⇒ y⊥x.

U V ⇐⇒ [x, y] = 0, ∀ x ∈ U, y ∈ V,

U ⊥V ⇐⇒ (x, y) = 0, x ∈ U, y ∈ V.

U V ⇐⇒ U ⊥JV,

U ⊥V ⇐⇒ U J −1 V.

Definition 3.1. V  = {x ∈ R2n | [x, y] = 0, ∀y ∈ V }. V ⊥ = {x ∈ R2n | (x, y) = 0, ∀y ∈ V }. By definition, we have V  = (JV )⊥ , V ⊥ = (J −1 V ) .

138

2. Symplectic Algebra and Geometry Preliminaries

Proposition 3.2. U ⊂ R2n , V ⊂ R2n . Then, U ⊂ V ⇐⇒ U  ⊃ V  ,

U ⊂ V ⇐⇒ U ⊥ ⊃ V ⊥ ;

(U ∩ V ) = U  + V  ,

(U ∩ V )⊥ = U ⊥ + V ⊥ ;

(U + V ) = U  ∩ V  ,

(U + V )⊥ = U ⊥ ∩ V ⊥ ;

dim U + dim U = 2n,

dim U + dim U⊥ = 2n;

U



⊥

U⊥ = U;

= U,

∃ U, U ∩ U  = {0},

U ∩ U ⊥ = {0};

U + U  = R2n ,

U + U ⊥ = R2n .

Definition 3.3. U ⊂ R2n , define U 0 = U ∩ U  , called as the radical of U . 2.

Special classes of subspaces[HW63,Tre75,LM87,Wei77] (1) V degenerate subspace : V 0

V ∩ V  = 0

=

⇐⇒ dim V is odd. (2) V isotropic subspace : V

⊂

V

⇐⇒ V ∩ V  = V 0 = V ⇐⇒ [x, y] = 0 on V =⇒ dim V ≤ n ⇐⇒ V  coisotropic ⇐= dim V = 1. (3) V coisotropic subspace : V 

⊂

V

⇐⇒ V ∩ V  = V 0 = V  ⇐⇒ [x, y] = 0 on V  =⇒ dim V ≥ n ⇐⇒ V  isotropic ⇐= dim V = 2n − 1.

2.3 Symplectic Space

(4) V Lagrangian : V 

=

139

V

⇐⇒ V is isotropic and coisotropic ⇐⇒ V is isotropic and dim V = n ⇐⇒ V is coisotropic and dim V = n ⇐⇒ V maximally isotropic ⇐⇒ V minimally coisotropic. (5) V non-degenerate : V ∩ V 

=

{0}

⇐⇒ V + V  = R2n ⇐⇒ [x, y] non-degenerate on V ⇐⇒ If [x, y] = 0 ∀ y ∈ V, then x = 0 ⇐⇒ V  non-degenerate =⇒ dim V is even. (6)

Coordinate subspaces.

Define ν = {1, 2, · · · , n} with the natural order. If α = {i1 , i2 , · · · , ik } ⊂ ν (i1 < · · · < ik ), the total number of α s is 2n . Denote α  = ν\α = {j1 , · · · , jn−k } (j1 < . . . < jn−k ), ν = ∅, α  ∪ α = ν, α  ∩ α = ∅. Definition 3.4. Define a coordinate subspace Rα,β = {ei , fj }i∈α,j∈β , where ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0 0 ⎢ 0 ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥ , · · · , en = ⎢ ⎥ , f1 = ⎢ ⎥ , · · · , fn = ⎢ 0 ⎥ . e1 = ⎢ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ 0

0

0

1 2n

The total number of coordinate subspaces Rα,β is 2 . Problem 3.5. We have the following issues to address: 1◦ Under what conditions about ' α, β is Rα,β isotropic, coisotropic, Lagrangian? 1, i ∈ α, 2◦ Define Iα = [di δij ], di = 0, i ∈ α . Proof. .

Iν = In , Iν = I∅ = On , Iα2 = Iα , Iα Iα = On , Iα Iβ = Iα∩β , Iα + Iα = I, Iα + Iβ = Iα∩β + Iα∪β = Iα∪β + Iα Iβ .

The proof can be obtained.



140

2. Symplectic Algebra and Geometry Preliminaries

3◦ Show that the subspaces D = {pi = ±qi , i = 1, 2, · · · , n} are Lagrangian and transversal to all coordinate Lagrangian subspaces.  where U "V  stands for Theorem 3.6. Let W be non-degenerate and W = U "V, the J-orthogonal sum. Then U, V both are non-degenerate. Proof. Let x ∈ U such that [x, y] = 0, ∀y ∈ U . Then, from U ⊂ V  , it follows that [x, z] = 0, ∀z ∈ V . Therefore, [x, w] = 0, ∀w ∈ W . By assumption, W is non-degenerate, and so x = 0. This shows U is non-degenerate. Similarly, V is nondegenerate too.  Theorem 3.7. If U is isotropic, then there exists a Lagrangian subspace V ⊃ U . Proof. Without loss of generality, we can assume that dim U = k < n. Therefore, there exists a vector ak+1 ∈ U  \U = ∅, V = U + {ak+1 } ⊃ U, V is isotropic.  By repeating this procedure, we can get a Lagrangian subspace V ⊃ U . Theorem 3.8. For a given Lagrangian subspace U = {a1 , a2 , · · · , an }, there exists another Lagrangian subspace V = {b1 , b2 , · · · , bn }, such that ˙  2"  · · · "P  n = U +V, R2n = P1 "P where Pi = {ai , bi }, [ai , aj ] = [bi , bj ] = 0, [ai , bj ] = δij . Proof. We proceed by induction with respect to dimension n. For n = 1, by nondegenerality of R2 , there is a vector b1 , satisfying [a1 , b1 ] = 1. Of course, {b1 } is Lagrangian. Thus, this theorem is true for n = 1. Assume that for n−1, the theorem is true. Then, for R2n , {a1 } ⊂ {a2 , a3 , · · · , an }, and so {a2 , a3 , · · · , an } \{a1 } = ∅, there is a vector b1 ∈ {a2 , a3 , · · · , an } \ {a1 } , such that [a1 , b1 ] = 1,

[a2 , b1 ] = · · · = [an , b1 ] = 0.

Set P1 = {a1 , b1 }. By above subspace(5) and Theorem 3.6,  1 , R2n = P1 "P

P1 ∩ P1 = {0},

dim P1 = 2(n − 1), {a2 , a3 , · · · , an } ⊂ P1 is maximally isotropic in P1 , i.e., a Lagrangian subspace of P1 . By inductive assumption, there exists b2 , · · · , bn and {b2 , · · · , bn } is Lagrangian space in P1 . Moreover, [ai , aj ] = [bi , bj ] = 0,

[ai , bj ] = δij ,

i, j = 2, · · · , n.

Therefore, for all i, j, we have [ai , aj ] = [bi , bj ] = 0,

[ai , bj ] = δij .

 2"  · · · "P  n. Set Pi = {ai , bi }, then R2n = P1 "P



2.3 Symplectic Space

141

Theorem 3.9. Let U, V be two Lagrangian subspaces, U = {a1 , a2 , · · · , an }, U ∩ V = {0}. Then, there exists a unique base b1 , b2 , · · · , bn , such that V = {b1 , b2 , · · · , bn }, and [ai , aj ] = [bi , bj ] = 0, [ai , bj ] = δij . Proof. Similarly manner, by induction with respect to dimension n, the proof can be obtained.  Theorem 3.10. If U is not isotropic, then there exists a non-degenerate subspace N ⊂ U, N = {0}, such that  N, U = U0 "

U 0 = U ∩ U .

Proof. By assumption U 0 = {x ∈ U |[x, y] = 0, ∀y ∈ U } = U . Therefore, there is a ˙ 0. subspace N ⊂ U, N = {0}, such that N ∩ U 0 = {0} and U = N +U 0 0 Since U U , of course U N . If x ∈ N and [x, y] = 0, ∀y ∈ N, then by U 0 N , we have [x, z] = 0, ∀z ∈ U 0 i.e., [x, y] = 0, ∀y ∈ U . Thus, x ∈ U 0 ∩ N = {0}, and x must be = 0. Therefore, N is non-degenerate.  Theorem 3.11. Let V1 , V2 be two disjoint isotropic subspaces. Then, there exist two Lagrangian subspaces W1 , W2 that are disjoint. W1 ∩ W2 = {0} such that W1 ⊃ V1 , W2 ⊃ V2 . ˙ 2 , U 0 = U ∩ U  = (V1 +V ˙ 2 ) ∩ (V1 +V ˙ 2 ) . Proof. Let U = V1 +V ◦ 1 If U is isotropic. Assume V1 = {a1 , · · · , ar },

V2 = {br+1 , · · · , br+s }.

By Theorem 3.7, there exists V3 = {ar+s+1 , · · · , an }, such that ˙ 2 +V ˙ 3 = {a1 , · · · , ar , br+1 , · · · , br+s , ar+s+1 , · · · , an } V1 +V is Lagrangian. Moreover, by Theorem 3.8, there exist b1 , · · · , br , ar+1 , · · · , ar+s , br+s+1 , · · ·, bn ∈ R2n , such that [ai , aj ] = [bi , bj ] = 0,

[ai , bj ] = δij .

Set W1 = {a1 , · · · , ar , ar+1 , · · · , ar+s , ar+s+1 , · · · , an }, W2 = {b1 , · · · , bn }. Obviously, W1 , W2 are Lagrangian and W1 ∩ W2 = {0}, V1 ⊂ W1 , V2 ⊂ W2 . 2◦

If U is not isotropic. Set U10 = V1 ∩ V2 , U20 = V2 ∩ V1 , then ˙ 20 , U 0 = U10 +U

Ui0 = Vi ∩ U 0 ,

i = 1, 2.

Evidently, Ui0

= Vi ∩ U 0 = Vi ∩ U ∩ U  ˙ 2 ) = Vi ∩ (V1 ∩ V2 ), = Vi ∩ (V1 +V

i = 1, 2.

142

2. Symplectic Algebra and Geometry Preliminaries

By assumption, Vi (i = 1, 2) are isotropic, i.e., Vi ⊂ Vi (i = 1, 2). By the following Lemma 3.12, ˙ 2 ∩ (V1 ∩ V2 ) ˙ 20 = V1 ∩ (V1 ∩ V2 )+V U10 +U ˙ 2 ) ∩ (V1 + V2 ) = (V1 +V = U ∩ U  = U 0.  i. Take Ni ⊂ Vi (i = 1, 2), such that Vi = Ui0 "N ˙ 2 , then Set N = N1 +N U = V1  V2 = U10  N1  U20  N2 = U 0  N. Similar to the proof of Theorem 3.10, we can see that N is non-degenerate and  . Of course, Ni = N ∩ Vi (i = 1, 2), and N1 ⊂ N ∩ V1 . U = U 0 "N If x ∈ N ∩ V1 , then x = x1 + x2 , x1 ∈ N1 , x2 ∈ N2 . Moreover, by x ∈ V1 , we have x2 = x − x1 ∈ V1 . Thus, x2 ∈ V1 ∩ V2 = {0}, x2 = 0. This shows that x = x 1 ∈ N1 .   , U 0 = U10  U20 ⊂ N  . By N = N1  As N is non-degenerate, R2n = N "N N2 and N1 , N2 being isotropic, we know that N1 , N2 both are maximal isotropic, i.e., Lagrangian in N . For the non-degenerate subspace N  , since U 0 = U10  U20 ⊂ N  is isotropic, applying results 1◦ of Theorem 3.11 , there exist two Lagrangian subspaces W1 , W2 , in N  , such that Wi ⊃ Ui0 (i = 1, 2) and W 1 ∩ W 2 = {0}. Set W1 = W1  N1 , W2 = W2  N2 . Then, Wi (i = 1, 2) are Lagrangian in R2n (i = 1, 2) and  Vi ⊂ Wi (i = 1, 2), W1 ∩ W2 = {0}. Lemma 3.12. If A ⊂ A1 , B ⊂ B1 , then (A + B) ∩ (A1 ∩ B1 ) = A ∩ (A1 ∩ B1 ) + B ∩ (A1 ∩ B1 ) = A ∩ B1 + B ∩ A1 . Proof. (A + B) ∩ (A1 ∩ B1 ) ⊂ A ∩ (A1 ∩ B1 ) + B ∩ (A1 ∩ B1 ) is trivial. Let x ∈ (A + B) ∩ (A1 ∩ B1 ). Since x ∈ A + B, then exists a decomposition x = a + b,

a ∈ A ⊂ A1 ,

b ∈ B ⊂ B1 .

Since x ∈ A1 ∩ B1 , a = x − b1 ∈ B1 , b = x − a ∈ A1 . Thus, a ∈ A ∩ B1 , b ∈ B ∩ A1 , i.e., x = a + b ∈ A ∩ B1 + B ∩ A1 .



Lemma 3.13. Let U, V, W ⊂ Rm , V  W = Rm . Define a linear projection πVW along W into V as πVW : Rm → Rm , ' x, ∀x ∈ V, πVW x = 0, ∀x ∈ W. Then, U  W = Rm , iff πVW : U → V is non-singular and onto.

2.3 Symplectic Space

143

Proof. Assume x ∈ U , πVW x = 0. From V  W = Rm , it follows that there exists a decomposition x = v + w, where v ∈ V and w ∈ W. Thus, 0 = πVW x = πVW (v + w) = πVW v + πVW w = πVW v = v. Then, x = w ∈ W , i.e., x ∈ U ∩ W = {0}. We get x = 0. Conversely, if x ∈ U ∩ W , then x ∈ W , x ∈ U . By the definition of π, we have πx = 0. π : U → V is non-singular and so x = 0, i.e., W ∩ U = {0},

U  W = Rm . 

Therefore, the lemma is completed.

Theorem 3.14. For every Lagrangian subspace L in R2n , there exists a coordinate Lagrangian subspace Rα,α transversal to L, i.e., ∃α ⊂ ν , such that Rα,α +L = R2n . Proof. Since (Rν,0 ∩ L) ⊂ Rν,0 , ∃α ⊂ ν, such that Rα,0 ∩ (Rν,0 ∩ L) = {0}, Rα,0 + (Rν,0 ∩ L) = Rν,0 . Lag.

Lag.

T T   Rν,0 = Rν,0 ⊂ Rα,α + L = Rα, α ∩ L) . α + L = (Rα,

Therefore,

Rα,α ∩ L = (Rα,α ∩ Rν,0 ) ∩ (Rν,0 ∩ L) = Rα,0 ∩ (Rν,0 ∩ L) = {0}.

Then,

Rα,α + L = R2n . 

The theorem is proved. 3.

Matrix representation of subspaces in R2n ⎤ ⎡ · · · a1k a11 ⎥ ⎢ .. A = ⎣ ... ⎦ = [a1 , a2 , · · · , ak ] = . a2n,1

· · · a2n,k

A1 A2

! ,

where A ∈ M (2n, k), ai ∈ M (2n, 1), Aj ∈ M (n, k). Definition 3.15. A ∈ M (2n, k) is non-singular, if rankA = k. Let A ∈ M (2n, k), B ∈ M (2n, l). Then, [A, B] ∈ M (2n, k + l). If [A, B] is non-singular, then both A and B are non-singular. G2n,k = {all k-dim subspaces in R2n }, called as Grassmann manifold. If A ∈ M (2n, k) is non-singular, we define {A} = {a1 , · · · , ak } to be a k-dim subspace in R2n generated by k column vectors a1 , · · · , ak of A.

144

2. Symplectic Algebra and Geometry Preliminaries

Proposition 3.16. Let A, B ∈ M (2n, k) be non-singular. {A} = {B}, iff ∃Q ∈ GL(k), such that AQ = B, i.e., A1 Q A2 Q

! =

B1 B2

! .

Definition 3.17. Let A, B ∈ M (2n, k). If there is a matrix Q ∈ GL(k), and AQ = B, then we say that A is equivalent to B, denoted by A ∼ B. Proposition 3.18. G2n,k consists of equivalent classes of non-singular elements under M (2n, k) i.e., G2n,k ≈ {equivalent classes of non-singular elements under “∼” in M (2n, k)}. Definition 3.19. Λn = { all Lagrangian subspaces in R2n } ⊂ G2n,n . Definition 3.20. A =

A1 A2

! ∈ M (2n, n) is called a symmetric pair of square

matrices, if A JA = On , i.e., A1 A2 − A2 A1 = On , where J =

0 −In

In 0

! .

! A1 is a symmetric pair iff A1 A2 ∈ SM (n), or A2 A1 ∈ SM (n). A2 All symmetric pairs of square matrices are denoted by SM 2n,n . ! ! S I In particular, A = or ∈ SM 2n,n , iff S ∈ SM (n). I S

Evidently, A =

A=

A1 A2

! ,

|A1 | = 0, A ∈ SM 2n,n ⇐⇒ A2 A−1 1 ∈ SM (n). |A2 | = 0, A ∈ SM 2n,n ⇐⇒ A1 A−1 2 ∈ SM (n).

Definition 3.21. Let A, B ∈ M (2n, n). A is conjugate to B if A JB = In . A is conformally conjugate to B if ∃μ = μ(A, B) = 0, such that A JB = μIn . Obviously, A is conjugate to B iff −B is conjugate to A; A is conformally conjugate to B iff B is conformally conjugate to A with μ(B, A) = −μ(A, B).

2.3.2 Symplectic Group Let

2.3 Symplectic Space

A=

A1 A2

B=

B1 B2

145

! = [a1 , · · · , an ] ∈ M (2n, n), ! = [b1 , · · · , bn ] ∈ M (2n, n),

M = [A, B] =

A1 A2

B1 B2

! = [a1 , · · · , an , b1 , · · · , bn ] ∈ M (2n),

M ∈ Sp(2n) ⇐⇒ M  JM = J ⇐⇒ A, B are a symmetric pair, A is conjugate to B ⇐⇒ A JA = On = B  JB, A JB = In ⇐⇒ A1 A2 − A2 A1 = On = B1 B2 − B2 B1 , A1 B2 − A2 B1 = In ⇐⇒ ai Jaj = bi Jbj = 0, ai Jbj = δij , i.e., a1 , · · · , an , ¯b1 , · · · , ¯bn is a symplectic basis ! A1 A2 ⇐⇒ M  = ∈ Sp(2n) B1 B2 ! ! A2 A1 , are symmetric pairs and are conjugate to each other ⇐⇒ B1 B2 ⇐⇒ A1 B1 − B1 A1 = On = A2 B2 − B2 A2 ; A1 B2 − B1 A2 = In . M ∈ CSp(2n)

⇐⇒ ∃μ = 0, M  JM = μJ ⇐⇒ A, B form a symmetric pair and A is conformally conjugate to B ⇐⇒ A JA = On = B  JB, ∃μ = 0, s.t. A JB = μIn ⇐⇒ A1 A2 − A2 A1 = On = B1 B2 − B2 B1 , A1 B2 − A2 B1 = μIn , μ = 0 ! A1 A2 ⇐⇒ M  = ∈ CSp(2n)   B1 B2 ! ! A1 A2 are symmetric pairs and are conformally ⇐⇒ , conjugate to each other B1 B2 ⇐⇒ A1 B1 − B1 A1 = On = A2 B2 − B2 A2 ; A1 B2 − B1 A2 = μIn , μ = 0.

Proposition 3.22. If M  JM = J, then M JM  = J. Generally speaking, if K 2 = ±I, then M  KM = K, iff M KM  = K. 

Proof. If M  KM = K, then K = M −1 KM −1 ,

146

2. Symplectic Algebra and Geometry Preliminaries 

K −1 = (M −1 KM −1 )−1 = M K −1 M  . By assumption, K 2 = ±I, and so K −1 = ±K. Therefore, M KM  = K.



If M ∈ Sp(2n), then A is a symmetric pair, iff M A is a symmetric pair; and A is conjugate to B, iff M A is conjugate to M B. If M ∈ CSp(2n), then A is a symmetric pair iff M A is a symmetric pair; A is conformally conjugate to B iff M A is conformally conjugate to M B. M M

= [A, B] ∈ O2n ⇐⇒ M  M = I2n =

⇐⇒ A A = B  B = In , A B = On . [A, B] ∈ Un = Sp(2n) ∩ O2n

=

GL(n, C) ∩ Sp(2n) = GL(n, C) ∩ O2n

⇐⇒ A JA = B  JB = On , A B = On , A JB = A A = B  B = In . ⇐⇒ A is non-singular symmetric pair, A A = In B = J −1 A (see Theorem 3.34). Theorem 3.23. Let M be non-singular. M ∈ CSp(2n) iff M Z ∈ SM 2n×n ,

∀Z ∈ SM 2n×n ,

Z non-singular.

A1 Proof. We only need to prove the sufficiency. Let M = [A, B] = A2 ! ! I I ∈ SM 2n×n , M Z = [A, B] = A. 1◦ Take Z = O O By assumption, M Z ∈ SM 2n×n , i.e., (M Z) JM Z = A JA = O. ! O ◦ 2 Take Z = ∈ SM 2n×n , similarly, B  JB = 0. I ! S 3◦ Take Z = , S  = S, and so Z ∈ SM 2n×n . I By (M Z) JM Z = 0, we have

B1 B2

! .

(AS + B) J(AS + B) = S  A JAS + B  JB + S  A JB + B  JAS = S  A JB + B  JAS = O. Let C = A JB. Then, S  C = C  S, ∀ S  = S. Take S = I. Then, C  = C, CS = SC, ∀ S  = S. This shows that C must be μI, i.e., A JB = μI. The μ = 0 follows from |M | = 0. In fact, if μ = 0, then A JB = O. Hence,  A JA = A JB = 0. Thus, A J[A, B] = 0. This leads to A = 0, and therefore, A is also equal to 0. This is a contradiction. Therefore, M ∈ CSp(2n).  Remark 3.24. If CS = SC, ∀S  = S, then C = μI.

2.3 Symplectic Space

147

2.3.3 Lagrangian Subspaces Theorem 3.25. {A} ∈ Λn ⇔ A is a non-singular symmetric pair. M ∈ Sp(2n) or CSp(2n) implies that {A} ∈ Λn , iff M {A} ∈ Λn . Examples of Lagrangian subspaces: Coordinate Lagrangian subspaces[Arn89,Wei77,AM78,HW63] : ! ! I I Rν,0 = Rν,ν = , in which is a non-singular symmetric pair; O O ! ! O O R0,ν = Rν ,ν = , in which is a non-singular symmetric pair; I I ! Iα is a non-singular symmetric pair. Rν,ν = {Iα,α } in which Iα,α = Iα Proposition 3.26. We have the following results: 1◦ Let {A} be k-dim, {B} be l-dim. Then, {A} ⊂ {B} , iff A JB = Ok×l . 2◦ Let A, B be non-singular. Then, {A} ∩ {B} = {0}, iff [A, B] is non-singular. 3◦ If dim{A} = dim{B} = n, then {A} ∩ {B} = {0}, iff det [A, B] = 0. 4◦ {A} is isotropic of k-dim ⇔ A JA = Ok . 5◦ {A} is Lagrangian ⇔ A JA = On . 6◦ k-dimensional subspaces {A} is non-degenerate iff |A JA| = 0, k = 2s, iff ∃ B, such that {B} = {A}, B  JB = J2s . 7◦ {A} is degenerate ⇔ |A JA| = 0. Theorem 3.7 to Theorem 3.14 can be restated in matrix language as follows: Theorem 3.27. If A JA = Ok , A is non-singular. Then, there exists B ∈ M (2n, n − k), such that [A, B] is a non-singular symmetric pair. Theorem 3.28. If A ∈ M (2n, n) is a non-singular symmetric pair, then there exists a matrix, B ∈ M (2n, n), such that [A, B] ∈ Sp(2n). Theorem 3.29. If A, C ∈ M (2n, n) are two non-singular symmetric pairs and det [A, C] = 0, then there exists uniquely a non-singular symmetric pair B such that B ∼ C and [A, B] ∈ Sp(2n). Theorem 3.30. Let A ∈ M (2n, k), B ∈ M (2n, l), A JA = Ok , B  JB = Ol , and [A, B] be non-singular. Then, there exist C, D, such that [A, C], [B, D] are nonsingular symmetric pairs and det [A, C, B, D] = 0.  A  1 Theorem 3.31. If A = is a non-singular symmetric pair, then ∃ α ⊂ ν, such A2 that |Iα A1 + Iα A2 | = 0.

148

2. Symplectic Algebra and Geometry Preliminaries

Theorem 3.32. For two mutually transversal Lagrangian subspaces {A}, {B}, there always exists a third Lagrangian subspace {C}, transversal to {A} and {B}. Proof. Take {a1 , a2 , · · · , an } = {A}, {B} = {b1 , b2 , · · · , bn }, such that A JA = O,

B  JB = O,

A JB = In ,

B  JA = −In .

Set C = A + B. Then, (A + B) J(A + B) = (A + B  )J(A + B) = A JA + A JB + B  JA + B  JB = O; and

det [A, A + B] = det [A, B] = 0, det [B, A + B] = det [A, B] = 0. 

The theorem is proved.

Theorem 3.33. For any two Lagrangian subspaces {A}, {B} there exists another Lagrangian subspace {C}, transversal to {A} and {B}. Proof. Assume U 0 = {A, B} = 0. Take {a1 , · · · , ak } = {A} ⊂ {A}, {¯b1 , · · · , ¯bk } = {B} ⊂ {B}, such that

{c1 , · · · , cn−k } = U 0 ,

{a1 , · · · , ak , c1 , · · · , cn−k } = {A}, {¯b1 , · · · , ¯bk , c1 , · · · , cn−k } = {B}, [ai , aj ] = 0 = [¯bi , ¯bj ], [ai , ¯bj ] = δij ,

i, j = 1, · · · , k.

Set C = {a1 + ¯b1 , · · · , ak + ¯bk , d1 , · · · , dn−k }, where [di , dj ] = 0,

[di , cj ] = δij ,

[di , aj ] = [di , ¯bj ] = 0. Then, {C} is what we want to find out.



2.3.4 Special Types of Sp(2n) Set M = [A, B] = Sp(2n).

A1 A2

B1 B2

!

∈ Sp(2n). The following are special types of

2.3 Symplectic Space

149

(0) M ∈ Sp(2n) ∩ O2n = Un =⇒ M = [A, J −1 A], A JA = On , A A = In . (I) Sp(2n)  M =

=⇒ M =

A1 O A1 O

(II) Sp(2n)  M =

I O

(II )

A1 A2

Sp(2n)  M =

(III) Sp(2n)  M = Jα = ! Iα Iα Jα = . −Iα Iα 1.

O B1

! , which are diagonal blocks

! O , A1 ∈ GL(n, R).  A−1 1 ! ! I S B1 =⇒ M = , S  = S. O I B2 O I

!

Iα −Iα

=⇒ M = Iα Iα

I S

O I

! , S  = S.

! , α ⊂ ν, and symplectic substitution

Several special types (1)

Sp2n (0).

Sp2n (0) = Sp2n ∩ O2n = Un .

Theorem 3.34. M = [A, B] ∈ Sp2n (0) = Un ⇔ B = J −1 A, A A = I, A JA = O. Proof. Evidently, if A A = I, A JA = O. Then, [A, J −1 A] ∈ Sp2n (0). Conversely, M = [A, B] ∈ Sp2n =⇒ A JA = O, A JB = I. M = [A, B] ∈ O2n =⇒ A B = O, B  B = I. 7 ⎫ A JA = O ⎪  ⎪ =⇒ A (JA + B) = O ⎪ ⎪ ⎬  AB=O =⇒ / ⎪ ⎪ ⎪ B  JA = −I ⎭ =⇒ B  (JA + B) = O ⎪ BB = I

A B

! (JA + B) = O

=⇒ M  (JA + B) = O =⇒ JA + B = O =⇒ B = J −1 A. The theorem is proved.



Lemma 3.35. There exist polynomials ϕn (A) = ϕn (ajk ), ψn (A) = ψn (ajk ) in 2n× n variables a11 , · · · , a2n,n with integer coefficients such that

150

2. Symplectic Algebra and Geometry Preliminaries



det [A J A] = det

−A2 A1

⎡

where

Proof.

A1 A2

a11 ⎢ A = ⎣ ... a2n,1   A1   A2

   A + i A2 −A2  =  1 A1  A2   A1 + i A2 =  A2   A1 + i A2 =  A2

!

= (ϕn (A))2 + (ψn (A))2 ≥ 0,

⎤ a1n .. ⎥ . . ⎦ · · · a2n,n ···

 −A2 + i A1   A1

 −A2 + i A1 − i (A1 + i A2 )   A1 − i A2   O  A1 − i A2 

= (A1 + i A2 ) (A1 − i A2 ) = (A1 + i A2 ) (A1 + i A2 ) = (Re|A1 + i A2 |)2 + (Im|A1 + i A2 |)2 = (ϕn (A))2 + (ψn (A))2 ≥ 0. 

Therefore, the lemma is completed. Theorem 3.36. M ∈ Sp2n (0) = Un ⇒ |M | = 1. Proof. We will prove that M  M = I =⇒ |M |2 = 1 =⇒ |M | = ±1 By Lemma 3.35,

M = [A, J −1 A] =⇒ |M | ≥ 0

7 =⇒ |M | = 1. 

The proof can be obtained. (2)

Sp2n (I).

!

M=

A1 O

O B1

M=

A1 O

O  A−1 1

, in which diagonal blocks ∈ GL(n, R), ! , A1 ∈ GL(n, R), |M | = |A1 ||A1 |−1 = 1,

A1 ∈ GL(n, R) =⇒ A1 = T P, ! ! p p R2n  −→ M = q q ! ! A1 O T O =  O T O A−1 1 ! T O ∈ On ⊂ Un , O T

T  T = In , P  = P > 0(positive definite), ! A1 p ∈ R2n ,  (A−1 1 ) q ! P O , O P −1

2.3 Symplectic Space

151

where '

T O

On = ' '

P O

O P −1

P O

O P −1

(3) Sp2n (II).

'

Sp2n (II) =

! / O  , T T = In = Un ∩ GL(n, R) = Sp2n (0) ∩ Sp2n (I), T / !  , P = P > 0 is not a group, !

/ , P = P > 0 ∩ Sp2n (0) = {I2n }. 

I O

S I

!

/ 

,S = S ' ! / I O   , S = S , which is a multiplicative group  Sp2n (II ) = S I  SM n = {S, S  = S}, which is an additive group.

(4)

Sp2n (I, II). Sp2n (I, II) ' Q = O

QS −1 Q

!

Q O

=

O −1 Q

!

I O

S I

!

/ , Q ∈ GL(n, R), S  = S .

M ∈ Sp2n (I,II) =⇒ |M | = 1. Sp2n (II) ∩ Sp2n (0) = Sp2n (II ) ∩ Sp2n (0) = {I}. (5) Sp2n (III). Jα =

Iα −Iα

Iα Iα

! ,

Jα = Jα−1 =

Iα Iα

−Iα Iα

! ,

α ⊂ ν.

In particular, Jν = J2n , J0 = I2n , |Jα | = 1. {Symplectic Substitution} = {Jα , Jα } is not a group. Total number of Jα = 2n , total number of {Jα , Jα } = 2n+1 − 1, and ! ! I S I O  J2n , = J2n O I −S I which is the bijection from Sp2n (II) to Sp2n (II ). 2. Some theorems about Sp(2n) Theorem 3.37. If {A}, {B} ∈ Λn , then there exists M ∈ Sp(2n), such that M A = B. Moreover, M {A} = {B}.

152

2. Symplectic Algebra and Geometry Preliminaries

Proof. By Theorem 3.28, there exist C, D ∈ M2n,n , such that [A, C] = MA ∈ Sp(2n),

[B, D] = MB ∈ Sp(2n).

Set M = MB MA−1 . Then, M MA = MB , i.e., M [A, C] = [B, D]. Therefore, M A = B.  Theorem 3.38. If {A}, {B} ∈ Λn , then there exists M ∈ Sp2n (0) = Un = Sp2n ∩ O2n , such that M {A} = {B}. Proof. From the procedure of Grass-Schmidt’s orthogonal normalization, we can get two matrices C and D of order 2n × n i.e., C, D ∈ M (2n, n), such that {C} = {A},

C  JC = O,

C  C = I.

{D} = {B},

D JD = O,

D D = I.

Set MC = [C, J −1 C], MD = [D, J  D]. Obviously, MC , MD ∈ Sp2n (0) = Un . Set M = MD MC−1 . Then, M MC = MD , i.e., M [C, J −1 C] = [D, J  D]. Thus, M C = D. We obtain M {C} = {M C} = {D}, i.e., M {A} = {B}.



Theorem 3.39. Let G = CSp(2n), Sp(2n), or Sp2n (0) = Un . Then, the following action of G on Λn : M {A} = {M A}(A ∈ Λn , M ∈ G) is 1◦ Transitive, i.e., ∀{A}, {B} ∈ Λn , ∃M ∈ G, such that M {A} = {B}. 2◦ If for any {A} ∈ Λn , M {A} = {A} then M = ±I2n when G = Sp(2n) or Sp2n (0); and M = μI2n when G = CSp(2n). Proof. 1◦ can be obtained by Theorem 3.37 and Theorem 3.38. ! ' / ' / A B I O 2◦ Assume M = . Taking {A} = , ∈ Λn , respecC D O I tively, we have ' / ' / ' / ' / I I O O M = and M = , O O I I i.e., for some Q ∈ GL(n), ! A B C D

I O

! =

A C

! =

I O

! Q=

Q O

! ,

2.3 Symplectic Space

A C

B D

!

!

O I

B D '

=

and so B = C = O. Again, take {A} = ' M it follows that A O

O D

!

I P

I P

!

O I

= I P

/

!

O Q

Q=

! ,

/ , P  = P . Then, from the equality

' =

A DP

=

!

153

I P

!

/ ,

I P

=

!

Q PQ

Q=

! ,

for some Q ∈ GL(n) and ∀P  = P , i.e., A = Q, and DP = P Q = P A. Set P = I, D = A. Then, we get AP = P A,

∀P  = P.

This implies that A = μI(μ = 0) (since |M | = 0). Besides, if M ∈ Sp(2n) or Sp2n (0) then μ2 I = I, i.e., μ2 = 1. Therefore, we have M = ±I.  Theorem 3.40. Λn  Un /On = Sp2n (0)/Sp2n (0) ∩ Sp2n (I), or ! I ϕ : Un / On −→ Λn , ϕ(M On ) = M , M ∈ Un , O is a bijection from Un /On to Λn . Proof. First, ϕ is well-defined. In fact, if M1 On = M2 On , or M1 ∈ M2 On , then ! T O . M1 = M 2 O T ' / ! ! ! I T O I T = M2 Q = M2 Q For some T’= T, M1 O O T O O ! ' / I I = M2 T Q = M2 . O O It follows from the Theorem 3.38 that ϕ is surjective. We will prove that ϕ is injective. ' / ' / ' / ' / I I I I −1 = M2 , then M2 M1 = . If M1 O O O O Set M = M2−1 M1 =

A C

A C

B D

B D !

!

' ∈ Sp2n (0). Then, M

I O

! =

A C

! =

I O

I O

! Q.

/

' =

I O

/ , i.e.,

154

2. Symplectic Algebra and Geometry Preliminaries

Thus, C = O. Since

A C

B D B D

! ∈ Sp2n (0), by Theorem 3.34: !

A O

= J −1

!

and 

[A , O] This means that M =

A O

O A

O I

=

A O

!

−I O

!

A O

! =

!

O A

= A A = I.

! , A A = I ∈ On .



Theorem 3.41. Λn  Sp(2n)/Sp2n (I,II), or the mapping ϕ : Sp(2n)/Sp2n (I, II) −→ Λn ,

ϕ(M Sp2n (II)) = M

I O

! , M ∈ Sp

is a bijection. Proof. If M1 Sp2n (I, II) = M2 Sp2n (I, II), then M1 ∈ M2 Sp2n (I, II), i.e., ! Q QS  M1 = M2 for some Q ∈ GL and S ∈ SM (n). O Q−1 Thus, ' M1

I O

/ = M2 = M2

! ! QS I  P = M2 O Q−1 ! ' / I I QP = M2 . O O Q O

This implies that ϕ is well-defined. By Theorem 3.37, we know that ϕ is surjective.

Q O

! P

' / ' / I I = M2 , then Last, ϕ is injective too. As a matter of fact, if M1 O O ' / ' / I I M2−1 M1 = . O O ! ' / ' / A B I I −1 ; then M = , i.e., Set M = M2 M1 = C D O O ! ! ! ! ! A B I A I P = = P = , for some P ∈ GL(n). C D O C O O Thus, C = O.

2.3 Symplectic Space

A C

M=

!

B D

A O

=

−1

O  A−1

!

I O

155

!

B1 D1

∈ Sp(2n), A O



O A −1

!

∈ Sp(2n), where B1 = A B, D1 = A D. Since M ∈ Sp(2n) and ! I B1 must be symplectic too. O D1 ! I B1  ∈ Sp2n (II). Therefore, M ∈ Sp2n (I, II). By definition, O D1

2.3.5 Generators of Sp(2n) Theorem 3.42. Every symplectic matrix M can be decomposed as the multiplication of three kinds of special symplectic matrices M = M0 M1 M2 , where M0 ∈ Sp2n (0),

M1 ∈ Sp2n (I),

M2 ∈ Sp2n (II).

Proof. Let M = [A, B] ∈ Sp2n ; then {A} ∈ Λn . M0 ∈ Sp2n (0) = Un , such that ! I A = M0 Q = M0 O ! Q O = M0 −1 O Q Let B1 =

Q O

!−1

O Q −1

i.e., Q O −1 O Q

B = M0

!

By Theorem 3.38, there exists Q O !

!

I O

.

M0−1 B, C1 C2

! .

Thus, M = [A, B] =

M0

= M0

Q O Q O

!

O Q −1 O Q −1

!

I O I O

! , M0 C1 C2

Q O

O Q −1

!

!

C1 C2

!!

∈ Sp(2n)

= M 0 M1 M2 , where M1 =

Q O

O −1 Q

! ∈ Sp2n (I),

M2 =

I O

C1 C2

! ∈ Sp2n (II).



156

2. Symplectic Algebra and Geometry Preliminaries

Proposition 3.43. |M | = 1, ∀ M ∈ Sp(2n). Theorem 3.44. The following decomposition of a symplectic matrix M ∈ Sp(2n) is unique. M = M0 M1 M2 , ! P O , P  = P > 0, M2 ∈ Sp2n (II). where M0 ∈ Sp2n (0), M1 = O P −1 Proof. A non-singular matrix Q ∈ GL(n) can be uniquely decomposed as a multiplication of an orthogonal matrix T and a positive definite matrix P : Q = T P . By Theorem 3.42, we have a decomposition ! Q O M2 M = M 0M 1M 2 = M 0 O Q −1 ! ! T O P O = M0 M2 O T O P −1 = M0 M1 M2 , where M0 = M 0 M1 =

! O ∈ Sp2n (0), M2 = M 2 ∈ Sp2n (II), T ! O , P  = P > 0. P −1

T O

P O



We need to prove that such a decomposition is unique. Suppose M = M01 M11 M21 is another decomposition, where ! P1 O I , P1 = P1 > 0, M21 = M11 = O O P1−1

S1 I

! .

By the equality M0 M1 M2 = M01 M11 M21 , we have −1 M0 = M11 M21 M2−1 M1−1 Sp2n (0)  M01 ! O P1 I = O O P1−1 ! P1 O I = −1 O O P1

= Thus,

P1 P −1 O

S1 I

!

I O !

−S I

!

S1 − S P −1 I O ! P1 (S1 − S)P ∈ Sp2n (0). −1 P1 P

P −1 O ! O P

O P

!

2.3 Symplectic Space

P1 (S1 − S)P P1−1 P

!

157

! P1 P −1 = J O ! ! O −I P1 P −1 = O I O ! O = . P1 P −1 

Then, P1 (S1 − S)P = O and P1−1 P = P1 P −1 , i.e., S1 = S and P12 = P 2 . −1 M0 = I, Therefore, P1 = P , since P1 , P are positive definite. It follows that, M01 i.e., M0 = M01 . This is what we need to prove. ! A1 B1 ∈ Sp(2n) with |A1 | = 0, then it can be deTheorem 3.45. If M = A2 B2 composed as M = M2 M1 M2 , where M2 ∈ Sp2n (II ), M1 ∈ Sp2n (I), M2 ∈ Sp2n (II). This leads to |M | = 1. Proof. By Gauss elimination, ! ! ! A1 O A1 B1 I −A−1 1 B1 = , A2 B2 O I A2 B2 − A2 A−1 1 B1 ! ! ! A1 I O A1 O O = . −A2 A−1 I A2 B2 − A2 A−1 O B2 − A2 A−1 1 1 B1 1 B1 M = [A, B] ∈ Sp(2n), and so A JB = I and A is a symmetric pair, i.e., A1 B2 − A2 B1 = I,

A2 A−1 1 ∈ SM .

Thus, 



−1  −1  = B2 − A−1 A−1 1 1 A2 B1 = B2 − (A2 A1 ) B1 = B2 − A2 A1 B1 .

Obvioualy, M1 = M2 = M2 =

! A1 O = −1 B2 − A2 A1 B1 O !−1 I O ∈ Sp2n (II ), I −A2 A−1 1 !−1 I −A−1 1 B1 ∈ Sp2n (II), O I A1 O

by which we get the decomposition M = M2 M1 M2 .

O  A−1 1

! ∈ Sp2n (I),



158

2. Symplectic Algebra and Geometry Preliminaries

Theorem 3.46. Every symplectic matrix M can be decomposed as M = M3 M2 M1 M2 , where M3 ∈ Sp2n (III), M2 ∈ Sp2n (II ), M1 ∈ Sp2n (I), M2 ∈ Sp2n (II), and it reduces |M | = 1 too. ! A1 B1 Proof. It follows from that there exists α ⊂ ν such that Jα−1 = A2 B2 ! C1 D1  with |C1 | = 0 and Theorem 3.45. C2 D2

2.3.6 Eigenvalues of Symplectic and Infinitesimal Matrices Definition 3.47. A real polynomial Pm (λ) = a0 λm + a1 λm−1 + · · · + am is called reflective if Pm (λ) = λm Pm (1/λ). It is easy to see that Pm is reflective if and only if ai = am−i (i = 0, · · · , m). Lemma 3.48. We have the following results: 1◦ Q(λ) = b0 λ2 + b1 λ + b2 is reflective iff b0 = b2 , i.e., Q(λ) = b0 (λ − α)(λ − 1/α). 2◦ L(λ) = c0 λ + c1 is reflective iff c0 = c1 , i.e., L(λ) = c0 (λ + 1). Property 3.49. We have the following properties: 1◦ P1 , P2 reflective ⇒ P1 · P2 reflective. 2◦ P = P1 · P2 , P, P1 reflective ⇒P2 reflective. n ; 3◦ m = 2n, Pm reflective ⇒ Pm = Qi (λ), Qi reflective of order 2. i=1

4

◦

m = 2n + 1, Pm reflective ⇒ Pm = L(λ)

n ;

Qi (λ), Qi , L reflective of

i=1

order 2 and 1, respectively. Lemma 3.50. The characteristic polynomial of a symplectic matrix M ∈ Sp(2n), P (λ) = |M − λI| is reflective. Theorem 3.51. [Arn89,AM78] λ0 is an eigenvalue of a symplectic matrix M with multi−1 plicity k. Then, λ−1 0 , λ0 , λ0 are also the eigenvalues of M with the same multiplicity. If ±1 are the eigenvalues of M , then their multiplicity is even. Possible cases of distribution of eigenvalues of a symplectic matrix of order 4 are depicted in Fig. 3.1.

Definition 3.52. A real polynomial P2n (λ) = a0 λ2n + a1 λ2n−1 + · · · + a2n is even if P (λ) = P (−λ).

2.3 Symplectic Space

complex saddle

saddle center

real saddle

generic center

159

(2) (4)

(2) (2)

degenerate saddle Fig. 3.1.

identity

(2)

degenerate center

Distribution of Eigenvalues of a symplectic matrix of Sp(4)

Obviously, P2n (λ) is even iff a2i+1 = 0 (i = 0, 1, · · · , n − 1). Every even polynomial P2n (λ) can be rewritten in the following form P2n (λ) = a0

n ;

(λ2 − ci ).

i=1

Lemma 3.53. The characteristic polynomial of every infinitesimal symplectic matrix is even. Theorem 3.54. [Arn89,AM78] If λ0 is an eigenvalue of on infinitesimal symplectic matrix B with multiplicity k, then −λ0 , λ0 and −λ0 are the eigenvalues of B with the same multiplicity. If 0 is an eigenvalue of B, then its multiplicity is even. The possible cases of distribution of eigenvalues of an infinitesimal symplectic matrix of order 4 are depicted in Fig. 3.2.

160

2. Symplectic Algebra and Geometry Preliminaries

(2) (4) (2)

Fig. 3.2.

(2)

(2)

Distribution of eigenvalues of an infinitesimal symplectic matrix of sp(4)

2.3.7 Generating Functions for Lagrangian Subspaces Special cases /    '   S  S p  L = = ∈ R2n  (p , q  )J = O (S  = S) I q I / / ' '     I   p p 2n    2n  =O = ∈ R  p = Sq = ∈ R  (p , q ) −S q q ' /  p  ∂ϕ 2n  = ∈ R p = , q ∂q 1

where we define ϕ(q) = q  Sq, called as a generating function [Wei77,FWQW89,Fen95,Ge91] 2 for L. Remark 3.55. There exists a generating function for Lagrangian subspace transversal to Rν,0 or R0,ν . Theorem 3.56. For a non-singular symmetric pair A, there is α ⊂ ν such that det(Jα−1 A)2 = 0 and det(Jα−1 A)1 =  0, where Jα−1 A and Jα−1 A are non-singular symmetric pairs. 2n ˙ Proof. By Theorem 3.14, there exists α ⊂ ν such that {A}+{R α, α } = R . This ! Iα A1 is non-singular. shows that the matrix Iα A2

Multiplying by Jα−1 , we have ! ! Iα A1 Iα −Iα −1 Jα = Iα A2 Iα Iα

Iα Iα

A1 A2

! =

O I

(Jα−1 A)1 (Jα−1 A)2

! .

2.3 Symplectic Space

161

Therefore, det (Jα−1 A)1 = 0. If replace Jα−1 with Jα−1 , then det (Jα−1 A)2 = 0. For a general case, we have the following theorem:

 '

/ A1 Theorem 3.57. For every Lagrangian subspace L = {A} = , there exist an A2 α ⊂ ν and a generating function ϕ, a quadratic form in n-variables {pi , qj }i∈α,j∈α such that / '% &  p ∂ϕ ∂ϕ 2n  , i ∈ α; qi = − , i∈α  . L= ∈ R  pi = q ∂qi ∂pi Proof. Taking α ⊂ ν in Theorem 3.56, the matrix ! ! B1 Iα A1 − Iα A2 = B = Jα−1 A = B2 Iα A1 + Iα A2 is Lagrangian with |B2 | = 0. Define ! u = Jα−1 v then,

p q

! =

Iα p − Iα q Iα p + Iα q

! ;

'

' pi , pi , i ∈ α i ∈ α, , vi = ui = , −qi , i ∈ α qi , i ∈ α, ! ! ! A1 p A1 [p , q  ]J Jα (Jα−1 JJα )Jα−1 = A2 A2 q !! ! B1 p J = Jα−1 q B2 ! S = [u , v  ] J B2 , I

where S = B1 B2−1 , S  = S, and Jα−1 = Jα ∈ Sp(2n). Thus, if we define ϕ(v) =

1  v Sv, then 2 ' p L = q ' u = v ' p = q ' p = q

 ! / ' !  u A1 =O = ∈ R2n ∈ R2n  [p , q  ]J v A2   ! / ! / '   ∂ϕ u ∈ R2n u = Sv = ∈ R2n u = v ∂v  / ! ! !  ∂ϕ p u = Jα ,u = ∈ R2n  q v ∂v  ! /  ∂ϕ ∂ϕ ∈ R2n  pi = , i ∈ α; qi = − ,i ∈ α  . ∂qi ∂pi !

Therefore, the theorem is completed.

     [u , v ]J 

S I

!

/ =O



162

2. Symplectic Algebra and Geometry Preliminaries

2.3.8 Generalized Lagrangian Subspaces In the previous sections, we have considered in detail the special symplectic space with the special symplectic structure ! O I ωJ (x, y) = ϕJ (x, y) = x Jy, J = . −I O For every non-singular anti-symmetric matrix K of order 2n, ωK (x, y) = ϕK (x, y) = x Ky is a symplectic structure on R2n , and so (R2n , ωK ) is also a symplectic space. Several previous results can be directly applied to this case. Here, we only give a few different theorems [FW91b,FWQW89] . Definition 3.58. {A}(or A) is called as a K-Lagrangian subspace, if A KA = 0, and A is non-singular. Let us denote: Λn (K) = {{A} | {A} is a K-Lagrangian subspace} , Sp(K, 2n) = {M ∈ M (2n) | M  KM = K} , CSp(K, 2n) = {M ∈ M (2n) | M  KM = μK, μ = 0} . Elements of Sp(K, 2n) or CSp(K, 2n) are called as K-symplectic matrices and conformally K-symplectic matrices respectively. Theorem 3.59. Let M be non-singular. 1◦ M ∈ CSp(K, 2n), iff A ∈ Λn (K) ⇒ M A ∈ Λn (K). 2◦ M ∈ Sp(K, 2n), iff (M A) K(M B) = A KB or ϕk (M A, M B) = ϕk (A, B),

∀A, B ∈ Λn (K).

Proof. 1◦

“⇒” is trivial. We know that for any non-singular matrix % anti-symmetric & O I . Thus, K, there exists Q ∈ GL such that K = Q JQ, where J = −I O A KA = A Q JQA = (QA) J(QA), (M A) K(M A) = A M  Q JQ(M A) −1

= A Q Q M  Q JQM Q−1 QA = (QA) (QM Q−1 ) J(QM Q−1 )(QA). A ∈ Λn (K) =⇒ M A ∈ Λn (K) ⇐⇒ QA ∈ Λn (J) =⇒ QM Q−1 ∈ QA ∈ Λn (J). By Theorem 3.23, we get QM Q−1 ∈ CSp(J), i.e., (QM Q−1 ) J(QM Q−1 ) = μJ,

μ = 0.

This leads to M  Q JQM = μQ JQ, i.e., M  KM = μK.

2.3 Symplectic Space

163

2◦ Similarly, we only need to prove the sufficiency. By 1◦ , we have M  KM = μK, μ = 0. Thus, ∀A, B ∈ Λn (K), we have A KB = μ−1 A M  KM B = μ−1 A KB. Taking A and B ∈ Λn (K) such that A KB = 0, we get μ = 1. Therefore, M  KM = K. 

The theorem is proved.

Definition 3.60. Let K1 and K2 be two non-singular, anti-symmetric matrices of order 2n. Define Sp(K1 , K2 ) = {M ∈ M (2n) | M  K1 M = K2 } , CSp(K1 , K2 ) = {M ∈ M (2n) | ∃ μ = 0, s.t. M  K1 M = μK2 } . Remark 3.61. Sp(K1 , K2 ) and CSp(K1 , K2 ) are not groups. However, Sp(K2 , K1 ) = Sp(J),

CSp(K2 , K1 ) = CSp(J)

have the same power. In fact, ∃ Q1 , Q2 ∈ GL, such that K1 = Q1 JQ1 , K2 = Q2 JQ2 , M  K2 M = K1 ⇐⇒ M  Q2 JQ2 M = Q1 JQ1 , i.e.,



−1   Q−1 1 M Q2 JQ2 M Q1 = J.

∈ Sp(J). Hence, the mapping: M ∈ Sp(K2 , K1 ) → It is equivalent to Q2 M Q−1 1 ∈ Sp(J) is a one-to-one correspondence. It is also a one-to-one corresponQ2 M Q−1 1 dence between CSp(K2 , K1 ) and CSp(J). In addition, M can be viewed as a mapping from Λn (K2 ) to Λn (K1 ): M : Λn (K2 ) −→ Λn (K1 ), A ∈ Λn (K2 ) −→ M A ∈ Λn (K1 ). We have the following theorem similar to Theorem 3.23 and Theorem 3.59. Theorem 3.62. Let M be non-singular. Then, 1◦ M ∈ CSp(K2 , K1 ), iff M A ∈ Λn (K1 ), ∀A ∈ Λn (K2 ). 2◦ M ∈ Sp(K2 , K1 ), iff (M A) K1 (M B) = A K2 B, ∀A, B ∈ Λn (K2 ). Proof. The proof is omitted, as it is similar to the proof of Theorem 3.59.



Bibliography

[AM78] R. Abraham and J. E. Marsden: Foundations of Mechanics. Reading, MA: AddisonWesley, Second edition, (1978). [Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Berlin Heidelberg, Second edition, (1989). [Art57] E. Artin: Geometrical Algebra. Interscience Publishers, New York, Second edition, (1957). [Car65] C. Carathe’odory: Calculus of Variation and Partial Differential Equations of First Order, Vol.I. Holden-Day, San Franscisco, (1965). [Fen95] K. Feng: Collected Works of Feng Kang. Volume I, II. National Defence Industry Press, Beijing, (1995). [FQ91] K. Feng and M.Z. Qin: Hamiltonian Algorithms for Hamiltonian Dynamical Systems. Progr. Natur. Sci., 1(2):105–116, (1991). [FW91] K. Feng and D.L. Wang: Symplectic Difference Schemes for Hamiltonian Systems in General Symplectic Structures. J. Comput. Math., 9(1):86–96, (1991). [FWQW89] K. Feng, H. M Wu, M.Z. Qin, and D.L. Wang: Construction of canonical difference schemes for Hamiltonian formalism via generating functions. J. Comput. Math., 7:71–96, (1989). [Ge91] Z. Ge: Equivariant symplectic difference schemes and generating functions. Physica D, 49:376–386, (1991). [HW63] L.G. Hua and Z.X. Wan: Classical Groups. Shanghai Science and Technology Publishing House, in Chinese, Shanghai, (1963). [LM87] P. Libermann and C.M. Marle: Symplectic Geometry and Analytical Mechanics. Reidel pub. company, Boston, First edition, (1987). [Tre75] F. Treves: Pseudo-Differential Operator. Acad.Press, New York, First edition, (1975). [Wei77] A. Weinstein: Lectures on symplectic manifolds. In CBMS Regional Conference, 29. American Mathematical Society, Providence, RI, (1977). [Wey39] H. Weyl: The Classical Groups. Princeton Univ. Press, Princeton, Second edition, (1939). [Wey40] H. Weyl: The method of orthogonal projection in potential theory. Duke Math. J., 7:411–444, (1940).

Chapter 3. Hamiltonian Mechanics and Symplectic Geometry

Hamiltonian mechanics is geometry in phase space. Phase space has the structure of a symplectic manifold.

3.1 Symplectic Manifold A symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form called the symplectic form. On a symplectic manifold, as on a Riemannian manifold, there is a natural isomorphism between the vector field and the 1-form. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e.g., in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field. The set of all possible configurations of a system is modelled as a manifold, and this manifold’s cotangent bundle describes the phase space of the system. Any real-valued differentiable function H on a symplectic manifold can serve as an energy function or Hamiltonian. Associated to any Hamiltonian is a Hamiltonian vector field; the integral curves of the Hamiltonian vector field are solutions to the Hamilton–Jacobi equations. The Hamiltonian vector field defines a flow on the symplectic manifold, called a Hamiltonian flow or symplectomorphism. By Liouville’s theorem, Hamiltonian flows preserve the volume form on the phase space. The vector fields on a manifold form a Lie algebra. The Hamiltonian vector fields on a symplectic manifold form a Lie algebra. The operation in this algebra is called Poisson bracket.

3.1.1 Symplectic Structure on Manifolds Definition 1.1. Let M2n be an even-dimensional differential manifold. A symplectic structure on M2n is a closed nondegenerate differential 2-form ω on M2n [AM78,Arn89] . 1◦ d ω = 0 is closed. 2◦ ∀ x ∈ M, ∃ ξ ∈ Tx M , s.t., if ω(ξ, η) = 0,

∀η ∈ Tx M,

then ξ = 0 (nondegenerate). The pair (M, ω) is called a symplectic manifold. We call it a presymplectic (almost symplectic) manifold, if only condition 1◦ (2◦ ) is satisfied. K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

166

3. Hamiltonian Mechanics and Symplectic Geometry

Example 1.2. Consider the vector space R2n with coordinates (pi , qi ). Let ω = n  d pi ∧ d qi . Then, ω defines a symplectic structure. Given two tangent vectors i=1

ξ j = (ξ1j , · · · , ξnj ; η1j , · · · , ηnj ), we have ω(ξ 1 , ξ 2 ) =

n 

j = 1, 2,

ηi1 · ξi2 − ηi2 · ξi1 .

i=1

This example shows that any symplectic manifold can have standard symplectic structure at least locally. Exercise 1.3. Verify that (R2n , ω) is a symplectic manifold. For n = 1, ω becomes an area measure on a plane.

3.1.2 Standard Symplectic Structure on Cotangent Bundles Let M be an n-dimensional differential manifold. A 1-form on Tx M , the tangent space to M at a point x, is called a cotangent vector to M at x. The set of all cotangent vectors to M at x forms an n-dimensional vector space, dual to the tangent space Tx M . We will denote this vector space of cotangent vectors by Tx∗ M and call it the cotangent space to M at x. The union of the cotangent space to the manifold at all of its points is called cotangent bundle, denoted as T ∗ M . Let (q1 , · · · , qn ) be the local coordinates of M and (p1 , · · · , pn ) be the coordinates of the fiber. Then (q, p) becomes a local coordinate of T ∗ M , and T ∗ M is equipped with a structure of differential manin  fold. Locally, ω = dpi ∧dqi is a natural symplectic structure of T ∗ M [AM78,Arn89] . In i=1

order to give a coordinate-free definition of the form ω, we first define a distinguished 1-form on T ∗ M . Let π : T ∗ M → M be a natural projection. Let ξ ∈ Tp (T ∗ M ) be a vector tangent to the cotangent bundle at the point p ∈ Tx∗ M . The derivative taken tangent mapping π∗ : Tp (T ∗ M ) → Tπ(p) M of the natural projection π : T ∗ M → M takes ξ to a vector π∗ (ξ) tangent to M at x = π(p). We define the 1-form σ on T ∗ M by the relation σ = p(π∗ (ξ)), which has a form under the local coordinate as follows ξ=

n 

ai

i=1

Therefore, p(π∗ (ξ)) =

n 

∂ ∂ + bi , ∂pi ∂qi

π∗ (ξ) =

n  i=1

bi

∂ . ∂qi

pi bi , which results in

i=1

σ=

n  i=1

pi dqi ,

ω = dσ =

n  i=1

dpi ∧ dqi .

3.1 Symplectic Manifold

167

3.1.3 Hamiltonian Vector Fields As we pointed out in Section 2.3 of Chapter 2, the symplectic structure of a symplectic space is similar to a Euclidean structure in some aspects. The symplectic structure on a symplectic manifold is also similar to a Riemann structure, which defines the Euclidean structure on a tangent space so that the tangent space becomes isomorphic to the cotangent space. The same is true for a symplectic structure. Let (M, ω) be a symplectic manifold; ∀η ∈ Tx M , there exists a linear form on Tx M : Tx M  ξ → ω(ξ, η); therefore, ω( · , η) defines an element of the cotangent space. Thus, we get a linear mapping Ω : Tx M → Tx∗ M , η → ω( · , η). The non-degeneracy of ω shows that Ω is an injective. Since Tx M and Tx∗ M have the same dimensions, Ω must be an isomorphism, i.e., ω(ξ, η) = (Ωη)ξ. Using the local coordinates, we can set (M, ω) = (R2n , d p ∧ d q). For ξ = (q1 , · · ·, n  pi qi − pi qi , qn ; p1 , · · · , pn ), η = (q1 , · · · , qn ; p1 , · · · , pn ), (Ωη)ξ = ω(ξ, η) = i=1 ! O −I Ω has a matrix representation as , i.e., Ω = −J, Ω−1 = J. I O Although the above results are defined for the tangent and cotangent space at a specific point x ∈ M , they can be easily extended to the entire tangent and cotangent bundle. Let θ be a 1-form on M , i.e., a C ∞ section on T ∗ M . Ω−1 θ should be on a vector field on M , i.e., a section on T M . One of the most important cases is when θ = dH is an exact differential form, i.e., θ is a C ∞ total differential on M . We denote it by Ω−1 = J. We often say that JdH is Hamiltonian vector field [AM78,Arn89] with Hamiltonian function H, which can be represented using the local coordinate as (M, ω) = (R2n , dp ∧ dq). We will use J : T ∗ M → T M∗ to denote the above isomorphism. Let H be a function on the symplectic manifold M 2n . Then, dH is a 1-form on M , and at every point there exists a tangent vector associated with it. Thus, we can obtain a vector field JdH on M . From 5 6 O I J= , d H = Hq d q + Hp d p, −I O we obtain the Hamiltonian vector field 5 65 6 5 6 O I Hq Hp JdH = = , −I O Hp −Hq which has an expression under local basis for the tangent field Hp

∂ ∂ − Hq . ∂q ∂p

168

3. Hamiltonian Mechanics and Symplectic Geometry

3.1.4 Darboux Theorem Symplectic geometry arises from the globalization of the symplectic algebra considered in the previous chapter. First, we prove Darboux’s theorem, according to which every symplectic manifold has local coordinates p, q in which the symplectic structure can be written in the simplest way ω = dp ∧ dq. Theorem 1.4 (Darboux theorem). Let ω be a closed non-degenerate 2-form on a manifold M 2n . Then, dω = 0 iff there exists a local coordinate system (u, ϕ) for every m ∈ M , such that ϕ(m) = 0,

ϕ(u) = (x1 (u), · · · , xn (u), y1 (u), · · · , yn (u)),

and ω|u =

n 

d xi ∧ d yi .

i=1

Proof. The sufficiency can be easily derived, since

n 

dxi ∧ dyi is a closed form.

i=1

Necessity. We first assume that M = E is a linear space, and m = 0 ∈ E. Let ω1 be the constant-form ω(0), ω " = ω1 − ω, ωt = ω + t" ω (0 ≤ t ≤ 1). For each t, ωt (0) = ω(0) is nondegenerate. Hence, by the openness of the set of isomorphisms from E to E ∗ , there exists a neighborhood of 0 on which ωt (0 ≤ t ≤ 1) is nondegenerate for all 0 ≤ t ≤ 1. We can assume that this neighborhood is a ball. Thus by Poincar´e lemma, there exists a 1-form θ s.t. ω " = dθ. Without loss of generality, we assume θ(0) = 0. Since ωt is nondegenerate, there exists a smooth vector field X, s. t. iX ωt = −θ. Since Xt (0) = 0, from the local existence theory of ODEs, there is a sufficiently small ball on which the integral curves of Xt are well defined for t ∈ [0, 1]. Let Ft be the flow starting at F0 = identity. By the Lie derivative formula for a time-dependent vector field, we have d d (F ∗ ωt ) = Ft∗ (LXt ωt ) + Ft∗ ωt dt t dt

= Ft∗ d iXt ωt + Ft∗ ω " = Ft∗ (−d θ + ω " ) = 0. Therefore, F1∗ ω1 = F0∗ ω = ω. So F1 provides the chart transforming ω to the constant  form ω1 .

3.2 Hamiltonian Mechanics on R2n Darboux theorem shows that every symplectic manifold of dimension 2n is locally identified with the standard symplectic manifold (R2n , ω). Thus, the results obtained in (R2n , ω) can be locally transferred to any finite-dimensional symplectic manifolds. Therefore, in this section, we only consider Hamiltonian systems in R2n with the standard symplectic structure ω = dp ∧ dq.

3.2 Hamiltonian Mechanics on R2n

169

3.2.1 Phase Space on R2n and Canonical Systems 1. 1-form and 2-form in R2n In R2n , we denote z = (z1 , · · · , zn , zn+1 , · · · , z2n ) = (p1 , · · · , pn , q1 , · · · , qn ) 5 6 p = ∈ R2n , q where the prime indicates the matrix transpose. Definition 2.1. A fundamental differential 1-form and 2-form in R2n are defined by the following formulae: n n   1-form : θ = pi d qi = zi d zn+i ; i=1

i=1

2-form : ω = d θ = d p ∧ d q =

n 

n 

d p i ∧ d qi =

i=1

d zi ∧ d zn+i .

i=1

Thus, it can be seen that ω satisfies the following propoties: 1◦ Closed: d ω = d d θ = 0. 2◦ Non-degenerate: ω(ξ, η) = 0, ∀ η ∈ Tz R2n ⇒ ξ = 0, ∀ z ∈ R2n . 3◦ Anti-symmetric: ω(ξ, η) = −ω(η, ξ), ∀ ξ, η ∈ Tz R2n , z ∈ R2n . Any differential 2-form satisfying the conditions 1, 2 and 3 is called a symplectic structure in R2n . ω is called the standard symplectic structure in R2n . θ is called the standard 1-form. R2n equipped with ω is called the symplectic space, or symplectic manifold, denoted by (R2n , ω), or briefly, R2n . 2n 2n   ∂ ∂ Let ξ = ξi , η= ηi ∈ Tz R2n . Then, i=1

∂ zi

ω(ξ, η) =

i=1 n 

∂ zi

d zi ∧ d zn+i (ξ, η) =

i=1

n  (ξi ηn+i − ηi ξn+i ) i=1

5 = (ξ1 , · · · , ξ2n )

O

In

−In

O

6

⎡

⎤ η1 ⎢ .. ⎥ ⎣ . ⎦ η2n

(2.1) = ξ  Jη, ! O In where J = , and (ξ1 , · · · , ξ2n ) represents the vector ξ ∈ Tz R2n con−In O sisting of components ξi . The Equation (2.1) is the matrix representation of the 2-form ω on Tz R2n . 2.

Hamiltonian vector fields on R2n

170

3. Hamiltonian Mechanics and Symplectic Geometry

To each vector ξ, tangent to the symplectic manifold (R2n , ω) at the point z, we associate a 1-form ωξ1 on Tz R2n by the formula ωξ1 (η) = ω 2 (η, ξ),

∀ η ∈ Tz R2n .

We denote this correspondence by Ω : Tz R2n −→ Tz∗ R2n , i.e.,

ωξ1 (η) = Ωξ(η) = ω 2 (η, ξ) = (−iξ ω)η,

or

∀ η ∈ Tz R2n ,

ωξ1 = Ωξ = −iξ ω. From the equation ω(η, ξ) = η Jξ = (Jξ) η, it follows that ωξ1 = Ωξ = Jξ

or ωξ1 = Ωξ =

2n 

(Jξ)i dzi

i=1

= ξn+1 d z1 + · · · + ξ2n d zn − ξ1 d zn+1 − ξn d z2n . Obviously, Ω is an isomorphism from the tangent space Tz R2n into the cotangent space Tz∗ R2n . This naturally induces a mapping from X (R2n ) into Ω1 (R2n ): 1 ωX = Ω(X)(z) = Ω(X(z)) = −iX ω,

∀ X ∈ X (R2n ).

In particular, if H ∈ C ∞ (R2n ), then dH ∈ Ω1 (R2n ), Ω−1 dH is a vector field on R2n , and we denote it as XH . Definition 2.2. The vector field XH = Ω−1 dH is called a Hamiltonian vector field, and H is called the Hamiltonian function. If we write d H = (Hz1 , · · · , Hz2n ) = Hz , then XH = J −1 Hz , d H = ΩXH = −iX ω. 3. Canonical systems Now, we consider a canonical equation in R2n . Definition 2.3. or

dz = J −1 Hz dt dp = −Hq , dt

dq = Hp . dt

(2.2) (2.3)

3.2 Hamiltonian Mechanics on R2n

171

Since J −1 Hz is the matrix representation of a Hamiltonian vector field Ω−1 d H = XH , the Equation (2.2) can be rewritten as dz = XH (z). dt

(2.4)

The phase flow of the Hamiltonian vector field is denoted as φtH , and called the Hamiltonian phase flow. Theorem 2.4. A Hamiltonian phase flow preserves the symplectic structure : (φtH )∗ ω = ω. Proof. Since

 d t ∗ d (φ ) ω = (φt+s )∗ s=0 ω dt H ds H =

 d (φtH )∗ · (φsH )∗ s=0 ω ds

= (φtH )∗ LXH ω, and

LXH ω = (iXH d + d iXH )ω = iXH d ω + d iXH ω = 0 + (−d (dH)) = 0,

we have i.e.,

d t ∗ (φ ) ω = 0, dt H

 (φtH )∗ ω = (φtH )∗ t=0 ω = ω. 

The theorem is proved. 4. Integral invariants[Arn89] Let g : R2n → R2n be a differentiable mapping.

Definition 2.5. A differential k-form ω k is called an integral invariant of the map g, if the integrals of ω on any k-chain c and on its image under g are the same, i.e., ω k = ωk . gc

c

Example 2.6. If n = 1, ω 2 = d p ∧ d q is the area element, then ω 2 is the integral invariant of the map whose Jacobian determinant is equal to 1. Theorem 2.7. A k-form ω k is an integral invariant of a map g if and only if g∗ ω k = ω k .

172

3. Hamiltonian Mechanics and Symplectic Geometry

The proof is left for the reader to derive as a separate exercise. Theorem 2.8. If the forms ω k and ω l are integral invariants of the map g, then the form ω k ∧ ω l is also an integral invariant of g. This follows immediately from the Theorem 2.7. Theorem 2.9. Let ω 2 be a standard symplectic structure. Then, ω 2 , (ω 2 )2 = ω 2 ∧ ω 2 , (ω 2 )3 = ω 2 ∧ ω 2 ∧ ω 2 , · · · are all the integral invariants of a Hamiltonian phase flow. We define a volume element on R2n using (ω 2 )n . Then, a Hamiltonian phase flow preserves volume, and we obtain Liouville’s theorem from the Theorem 2.4. Since the form (ω 2 )k is proportional to  dpi1 ∧ · · · ∧ dpik ∧ dqi1 ∧ · · · ∧ dqik , ω 2k = i1 <···
the integral of ω 2k is equal to the sum of the oriented volume of projections onto the coordinate planes (pi1 , · · · , pik , qi1 , · · · , qik ). Therefore, a Hamiltonian phase flow preserves the sum of the oriented area as projections onto the coordinate planes (pi1 , · · · , pik , qi1 , · · · , qik ) (1 ≤ k ≤ n).

3.2.2 Canonical Transformation Definition 2.10.

[Arn89]

A diffeomorphism g : R2n → R2n , z = g(z) is called a

canonical transformation on R2n , if for every z ∈ R2n , M =

∂ z ∈ Sp(2n). ∂z

It is easy to see that a linear canonical transformation is a symplectic transformation. Theorem 2.11. A diffeomorphism g is canonical if and only if g preserves ω, i.e.,   P (z) g∗ ω = ω. In other words, if we denote z = g(z) = , i.e., Q(z) 5 6 5 6 P (z) p g −→ = z, z= Q(z) q then g is canonical iff d P ∧ d Q = d p ∧ d q. Thus, a Hamiltonian phase flow φtH is a one-parameter group of canonical transformations on R2n . Proof. For every ξ, η ∈ Tz R2n , (g ∗ ω)(ξ, η) = ω(g∗ ξ, g∗ η) = ξ  M  JM η, where M = g∗ =

∂g is Jacobian of g. ∂z

3.2 Hamiltonian Mechanics on R2n

g canonical

⇐⇒ M  JM = J,

∀ z ∈ R2n ,

⇐⇒ ξ  M  JM η = ξ  Jη,

∀ ξ, η ∈ Tz R2n , z ∈ R2n ,

173

⇐⇒ g ∗ ω(ξ, η) = ω(ξ, η), ∀ ξ, η ∈ Tz R2n , ⇐⇒ g ∗ ω = ω. 

Therefore, the theorem is completed.

Definition 2.12. A diffeomorphism g : R2n → R2n is conformally canonical if its Jacobian M (z) =

∂ g(z) ∈ CSp(2n), ∀ z ∈ R2n . ∂z

Besides the parameters above, a canonical transformation g(z) can be determined by whether or not it transforms every canonical equation into a canonical equation. We first consider a conformally canonical transformation. Let z = g(z, t) be a time-dependent transformation and M (z, t) = ∂ g(z, t) , the Jacobian of g(z, t) with respect to z. ∂z

∂ z = ∂z

z , t) : R2n → R2n Theorem 2.13. The time-dependent transformation z = g −1 ( transforms every canonical equation d z  z ( = J −1 H z ), dt

 z ) into a canonical equation with the Hamiltonian H( dz = J −1 Hz (z), dt

with some Hamiltonian H(z), iff M (z, t) =

∂ z satisfies ∂z

M  JM = μJ, where μ = 0, independent of z and t. Proof.

d z ∂ z d z ∂g dz ∂g = + =M + . dt ∂ z dt ∂t dt ∂t

  ◦ g; then H z = t), t) = H Set H(z) = H(g(z,



∂ z ∂z



 z . Thus, from the equation H

d z  z , = J −1 H dt

we have M

 dz ∂g + = J −1 M −1 H z , dt ∂t

174

3. Hamiltonian Mechanics and Symplectic Geometry

i.e.,



dz ∂g  = M −1 J −1 M −1 H z − dt ∂t



  ∂g  = J −1 JM −1 J −1 M −1 H z − JM −1 ∂t

= J −1 (u + v), 

∂g

where u = BH z , B = JM −1 J −1 M −1 , v = C , C = −JM −1 , and u depends ∂t  as well as on z, and v depends only on z. on the Hamiltonian H  ∈ C ∞ (R2n ), there exists another function H(z), such that For every H dz = J −1 Hz , dt

iff there exists a function H(z) ∈ C ∞ (R2n ), such that ∂H , ∂z

u+v =

i.e., u + v is a gradient transformation. We know that a Jacobian matrix which equals to u + v and

∂ (u + v) is symmetric, i.e., ∂z ∂ui ∂ vi ∂ uk ∂v + = + k, ∂ zk ∂ zk ∂ zi ∂ zi

 ∀ H(z),

i, k = 1, · · · , 2n.

 as a constant, we get In the above equation , taking H ∂ vi ∂ vk = , ∂ zk ∂ zi

i, k = 1, · · · , 2n.

(2.5)

Consequently, ∂ ui ∂ uk = , ∂ zk ∂ zi

Notice that ui = (BH z )i =

2n 

 z ), ∀ H(

i, k = 1, · · · , 2n.

(2.6)

Bij H zj , (2.6) becomes

j=1

   ∂  ∂H ∂ ∂H = . Bij Bkj ∂ zj ∂ zi ∂ zj ∂ zk j=1 j=1 2n

2n

(2.7)

Expanding it, we get 2n  ∂ Bij ∂ H j=1

∂ z k ∂ zj

+

2n  j=1

2n 2n   ∂2 H ∂Bkj ∂H ∂2 H Bij = + Bkj . ∂ zk ∂ zj ∂ zi ∂zj ∂ zi ∂ zj j=1

(2.8)

j=1

 z ) = zl ◦ g −1 (l = 1, · · · , 2n), then H(z)  Take H( = zl (l = 1, · · · , 2n). By this, Equation (2.8) gets split into classes of equations:

3.2 Hamiltonian Mechanics on R2n ∂ Bij ∂ Bkj = , ∂ zk ∂ zi 2n  j=1

Set A =

Bij

i, k, j = 1, · · · , 2n,

2n  ∂2 H ∂2 H = Bkj , ∂ zk ∂ zj ∂ zi ∂ zj

i, k = 1, · · · , 2n.

175

(2.9) (2.10)

j=1

∂2 H . Obviously, A is symmetric, i.e., A = A. Then, (2.10) indicates ∂ zk ∂ zj

BA = (BA) = A B  = AB  ,

∀ A = A.

This implies B = μ(z, t)I,

(2.11)

where μ = 0. Since |B| = 0, or Bij = μ(z, t)δij . Substituting it into (2.9), we get ∂μ ∂μ δij = δkj , ∂ zk ∂ zi

i, j, k = 1, · · · , 2n.

(2.12)

From this, it follows that ∂μ = 0, ∂ zi

i = 1, · · · , 2n, 

i.e., μ = μ(t) is independent of z. Thus, JM −1 J −1 M −1 = B = μ(t)I, i.e., M  JM = μ−1 (t)J, M is conformally symplectic with μ−1 (t). We now prove that μ is independent of t. Since 

JM −1 J −1 M −1 = μ(t)I, Cij = −μ(t)

∂ zl Jlj , ∂ zi

we have vi = Cij

C = −JM −1 = −μ(t)M  J, v=C

∂g , ∂t

∂ gj ∂ z ∂g = −μ(t) l Jlj j , ∂t ∂ zi ∂t

∂ vi ∂ = ∂ zk ∂ zk



− μ(t)

∂ zl ∂g Jlj j ∂ zi ∂t



 2  ∂ zl ∂g ∂ z ∂ 2 gj Jlj j + l Jlj = −μ(t) . ∂ zk ∂ zi

∂t

∂ zi

Then, the system (2.5) ∂ vi ∂ vk = , ∂ zk ∂ zi

is equivalent to

i, k = 1, · · · , 2n

∂ zk ∂ t

176

3. Hamiltonian Mechanics and Symplectic Geometry



−μ(t)

∂ 2 zl ∂g ∂ z ∂ 2 gj Jlj j + l Jlj ∂ zk ∂ zi ∂t ∂ zi ∂ zk ∂ t



= μ(t) i.e.,



∂ 2 zl ∂g ∂ zl ∂ 2 gj , Jlj j + Jlj ∂ zi ∂ zk ∂t ∂ zk ∂ zi ∂ t



∂ zl ∂ zj Jlj ∂ zi ∂ zk

i.e.,







= t

∂ zl ∂ zj Jlj ∂ zk ∂ zi

 , t

(M  JMt )ik = (M  JMt )ki ,

which shows show that M  JMt is a symmetric matrix. Therefore, M  JMt = (M  JMt ) = Mt J  M = −Mt JM. Then, we have (M  JM )t = Mt JM + M  JMt = Mt JM − Mt JM = 0. However, M  JM = μ(t)J, and so it follows that μ(t) = μ =constant. Consequently, M  JM = μJ, μ independent of z and t.  In particular, if g is independent of t, then v = 0 and u = μH z (z). Thus, we obtain the following Theorem 2.14. Theorem 2.14. A transformation g(z) = z : R2n → R2n is conformally canonical with μ independent of z iff z = g−1 ( z ) transforms every canonical system d z  z = J −1 H dt

 z ) into a canonical system with the Hamiltonian H( dz = J −1 Hz (z), dt

  ◦ g. with the Hamiltonian H(z) = μH(g(z)) = μH For a further transform, we obtain Theorem 2.15. Theorem 2.15. A transformation z = g(z) : R2n → R2n is canonical iff g −1 transforms a canonical system d z  z = J −1 H dt

 z ) into a canonical system with the Hamiltonian H( dz = J −1 Hz , dt

  ◦ g. with Hamiltonian H(z) = H(g(z)) =H

3.2 Hamiltonian Mechanics on R2n

177

3.2.3 Poisson Bracket 1. Poisson bracket Definition 2.16. The Poisson bracket {φ(z), ψ(z)} of smooth functions φ(z) and ψ(z) on R2n is also a smooth function on R2n , defined by the formula 5 65 6 O −I ψp  −1   {φ, ψ}(z) = φz J ψz = [φp , φq ] I O ψq = −(φp ψq − φq ψp ). Property 2.17. Let φ, ψ, χ be smooth functions on R2n , then the Poisson bracket has following basic properties: 1◦ anti-symmetric: {φ, ψ} = −{ψ, φ}. 2◦ bilinear: {αφ + βψ, χ} = α{φ, χ} + β{ψ, χ}, α, β ∈ R. 3◦ Jacobi identity: {{φ, ψ}, χ} + {{ψ, χ}, φ} + {{χ, φ}, ψ} = 0. 1◦ and 2◦ are self-evident. The Jacobi identity can be proved by direct computation, but it also follows from the following proposition and the corresponding Jacobi identity of the vector field. Proposition 2.18. Let φ and ψ be smooth functions on R2n . Then, 1◦ {φ, ψ} = −ω(Xφ , Xψ ). 2◦ {φ, ψ} = d φ(Xψ ) = iXφ ω(Xψ ). 3◦ {φ, ψ} = iXφ iXψ ω .  d  4◦ {φ, ψ}(z) =  φ(φtψ z) = LXψ φ(z). dt

t=0

5◦ Ω−1 d {φ, ψ} = −[Ω−1 d φ, Ω−1 d ψ] = −[Xφ , Xψ ] ⇔ X{φ,ψ} = −[Xφ , Xψ ], where Xφ = Ω−1 d φ is the Hamiltonian vector field of a Hamiltonian function φ. Each of the equalities 1◦ , 2◦ , 3◦ and 4◦ can be a definition of the Poisson bracket of functions. Proof. By definition, {φ, ψ}(z) = φz J −1 ψz = φz J −1 JJ −1 ψz = −(J −1 φz ) J(J −1 ψz ) = −ω(Xφ , Xψ ) = −iXφ ω(Xψ ) = d φ(Xψ ), where Xφ = Ω−1 d φ =

2n 

(2.13) (2.14)

(J −1 φz )i d zi . The Equations (2.13) and (2.14) are just

i=1

1◦ and 2◦ of Proposition 2.18 respectively. However, −ω(Xφ , Xψ ) = ω(Xψ , Xφ ) = iXψ ω(Xφ ) = iXφ iXψ ω, and so {φ, ψ} = iXφ iXψ ω, which is 3◦ . For 4◦ , by Equation (2.14),

178

3. Hamiltonian Mechanics and Symplectic Geometry

{φ, ψ} = d φ(Xψ ) = iXψ d φ = LXψ φ, ∞

2n

since for φ ∈ C (R ), LX φ = iX dφ. Finally, for 5◦ , we have [Xφ , Xψ ] = (J −1 ψz )z J −1 φz − (J −1 φz )z J −1 ψz = J −1 ψzz J −1 φz − J −1 φzz J −1 ψz , and {φ, ψ}z = (φ z J −1 ψz )z = φzz J −1 ψz − ψzz J −1 φz , Ω−1 d {φ, ψ} = J −1 {φ, ψ}z = J −1 φzz J −1 ψz − J −1 ψzz J −1 φz = −[Xφ , Xψ ]. 

Therefore, the proposition is completed.

Exercise 2.19. Show that the map g : R2n → R2n , sending(p, q) → (P (p, q), Q(p, q)) is canonical, iff the Poisson bracket of any two functions in variables (p, q) and (P, Q) coincide: {φ ◦ g −1 , ψ ◦ g −1 } = {φ, ψ} ◦ g −1 , i.e., {φ, ψ}p,q = =

∀ φ, ψ ∈ C ∞ (R2n ).

∂ ψ ∂φ ∂ψ∂φ − ∂ p ∂q ∂q ∂p ∂ψ ∂φ ∂ψ ∂φ − ∂P ∂Q ∂Q∂P

= {φ, ψ}P,Q . Theorem 2.20. A function F is a first integral of the phase flow with the Hamiltonian H iff its Poisson bracket with H is identically zero: {F, H} = 0. Proof. By the 4◦ of proposition above,  d  LXH F =  (φtH )∗ F = {F, H} = 0. dt

t=0

Thus,



d d t ∗ d  F (φtH (z)) = (φ ) F (z) =  (φt+s )∗ F (z) dt dt H d s s=0 H



=

∗ ∗  d (φt φs ) F (z) d s H H s=0 ∗

= φtH



∗ d  (φs )∗  F (z) = φtH LXH F (z) = 0, d s H s=0

i.e., F is a first integral of the phase flow with the Hamiltonian H. The necessary condition is evident. 

3.2 Hamiltonian Mechanics on R2n

179

From the Theorem 2.20, we immediately obtain the following. Theorem 2.21. H is a first integral of the phase flow with Hamiltonian function H. Theorem 2.22 (E. Noether theorem). If a Hamiltonian H is a first integral of the phase flow with a Hamiltonian function H, then F is also a first integral of the phase flow with the Hamiltonian function H. This follows immediately from the Theorem 2.21 and the fact that {F, H} = −{H, F }. Theorem 2.23 (Poisson theorem). The Poisson bracket of the two first integrals F1 , F2 of a system with a Hamiltonian function H is again a first integral. Proof. By the Jacobi identity, (

{F1 , F2 }, H

)

=

(

) ( ) F1 , {F2 , H} + F2 , {H, F1 }

= 0 + 0 = 0, which is what we require. 2.



Lie algebras of Hamiltonian vector fields and functions

Definition 2.24. A Lie algebra is a vector space L, together with a bilinear skewsymmetric operation [ , ] : L × L → L, which satisfies the Jacobi identity. The operation [ , ] is usually called the commutator. Therefore, the set of all vector fields on R2n , X (R2n ), together with the Poisson bracket [ , ], forms a Lie algebra; the set of all smooth functions on R2n , C ∞ (R2n ), together with the Poisson bracket { , }, forms a Lie algebra too. Definition 2.25. A linear subspace of a Lie algebra is called a subalgebra if the subspace is closed under the commutator, i.e., the commutator of any two elements of the subspace belongs to it. Evidently, a subalgebra of a Lie algebra is itself a Lie algebra with the original commutator. By the proposition and theorems above, we have: Corollary 2.26. The Hamiltonian vector fields on R2n form a subalgebra of the Lie algebra of all vector fields. Corollary 2.27. The commutator of the Hamiltonian phase flow with a Hamiltonian form a subalgebra of the Lie algebra of all functions.

180

3. Hamiltonian Mechanics and Symplectic Geometry

3.2.4 Generating Functions Let a subset S ⊂ R2n be an r-dimensional submanifold in R2n . For any fixed point s ∈ S, there exists an open set U ⊂ Rr and a diffeomorphism ϕ : U → S such that s ∈ ϕ(U ). For simplicity, we consider the case only locally. Definition 2.28. A subset S ⊂ R2n is an r-dim submanifold if there exists a one-toone smooth map Z : U ⊂ Rr → R2n such that S = {z = Z(x) ∈ R2n | x ∈ U ⊂ Rr }. The tangent space Tz S to S at z = Z(x) is ⎡ ∂ Z1 ∂ Z1 ··· ∂ x ∂ xr 1  ⎢  ⎢ ∂Z . .. .. =⎢ Tz S = . ⎢ ∂x ⎣ ∂ Z2n ∂ Z2n ··· ∂ x1

⎤

⎤ ⎡ ∂P ⎥ ⎥ ⎢ ∂x ⎥ ⎥=⎣ ⎦ ⊂ Tz R2n , ⎥ ∂ Q ⎦ ∂x

∂ xr

 P (x)  . Q(x) Let f : Rn → Rn be a smooth function. The graph of f in the Rn × Rn = R2n , 5 6 7 f (q) 2n n gr (f ) := Gf = ∈R | q∈R q ⎧ ⎨ ∂f 2n ∂q is an n-dim submanifold in R , and its tangent space Tz Gf = ⎩ I   I n 2n in Tz R for any z ∈ Gf . sal to Rp = O where Z(x) =

product space

⎫ ⎬ ⎭

is transver-

Theorem 2.29. Let S be an n-dim submanifold in R2n . S is a graph of some function f iff for any z ∈ S, its tangent space Tz S to S at z is transversal to Rnp in Tz R2n . Proof. We need to only prove the sufficiency. By definition, there are two functions P (x) and Q(x), such that 5 6 5 6 7 p P (x) n S= = | x∈U ⊂R . q Q(x) ⎤ ⎡ ∂P    I  ⎢ ∂x ⎥ ∂ Q n 2n Since Tz S = ⎣ is transversal to R = in T R , we have  = 0.  ⎦ z p O ∂x ∂Q ∂x

By the inverse function theorem, q = Q(x) has an inverse function x = X(q) with   ∂X ∂ Q −1 the Jacobian = . This implies that ∂q

∂X

3.2 Hamiltonian Mechanics on R2n

181

⎡ ∂P ⎤ ⎡ % &−1 ⎤ ⎡ ⎤ ∂P ∂X ⎤ ⎡ ∂ ∂P ∂Q (P ◦ X) ⎥ ⎣ ∂x ∂q ⎦ ⎣ ∂q ⎢ ∂x ⎥ ⎢ ∂x ∂q ⎦. Tz S = ⎣ = ⎦=⎣ ⎦= ∂Q I I I ∂x

Setting f = P ◦ X, we get 5 S=

P (x) Q(x)

7

6 ∈ R2n



P ◦ X(q)

=

7



f (q)

=

q

q

⎤

⎡

7

∂f Tz S = ⎣ ∂ q ⎦ ,

,

I

i.e., S is the graph of the function f (q) = P ◦ X(q).  p   U (p, q)  , z= be a diffeomorLet W : R2n → R2n , z = W (z) = q V (p, q) phism with the Jacobian ⎤ ⎡ ∂U ∂U 5 6 A B ⎢ ∂p ∂q ⎥ ∂W ⎥ . =⎢ ⎣ ∂V ∂V ⎦ = ∂z C D ∂p

∂q

If f (q) is a function with Jacobian M =

∂f , and S = Gf , the graph of f , then ∂q

 W (S) =

5 z = W (z) | z =

f (q) q

6

7 ∈S

is an n-dim submanifold in R2n with Jacobian ⎡ ⎤ 5 65 6 5 6 ∂f A B M AM + B ∂W ⎣ ∂q ⎦ Tz W (S) = = = . ∂z C D I CM + D I 

Therefore, the theorem is completed. 

AM + B CM + D  0. Thus, we obtain the following is transversal to Rnp on W (S), i.e., |CM + D| = theorem. By Theorem 2.29, W (s) is a graph of some function g iff Tz W (s) =

∂W

Theorem 2.30. Let W (z) : R2n → R2n be a diffeomorphism with Jacobian = ∂z  A B  ∂f and f (q) be a function with Jacobian M = . Then, M satisfies the C D ∂q transversality condition: |CM + D| = 0, iff there exists a function g(q) with Jacobian N =

∂g = (AM + B)(CM + D)−1 such that W (Gf ) = Gg , i.e., W transforms ∂q

the graph of f into the graph of g.



182

3. Hamiltonian Mechanics and Symplectic Geometry

Definition 2.31. Let f : Rn → Rn be a transformation and ϕ : Rn → R be a scalar function; if f =grad ϕ = ϕq (q), then ϕ is called a generating function of f and f called a gradient transformation[AM78,Fen86,FWQW89] . Given an n-value function f on Rn , we may construct a differential 1-form ω 1 = f d q = f1 d q1 + · · · + fn d qn . If there exists a 0-form ϕ such that ω 1 = f d q = d ϕ, i.e., ω 1 is exact, then f = ϕq . In Rn , by Poincare lemma 4.15 in Subsection 1.4.4, the only requirement is that ω 1 is closed, i.e.,

∂f is symmetric. Thus, any ∂g

transformation from Rn into itself with a symmetric Jacobian may be called a locally gradient transformation.

Definition 2.32. Let S be an r-dim submanifold in R2n . S is called an isotropic, coisotropic, Lagrangian, or K-Lagrangian submanifold if for any z ∈ S, Tz S is an isotropic, coisotropic, Lagrangian, or K-Lagrangian subspace of Tz R2n respectively. It is obvious that the graph of any gradient transformation is Lagrangian. Corollary 2.33. A Lagrangian submanifold S in R2n is the graph of some gradient transformation f : Rn → Rn , S = Gf , iff its tangent space Tz S is transversal to Rnp in Tz R2n for any z ∈ S. Corollary 2.34. A transformation W (z) : R2n → R2n is a conformally canonical transformation iff W (s) is Lagrangian for any Lagrangian submanifold S.

3.2.5 Hamilton–Jacobi Equations Consider a canonical system d z(t) = J −1 Hz , dt

z(0) = z0 ,

(2.15)

with the Hamiltonian H(z) = H(p, q). Let z(t) = (p(t), q(t)) be its solution and G(t) the 1-parameter group of diffeomorphisms in R2n . 5 5 6 6 p(t) p0 G(t) : z0 = , G(0) = I. −→ z(t) = G(t)z0 = q0 q(t) 

p0  be an n-dim initial manifold, p0 a function of q0 , and M0 q0  (q )  p0 0 ∂ p0 form a Lagrangian manifold, i.e., M0 = ∈ Sm. Since G(t) : and q0 ∂ q0  p    p(t) 0 → is a canonical transformation for a fixed t in some neighbourhood q0 q(t) 2n of R , and Let M0 =

3.2 Hamiltonian Mechanics on R2n

5 G∗ (t) =

A

B

C

D

⎤

⎡

∂p ⎢ ∂ p0

6

=⎢ ⎣

183

∂p ∂ q0 ⎥

⎥

∂q ⎦ ∂ q0

∂q ∂ p0

is a symplectic matrix, 5

Y1 Y2

6

⎡ ∂p ⎡ ∂ p0 ⎤ A 0 +B ⎢ ∂ q0 = G∗ ⎣ ∂ q0 ⎦ = ⎢ ⎣ ∂ p0 +D C I

⎤ ⎥ ⎥ ⎦

∂ q0

   Y   N   ∂p  1 is a symmetric pair. If C 0 + D = 0, then ∼ , where Y2 I ∂q0   −1 ∂p ∂p N = A 0 +B C 0 +D ∈ Sm(n). ∂ q0

∂ q0

By Theorem 2.29, p can be represented as a function of q, i.e., p(t) = p(q, t). Let H(q) = −H(p, q)|p=p(q) = −H(p(q), q). Consider a 1-form in Rn+1 : ω 1 = p d q + H d t. There is a scalar function ϕ(q, t) such that ω 1 = p d q + H d t = d ϕ, iff ω 1 is closed, i.e., the following matrix ⎡ ∂ p1 ∂ p1 ∂ p1 ··· ∂ q ∂ q ∂t 1 n ⎢ ⎢ . . .. ⎢ . .. ⎢ . . ⎢ ⎢ ∂ pn ∂ pn ∂ pn ⎢ ⎢ ∂ q1 · · · ∂ qn ∂t ⎢ ⎢ ∂H ∂ H ∂ H ⎣ ··· ∂ q1

∂ qn

∂t

is symmetric. We know that the matrix

⎤ ⎥ ⎥ ⎡ ⎥ ⎥ ⎥ ⎢ ⎥=⎢ ⎥ ⎣ ⎥ ⎥ ⎥ ⎦

⎤

∂p ∂t ⎥

∂p ∂q ∂H ∂q



∂H ∂t

⎥ ⎦

∂p is symmetric, and so we need to only prove ∂q

∂p ∂H = . ∂t ∂q

By canonical Equations (2.15), −Hq = In addition,

d p(q, t) ∂p∂q ∂p ∂p ∂p = + = Hp + . dt ∂q ∂t ∂t ∂q ∂t

184

3. Hamiltonian Mechanics and Symplectic Geometry



∂ H(p(q, t), q) ∂H ∂ p  =− =− Hp − Hq . ∂q ∂q ∂t

Thus, ∂p ∂p ∂H = −Hq − Hp = . ∂t ∂q ∂q

Consequently, there exists a scalar function ϕ(q, t), s.t. p d q + H d t = d ϕ. Thus, it follows that p=

∂ϕ , ∂q

ϕt = H = −H(p, q) = −H(ϕq , q), or ϕt + H(ϕq , q) = 0, which is called the Hamilton–Jacobi equation.

Bibliography

[AA88] D.V. Anosov and V.I. Arnold: Dynamical Systems I. Springer, Berlin, (1988). [AA89] V. I. Arnold and A. Avez: Ergodic Problems of Classical Mechanics. Addison-Wesley, New York, (1989). [Abd02] S. S. Abdullaev: The Hamilton-Jacobi method and Hamiltonian maps. J. Phys. A: Math. Gen., 35(12):2811–2832, (2002). [AKN78] V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt: Mathematical Aspects of Classical and Celestial Mechanics. Springer, Berlin, Second edition, (1978). [AM78] R. Abraham and J. E. Marsden: Foundations of Mechanics. Reading, MA: AddisonWesley, Second edition, (1978). [AMR88] R. Abraham, J. E. Marsden, and T. Ratiu: Manifolds, Tensor Analysis, and Applications. AMS 75. Springer-Verlag, Berlin, Second edition, (1988). [AN90] A. I. Arnold and S.P. Novikov: Dynamical System IV. Springer Verlag, Berlin, (1990). [Arn88] V. I. Arnold: Geometrical Methods in The Theory of Ordinary Differential Equations. Springer-Verlag, Berlin, (1988). [Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Berlin Heidelberg: Springer-Verlag, GTM 60, Second edition, (1989). [Ber00] R. Berndt: An Introduction to Symplectic Geometry. AMS Providence, Rhode Island, (2000). [Bir23] G. D. Birkhoff: Relativity and Modern Physics. Harvard Univ. Press, Cambridge, Mass., Second edition, (1923). [BK89] G.W. Bluman and S. Kumei: Symmetries and differential equations. AMS 81. Springer-Verlag, New York, (1989). [Car65] C. Carathe’odory: Calculus of Variation and Partial Differential Equations of First Order, Vol.1. Holden-Day, San Franscisco, (1965). [Car70] H. Cartan: Differential Forms. Houghton-Mifflin, Boston, (1970). [CH53] R. Courant and D. Hilbert: Methods of Mathematical Physics. Interscience, New York, Second edition, (1953). [Che53] S. S. Chern: Differential Manifolds. University of Chicago, (1953). Lecture notes. [Fen86] K. Feng: Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math., 4:279–289, (1986). [Fla] H. Flanders: Differential Forms. Academie Press, New York, Second edition, (1963). [FQ87] K. Feng and M.Z. Qin: The symplectic methods for the computation of Hamiltonian equations. In Y. L. Zhu and B. Y. Guo, editors, Numerical Methods for Partial Differential Equations, Lecture Notes in Mathematics 1297, pages 1–37. Berlin, Springer, (1987). [FQ91a] K. Feng and M.Z. Qin: Hamiltonian Algorithms for Hamiltonian Dynamical Systems. Progr. Natur. Sci., 1(2):105–116, (1991). [FQ91b] K. Feng and M.Z. Qin: Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study. Comput. Phys. Comm., 65:173–187, (1991). [FQ03] K. Feng and M. Z. Qin: Symplectic Algorithms for Hamiltonian Systems. Zhejiang Science and Technology Publishing House, Hangzhou, in Chinese, First edition, (2003).

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Chapter 4. Symplectic Difference Schemes for Hamiltonian Systems

The canonicity of the phase flow for time-independent Hamiltonian systems is one of the most important properties. It ensures the preservation of phase areas and the phase volume. Thus, preserving the canonicity of transition of difference schemes from one time step to the next is also important in the numerical solutions of Hamiltonian systems. The goal of this chapter is to find some simple symplectic schemes, i.e., to identify which one, among the existing difference schemes, is symplectic.

4.1 Background It is well known that Hamiltonian systems have many intrinsic properties: the preservation of phase areas of even dimension and the phase volume, the conservation laws of energy and momentum, and other symmetries.

4.1.1 Element and Notation for Hamiltonian Mechanics Let H be a smooth function of 2n variables p1 , · · · , pn , q1 , · · · , qn . Then, the Hamiltonian canonical systems are of the form : p˙ = −Hq ,

q˙ = Hp ,

where p = (p1 , · · · , pn )T , q = (q1 , · · · , qn )T . Let z = symplectic matrix be:

5 J=

O

In

−In

O

(1.1) p q

! , and the standard

6 ,

(1.2)

where In is the n × n identity matrix, and J has property J −1 = J  = −J. Then, system (1.1) can be written in a compact form: z˙ = J −1 Hz ,

(1.3)

! Hq ; H is called the Hamiltonian function of the system. The phase Hp t flow of system (1.1) can be represented as gH . According to the fundamental theorem

where Hz =

K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

188

4. Symplectic Difference Schemes for Hamiltonian Systems

of a Hamiltonian system, the solution of a canonical system is a one-parameter symplectic group Gt , denoted by Sp(2n). Therefore, symplectic geometry serves as the mathematical foundation of Hamiltonian mechanics. For simplicity, we consider only the classical phase space R2n = Rnp × Rnq , where Rnp is called the momentum space, and Rnq the configuration space. Locally, every 2n-dimensional manifold is diffeomorphic to a neighborhood of a point on R2n . The phase space R2n is equipped with a standard symplectic structure defined by ωJ =

n 

d zi ∧ d zn+i =

i=1

n 

d pi ∧ d qi ,

(1.4)

i=1

i.e., for each z of R2n , it is a bilinear antisymmetric form: ωJ (ξ, η) = ξ  Jη,

∀ ξ, η ∈ Tz R2n ,

for each pair of tangent vector ξ, η at point z ∈ Tz R2n , where J is the standard symplectic structure Equation (??). Let w : R2n → R2n be a differential mapping, z ∈ R2n → w(z) ∈ R2n ; the corresponding Jacobian matrix is denoted by ⎡ ∂w ∂ w1 ⎤ 1 ··· ∂ z2n ⎥ ⎢ ∂ z1 ⎥ ⎢ ⎥ ⎢ .. .. ∂w ⎥ ⎢ . . =⎢ ⎥. ∂z ⎥ ⎢ ⎥ ⎢ ∂w ∂ w2n ⎦ 2n ⎣ ··· ∂ z1

∂ z2n

The mapping w induces, for each z ∈ R2n , a linear mapping w∗ (z) from the tangent space at z into the the tangent space at w(z): ξ = (ξ1 , · · · , ξ2n ) −→ w∗ ξ =

∂w ξ. ∂z

Each 2-form w on R2n also induces a 2-form w∗ ω on R2n by the formula   ∂w ∂w ξ, η . w ∗ ω(ξ, η)z ≡ ω ∂z

∂z

w(z)

If ω(ξ, η)z = ξ A(z) η, A (z) = −A(z), then w ∗ ω(ξ, η) = ξ B(z) η, i.e.,  B(z) =

∂w ∂z



 A(w(z))



∂w . ∂z

Refer to Definition 4.7 from Chapter 1. Definition 1.1 (Diff). A diffeomorphism (differentiable one to one onto mapping ) of R2n is called a canonical transformation if w preserves the standard symplectic structure. i.e., w∗ ωJ = ωJ ,

4.1 Background

∂w ∂z i.e., Jacobian

!

! ∂w = J, J ∂z

189

(1.5)

∂w is a symplectic matrix for every z. ∂z

Its geometric meaning is depicted in Fig. 1.1. According to the general theory of ODE, for each Hamiltonian system (1.1), there corresponds a one parameter group of diffeomophisms gt , at least locally in t and z, of R2n such that g0 = id, q

g t1 +t2 = g t1 · g t2 . q

6 ξ

[ ∂∂ wz ]



-

1

6

∂w ξ ∂z

 :

η

Fig. 1.1.

- p O O Geometric meaning of preserving symplectic structure

∂w η ∂z

- p

4.1.2 Geometrical Meaning of Preserving Symplectic Structure ω If z(0) is taken as an initial value, then the solution of (1.1) can be written as z(t) = g t z(0). The basic property of a Hamiltonian system is that g t is a canonical transformation, i.e., (g t )∗ ωJ = ωJ , for all t. This leads to the following class of phase-area conservation law: ωJ = ωJ , every 2-chain σ 2 ⊂ R2n , gt σ2

σ2

-

gt σ4

ωJ ∧ ωJ =

g t σ 2n

σ4

ωJ ∧ ωJ ,

···ω J ∧ · · · ∧ ωJ =

every 4-chain σ 4 ⊂ R2n , ···

σ2n

ωJ ∧ · · · ∧ ωJ ,

every 2n-chain σ 2n ⊂ R2n ,

where the last one is the Liouville’s phase-volume conservation law. Another class of conservation law is related to the energy and all the first integrals. A smooth function ϕ(x) is said to be a first integral if ϕ(g t z) = ϕ(z), for all t, z. The latter is equivalent to the condition {ϕ, H} = 0; H usually represents the energy, which is a first integral of itself.

190

4. Symplectic Difference Schemes for Hamiltonian Systems

The above situations can be generalized. A symplectic structure in R2n is specified by a non-degenerate, closed 2-form ωK =



Kij (z) d zi ∧ d zj ,

(1.6)

i<j

i.e., ωK (ξ, η)z =

1  ξ K(z)η, 2

K  (z) = −K(z),

det K(z) = 0.

A differentiable mapping w : R2n → R2n called K-symplectic, if w∗ ωK = ωK , i.e., & %  ∂ w ∂w = K(z). (1.7) K w(z) ∂z ∂z The Darboux theorem establishes the equivalence between all symplectic structures. Every non-singular closed 2-form ωK can be brought to the standard form 

Kij (z) d zi ∧ d zj =

i<j



d ωi ∧ d wn+j

i<j

locally by suitable coordinate transformation z → w(z).

4.1.3 Some Properties of a Symplectic Matrix From Subsection 2.3.2, a matrix S of order 2n is called a symplectic matrix if it satisfies: S  JS = J, (1.8) where S  is the transpose of S. All symplectic matrices form a symplectic group Sp(2n). Definition 1.2. A matrix B of order 2n is called infinitesimal symplectic, if JB + B  J = O.

(1.9)

All infinitesimal symplectic matrices form a Lie algebra with commutation operation [A, B] = AB − BA, denoted as sp(2n). sp(2n) is the Lie algebra of the Lie group Sp(2n). We have the following well-known proposition[FWQ90] , which can be found in Chapter 2. Here, we omit the proof. Proposition 1.3. det S = 1, if S ∈ Sp(2n). Proposition 1.4. S −1 = −JS  J = J −1 S  J, if S ∈ Sp(2n). Proposition 1.5. SJS  = J, if S ∈ Sp(2n).

4.1 Background

A C

Proposition 1.6. Let S =

B D

191

! , A, B, C, D be an n × n matrix; then S ∈

Sp(2n) iff: AB  − BA = O,

CD − DC  = O,

AD − BC  = I,

A C − C  A = O,

B  D − D B = O,

A D − C  B = I.

Proposition 1.7. Matrices 5

I

B

O

I

6

5 ,

I

O

D

I

6

are symplectic, iff B  = B, D = D. Proposition 1.8. Matrices 5

A

O

O

D

6 iff A = (D )−1 .

∈ Sp(2n),

Proposition 1.9. Matrices S = M −1 N ∈ Sp(2n),

iff M JM  = N JN  .

Proposition 1.10. Matrices 5

Q

I −Q

−(I − Q)

Q

6 ∈ Sp(2n),

iff Q2 = Q,

Q = Q.

Proposition 1.11. If B ∈ sp(2n), then exp (B) ∈ Sp(2n). Proposition 1.12. If B ∈ sp(2n), and |I + B| = 0, then F = (I + B)−1 (I − B) ∈ Sp(2n), the B Cayley transform of B. Proposition 1.13. If B ∈ sp(2n), then (B 2m ) J = J(B 2m ). Proposition 1.14. If B ∈ sp(2n), then (B 2m+1 ) J = −J(B 2m+1 ). Proposition 1.15. If f (x) is an even polynomial, and B ∈ sp(2n), then f (B  )J = Jf (B). Proposition 1.16. If g(x) is an odd polynomial, and B ∈ sp(2n), then g(B) ∈ sp(2n), i.e., g(B  )J + Jg(B) = O.

192

4. Symplectic Difference Schemes for Hamiltonian Systems

4.2 Symplectic Schemes for Linear Hamiltonian Systems A Hamiltonian system (1.1 ) is called linear, if the Hamiltonian is a quadratic form of z: 1 H(z) = z  Cz, C  = C, 2 and J is a standard antisymmetric matrix: ! O In , J  = −J = J −1 , det J = 1. J= −In O Then, the canonical system (1.1), (1.3) become: dz = Bz, dt

B = J −1 C,

C  = C,

(2.1)

where B = J −1 C is infinitesimal symplectic. The solution of (1.1 ) is: z(t) = g t z(0),

g t = exp (tB),

(2.2)

where g t , as the exponential transformation of infinitesimal symplectic tB, is symplectic (Proposition 1.11 ). 1 Consider now a quadratic form F (z) = z  Az. The Poisson bracket of two 2 quadratic forms H, F is also a quadratic form: {H, F } =

1  z (AJC − CJA)z. 2

Theorem 2.1. The condition for the quadratic form F to be an invariant integral of the linear Hamiltonian (2.1) can be expressed in any one of the following equivalent ways:   F (exp (tJ −1 C))z ≡ F (z), (2.3) 

exp (tJ

−1

{H, F } = 0,    C) A exp (tJ −1 C) = A, AJC = CJA.

(2.4) (2.5) (2.6)

4.2.1 Some Symplectic Schemes for Linear Hamiltonian Systems Some types of the symplectic schemes for system (1.1 ) are proposed[Fen85] , the first of which is called the time-centered Euler schemes (or midpoint Euler) z n+1 − z n z n+1 + z n =B . τ 2 The transition z n → z n+1 is given by

(2.7)

4.2 Symplectic Schemes for Linear Hamiltonian Systems

  τ Fτ = φ − B ,

z n+1 = Fτ z n ,

φ(λ) =

2

193

1−λ , 1+λ

(2.8) τ

where Fτ is a Cayley transformation of the infinitesimal symplectic − B, and is 2 symplectic according to Proposition 1.12. The second scheme we consider is the staggered explicit scheme for a separable Hamiltonian. For a separable Hamiltonian H(p, q) = U (p) + V (q), 1 2

H(p, q) = [p , q  ] S

 p  1 1 = p U p + q  V q = U (p) + V (q), q 2 2 5

where S=

U

O

O

V

(2.9)

6 .

U  = U is positive definite and V  = V , the canonical Equation (1.1), becomes: dp = −Vq , dt

dq = Up . dt

(2.10)

The staggered explicit scheme is: 1 n+1 n+ 1 (p − pn ) = −Vq 2 , τ   1 1 1 q n+ 2 +1 − q n+ 2 = Upn+1 . τ

(2.11) (2.12)

  1 τ. pT s are defined at integer time t = nτ , and q T s at half-integer times t = n + 2 The transition 6 5 6 5 pn+1 pn n −→ = w n+1 w = 1 1 q n+ 2 q n+ 2 +1 is given by the following: wn+1 = Fτ wn , where

5 Fτ =

I

O

−τ U

I

6−1 5

I

−τ V

O

I

6 ,

(2.13)

as the product of two symplectic matrices, is symplectic (Proposition 1.7 ), and the scheme has second order of accuracy.

4.2.2 Symplectic Schemes Based on Pad´e Approximation We know that the trajectory z(t) = g t z0 is the solution satisfying the initial condition z(0) = z0 . In a linear system, g t coincides with its own Jacobian. One might asks how to approximate to of exp (tB). This can be simply described in terms of Pad´e rational approximation [FWQ90,Qin89] . Here, we consider the rational approximation to exp (x) defined by

194

4. Symplectic Difference Schemes for Hamiltonian Systems

nlm (x) = glm (x), dlm (x)

(2.14)

(l + m − k) ! m ! xk , (l + m) ! k ! (m − k) !

(2.15)

(l + m − k) ! l ! (−x)k . (l + m) ! k ! (l − k) !

(2.16)

exp (x) ∼ where nlm (x) =

m  k=0

dlm (x) =

l  k=0

For each pair of nonnegative integers l and m, the Taylor series expansion of nlm (x) about the origin point is: dlm (x)

exp (x) −

nlm (x) = o(|x|m+l+1 ), dlm (x)

|x| −→ 0,

(2.17)

and the resulting (l + m)-th order Pad´e approximation of exp (x) is denoted by glm . Theorem 2.2. Let B be an infinitesimal symplectic; then, for sufficiently small |t|, glm (tB) is symplectic iff l = m, i.e., gll (x) is the (l, l) diagonal Pad´e approximant to exp (x). Proof. Sufficiency. Let nll (x) = f (x) + g(x), dll = f (x) − g(x), where f (x) is an even polynomial, g(x) is an odd one. In order to prove gtt (tB) ∈ Sp(2n), we only need to verify Proposition 1.9 .         f (tB)+g(tB) J f (tB)+g(tB) = f (tB)−g(tB) J f (tB)−g(tB) . (2.18) By Propositions 1.15 and 1.16, the L.H.S of Equation (2.18) is:        f (tB)+g(tB) J f (tB  )+g(tB  ) = f (tB)+g(tB) f (tB)−g(tB) J. (2.19) Similarly for the R.H.S of Equation (2.18), we have:        f (tB)−g(tB) J f (tB  )−g(tB  ) = f (tB)−g(tB) f (tB)+g(tB) J. (2.20) Comparing Equations (2.19) and (2.20) completes the proof of the “if” part of the theorem. The “only if” part. Without loss of generality, we may take l > m. We only need to notice that in Equation (2.18), the order of the polynomial on the right hand is higher than that on the left hand.  From Theorem 2.2, we can obtain a sequence of symplectic difference schemes based on the diagonal (k, k) Pad´e table. In Table 2.1, the element of l-th row, m-th column is denoted by (l, m). For the (1,1) approximation (i.e., l = 1, m = 1), we have the Euler centered scheme

4.2 Symplectic Schemes for Linear Hamiltonian Systems

z n+1 = z n +

τB n (z + z n+1 ), 2

195

(2.21)

Fτ(1,1) = φ(1,1) (τ B), φ

(1,1)

λ 2. (λ) = λ 1− 2 1+

(2.22)

This scheme has second order accuracy. For the (2,2) Pad´e approximation, we have: z n+1 = z n + whose transition is

τB n τ 2B2 n (z + z n+1 ) + (z − z n+1 ), 2 12 (2,2)

Fτ

(2.23)

= φ(2,2) (τ B),

λ2 λ + 2 12 . φ(2,2) (λ) = λ λ2 1− + 2 12 1+

(2.24)

This scheme has fourth order accuracy. For the (3,3) approximation, we have: z n+1 = z n +

τB n τ 2B2 n τ 3B3 n (z + z n+1 ) + (z − z n+1 ) + (z + z n+1 ). (2.25) 2 10 120

Fτ(3,3)

=φ

(3,3)

(τ B),

(3,3)

φ

λ2 λ3 λ + + 2 10 120 . (λ) = λ λ2 λ3 1− + − 2 10 120 1+

(2.26)

This scheme has sixth order accuracy. For the (4,4) approximation, we have: τB n 3τ 2 B 2 n (z + z n+1 ) + (z − z n+1 ) 2 28 τ 3B3 n τ 4B4 n + (z + z n+1 ) + (z − z n+1 ), 84 1680 λ 3λ2 λ3 λ4 1+ + + + (4,4) (4,4) 2 28 84 1680 . = φ (τ B), φ (λ) = 3λ2 λ3 λ4 λ − + 1− + 2 28 84 1680

z n+1 = z n +

Fτ(4,4)

This scheme has eighth order accuracy. Theorem 2.3. The difference schemes z n+1 = gll (τ B)z k ,

l = 1, 2, · · ·

for a linear Hamiltonian system (2.1) are symplectic of 2l-th order accuracy .

(2.27) (2.28)

4

3

2

1+x+

x3 x2 + 2 6 1

1+

x2 x3 3x + + 4 4 24 x 1− 4

x2 2x + 3 6 x 1− 3

1+

2x x3 + 3 6

x 3

3x2 x3 3x + + 5 20 60 2x x2 1− + 5 20

x2 x + 2 12 x x2 1− + 2 12 1+

1−

1+

x2 1−x+ 2

1

2

x2 x3 x + + 2 10 120 x x2 x3 1− + − 2 10 120 1+

x2 2x + 5 20 3x x2 x3 1− + − 5 20 60 1+

x 4 3x x2 x3 1− + − 4 4 24 1+

1 x2 x3 1−x+ − 2 6

3

x2 x3 3x + + 7 14 210 4x x2 4x3 x4 1− + − + 7 7 210 840 1+

1+

4x 3x2 x3 x4 + − + 5 10 15 120 .

x 5

x2 x + 3 30 2x x2 x3 x4 1− + − + 3 20 30 360

1−

1+

1 x2 x3 x4 1−x+ − + 2 6 24

4

x3 x4 3x2 x3 x4 x2 x3 x4 x2 4x3 x4 3x2 x3 x4 x2 4x 2x 4x x + + + + + + + + + + + + + 1+ 1+ 1+ 1+ + 2 6 24 5 10 15 120 3 20 30 360 7 7 210 840 2 28 84 1680 x 1 x 3x x x2 x2 x3 3x2 x3 x4 1− 1− + 1− 1− + + − − + 5 3 30 7 14 210 2 28 84 1680

1+x+

1

1+

1

x2 2

x 2 x 1− 2

1+x 1

1+x+

1 1−x

1 1

0

1+

1

0

Pad´e approximation table (l, m)

l m

Table 2.1.

196 4. Symplectic Difference Schemes for Hamiltonian Systems

4.2 Symplectic Schemes for Linear Hamiltonian Systems

197

4.2.3 Generalized Cayley Transformation and Its Application Definition 2.4. A matrix B is called non-exceptional, if det (I + B) = 0.

(2.29)

Let B be non-exceptional; let us introduce a matrix S by I + S = 2(I + B)−1 , whose inversion is

I + B = 2(I + S)−1 .

Therefore S is non-exceptional, and we have the Cayley transformation

and

(2.30) (2.31) [FWQ90]

:

S = (I − B)(I + B)−1 = (I + B)−1 (I − B),

(2.32)

B = (I − S)(I + S)−1 = (I + S)−1 (I − S).

(2.33)

Let A be an arbitrary matrix. The equation S  AS = A

(2.34)

expresses the condition that the substitution of S into both variables z, w leaves invariant the bilinear form z  Aw. Lemma 2.5. [Wey39] If the non-exceptional matrices B and S are connected by (2.32) and (2.33), and A is an arbitrary matrix, then

iff

S  AS = A

(2.35)

B  A + AB = O.

(2.36)

Proof. Taking the transpose of (2.33), we obtain B  (I + S  ) = I − S  . Right multiplying by AS on both sides, and from (2.35), we obtain A(S − I) = B  A(S + I). Right multiplying by (S + I)−1 again on both sides, we obtain −AB = B  A. Conversely, by assuming (2.36) and right multiplying the transposed equation S  (I + B  ) = I − B 

198

4. Symplectic Difference Schemes for Hamiltonian Systems

of (2.33) by A on both sides, we have S  A(I − B) = A(I + B), which yields (2.35) on post-multiplication by (I + B)−1 . Let φ(λ) = (1 − λ)/(1 + λ); then the Cayley transform of B is denoted by φ(B) = (I + B)−1 /(I − B). By taking successively A = J and A = A in Lemma 2.5, this lemma is proved.  Theorem 2.6. The Cayley Transform of a non-exceptional infinitesimal symplectic (symplectic) matrix is a non-exceptional symplectic (infinitesimal symplectic) matrix. If B = J −1 C, C  = C, B ∈ sp(2n), det (I + τ B) = 0, A = A, then     (2.37) φ(τ B) A φ(τ B) = A iff

B  A + AB = O. 1

In other words, a quadratic form F (z) = z  Az is invariant under the symplectic 2 transformation φ(τ B) iff F (z) is an invariant integral of the Hamiltonian system (2.1). Theorem 2.7. [FWQ90] Let ψ(λ) be a function of a complex variable λ, satisfying: 1◦ ψ(λ) is analytic with real coefficients in a neighborhood D of λ = 0. 2◦ ψ(λ)ψ(−λ) = 1 in D. 3◦ ψλ (0) = 0. Let A, B be matrices of order 2n. Then,     ψ(τ B) A ψ(τ B) = A, for all τ with sufficient small |τ |, iff B  A + AB = O. We call these ψ(λ) the generalized Cayley transformation. Proof. Condition 2◦ implies ψ 2 (0) = 0. Thus, ψ(0) = 0, if     ψ(τ B) A ψ(τ B) = A, for all τ with |τ | sufficiently small. Then, differentiating both sides of the above equation with respect to τ , we get     B  ψλ (τ B) Aψ(τ B) + ψ(τ B) ABψλ (τ B) = O. Setting τ = 0, it becomes (B  A + AB)ψ(0) ψλ (0) = O. From condition 3◦ , we get

B  A + AB = O.

4.2 Symplectic Schemes for Linear Hamiltonian Systems

199

Conversely, if B  A + AB = 0, then it is not difficult to verify that the equations ψλ (τ B  )A = Aψλ (τ B),

ψ(τ B  )A = Aψ(−τ B)

hold good for any analytic function ψ. From condition 2◦ , it follows that ψλ (λ) ψ(−λ) − ψ(λ) ψλ (−λ) = 0. Therefore,  d  ψ(τ B) A ψ(τ B) dτ =

 d  ψ(τ B  ) A ψ(τ B) dτ

= B  ψλ (τ B  ) Aψ(τ B) + ψ(τ B  ) ABψλ (τ B) = B  Aψλ (−τ B) ψ(τ B) + ABψλ (−τ B) ψ(τ B) = (B  A + AB) ψλ (−τ B) ψ(τ B) = O, i.e.,

ψ(τ B  ) Aψ(τ B) = ψ(0) Aψ(0) = Aψ 2 (0) = A. 

The proof is completed.

By taking successively A = J and A = A in Theorem 2.8 and using (2.3) – (2.6), we obtain the following theorems. Theorem 2.8. Take |τ | sufficiently small so that τ B has no eigenvalue at the pole of the function φ(λ) in Theorem 2.7. Then, ψ(τ B) ∈ Sp(2n) iff B ∈ sp(2n). Let B = J −1 C, C  = C, A = A; then, ψ(τ J −1 C) Aψ(τ J −1 C) = A,

(2.38)

iff AJC = CJA. 1

In other words, a quadratic form F (z) = z  Az, is invariant under the symplectic 2 transformation ψ(τ B), iff F (z) is an invariant integral of the system (2.1). The transformation φ(τ B) based on Theorem 2.7 includes exponential transformation exp(τ B), Cayley transformation ψ(−τ B/2), and diagonal Pad´e transformation as special cases. Taking φ(λ) in Theorem 2.7 as a rational function, then necessarily ψ(λ) =

P (λ) , P (λ) is a polynomial, and is often normalized by setting P (−λ)

P (0) = 1, P  (0) = 0.

Theorem 2.9. Let P (λ) be a polynomial P (0) = 1, P  (0) = 0, and exp (λ) −

P (λ) = O (|λ|2k+1 ). P (−λ)

(2.39)

200

4. Symplectic Difference Schemes for Hamiltonian Systems

Then, P (−τ B)z m+1 = P (τ B)z m , i.e., z m+1 =

P (τ B) m z P (−τ B)

(2.40)

is a symplectic scheme of order 2k for a linear system (2.1). This difference scheme and the original system (2.1) have the same set of quadratic invariants. In order to find the approximate

P (x) to exp (x) , we may express exp(x) in various P (−x)

rational fraction ways. The following are examples: (1) (3)

(5)

nll (x) d (−x) = ll . nll (−x) dll (x) x 1 + tanh 2 exp (x) = x. 1 − tanh 2 1 x (1 + e ) exp (x) = 12 . (1 + e−x ) 2

exp (x) ∼

(2)

1 2

1 2

exp (x) ∼ glm (x) · gml (x). x

(4) exp (x) =

e2 x . e− 2

Each denominator and numerator in the above expressions can be expanded about the origin Taylor The first term of the approximation gives the function < series.  in x x 1 − , which yields the Euler centered scheme. Keeping m(> 1) ψ(x) = 1 + 2 2 terms in the expansions for both the denominator and numerator, we will get a function ψ(x) that will extend the Euler centered schemes. The schemes obtained in this way are all symplectic; however, the order of accuracy of the first and third schemes is higher than that of the last two kinds. For example, if in the formula (5) the first three terms of the expansions of the denominator and numerator are retained, then the 4-th order symplectic scheme is obtained. However, the same kind of truncation gives 6-th order schemes from (1) and (3).

4.3 Symplectic Difference Schemes for a Nonlinear Hamiltonian System For a nonlinear Hamiltonian system, we give some simple symplectic difference schemes. Centered Euler scheme. For Equation (1.3), we give Euler centered schemes [Fen85] : % m+1 & 1 m+1 + zm z m −1 (z , (3.1) − z ) = J Hz τ 2 where the mapping Fτ : z m → z m+1 is nonlinear. By differentiation,

4.3 Symplectic Difference Schemes for a Nonlinear Hamiltonian System

201

% m+1 &% & z ∂z m+1 + zm 1 ∂z m+1 1 −1 = I + τ J Hzz + I , ∂z m 2 2 ∂z m 2 % m+1 & z + zm is the Hessian matrix of the function H(z) at point z = where Hzz 2 z m+1 + z m ∂z m+1 , and is the Jacobian matrix of Fτ . We have 2 ∂z m

!−1

 z m+1 + z m  τ Fτ = I − J −1 Hzz 2 2

 z m+1 + z m ! τ −1 . I + J Hzz 2 2

When z remains bounded and by taking τ to%be sufficiently & small, we can keep the τ −1 z m+1 + z m infinitesimally symplectic matrix J Hzz non-exceptional. Then, 2

2

Fτ as a Cayley transform is symplectic. Thus, all the conservation laws for phase areas remain true. However, unlike the linear case, the first integral φ(x) including H itself are not conserved exactly. Indeed, it satisfies conservation law only nearby: ϕ(z m+1 ) = ϕ(z m )

mod o(τ 3 ).

1

Property 3.1. Let f (z) = z  Bz be a conservation law for the Hamiltonian system 2 (1.3). Then, it is also a conservation law of the Euler centered scheme for system (1.3). Proof.

= > z k+1 − z k B(z k+1 + z k ), =

τ

= B(z k+1 + z k ), J −1 Hz



z k+1 + z k 2

>

 k+1 > = z + zk = 0, = (z k+1 + z k ), BJ −1 Hz 2

and so Bz k , z k  = Bz k+1 , z k+1 . The proof is proved.



The last equation comes from the conservation law of original system. Remark 3.2. As Euler centered schemes, high-order schemes constructed by the diagonal element in the Pad´e table preserve all quadratic first integrals of the original Hamiltonian system. It is worth to point out that the trapezoidal scheme:  1 m+1 1 − z m ) = J −1 Hz (z m+1 ) + Hz (z m ) (z τ 2 is non-symplectic, because the transition

(3.2)

202

4. Symplectic Difference Schemes for Hamiltonian Systems

 −1   τ τ Fτ = I − J −1 Hzz (z m+1 ) I + J −1 Hzz (z m ) 2 2 is non-symplectic in general. By a nonlinear transformation [Dah59,QZZ95] , ξ k = ρ(z k ) = z k +

h f (z k ), 2

ξ k+1 = ρ(z k+1 ) = z k+1 +

h f (z k+1 ), 2

(3.3)

and the trapezoidal scheme can be transformed into a symplectic Euler centered scheme  h k ξ k + ξ k+1 = z k + z k+1 + f (z ) + f (z k+1 ) . 2 Applying (3.2) to the above formula, we get ξ k + ξ k+1 = z k + z k+1 + z k+1 − z k = 2z k+1 . By taking z k+1 =

ξ k + ξ k+1 in the second equation of (3.3), we obtain 2

ξ k+1 =

h ξ k + ξ k+1 + f 2 2

i.e.,

% ξ k+1 = ξ k + hf

%

ξ k + ξ k+1 2

ξ k + ξ k+1 2

& ,

& ,

which is a Euler centered scheme. Theorem 3.3. The trapezoidal scheme (3.2) preserves the following symplectic structure[WT03] : h2 J + Hzz (z)JHzz (z), (3.4) 4 i.e., & % k+1 & % ∂ z k+1 h2 h2 ∂z k+1 k+1 (z )JH (z ) = J+ Hzz (z k )JHzz (z k ). J + H zz zz k k ∂z 4 ∂z 4 Proof. The proof can be easily obtained by direct calculation using nonlinear transform of (3.3) to (1.7).  Remark 3.4. For the canonical system with general separable Hamiltonian, H(p, q) = U (p) + V (q), and we have dq = −Vq (q), dt

dp = Up (p), dt

1 1 m+1 (p − pm ) = −Vq (qm+ 2 ), τ 1 1 m+1+ 1 2 − q m+ 2 ) = U (pm+1 ). (q p τ

(3.5)

(3.6)

4.4 Explicit Symplectic Scheme for Hamiltonian System

The transition Fτ :

!

pm

→

1

q m+ 2 5 Fτ =

!

pm+1 1

q m+1+ 2

I

O

−τ M

I

203

6−1 5

has the Jacobian: I

−τ L

O

I

6 . 1

From Proposition 1.7, it is symplectic, but with M = Upp (pm+1 ), L = Vqq (q m+ 2 ). 3

Property 3.5. Let f (p, q) = p Bq be a conservation law of (3.5). Then, (pk+1 ) Bq k+ 2 1 = (pk ) Bq k+ 2 is a conservation law of the difference scheme (3.6) also. Proof. Indeed, because f (p, q) is a conservation law of the original Hamiltonian system     Bp, Up (p) = 0, Bq, Vq (q) = 0, we get

%

3

1

q k+ 2 − q k+ 2 , Bpk+1 τ

%

1 pk+1 − pk , Bq k+ 2 τ

&

&

  = Up (pk+1 ), Bpk+1 = 0,

 1 1 = Vq (q k+ 2 ), Bqk+ 2 = 0.

Subtracting the two equations above, we get 3

1

(Bpk+1 , q k+ 2 ) = (Bpk , q k+ 2 ). The proof can be obtained.



4.4 Explicit Symplectic Scheme for Hamiltonian System The oldest and simplest difference scheme is the explicit Euler method. Usually, it is not symplectic for general Hamiltonian systems. It is interesting to ask: under what condition of Hamiltonian systems, can the explicit Euler method become symplectic? In fact, the explicit Euler scheme should be the phase flow of a system (i.e., exact solution) to be symplectic. Most of the important Hamiltonian systems can be decomposed into the sum of these simple systems. Then, the composition of the Euler method acting on these systems yields a symplectic method, which is also explicit. These systems are called symplectically separable. So classical separable Hamiltonian systems are symplectically separable. In this section, we will prove that any polynomial Hamiltonian is symplectically separable.

204

4. Symplectic Difference Schemes for Hamiltonian Systems

4.4.1 Systems with Nilpotent of Degree 2 For a Hamiltonian system (1.3), the oldest and simplest is the explicit Euler scheme: τ z = EH z := z + τ JHz (z),

(4.1)

τ where EH = 1 + τ JHz . Usually, the scheme (4.1) is non-symplectic. However, it is symplectic for a specific kind of Hamiltonian system, called a system with nilpotent of degree 2.

Definition 4.1.

[FW98]

A Hamiltonian system is nilpotent of degree 2 if it satisfies ∀z ∈ R2n .

JHzz (z)JHz (z) = 0,

(4.2)

Evidently, H(p, q) = φ(p) or H(p, q) = ψ(q), which represents inertial flow and stagnant flow, are nilpotent of degree 2 since for H(p, q) = φ(p), 6 5 65 65 65 6 5 φpp O O −I φp O φpp O = O, Hzz (z)JHz (z) = = O O I O O O φp O and for H(p, q) = ψ(q), 5 Hzz (z)JHz (z) =

O

O

O

ψqq

65

O

−I

I

O

65

O ψq

6

5 =

O

O

O

ψqq

65

−ψq O

6 = O.

τ Theorem 4.2. If H is nilpotent of degree 2, then the explicit Euler scheme EH is the exact phase flow of the Hamiltonian system, and hence symplectic.

Proof. Let z = z(0). From the condition (4.2), it follows that z¨(t) =

d ˙ = JHzz (z(t))JHz (z(t)) = 0, JHz (z(t)) = (JHz (z(t)))z z(t) dt

and therefore, z(t) ˙ = z(0) ˙ = JHz (z(0)). Hence, t z(t) = z(0) + tJHz (z(0)) = z + tJHz (z) = EH (z). t This is just the explicit Euler scheme EH . This shows that for such a system, the τ is the exact phase flow, and therefore symplectic.  explicit Euler scheme EH

Theorem 4.3. Let φ(u) : Rn → R be a function on n variables u, φ(u) = φ(u1 , u2 , · · ·, un ). Let Cn×2n = (A, B) be a linear transformation from R2n to Rn . Then, the Hamiltonian H(z) = φ(Cz) satisfies JHzz (z)JHz (z) = O, iff

CJC T = O.

∀ φ, z,

(4.3) (4.4)

4.4 Explicit Symplectic Scheme for Hamiltonian System

Proof. Since

JHzz (z)JHz (z) = JC T φuu (CJC T φu (Cz)),

205

(4.5)

the sufficient condition is trivial. We now prove the necessity. If JHzz (z)JHz (z) = O,

∀φ, z,

then from (4.5) it follows that JC T φuu (Cz)(JC T φu (Cz) = O,

∀φ, z.

1 2

Especially take φ(u) = uT u, then JC T CJC T Cz = O, i.e.,

∀ z,

JC T CJC T C = O.

Left multiplying by C and right multiplying JC T by this equation, we get: (CJC T )3 = O. The anti-symmetry of CJC T implies CJC T = O.



Lemma 4.4. Let C = (A, B); then CJC T = O, if and only if AB T = BAT . Theorem 4.5. For any Hamiltonian system: H(z) = H(p, q) = φ(Cz) = φ(Ap + Bq),

AB T = BAT ,

where φ(u) is any n variable function. The explicit Euler method τ z = EH z = Eφτ z = z + τ JHz (z) = z + τ JC T φu (Cz)

is the exact phase flow, i.e., eτφ := Eφτ = 1 + τ JHz = 1 + τ JC T φu ◦ C, hence, Eφτ is symplectic.

4.4.2 Symplectically Separable Hamiltonian Systems Definition 4.6.

[FW98,FQ91]

H(z) =

m  i=1

Hamiltonian H(z) is separable, if

Hi (z),

Hi (z) = φi (Ci z) = φ(Ai p + Bi q),

(4.6)

206

4. Symplectic Difference Schemes for Hamiltonian Systems

where φi are functions of n variables and Ci = (Ai , Bi ) satisfies the condition Ai BiT = Bi AT i (i = 1, · · · , m). Obviously, we have the following proposition. Proposition 4.7. A linear combination of a symplectic separable Hamiltonian is symplectically separable. For a symplectically separable Hamiltonian (4.6), the explicit composition scheme τ τ τ gH = Em ◦ Em−1 ◦ · · · ◦ E2τ ◦ E1τ τ τ τ τ := EH ◦ EH ◦ · · · ◦ EH ◦ EH m m−1 2 1

(4.7)

is symplectic and of order 1. As a matter of fact: τ τ EH ◦ EH = (1 + τ JH2,z ) ◦ (1 + τ JH1,z ) 2 1

= 1 + τ JH2,z + τ JH1,z + O(τ 2 ) = 1 + τ J(H2,z + H1,z ) + O(τ 2 ), τ τ τ gH = Em ◦ Em−1 ◦ · · · ◦ E2τ ◦ E1τ m−1    = (1 + τ JHm,z ) ◦ 1 + τ J Hi,z + O(τ 2 ) i=1

= 1 + τJ

m 

Hi,z + O(τ 2 )

i=1

= 1 + τ JHz + O(τ 2 ). τ The symplecticity of gH follows from the fact that symplectic maps on R2n form a group under composition. Similarly, τ τ τ = E1τ ◦ E2τ ◦ · · · ◦ Em−1 ◦ Em gH

is symplectic and of order 1. More discussion on how to construct separable schemes with high order is provided in Chapter 8. Example 4.8. The Hamiltonian [FW98,FQ91] & % k−1  2π i 2π i + q sin Hk (p, q) = cos p cos k k i=0

with k-fold rotational symmetry in a phase plane[2,4] are not separable in the conventional sense they are symplectically separable, since every  if k = 1, 2, 4. Otherwise  2π i 2π i term cos p cos + q sin is nilpotent of degree 2 according to Theorem 4.3. k k For example, for k = 3,     2π 2π 4π 4π H3 (p, q) = cos p + cos p cos + q sin + q sin + cos p cos 3 3 3 3 % % √ & √ & 1 3 1 3 p− q + cos − p − q , = cos p + cos 2

2

2

2

4.4 Explicit Symplectic Scheme for Hamiltonian System

207

and the explicit symplectic schemes of order 1 are % √ & 1 1 3 q 1 = q − τ sin p+ q , 2 2 2 % √ √ & 3 1 3 p1 = p + τ sin p+ q , 2

%

2

2 √

1 1 1 p − q = q − τ sin 2 2 % √ 3 1 τ sin p− p = p1 − 2 2 2

1

&

3 1 q , 2 √ & 3 q , 2

q = q 2 − τ sin p. Using the composition theory discussed in Chapter 8 , we can construct an explicit symplectic scheme with higher order accuracy.

4.4.3 Separability of All Polynomials in R2n Theorem 4.9. [FW98] Every monomial xn−k y k of degree n in 2 variables x and y, n ≤ 2, 0 ≤ k ≤ n can be expanded as a linear combination of n + 1 terms: {(x + y)n , (x + 2y)n , · · · , (x + 2n−1 y)n , xn , y n }. Proof. Using binomial expansion, (x + y)n = xn + C1n xn−1 y 1 + C2n xn−2 y 2 + · · · + Cn−2 x2 y n−2 + C1n x1 y n−1 + y n . n Define P1 (x, y) : = (x + y)n − xn − y n = C1n xn−1 y 1 + C2n xn−2 y 2 + · · · + C2n x2 y n−2 + C1n x1 y n−1 , which is separable, and the right side consists of mixed terms; P1 is a linear combination of 3 terms (x + y)n , xn , and y n . P1 (x, 2y) = 2C1n xn−1 y 1 + 22 C2n xn−2 y 2 + · · · + 2n−2 C2n x2 y n−2 + 2n−1 C1n x1 y n−1 , 2P1 (x, 1y) = 2C1n xn−1 y 1 + 2C2n xn−2 y 2 + · · · + 2C2n x2 y n−2 + 2C1n x1 y n−1 . Define P2 (x, y) : = P1 (x, 2y) − 2P1 (x, y) = (22 − 2)C2n xn−2 y 2 + · · · + (2n−2 − 2)C2n x2 y n−2 +(2n−1 − 2)C1n x1 y n−1 , which is separable in 4 terms (x + y)n , (x + 2y)n , xn , and y n .

208

4. Symplectic Difference Schemes for Hamiltonian Systems

P3 (x, y) = P2 (x, 2y) − 22 P2 (x, y) = (23 − 22 )(23 − 2)C3n xn−3 y 3 + · · · + (2n−2 − 22 )(2n−2 − 2)C2n x2 y n−2 +(2n−1 − 22 )(2n−1 − 2)C1n x1 y n−1 , which is separable in 5 terms (x + y)n , (x + 2y)n , (x + 22 y)n , xn , and y n . Define: Pn−2 (x, y) : = Pn−3 (x, 2y) − 2n−3 Pn−3 (x, y) = (2n−2 − 2n−3 ) · · · (2n−2 − 2)C2n x2 y n−2 +(2n−1 − 2n−3 ) · · · (2n−1 − 2)C1n x1 y n−1 , which is separable in n terms (x + y)n , (x + 2y)n , · · · , (x + 2n−3 y)n , xn , and y n . Finally, we get: Pn−1 (x, y) = Pn−2 (x, 2y) − 2n−2 Pn−2 (x, y) = (2n−1 − 2n−2 )(2n−1 − 2n−3 ) · · · (2n−1 − 2)C1n x1 y n−1 = γn−1 x1 y n−1 ,

γn−1 = 0.

The separable n + 1 terms are (x + y)n , (x + 2y)n , · · · , (x + 2n−2 y)n , xn , and y n . Hence, the mixed term xyn−1 is separable into n + 1 terms. Then, from the separability of Pn−2 (x, y) and xy n−1 , we know that x2 y n−2 is separable into n + 1 terms. Similarly, x3 y n−3 , x4 y n−4 , · · · , xn−2 y 2 , and xn−1 y is separable into n + 1 terms.  Remark 4.10. We can also work with the following formulae: 1 1 (x + y)2m+1 + (x − y) − x2m+1 2 2

2m = C22m+1 x2m−1 y 2 + C42m+1 x2m−3 y 4 + · · · + C2m , 2m+1 xy

1 1 (x + y)2m+1 − (x − y)2m+1 − y 2m+1 2 2 2 2m−1 = C12m+1 x2m y + C32m+1 x2m−2 y 3 + · · · + C2m−1 , 2m+1 x y

1 1 (x + y)2m + (x − y)2m − x2m − y2m 2 2

= C22m x2m−2 y 2 + C42m x2m−4 y 4 + · · · + C2m−2 x2 y 2m−2 , 2m 1 1 (x + y)2m − (x − y)2m 2 2

= C12m x2m−1 y + C32m x2m−3 y 3 + · · · + C2m−1 xy 2m−1 , 2m by means of elimination to get more economic expansions, e.g., xy =

1 1 1 1 1 (x + y)2 − x2 − y 2 = (x + y)2 − (x − y)2 . 2 2 2 4 4

4.5 Energy-conservative Schemes by Hamiltonian Difference

209

Theorem 4.11. Every polynomial P (x, y) of degree n in variables p and q can be expanded as n + 1 terms P1 (x, y), P2 (x, y), · · · , Pn−1 (x, y), Pn (x), Pn+1 (y), where each Pi (u) is a polynomial of degree n in one variable or more. Generally, every polynomial P (p, q) can be expanded as P (p, q) =

m 

Pi (ai p + bi q),

m ≤ n + 1,

i=1

where Pi (u) are polynomials of degree n in one variable. Theorem 4.12. Every monomial in 2n variables is of the form 1 −k1 k1 2 −k2 k2 n −kn kn f (p, q) = (pm q1 )(pm q2 ) · · · (pm qn ) n 1 2

and can be expanded as a linear combination of the terms in the form: φ(Ap + Bq) = (a1 p1 + b1 q1 )m1 (a2 p2 + b2 q2 )m2 · · · (an pn + bn qn )mn , mn 1 m2 where φ(u) = φ(u1 , · · · , un ) = um 1 u2 · · · un is the monomial in n with total dem  gree m = mi and with degree mi in variable ui . A and B are diagonal matrices i=1

of order n:

⎛

⎜ ⎜ A=⎜ ⎝

a1 0 .. .

0 a2 .. .

··· ···

0

0

· · · an

0 0 .. .

⎞ ⎟ ⎟ ⎟, ⎠

⎛ ⎜ ⎜ B=⎜ ⎝

b1 0 .. .

0 b2 .. .

··· ···

0

0

· · · bn

0 0 .. .

⎞ ⎟ ⎟ ⎟, ⎠

which automatically satisfies AB T = BAT . The elements ai , bi can be chosen as integers. Theorem 4.13. Every polynomial P (p1 , q1 , · · · , pn , qn ) of degree m in 2n variables can be expanded as [FW98] P (p, q) =

m 

Pi (Ai p + Bi q),

i=1

where each Pi is a polynomial of degree m in n variables, and Ai , Bi are diagonal matrices (satisfying Ai BiT = Bi AT i ). Thus, for polynomial Hamiltonian, the symplectic explicit Euler composite schemes of order 1, 2, or and 4 can be easily constructed.

4.5 Energy-conservative Schemes by Hamiltonian Difference Now, we consider energy-conservative schemes by Hamiltonian differencing, which was first proposed by A.J. Chorin[CHMM78] , and later considered by K. Feng[Fen85] .

210

4. Symplectic Difference Schemes for Hamiltonian Systems

However, these schemes are not symplectic. For simplicity, we illustrate the cases only when n = 2. Let z = z m , z¯ = z m+1 . 1 (¯ p1 − p1 ) τ 1 (¯ p2 − p2 ) τ 1 (¯ q1 − q 1 ) τ 1 (¯ q2 − q 2 ) τ

1 {H(p1 , p2 , q¯1 , q2 ) − H(p1 , p2 , q1 , q2 )}, q¯1 − q1 1 = − {H(¯ p1 , p2 , q¯1 , q¯2 ) − H(¯ p1 , p2 , q¯1 , q2 )}, q¯2 − q2 1 = {H(¯ p1 , p2 , q¯1 , q2 ) − H(p1 , p2 , q¯1 , q2 )}, p¯1 − p1 1 = {H(¯ p1 , p¯2 , q¯1 , q¯2 ) − H(¯ p1 , p2 , q¯1 , q¯2 )}. p¯2 − p2

= −

(5.1)

By addition and cancellation, we have energy conservation for the arbitrary Hamiltonian H(¯ p1 , p¯2 , q¯1 , q¯2 ) = H(p1 , p2 , q1 , q2 ). Since the proposed energy conservative schemes based on Hamiltonian differencing only have the first order accuracy, Qin[Qin87] first proposed another more symmetric form in 1987, which possesses the second order accuracy. Independently, Itoh and Abe[IA88] also proposed the same schemes in 1988. For simplicity, we consider only the case n = 2, and the following difference schemes are given: d p1 1 H(p1 , p2 , q¯1 , q2 ) − H(p1 , p2 , q1 , q2 ) 1 H(p1 , p¯2 , q¯1 , q¯2 ) − H(p1 , p¯2 , q1 , q¯2 ) =− − dt 4 Δq1 4 Δq1 1 H(¯ p1 , p2 , q1 , q2 ) 1 H(¯ p1 , p¯2 , q1 , q¯2 ) p1 , p2 , q¯1 , q2 ) − H(¯ p1 , p¯2 , q¯1 , q¯2 ) − H(¯ − , 4 Δq1 4 Δq1 p1 , p2 , q¯1 , q2 ) − H(p1 , p2 , q¯1 , q2 ) p1 , p¯2 , q¯1 , q¯2 ) − H(p1 , p¯2 , q¯1 , q¯2 ) d q1 1 H(¯ 1 H(¯ = + dt 4 Δp1 4 Δp1

−

p1 , p2 , q1 , q2 ) − H(p1 , p2 , q1 , q2 ) p1 , p¯2 , q1 , q¯2 ) − H(p1 , p¯2 , q1 , q¯2 ) 1 H(¯ 1 H(¯ + , 4 Δ p1 4 Δp1 d p2 1 H(¯ p1 , p2 , q¯1 , q2 ) 1 H(p1 , p2 , q1 , q¯2 ) − H(p1 , p2 , q1 , q2 ) p1 , p2 , q¯1 , q¯2 ) − H(¯ =− − dt 4 Δ q2 4 Δq2

+

1 H(¯ p1 , p¯2 , q¯1 , q2 ) 1 H(p1 , p¯2 , q1 , q¯2 ) − H(p1 , p¯2 , q1 , q2 ) p1 , p¯2 , q¯1 , q¯2 ) − H(¯ − , 4 Δq2 4 Δ q2 p1 , p2 , q¯1 , q¯2 ) p1 , p¯2 , q¯1 , q¯2 ) − H(¯ d q2 1 H(¯ 1 H(p1 , p¯2 , q1 , q¯2 ) − H(p1 , p2 , q1 , q¯2 ) = + dt 4 Δp2 4 Δ p2

−

+

1 H(¯ p1 , p2 , q¯1 , q2 ) 1 H(p1 , p¯2 , q1 , q2 ) − H(p1 , p2 , q1 , q2 ) p1 , p¯2 , q¯1 , q2 ) − H(¯ + . 4 Δ p2 4 Δ p2

From the above first two equations, we have: 1 1 (H(¯ p1 , p2 , q¯1 , q2 ) + H(¯ p1 , p¯2 , q¯1 , q¯2 )) = (H(p1 , p¯2 , q1 , q¯2 ) + H(p1 , p2 , q1 , q2 )). 2 2 From the last two equations, we have: 1 1 (H(¯ p1 , p¯2 , q¯1 , q¯2 ) + H(p1 , p¯2 , q1 , q¯2 ) = (H(¯ p1 , p2 , q¯1 , q2 ) + H(p1 , p2 , q1 , q2 )). 2 2 Combining these equations, we observe that these schemes have exact conservation of the Hamiltonian H. Further research about conservative energy scheme can be referred in recent studies[WWM08] .

Bibliography

[Car65] C. Carathe’odory: Calculus of Variation and Partial Differential Equations of First Order, Vol.1. Holden-Day, San Franscisco, (1965). [CHMM78] A. Chorin, T. J. R. Huges, J. E. Marsden, and M. McCracken: Product formulas and numerical algorithms. Comm. Pure and Appl. Math., 31:205–256, (1978). [Dah59] G. Dahlquist: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. of the Royal Inst. of Techn., Stockholm, Sweden, 130:87, (1959). [Fen85] K. Feng: On difference schemes and symplectic geometry. In K. Feng, editor, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, pages 42–58. Science Press, Beijing, (1985). [FQ87] K. Feng and M.Z. Qin: The symplectic methods for the computation of Hamiltonian equations. In Y. L. Zhu and B. Y. Guo, editors, Numerical Methods for Partial Differential Equations, Lecture Notes in Mathematics 1297, pages 1–37. Springer, Berlin, (1987). [FQ91] K. Feng and M.Z. Qin: Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study. Comput. Phys. Comm., 65:173–187, (1991). [FW98] K. Feng and D.L. Wang: On variation of schemes by Euler. J. Comput. Math., 16:97– 106, (1998). [FWQ90] K. Feng, H.M. Wu, and M.Z. Qin: Symplectic Difference Schemes for Linear Hamiltonian Canonical Systems. J. Comput. Math., 8(4):371–380, (1990). [IA88] T. Itoh and K. Abe: Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J. Comp. Phys., 76:85–102, (1988). [Men84] C.R. Menyuk: Some properties of the discrete Hamiltonian method. Physica D, 11:109–129, (1984). [Qin87] M. Z. Qin: A symplectic scheme for the Hamiltonian equations. J. Comput. Math., 5:203–209, (1987). [Qin89] M. Z. Qin: Cononical difference scheme for the Hamiltonian equation. Mathematical Methods and in the Applied Sciences, 11:543–557, (1989). [QZZ95] M. Z. Qin, W. J. Zhu, and M. Q. Zhang: Construction of symplectic of a three stage difference scheme for ODEs. J. Comput. Math., 13:206–210, (1995). [Wey39] H. Weyl: The Classical Groups. Princeton Univ. Press, Princeton, Second edition, (1939). [WT03] D. L. Wang and H. W. Tam: A symplectic structure preserved by the trapezoidal rule. J. of Phys. Soc. of Japan, 72(9):2193–2197, (2003). [WWM08] Y. S. Wang, B. Wang, and M. Z.Qin: Local structure-preserving algorithms for partial differential equations. Science in China (Series A), 51(11):2115–2136, (2008).

Chapter 5. The Generating Function Method

This chapter discusses the construction of the symplectic difference schemes via generating function and their conservation laws.

5.1 Linear Fractional Transformation Definition 1.1. Let α =

!

Bα Dα

Aα Cα

∈ GL(2m). A linear fractional transforma-

tion is defined by[Sie43,Hua44,FWQW89,Fen86] σα : M (m) −→ N (m), M −→ N = σα (M ) = (Aα M + Bα )(Cα M + Dα )−1 ,

(1.1)

under the transversality condition |Cα M + Dα | = 0.

(1.2)

Proposition 1.2. Let α ∈ GL(2m), and the inverse α−1 = |Cα M + Dα | = 0

Aα Cα

Bα Dα

! , then

iff |M C α − Aα | = 0, (1.3)

|Aα M + Bα | = 0 iff |B α − M Dα | = 0. Thus the linear fractional transformation σα in (1.1) can be represented as σα (M ) = (M C α − Aα )−1 (B α − M Dα ). Proof. From the relation 5 65 α Aα Bα A

Bα

Cα

Dα

Cα

Dα

6

5 =

Aα

Bα

Cα

Dα

65

Aα

Bα

Cα

Dα

(1.4) 6 = I2m ,

i.e., K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

214

5. The Generating Function Method

Aα Aα + Bα C α = Aα Aα + B α Cα = Im , Cα B α + Dα Dα = C α Bα + Dα Dα = Im , (1.5)

Aα B α + Bα Dα = Aα Bα + B α Dα = O, Cα Aα + Dα C α = C α Aα + D α Cα = O, we obtain the following identities: 5 65 α 6 I −M Bα A 5

Cα

Dα

I

−M

Aα

Bα

65

Cα

Dα

Aα

Bα

Cα

Dα

In addition, we have: 5 I

−M

Cα

Dα

I

−M

Aα

Bα

5

6

5 =

6

5 =

6

5 =

B α − M Dα

O

I

Aα − M C α

B α − M Dα

I

O

I

O

Cα

I

I

O

Aα

I

5 =

Aα − M C α

65

65

I

−M

O

Cα M + Dα

I

−M

O

Aα M + Bα

6 , 6

(1.6) .

6 , 6

(1.7) .

Inserting (1.7) into (1.6), taking their determinate, we obtain |Cα M + Dα | |α|−1 = |Aα − M C α |, |Aα M + Bα | |α|−1 = (−1)m |B α − M Dα |.

(1.8)

Note that since α is a non-singular matrix, (1.3) is valid. By (1.8), Equation (1.4) is well defined. The only remaining step is to verify the equation (M C α − Aα )−1 (B α − M Dα ) = (Aα M + Bα )(Cα M + Dα )−1 , i.e., (B α − M Dα )(Cα M + Dα ) = (M C α − Aα )(Aα M + Bα ). Expanding it and using the conditions (1.5), we know that it holds.



Proposition 1.3. We have the following well-known relation (C α N + D α )(Cα M + Dα ) = I, hence |C α N + Dα | = 0

iff |Cα M + Dα | = 0,

(1.9)

5.2 Symplectic, Gradient Mapping and Generating Function

215

where N = σα (M ). Under the transversality condition (1.2), σα has an inverse linear fractional transformation σα−1 = σα−1 , M = σα−1 (N ) = (Aα N + B α )(C α N + D α )−1 = (N Cα − Aα )−1 (Bα − N Dα ).

(1.10)

Proof. (C α N + D α )(Cα M + Dα ) = (C α (Aα M + Bα )(Cα M + Dα )−1 + D α )(Cα M + Dα ) = (C α Aα + Dα Cα )M + C α Bα + D α Dα =I

(by (1.5)),

which is (1.9). The first equation of (1.10) can be obtained from (1.4) and the second equation can be derived from (1.1).  Combining (1.2) and (1.3) together, we obtain the following four mutually equivalent transversality conditions: |Cα M + Dα | = 0, |M C α − Aα | = 0, |C α N + Dα | = 0, |N Cα − Aα | = 0, where

(1.11) (1.12) (1.13) (1.14)

N = σα (M ) = (Aα M + Bα )(Cα M + Dα )−1 , M = σα−1 (N ) = (Aα N + B α )(C α N + D α )−1 .

Moreover, the linear fractional transformation σα from {M ∈ M (m) | |Cα M + Dα | = 0} to {N ∈ M (m) | |C α N + Dα | = 0} is 1-1 surjective.

5.2 Symplectic, Gradient Mapping and Generating Function To study the symplectic structure and Hamiltonian system in R2n phase space, we need R4n symplectic structure as a product of R2n space. Its symplectic structure comes from the product of original symplectic structure in R2n "= Ω

n  i=1

d zi ∧ d zi+n −

n  i=1

d zi ∧ d zi+n ,

(2.1)

216

5. The Generating Function Method

where the corresponding matrix is given by J"4n =

 J 2n O

O −J2n

 . We denote

" 4n = (R4n , J"4n ). R On the other hand, R4n has its standard symplectic structure: Ω=

2n 

d wi ∧ d w i+2n ,

(2.2)

i=1

1 , · · · , w 2n )T represents its coordinate. The corresponding mawhere (w1 , · · · , w2n , w trix is given by 5 6 O I2n J4n = . −I2n O We denote manifold R4n = (R4n , J4n ). Now we first review some notations and facts of the symplectic algebra. Every 4n × 2n matrix of rank 2n can be represented as: 5 6 A1 A= ∈ M (4n, 2n), A1 , A2 ∈ M (2n), A2 defines a 4n-dim subspace {A} spanned by its 2n column vectors. Evidently, {A} = {B} iff ∃P ∈ GL(2n) such that 5 6 5 6 A1 P B1 AP = B, i.e., = . A2 P B2 A 2n-dim subspace {X} = if i.e.,

 X  1 of R4n , X1 , X2 ∈ M (2n), is a J4n -Lagrangian, X2 X T J4n X = O,

X1T X2 − X2T X1 = O

X1T X2 ∈ Sm(2n).  X  1 a symmetric According to Siegel[Sie43] , we call such a 4n × 2n matrix X = X2  X1   N  pair. Moreover, if |X2 | = 0, then X1 X2−1 = N ∈ Sm(2n) and = . X2 I  Y  1 Similarly, a 2n-dim subspace {Y } = is J˜4n -Lagrangian, if Y2 or

Y T J˜4n Y = O, i.e.,

Y1T J2n Y1 = Y2T J2n Y2 ,

5.2 Symplectic, Gradient Mapping and Generating Function

217

 Y  1 is called a symplectic pair. |Y2 | = 0 implies Y1 Y2−1 = the 4n×2n matrix Y = Y2   M   Y1 = . M ∈ Sp(2n), and Y2 I 

Aα Bα  ∈ GL(4n) carries every J˜4n Cα Dα Lagrangian subspace into a J4n -Lagrangian subspace if and only if α ∈ CSp(J˜4n , J4n ), i.e., αT J4n α = μJ˜4n , for some μ = μ(α) = 0. (2.3) Theorem 2.1. A transformation α =

Proof. The “if” part is obvious, we need only to prove the “only if” 5 part. J2n Taking α0 ∈ Sp(J˜4n , J4n ) (which always exists), e.g., α0 = 1 I2n 2 we have CSp(J˜4n , J4n ) = CSp(4n) · α0 .

J2n

1 I2n 2

6 ,

Therefore, it suffices to show that if α carries every J4n - Lagrangian subspace into J4n -Lagrangian subspace, then α ∈ CSp(4n), i.e., αT J4n α = μJ4n 1◦

Take the symmetric pair X = 5 αX =

for some μ = 0.

  I 2n . By assumption, O2n

Aα

Bα

Cα

Dα

65

I

6

5 =

O

Aα

6

Cα

T T T is also a symmetric pair, i.e., AT α Cα − Cα Aα = O. Similarly, Bα Dα − Dα Bα = O.   S 2◦ Take the symmetric pair X = , S ∈ Sm(2n). Then every I

5 αX =

Aα

Bα

Cα

Dα

65

S I

6

5 =

Aα S + Bα

6

Cα S + Dα

is also a symmetric pair, i.e., O = (αX)T J4n (αX) =



T T T T S T AT α + Bα , S C α + D α



O −I

I O

!

A α S + Bα Cα S + Dα

!

T T T = S(AT α Cα − Cα Aα )S + S(Aα Dα − Cα Bα )

−(DαT Aα − BαT Cα )S + BαT Dα − DαT Bα T T T T = S(AT α Dα − Cα Bα ) − (Aα Dα − Cα Bα ) S,

∀ S ∈ Sm(2n).

218

5. The Generating Function Method

T Set P = AT α Dα − Cα Bα , then the above equation becomes

SP = P T S,

∀ S ∈ Sm(2n).

It follows that P = μI, i.e., T AT α Dα − Cα Bα = μI.

So

5 T

α J4n α = 5 =

Aα

Bα

Cα

Dα

6T 5

O

I

−I

O

T AT α Cα − Cα Aα

65

Aα

Bα

Cα

Dα 6

6

T AT α Dα − Cα Bα

BαT Cα − DαT Aα BαT Dα − DαT Bα 5 6 O I = μ = μJ4n , −I O

α ∈ GL(4n) implies μ = 0.



The inverse matrix of α is denoted by α−1 =

 Aα Cα

Bα Dα

 . By (2.3), we have

T T T AT α Cα − Cα Aα = μJ, Aα Dα − Cα Bα = O,

BαT Cα − DαT Aα = O,

BαT Dα − DαT Bα = −μJ,

Aα = μ−1 JCαT ,

B α = −μ−1 JAT α,

C α = −μ−1 JDαT ,

Dα = μ−1 JBαT .

(2.4)

(2.5)

 A Bα  α ∈ CSp(J˜4n , J4n ). The linear fractional transCα Dα formation σα :{M ∈ Sp(2n) | |Cα M +Dα | = 0}→ {N ∈ Sm(2n) | |C α N +Dα | = 0} is one to one and onto. Theorem 2.2. Let α =

Proof. From above we know that |Cα M + Dα | = 0, iff |C α N + D α | = 0. Now we need only to prove M ∈ Sp(2n) iff N = σα (M ) ∈ Sm(2n). It is derived from direct calculation, since N ∈ Sm(2n) iff 

(Aα M + Bα )(Cα M + Dα )−1

T

= (Aα M + Bα )(Cα M + Dα )−1 ,

i.e., T T T T T T (M T AT α + Bα )(Cα M + Dα ) = (M Cα + Dα )(Aα M + Bα ).

Expanding and combining them together, we obtain

5.2 Symplectic, Gradient Mapping and Generating Function

219

T T T O = M T (AT α Cα − Cα Aα )M + Bα Dα − Dα Bα T T T +M T (AT α Dα − Cα Bα ) + (Bα Cα − Dα Aα )M

= M T JM − J, then (2.4) holds, iff M ∈ Sp(2n).



Definition 2.3. A mapping w → w  = f (w) : R2n → R2n is called a gradient, if its Jacobian N (w) = fw (w) ∈ Sp(2n) everywhere. Definition 2.4. A 2n-dim submanifold U of R4n is a J"4n -Lagrangian submanifold or J4n -Lagrangian submanifold if its tangent plane Tz U at z for any z ∈ U is a 4n tangent space of J"4n -Lagrangian subspace or J4n -Lagrangian subspace. For a symplectic mapping z → z = g(z), the graph[Fen86,FWQW89,Ge91] 7   z  4n 2n Γg = gr(g) := ∈ R | z = g(z), z ∈ R z is always a J"4n -Lagrange submanifold. For every gradient mapping w → w  = f (w), its graph  7  w   4n 2n Γf = gr(f ) := ∈R |w  = f (w), w ∈ R w is always a J4n -Lagrange submanifold.   A Bα α ∈ CSp(J"4n , J4n ), it defines a linear fractional transforLet α = C α Dα mation  w  w  z   z      , =α , = α−1 w w z z i.e.,

w  = Aα z + Bα z,

z = Aα w  + B α w,

w = Cα z + Dα z,

z = Cαw  + Dα w.

(2.6)

Theorem 2.5. Let α ∈ CSp(J˜4n , J4n ). Let z → z = g(z) : R2n → R2n be a canonical mapping with Jacobian M (z) = gz (z) ∈ Sp(2n) satisfying (1.2) in (some  = f (w) in neighborhood of) R2n . Then there exists a gradient mapping w → w (some neighborhood of) R2n with Jacobian N (w) = fw (w) ∈ Sm(2n) and a scalar function —generating function— φ(w) (depending on α and g) such that 1◦ 2◦ 3◦ 4◦

f (w) = ∇φ(w); Aα g(z) + Bα z = f (Cα g(z) + Dα z) = ∇φ(Cα g(z) + Dα z); N = σα (M ) = (Aα M + Bα )(Cα M + Dα )−1 , M = σα−1 (N ) = (Aα N + B α )(C α N + Dα )−1 ; Γf = α(Γg ), Γg = α−1 (Γf ).

(2.7) (2.8) (2.9) (2.10)

220

5. The Generating Function Method

Proof. Under the linear transformation α, the image of Γg is ' / w   4n ∈R |w  = Aα g(z) + Bα z, w = Cα g(z) + Dα z . α(Γg ) = w Since Γg is a J˜4n -Lagrangian submanifold and α ∈ CSp(J˜4n , J4n ), the tangent plane of α(Γg ) defined by 7  Aα M (z) + Bα Cα M (z) + Dα is a J4n -Lagrangian subspace. So α(Γg ) is a J4n -Lagrangian submanifold. By assumption, |Cα M + Dα | = 0, and by the implicit function theorem, w = Cα g(z) + Dα z is invertible and its inverse is denoted by z = z(w). Set:  (2.11) w  = f (w) = (Aα g(z) + Bα z) z=z(w) = Aα g(z(w)) + Bα z(w), obviously, such a f (w) satisfies the identity Aα g(z) + Bα z ≡ f (Cα g(z) + Dα z).

(2.12)

The Jacobian of f (w) is 

N (w) = fw (w) =

∂w  ∂w  ∂w  ∂w = = ∂w ∂z ∂z ∂z

−1

= (Aα M (z) + Bα )(Cα M (z) + Dα )−1 = σα (M (z)).

(2.13)

By Theorem 2.2 it is symmetric. So f (w) is a gradient map. By the Poincar´e lemma, there exists a scalar function φ(w), such that f (w) = ∇φ(w). In addition, we have  w    Γf = ∈ R4n | w  = f (w) = Aα g(z(w)) + Bα z(w) = α(Γg ). w Therefore, the theorem is completed.



This theorem tells us that for a fixed α, the corresponding symplectic mapping determines only one gradient mapping with accuracy up to a constant factor. Theorem 2.6. Let α ∈ CSp(J˜4n , J4n ). Let φ(w) be a scalar function and w → w = f (w) = ∇φ(w) be its induced gradient mapping and N (w) = fw (w) = φww (w), the Hessian matrix of φ(w), satisfy (1.13) in (some neighborhood of) R2n . Then, there exists a canonical map z → z = g(z) with Jacobian M (z) = gz (z) satisfying (1.11) such that 1◦ Aα f (w) + B α w = g(C α f (w) + Dα w), identically in w . 2◦ M = σα−1 (N ) = (Aα N + B α )(C α N + Dα )−1 , N = σα (M ) = (Aα M + Bα )(Cα M + Dα )−1 . ◦ 3 Γg = α−1 (Γf ), Γf = α(Γg ). By the way, we can get the original symplectic mapping, if f (w) or φ(w) is obtained from Theorem 2.5.

5.3 Generating Functions for the Phase Flow

221

5.3 Generating Functions for the Phase Flow Consider the Hamiltonian system dz −1 ∇H(z), = J2n dt

z ∈ R2n ,

(3.1)

where H(z) is a Hamiltonian function. Its phase flow is denoted as g t (z) = g(z, t) = gH (z, t), being a one-parameter group of canonical maps, i.e., g 0 = identity,

g t1 +t2 = g t1 ◦ g t2 ,

and if z0 is taken as an initial condition, then z(t) = gt (z0 ) is the solution of (3.1) with the initial value z0 . Theorem 3.1. Let α ∈ CSp(J˜4n , J4n ). Let z → z = g(z, t) be the phase flow of the Hamiltonian system (3.1) and M0 ∈ Sp(2n). Set G(z, t) = g(M0 z, t) with Jacobian M (z, t) = Gz (z, t). It is a time-dependent canonical map. If M0 satisfies the transversality condition (1.2), i.e., |Cα M0 + Dα | = 0,

(3.2)

then there exists, for sufficiently small |t| and in (some neighborhood of) R2n , a timedependent gradient map w → w  = f (w, t) with Jacobian N (w, t) = fw (w, t) ∈ Sm(2n) satisfying the transversality condition (1.13) and a time-dependent generating function φα,H (w, t) = φ(w, t), such that 1◦ f (w, t) = ∇φ(w, t). 2◦ ◦

(3.3)

∂ φ(w, t) = −μH(Aα ∇φ(w, t) + B α w). ∂t

(3.4)

3 Aα G(z, t) + Bα z ≡ f (Cα G(z, t) + Dα z, t) ≡ ∇φ(Cα G(z, t) + Dα z, t).(3.5) 4◦ N = σα (M ) = (Aα M + Bα )(Cα M + Dα )−1 , (3.6) M = σα−1 (N ) = (Aα N + B α )(C α N + D α )−1 . (3.4) is the most general Hamilton–Jacobi equation for the Hamiltonian system (3.1) with the linear transformation α. Proof. Since g(z, t) is differentiable with respect to z and t, so is G(z, t). Condition (3.2) implies that for sufficiently small |t| and in some neighborhood of R2n , |Cα M (z, t) + Dα | = 0.

(3.7)

Thus, by Theorem 2.5, there exists a time-dependent gradient map w  = f (w, t), such that it satisfies (3.2) and (3.6). Set:  H(w, t) = −μH( z ) z=Aα w(w,t)+B αw  = −μH(Aα w(w,  t) + B α w).

(3.8)

222

5. The Generating Function Method

Consider the differential 1-form 1

ω =

2n 

w i d wi + H(w, t) d t,

then

i=1

d ω1 =

2n  ∂w i i,j=1

=

d wj ∧ d wi +

∂ wj

i=1

  ∂ wi ∂ wj

i<j

2n  ∂w i

−

∂t



d t ∧ d wi +

2n  ∂H i=1

2n

∂ wi



d wi ∧ d t



 ∂ wi ∂w j ∂H d w j ∧ d wi + d t ∧ d wi . (3.9) − ∂ wi ∂t ∂ wi i=1

∂w 

is symmetric, the first term of (3.9) is zero. Since N (w, t) = fw (w, t) = ∂w Notice that z = G(z, t) = g(M0 z, t), d z d g(M0 z, t) = = J −1 ∇ H(G(z, t)). dt dt

(3.10)

So G(z, t) is the solution of the following initial-value problem: ⎧ ⎨ d z = J −1 ∇ H( z ), dt ⎩ z(0) = M z. 0

Therefore, from the equations w  = Aα G(z, t) + Bα z,

w = Cα G(z, t) + Dα z,

dw  = Aα J −1 ∇H( z ), dt

dw = Cα J −1 ∇H( z ). dt

it follows that

Since

dw  ∂w  dw ∂w  = + , combining these equations, we obtain dt ∂ w dt ∂t





∂w  ∂w  = Aα − Cα J −1 ∇H( z ). ∂t ∂w

On the other hand,   T  T ∂w  H w (w, t) = μ − Hz · Aα + Bα     ∂w ∂w  T α T α T = −μ (B ) + (A ) ∇H( z) ∂w     ∂ w  Cα J −1 ∇ H( z ) by (2.5) and N ∈ Sm(2n) = Aα J −1 −

∇w H(w, t) =

=

∂w  . ∂t

∂w

So d ω 1 = 0. By Poincar´e lemma, there exists, in some neighborhood of R2n+1 , a scalar function φ(w, t), such that

5.3 Generating Functions for the Phase Flow

223

ω1 = w  d w + H d t = d φ(w, t), i.e.,

f (w, t) = ∇w φ(w, t),   ∂ φ(w, t) = −μH Aα ∇w φα,H (w, t) + B α w . ∂t



Therefore, the theorem is completed. Examples of generating functions are: ⎤ ⎡ O O −In O ⎢ In O O O ⎥ ⎥ , μ = 1, M0 = J, (I) α=⎢ ⎣ O O O In ⎦ O In O O  q  w= , φ = φ(q, q, t); q  −p   φ  q w = = , φt = −H(φq, q). φq p

|Cα M0 + Dα | = 0;

This is the generating function and H.J. equation of the first kind. ⎡ ⎤ O O −In O ⎢ O −In O O ⎥ ⎥ , μ = 1, M0 = I, |Cα M0 + Dα | = 0; (II) α=⎢ ⎣ O O O In ⎦ In O O O  q  w= , φ = φ(q, p, t); p  p   φ  q w =− = , φt = −H( p, −φp ). q φp This is the generating function and H.J. equation of the second kind. ⎤ ⎡ −J2n J2n ⎦, μ = 1, M0 = I, |Cα M0 + Dα | = 0; (III) α=⎣ 1 1 I2n I2n 2 2 1 w = (z + z), 2

w  = J(z − z) = ∇φ,

φ = φ(w, t);   1 φt = −H w − J −1 ∇φ . 2

This is the Poincar´e’s generating function[Wei72] and H.J. equation. If the Hamiltonian function H(z) depends analytically on z then we can derive the explicit expression of the corresponding generating function via recursions . Theorem 3.2. Let H(z) depend analytically on z. Then φα,H (w, t) is expressible as a convergent power series in t for sufficiently small |t|, with recursively determined coefficients:

224

5. The Generating Function Method

φ(w, t) =

∞ 

φ(k) (w)tk ,

(3.11)

k=0

φ(0) (w) =

1 T w N0 w, 2

N0 = (Aα M0 + Bα )(Cα M0 + Dα )−1 ,

φ(1) (w) = −μ(α)H(E0 w), E0 = Aα N0 + B α = M0 (Cα M0 + Dα )−1 .

(3.12) (3.13)

If k ≥ 1, (k+1)

φ

k μ(α)  1 (w) = − k+1 m! m=1

2n 



i1 ,···,im =1

j1 +···+jm =k jl ≥1

Hzi1 ,···,zim (E0 w)

· (Aα ∇φ(j1 ) )i1 , · · · , (Aα ∇φ(jm ) )im ,

(3.14)

where Hzi1 ,···,zim (E0 w) is the m-th partial derivative of H(z) w.r.t. zi1 , · · · , zim ,   evaluated at z = E0 w and Aα ∇φ(jl ) (w) i is the il -th component of the column l vector Aα ∇φ(jl ) (w). Proof. Under our assumption, the generating function φα,H (w, t) depends analytically on w and t in some neighborhood of R2n and for small |t|. Expand it as a power series as follows: ∞  φ(w, t) = φ(k) (w)tk . k=0

Differentiating it with respect to w and t, we get ∇φ(w, t) = ∂ φ(w, t) = ∂t

∞ 

∇φ(k) (w)tk ,

(3.15)

(k + 1)tk φ(k+1) (w).

(3.16)

k=0 ∞  k=0

By (3.15), ∇φ(0) (w) = ∇φ(w, 0) = f (w, 0) = N0 w. 1 2

So we can take φ(0) (w) = wT N0 w. We denote E0 = Aα N0 + B α . Then Aα ∇φ(w, t) + B α w = E0 w +

∞ 

Aα ∇φ(k) (w)tk .

k=1

  Substitutes it in H Aα ∇φ(w, t) + B α w and expanding at z = E0 w, we get

5.3 Generating Functions for the Phase Flow

225

H(Aα ∇φ(w, t) + B α w) + * ∞  α (k) k A ∇φ (w)t = H E0 w + k=1

= H(E0 w) +

∞  1 m! m=1

2n 

∞ 

i1 ,···,im =1 j1 ,···,jm =1

tj1 +···+jm Hzi1 ,···,zim

· (E0 w)(Aα ∇φ(j1 ) (w))i1 · · · (Aα ∇φ(jm ) (w))im = H(E0 w) +

∞ 2n    1 tk m ! m=1 i ,···,i =1 1

m

k≥m

 j1 +···+jm =kjl ≥1

Hzi1 ,···,zim

· (E0 w)(Aα ∇φ(j1 ) (w))i1 · · · (Aα ∇φ(jm ) (w))im = H(E0 w) +

∞  k=1

tk

2n 

k  1 m=1

m!



i1 ,···,im =1 j1 +···+jm =kjl ≥1

Hzi1 ,···,zim

· (E0 w)(Aα ∇φ(j1 ) )i1 · · · (Aα ∇φ(jm ) )im . Substituting this formula into the R.H.S. of (3.4), and (3.5) into the L.H.S. of (3.4), then comparing the coefficients of tk on both sides, we obtain the recursions Equations (3.13) and (3.14).  In the next section when we use generating functions φα,H to construct difference schemes we always assume M0 = I. For the sake of convenience, we restate Theorem 3.1 and Theorem 3.2 as follows. Theorem 3.3. Let α ∈ CSp(J˜4n , J4n ). Let z → z = g(z, t) be the phase flow of the Hamiltonian system (3.1) with Jacobian M (z, t) = gz (z, t). If |Cα + Dα | = 0, then there exists, for sufficiently small |t| and in (some neighborhood of) R2n , a timedependent gradient map w → w  = f (w, t) with Jacobian N (w, t) = fw (w, t) ∈ Sm(2n) satisfying the transversality condition (1.13) and a time-dependent generating function φα,H (w, t) = φ(w, t) such that f (w, t) = ∇ φ(w, t);

(3.17)

∂φ = −μH(Aα ∇ φ(w, t) + B α w); ∂t

(3.18)

Aα g(z, t) + Bα z ≡ f (Cα g(z, t) + Dα z, t) ≡ ∇φ(Cα g(z, t) + Dα z, t); N = σα (M ) = (Aα M + Bα )(Cα M + Dα )−1 ; M = σα−1 (N ) = (Aα N + B α )(C α N + Dα )−1 .

(3.19) (3.20) (3.21)

226

5. The Generating Function Method

Theorem 3.4. Let H(z) depend analytically on z. Then φα,H (w, t) is expressible as a convergent power series in t for sufficiently small |t|, with the recursively determined coefficients: φ(w, t) =

∞ 

φ(k) (w)tk ;

(3.22)

k=0

1 T w N0 w, 2 φ(1) (w) = −μ(α)H(E0 w),

φ(0) (w) =

N0 = (Aα + Bα )(Cα + Dα )−1 ;

(3.23)

E0 = (Cα + Dα )−1 .

(3.24)

If k ≥ 1, (k+1)

φ

k μ(α)  1 (w) = − k+1 m! m=1

2n 



i1 ,···,im =1

j1 +···+jm =k jl ≥1

Hzi1 ,···,zim (E0 w)

· (Aα ∇φ(j1 ) )i1 · · · (Aα ∇φ(jm ) )im .

(3.25)

5.4 Construction of Canonical Difference Schemes In this section, we consider the construction of canonical difference schemes for the Hamiltonian system (3.1). By Theorem 3.1, for a given time-dependent scalar function ψ(w, t) : R2n × R → R, we can get a time-dependent canonical map g˜(z, t). If ψ(w, t) approximates some generating function φα,H (w, t) of the Hamiltonian system (3.1), then g˜(z, t) approximates the phase flow g(z, t). Then, fixing t as a time step, we can get a difference scheme —the canonical difference scheme—whose transition from one time-step to the next is canonical. By Theorem 3.4, the generating functions φ(w, t) can be expressed as a power series. So a natural way to approximate φ(w, t) is to take the truncation of the series. More precisely, we have: Theorem 4.1. Using Theorems 3.3 and 3.4, for sufficiently small τ > 0 as the timestep, we define ψ (m) (w, τ ) =

m 

φ(i) (w)τ i ,

m = 1, 2, · · · .

(4.1)

i=0

Then the gradient mapping w→w  = f˜(w, τ ) = ∇ψ (m) (w, τ )

(4.2)

defines an implicit canonical difference scheme z = z k → z k+1 = z, Aα z k+1 + Bα z k = ∇ψ (m) (Cα z k+1 + Dα z k , τ ) of m-th order of accuracy.

(4.3)

5.4 Construction of Canonical Difference Schemes

227

(m)

Proof. Since ψ (m) (w, 0) = φ(w, 0), so ψww (w, 0) = φww (w, 0) = fw (w, 0) = N (w, 0) satisfies the transversality condition (1.13), i.e., |C α N (w, 0) + Dα | = 0. Thus for sufficiently small τ and in some neighborhood of R2n , N (m) (w, τ ) = (m) ψww (w, τ ) satisfies the transversality condition (1.13), i.e., |C α N (m) (w, τ ) + Dα | = 0. By Theorem 4.1, the gradient mapping w → w  = f˜(w, τ ) = ∇ψ (m) (w, τ ) defines implicitly a time-dependent canonical mapping z → z = g˜(z, τ ) by the equation Aα z + Bα z = ∇ψ (m) (Cα z + Dα z, τ ). Thus, the equation Aα z k+1 + Bα z k = ∇ψ (m) (Cα z k+1 + Dα z k , τ ) is an implicit canonical difference scheme. Since ψ (m) (w, τ ) is the m-th order approximation to φ(w, τ ), so is f˜(w, τ ) = ∇ψ (m) (w, τ ) to f (w, τ ), it follows that the canonical difference scheme given by (4.3) is of m-th order of accuracy.  Therefore, for every α ∈ CSp(J"4n , J4n ), we can construct a series of symplectic schemes for arbitrary order accuracy. Examples of the canonical difference scheme: Type (I). Constructing symplectetic scheme by the first kind of the generating function. From Theorem 3.2, as μ = 1, φ(0) (w) =

1 T w N0 w, 2

φ(1) (w) = −H(E0 w), φ(2) (w) =

N0 = (Aα + Bα )(Cα + Dα )−1 , E0 = (Cα + Dα )−1 ,

1 (∇H)T Aα E0T (∇H)(E0 w), 2 1 3

1 6

φ(3) (w) = − (∇H)T Aα ∇w φ(2) − (Aα ∇φ(1) )T Hzz (Aα ∇φ(1) ) 1 6

= − (∇H)T Aα (E0T Hzz Aα E0T ∇H + E0T Hzz E0 AαT ∇H) 1 6

− (∇H)T E0 AαT Hzz Aα E0T ∇H 1 6

= − {(∇H)T Aα E0T Hzz (Aα E0T + E0 AαT )∇H +(∇H)T E0 AαT Hzz Aα E0T ∇H}. Here we use the matrix notation instead of the component notation in Theorem 3.4. Hzz denotes the Hessian matrix of H, and all derivatives of H are evaluated at z = E0 w. Type (II). Constructing symplectetic scheme by the second kind of the generating function

228

5. The Generating Function Method

⎤ ⎡ O O −In O O O O ⎥ ⎢ O −In ⎢ O O O −I O n ⎥ , αT = α−1 = ⎢ α=⎢ ⎣ O ⎣ −In O O In ⎦ O O In O O O O O In     q p w= , w =− , p q  O I   O I   O O  N0 = − , E0 = , Aα E0T = − , I O I O I O ⎡

⎤ In O ⎥ ⎥. O ⎦ O

p, q), φ(1) (w) = −H( n 1 φ(2) (w) = − (Hqi Hpi )( p, q), 2

φ(3) (w) = −

1 6

i=1 n 

(Hpi pj Hqi Hqj + Hqi qj Hpi Hpj + Hqi pj Hpi Hqj ),

i,j=1

∂H

. where H(z) = H(p1 , · · · , pn , q1 , · · · , qn ), Hzi = ∂ zi a. The first order scheme. ψ(1) (w, τ ) = φ(0) (w) + τ φ(1) (w). The equation w  = ∇ψ(1) (w, τ ) defines a first order canonical difference scheme  k+1 = pki − τ Hqi (pk+1 , q k ), pi i = 1, · · · , n. (4.4) qik+1 = qik + τ Hpi (pk+1 , q k ), When H is separable, H = U (p) + V (q). So Hqi (pk+1 , q k ) = Vqi (qk ),

Hpi (pk+1 , q k ) = Upi (pk+1 ).

At this time, (4.4) becomes  k+1 pi = pki − τ Vqi (qk ), qik+1 = qik + τ Upi (pk+1 ),

i = 1, · · · , n.

(4.5)

Evidently, (4.4) is an explicit difference scheme of 1-st order of accuracy. If we set q’s  1 at half-integer times t = k + τ , then (4.4) becomes 2



1

pk+1 = pki − τ Vqi (q k+ 2 ), i k+ 12 +1

qi

k+ 12

= qi

+ τ Upi (pk+1 ),

i = 1, · · · , n.

(4.6) is a staggered explicit scheme of 2-nd order accuracy. b. The second order scheme. ψ (2) (w, τ ) = ψ (1) (w) + τ 2 φ(2) (w).

(4.6)

5.4 Construction of Canonical Difference Schemes

229

The induced gradient map is ⎡ w  = ∇w ψ (2) = −

p q

!

∇q H ∇p H

−τ

! −

τ2 2

⎢ ⎢ ⎢ ⎣

 ⎤ ∇q Hqi Hpi ⎥ ⎥ i=1 . n   ⎥ ⎦ ∇p Hqi Hpi n 

i=1

So the second order scheme is ⎧  n  ⎪ τ2  k+1 ⎪ k k+1 k ⎪ pi = pi − τ Hqi (p ,q ) − Hqj Hpj (pk+1 , q k ), ⎪ ⎨ 2 qi j=1 n   ⎪ τ2  ⎪ k+1 ⎪ = qik + τ Hpi (pk+1 , q k ) + Hqj Hpj (pk+1 , q k ), ⎪ qi ⎩ 2 j=1

i = 1, · · · , n.

pi

This scheme is already implicit even when H(z) is separable. c. The third order scheme is pk+1 = pki − τ Hqi (pk+1 , q k ) − i −

3

τ 6

n  τ 2  Hqj Hpj qi (pk+1 , q k ) 2 j=1

n  

Hpl pj Hql Hqj + Hql qj Hpl Hpj + Hpl qj Hql Hpj

l,j=1

qik+1 = qik + τ Hpi (pk+1 , q k ) +

 qi

(pk+1 , q k ),

n  τ 2  Hqj Hpj p (pk+1 , q k ) i 2 j=1

n  τ3   + Hpl pj Hql Hqj + Hql qj Hpl Hpj + Hpl qj Hql Hpj pi (pk+1 , q k ), 6 l,j=1

where i = 1, · · · , n. Type (III). Constructing symplectetic scheme by Poincar´e type generating function ⎡ ⎤ ⎡ ⎤ 1 −J2n J2n J2n I2n ⎥ 2 ⎦ , α−1 = ⎢ α=⎣ 1 (4.7) ⎣ ⎦. 1 1 I2n I2n − J I 2n 2n 2 2 2

1 z + z), w = ( 2

w  = J(z − z).

N0 = 0, E0 = I, Aα E0T + E0 AαT = 0. φ(0) = φ(2) = φ(4) = 0,  1 φ(1) (w) = −H ( z + z) , 2 φ(3) (w) =

1 (∇H)T JHzz J∇H, 24

(4.8) (4.9) (4.10) (4.11) (4.12)

ψ(2) (w, τ ) = −τ H,

(4.13)

τ3 ψ(4) (w, τ ) = −τ H + (∇H)T JHzz J∇H. 24

(4.14)

230

5. The Generating Function Method

a.

The second order scheme is J(z − z) = w  = ∇w ψ (2) (w, t) = −τ ∇H

i.e., z k+1 = z k + τ J −1 ∇H







1 (z + z) , 2



1 k+1 (z + zk ) . 2

(4.15)

It is centered Euler scheme. b. The 4-th order scheme is J(z − z) = w  = ∇w ψ (4) (w, t) = −τ ∇H





  1 τ3 (z + z) + ∇z (∇H)T JHzz J∇H , 2 24 (4.16)

i.e., z k+1 = z k +τ J −1 ∇H

%

& & % 1  1 k+1 k τ3 (z (z k+1 +z k ) . +z ) − J −1 ∇z (∇H)T JHzz J∇H 2 24 2

It is not difficult to show that the generating function φ(w, t) of type (III) is odd in t. Hence, Theorem 4.1 leads to a family of canonical difference schemes of arbitrary even order accuracy. 6 5 −J2n J2n . For sufficiently small τ > 0 as the timeTheorem 4.2. Let α = 1 1 I2n I2n 2 2 step, we define ψ (2m) (w, τ ) =

m 

φ(2i−1) (w)τ 2i−1 ,

m = 1, 2, · · · .

(4.17)

i=1

Then the gradient map w −→ w  = f˜(w, τ ) = ∇ψ (2m) (w, τ ) defines implicitly canonical difference schemes z = z k → z k+1 = z,   1 z k+1 = z k − J −1 ∇ψ (2m) (z k+1 + z k ), τ 2

of 2m-th order of accuracy. The case m = 1 is the Euler centered scheme. Remark 4.3. We have following diagram commutes: phase flow g(z, t)

gradient transf. α

-

f (w, t)

generating function ∇φ

6 −1 g m (z, t)  α

∇ψ f˜(w, t) 

- φ(w, t) o(tm+1 ) ? ψ(w)

(4.18)

5.5 Further Remarks on Generating Function

231

5.5 Further Remarks on Generating Function Now we want to construct unconditional Hamiltonian algorithms, i.e., they are symplectic for all Hamiltonian systems. First we consider the one-leg weighted Euler schemes , i.e., s z = EH,c z:

z = z + sJHz (c z + (1 − c)z),

(5.1) 1

with real number c being unconditionally symplectic if and only if c = , which 2 corresponds to the centered Euler scheme   z + z . (5.2) z = z + sJHz 2

These simple propositions illustrate a general situation: apart from some very rare exceptions, the vast majority of conventional schemes are non-symplectic. However, if we allow c in (5.1) to be a real matrix of order 2n, we get a far-reaching generalization: (5.1) is symplectic iff c=

1 (I2n + J2n B), 2

B T = B,

cT J + Jc = J.

(5.3)

The simplest and important cases are[FQ91] : C:

1 2

z = z + sJHz

c = I2n , %

P :

c=

Q:

c=

%

I O

O O

O O

O I

& , & ,





z + z , 2

p, q), p = p − sHq ( (5.4)

q = q + sHp ( p, q), p = p − sHq (p, q), q = q + sHp (p, q).

For H(p, q) = φ(p) + ψ(q), the above schemes P and Q reduce to explicit schemes. A matrix α of order 4n is called a Darboux matrix if αT J4n α = J˜4n , % & O −I2n , J4n = I2n O % & a b α= , c d

% J˜4n = % α−1 =

J2n O a1 c1

O −J2n & b1 . d1

& ,

Every Darboux matrix induces a (linear) fractional transform between symplectic and symmetric matrices σα :

Sp(2n) −→ Sm(2n), σα (S) = (aS + b)(cS + d)−1 = A

for

|cS + d| = 0

232

5. The Generating Function Method

with the inverse transform σα−1 = σα−1 σα−1 :

Sm(2n) −→ Sp(2n), σα−1 (A) = (a1 A + b1 )(c1 A + d1 )−1 = S

for

|c1 A + d1 | = 0,

where Sp(2n) = {S ∈ GL(2n, R) | S T J2n S = J2n } is the group of symplectic matrices. The above mechanism can be extended to generally non-linear operators on R2n . Let totally symplectic operators be denoted by SpD2n , and symm(2n) the totality of symmetric operators (not necessary one-one). Every f ∈ symm(2n) corresponds, at least locally, to a real function φ (unique up to a constant) such that f is the gradient of φ : f (w) = ∇φ(w), where ∇φ(w) = (φw1 (w), · · · , φw2n (w)) = φw (w). Then we have σα : SpD 2n −→ symm (2n), σα (g) = (a ◦ g + b) ◦ (c ◦ g + d)−1 = ∇φ or alternatively

for

|cgz + d| = 0

  ag(z) + bz = (∇φ) cg(z) + dz ,

where φ is called the generating function of Darboux type α for the symplectic operator g.[FQ91] Then σα−1 : symm(2n) −→ SpD 2n , σα−1 (∇φ) = (a1 ◦ ∇φ + b1 ) ◦ (c1 ◦ ∇φ + d1 )−1 = g, for |c1 φww + d1 | = 0

(5.5)

or alternatively a1 ∇φ(w) + b1 (w) = g(c1 ∇φ(w) + d1 w),

(5.6)

where g is called the symplectic operator of Darboux type α for the generating function φ. For the study of symplectic difference scheme, we may narrow down the class of Darboux matrices to the subclass of normal Darboux matrices, i.e., those satisfying a + b = 0, c + d = I2n . The normal Darboux matrices α can be characterized as * + * + a b J −J 1 α= = , c = (I + JB), B T = B, (5.7) 2 c d c I −c * + * + a1 b1 (c − I)J I −1 = . (5.8) α = cJ I c1 d1 The fractional transform induced by a normal Darboux matrix establishes a 1-1 correspondence between symplectic operators near identity and symmetric operators near nullity. Then the determinantal conditions could be taken for granted. Those B’s listed in section 5 correspond to the most important normal Darboux matrices. For

5.5 Further Remarks on Generating Function

233

every Hamiltonian H with its phase flow etH and for every normal Darboux matrix α, we get the generating function φ(w, t) = φtH (w) = φtH,α (w) of normal Darboux type α for the phase flow of H by ∇φtH,α = (JetH − J) ◦ (cetH + I − c)−1

for small |t|.

(5.9)

φtH,α satisfies the Hamilton–Jacobi equation ∂ φ(w, t) = −H(w + a1 ∇φ(w, t)) = −H(w + c1 ∇φ(w, t)) ∂t and can be expressed by Taylor series in |t|: φ(w, t) =

∞ 

φ(k) (w)tk ,

|t| small enough.

(5.10)

(5.11)

k=1

The coefficients can be determined recursively φ(1) (w) = −H(w), and for k ≥ 0, a1 = (c − I)J; k  −1  1 Dm H(w) φ(k+1) (w) = · k+1 m! j +j +···+j =k m m=1 1 2 jl 1  (j1 ) · (a1 ∇φ (w), · · · , a1 ∇φ(jm ) (w)),

(5.12)

where we use the notation of the m-linear form D m H(w)(a1 ∇ φ(j1 ) (w), · · · , a1 ∇ φ(jm ) (w)) 2n  := Hzi1 ···zim (w)(a1 ∇ φ(j1 ) (w))i1 · · · (a1 ∇ φ(jm ) (w))im . i1 ,···,im =1

By (5.9), the phase flow z = etH z satisfies

  z + (I − c)z z − z = −J∇ φtH,α c ∞    z + (I − c)z . = − tj J∇ φ(j) c

(5.13)

j=1

Let ψ s be a truncation of φsH,α up to a certain power, e.g., sm . Using the inverse transformation σα−1 , we obtain the symplectic operator g s = σα−1 (∇ψ s ),

|s| small enough,

(5.14)

which depends on s, H, α (or equivalently B) and the mode of truncation. It is a symplectic approximation to the phase flow esH and can serve as the transition operator of a symplectic difference scheme for the Hamiltonian system (3.1) 1 (I + JB). (5.15) 2 Thus, using the technique of the phase flow generating functions, we have constructed, for every H and every normal Darboux matrix, a hierarchy of symplectic schemes by truncation. The simple symplectic schemes (5.4) correspond to the lowest truncation. z −→ z = g s z :

z = z − J∇ψ s (c z + (I − c)z),

c=

234

5. The Generating Function Method

5.6 Conservation Laws The conservation laws we refer to here[FQ91,FW91a,GF88,Ge91] have two meanings. As it is well known, the Hamiltonian system (3.1) itself has first integrals which are conserved in time evolution, e.g., the Hamiltonian is always a first integral. Hence, the first question is how many first integrals of Hamiltonian system (3.1) can be preserved by symplectic algorithms. The second question is whether or not there exist their own first integrals in case the original first integrals can not be preserved by symplectic algorithms. We first consider preservation of the first integrals of Hamiltonian systems by symplectic algorithms. The detailed discussion is referred to references[FQ91,Fen93b,GF88,Wan94] . Consider the Hamiltonian system dz = J∇H(z). dt

(6.1)

s (z) z = gH

(6.2)

Suppose

is a symplectic algorithm. Under a symplectic transformation z = S(y), system (6.1) can be transformed into dy ˜ = J∇H(y), (6.3) dt ˜ where H(y) = H(S(y)) and scheme (5.6) can be transformed into s ◦ S(y). y = S −1 ◦ gH

(6.4)

On the other hand, the algorithm g s can be applied to system (6.3) directly and the corresponding scheme is s (6.5) y = gH ˜ (y). Naturally, one can ask if (6.4) and (6.5) are the same. This introduces the following concept. Definition 6.1. A symplectic algorithm g s is invariant under the group G of symplectic transformations, or G-invariant, for Hamiltonian H if s s ◦ S = gH◦S , S −1 ◦ gH

∀ S ∈ G;

gs is symplectic invariant for Hamiltonian H, if s s S −1 gH ◦ S = gH◦S ,

∀ S ∈ Sp(2n).

In practice, the second case is more common. Generally speaking, numerical algorithms depend on the coordinates, i.e., they are locally represented. But many numerical algorithms may be independent of the linear coordinate transformations.

5.6 Conservation Laws

235

Theorem 6.2. [FW91a,GF88,Coo87] Suppose F is a first integral of the Hamiltonian system (6.1) and etF is the corresponding phase flow. Then F is conserved up to a constant s by the symplectic algorithm gH , s = F + c, F ◦ gH

c is a constant

(6.6)

s is etF -invariant. if and only if gH s is etF -invariant, i.e., Proof. We first assume that the symplectic algorithm gH s t s e−t F ◦ gH ◦ eF = gH◦et , F

∀ t ∈ R.

(6.7)

Since F is a first integral of the Hamiltonian system (6.1) with the Hamiltonian H, H is also the first integral of the Hamiltonian system (5.6) with the Hamiltonian F , i.e., H ◦ etF = H. (6.8) It follows from (5.6) and (6.8) that s t s e−t F ◦ gH ◦ eF = gH ,

i.e.,

s −1 s etF = (gH ) ◦ etF ◦ gH .

(6.9)

Differentiating (6.9) with respect to t at point 0 and noticing that  d etF  = J∇F,  dt

we get

t=0

s −1 s )∗ J∇F ◦ gH . J∇F = (gH

(6.10)

s Since gH is symplectic, i.e., s −1 s T )∗ J = J(gH )∗ , (gH

we have then

s T s s )∗ ∇F ◦ gH = J∇(F ◦ gH ), J∇F = J(gH s T s s ∇F = (gH )∗ ∇F ◦ gH = ∇(F ◦ gH ).

It follows that s F ◦ gH = F + c.

(6.11)

s , i.e., (6.6) is valid. Then noticing that We now assume that F is conserved by gH s −1 s s −1 the phase flows of the vector fields J∇F and (gH )∗ J∇F ◦ gH are etF and (gH ) ◦ t s s t eF ◦ gH respectively, we can get (5.6) similarly, i.e., gH is eF -invariant. Symplectic invariant algorithms are invariant under the symplectic group Sp(2n) and hence invariant under the phase flow of any quadratic Hamiltonian. 

236

5. The Generating Function Method

Corollary 6.3. Symplectic invariant algorithms for Hamiltonian systems preserve all quadratic first integrals of the original Hamiltonian systems up to a constant. s If a symplectic scheme has a fixed point, i.e., there is a point z such that gH (z) = z, then the constant c = 0 and the first integral is conserved exactly. Since linear schemes always have the fix point 0, we have the following result.

Corollary 6.4. Linear symplectic invariant algorithms for linear Hamiltonian systems preserve all quadratic first integrals of the original Hamiltonian systems. Example 6.5. Centered Euler scheme and symplectic Runge–Kutta methods are symplectic invariants. Hence they preserve all quadratic first integrals of system (6.1) up to a constant. Example 6.6. Explicit symplectic scheme (4.5), and other explicit symplectic schemes (2.1) – (2.4) considered in Chapter 8 are invariant under the linear symplectic transformations of the form diag (A−T , A), A ∈ GL(n). Thus they preserve angular momentum pT Bq of the original Hamiltonian systems, since their infinitesimal symplectic matrices are diag (−B T , B), B ∈ gl(n). In fact, these results can be improved. Symplectic Runge–Kutta methods preserve all quadratic first integrals of system (6.1) exactly. For generating function methods, we have the following result[FW91a,GF88,FQ87] . s be a symplectic method constructed by the generating funcTheorem 6.7. Let gH,α

1

tion method with the Darboux type α. If F (z) = z T Az, A ∈ Sm(2n), is a quadratic 2 first integral of the Hamiltonian system (6.1) and AJB − BJA = O,

(6.12)

s then F (z) is conserved by gH,α , i.e.,

F ( z ) = F (z),

or

s = F. F ◦ gH,α

(6.13)

For B = O, i.e., the case of centered symplectic difference schemes, (6.12) is always valid. So all centered symplectic difference schemes preserve all quadratic first integrals of the Hamiltonian system (6.1) exactly. Proof. Since F (z) is the first integral of system (6.1), 1 T 1 z A z = z T Az, 2 2

z = etH .

It can be rewritten as 1 ( z + z)T A( z − z) = 0, 2

From (6.12), it follows that

z = etH .

(6.14)

5.6 Conservation Laws

T 1 1 z − z) = ( z − z)T (AJB − BJA)( z − z) = 0, JB( z − z) A( 2 4

237

∀ z, z ∈ R2n .

Combining it with (6.14), we have  T c z + (I − c)z A( z − z) = 0. Using (5.13), it becomes 

c z + (I − c)z

T

AJ

∞ 

  z + (I − c)z = 0. tj ∇φ(j) c

j=1

From this, we get wT AJ∇φ(j) (w) = 0,

∀ w ∈ R2n .

∀ j ≥ 1,

Taking w = c z + (I − c)z, where m      s z + (I − c)z = z − z + (I − c)z , z = gH,α z = z − J∇ψ (m) c sj J∇φ(j) c j=1

we have wT A( z − z) = −

m 

sj wT AJ∇φ(j) (w) = −AJ∇ψ(w) = 0,

j=1

since 1 T z A z− 2 1 = zT A z− 2

w T A( z − z) =

1 T 1 z Az + ( z − z)T (AJB − BJA)( z − z) 2 2 1 T z Az. 2



Therefore, the theorem is completed.

We list some of the most important normal Darboux matrices c, the type matrices B, together with the corresponding form of symmetric matrices A of the conserved 1 quadratic invariants F (z) = z T Az: 2

1 c = I − c = I, 2

B = O,

c=

B=

 I  n O , O O  O O  , c= O In c=

1 2

c=

1 2



In ∓In

 I ±I

A arbitrary,

  O −In ,  O −In O A =  O I  bT n B= , In O

 ±In , B = ∓I2n , In  I ±I  n , B=± O O

 A=

b O



aT = a, bT = −b; Hermitian type.  b aT = a, , T −a b = −b.

a b −b a

 a O  ,A= −In −b

 b arbitrary; , angular momemtum.

,

238

5. The Generating Function Method

Apart from the first integrals of the original Hamiltonian systems, a linear symplectic algorithm has its own quadratic first integrals. For the linear Hamiltonian system dz = Lz, L = JA ∈ sp(2n) (6.15) dt

1

with a quadratic Hamiltonian H(z) = z T Az, AT = A, let us denote its linear 2 symplectic algorithm by s z = gH (z) = G(s, A)z,

G ∈ Sp(2n).

(6.16)

Let us assume that the scheme (6.16) is of order r. Then G(s) has the form G(s) = I + sL(s), L(s) = L +

s 2 s2 sr−1 r L + L3 + · · · + L + O(sr ). 2! 3! r!

For sufficiently small time step size s, G(s) can be represented as "

" L(s) = L + O(sr ),

G(s) = esL(s) , So (6.16) becomes

" L(s) ∈ sp(2n).

"

z = esL(s) z. This is the solution z(t) of the linear Hamiltonian system dz " = L(s)z, dt

" L(s) ∈ sp(2n),

(6.17)

with the initial value z(0) = z 0 evaluated at time s. The symplectic numerical solution "

z k = Gk (s)z 0 = eksL(s) z 0 is just the solution of system (6.17) at discrete points ks, k = 0, ±1, ±2, · · ·. Hence, for sufficiently small s, scheme (6.16) corresponds to a perturbed linear Hamiltonian system (6.17) with the Hamiltonian  1 T −1  " " s) = 1 z, J −1 L(s)z = z J Lz + O(sr ) = H(z) + O(sr ). H(z, 2

2

(6.18)

It is well-known that the linear Hamiltonian system has n functionally independent quadratic first integrals. So does the scheme (6.15). The following " i (z, s) = 1 z T J −1 L " 2i−1 (s)z, H 2

i = 1, 2, · · · , n

(6.19)

are the first integrals of the perturbed system (6.17), therefore, of scheme (6.16), which approximate the first integrals of system (6.15) 1 2

Hi (z) = z T J −1 L2i−1 z,

i = 1, 2, · · · , n

5.7 Convergence of Symplectic Difference Schemes

239

up to O(sr ). Another group of first integrals of (6.16) is  i (z, s) = z T J −1 Gi (s)z, H

i = 1, 2, · · · , n.

They can be checked easily. The first one is[FW94]  1 (z, s) = z T J −1 G(s)z = z T J −1 (I + sL(s))z H = sz T J −1 L(s)z = 2sH(z) + O(s3 ).

5.7 Convergence of Symplectic Difference Schemes We considered Hamiltonian systems dz = JHz, dt

z ∈ U ⊂ R2n .

(7.1)

In this section, we shall prove that all symplectic schemes for Hamiltonian systems constructed by generating functions are convergent, if τ → 0. A normal Darboux matrix, which will be introduced in the next chapter, has the form ⎤ 5 6 ⎡ J −J Aα Bα ⎦ , B T = B, α= =⎣ 1 1 Cα Dα (I + JB) (I − JB) 2 2 ⎤ ⎡ (7.2) 1 5 6 α α (JBJ − J) I A B ⎥ ⎢ 2 −1 =⎣ α = ⎦, 1 C α Dα (JBJ + J) I 2

which defines a linear transformation in the product space R2n × R2n : 5 6 5 6 5 6 5 6 w  w  z z , =α , = α−1 w w z z i.e., w  = J z − Jz,

1 2

1 2

w = (I + JB) z + (I − JB) z,

B T = B.

(7.3)

Let z → z = g(z, t) be the phase flow of the Hamiltonian systems (5.7); it is a time dependent canonical map. There exist, for sufficiently small |t| and in (some neighborhood of) R2n , a time-dependent gradient map w → w  = f (w, t) with Jacobian fw (w, t) ∈ Sm(2n) (i.e.,everywhere symmetric) and a time-dependent generating function φ = φα,H , such that f (w, t) = ∇ φα,H (w, t),

Aα g(z, t) + Bα z = ∇ φ(Cα g(z, t) + Dα z, t).

(7.4)

240

5. The Generating Function Method

On the other hand, for a given time-dependent scalar function ψ(w, t) : R2n × R → R, we can obtain a time-dependent canonical map g"(z, t). If ψ(w, t) approximates the generating function φα,H (w, t) of the Hamiltonian system (5.7), then g"(z, t) approximates the phase flow g(z, t). For sufficiently small τ > 0 as the time step, define φ(m) =

m 

φ(k) (w)τ k ,

(7.5)

k=1

1 2

where φ(1) (w) = −H(w), and for k ≥ 0, Aα = (JBJ − J), φ(k+1) (w) =

2n 

k −1  1 k+1 m! m=1

·



Hzi1 · · · zim (w)



i1 ,···,in =1



Aα ∇ φ(j1 ) (w)

j1 +···+jm =k

i1

  · · · Aα ∇ φ(jm) (w) i . (7.6) m

Then, ψ (m) (w, τ ) is the m-th approximation of φα,H (w, τ ) , and the gradient map, w −→ w  = f"(w, τ ) = ∇ ψ (m) (w, τ ).

(7.7)

Define a canonical map z → z = g"(z, τ ) implicitly by equation Aα z + Bα z = (∇ ψ (m) )(Cα z + Dα z, z).

(7.8)

The implicit canonical difference scheme of m-th order accuracy z = z k −→ z = z k+1 = g"(z k , τ ), for system (5.7) is obtained. For the sake of simplicity, we denote g"τ (z) = g"(z, τ ). Then   di g"τ (z)  di gτ (z)  g"0 (z) = z, = ,   i i dτ

τ =0

dτ

τ =0

(7.9)

(7.10)

where gτ (z) is the phase flow of g(z, τ ). Theorem 7.1. If H is analytical in U ⊂ R2n , then the scheme (7.9) is convergent with m-th order accuracy[CHMM78,QZ93] . Proof. For the step-forward operator g"τ , we set z1 = g"τ (z), z2 = g"τ (z1 ), · · · , zk = g"τ (zk−1 ), we have z k = g"τk . First, we prove that the convergence holds locally. We begin by showing that for n (n ≤ k), if t is sufficiently small. Indeed, any z0 , the iterations are defined for z"t/k in the neighborhood of z0 , g"τ (z) = z + o(τ ), thus, if g"lt (z) (l = 1, 2, · · · , n − 1) is k defined for z in the neighborhood of z0 ,

5.7 Convergence of Symplectic Difference Schemes

241



     g"nt (z) − g"n−1 (z) + g"n−1 (z) − g"n−2 (z) + · · · + g" kt (z) − z t t t k k k k     t t = o(t). + ··· + o = o k 0 k 12 3

g"nt (z) − z = k

n n which is small and independent of k for sufficiently small t. So g"t/k (n ≤ k) is defined and remains in Uz0 for z near z0 . Since H is analytical, for any z1 , z2 ∈ Uz0 , there exists a constant C, such that

JHz (z1 ) − JHz (z2 ) ≤ J Hz (z1 ) − Hz (z2 ) ≤ Cz1 − z2 . - t Let F (t) = g(z1 , t)−g(z2 , t), where g(zi , t) = zi + JHz (g(zi , s)) d s (i = 1, 2), 0

?- t ? - t ? ?     ? F (t) = ? JH , s) d s − JH , s) d s + z − z g(z g(z z 1 z 2 1 2? ? 0 0 - t F (s) d s, ≤ z1 − z2  + C 0

using Gronwall inequality, we have F (t) = g(z1 , t) − g(z2 , t) ≤ eC|t| z1 − zn , gt (z) − g"kt = g kt (z) − g"kt k

k

k

k−1

= gt

k

g kt (z) − g k−1 g" kt (z) + g k−2 g kt (y1 ) t t k

k

g" kt (y1 ) + · · · + g k−1 g kt (yl−1 ) −gk−2 t t k

−g t

k−1 k

k

g" kt (yl−1 ) + · · · + g kt (yk−1 ) − g" kt (yk−1 ),

where yl = g"lt (z). Then we have k

gt (z) − g"kt (z) ≤ k

k 



exp

e=1



 C k − l |t| 

g kt (yl−1 ) − g"kt (yl−1 )

k

 m t ≤ k exp (C|t|) o −→ 0, k

if k −→ ∞.

Here, we use consistent supposition gτ (z) − g"τ (z) = o(τ )m . Now, we assume that g(z, t) is defined for 0 ≤ t ≤ T . We shall show that g"kt conk verges to g(z, t). By the above proof and domain compactness, if N is large enough, g Nt = lim g"kt N uniformly convergent on a neighborhood of the curve t → gt (z). k→∞

k

g kt N )N (z). By the uniformity of t, Thus, for 0 ≤ t ≤ T, gt (z) = gNt = lim (" N

k→∞

k

gt (z) = lim g"kt (z). k→∞

k

From this proof, one can see that if H is not analytical but Hz satisfies the local Lipschitz condition, then the scheme (7.9) is convergent with order m = 1 . 

242

5. The Generating Function Method

5.8 Symplectic Schemes for Nonautonomous System We consider the following system of canonical equations: d pi ∂H =− , dt ∂qi d qi ∂H = , dt ∂pi

i = 1, 2, · · · , n

(8.1)

with Hamiltonian function H(p1 , p2 , · · · , pn , q1 , q2 , · · · , qn , t). This is a nonautonomous Hamiltonian system. This approach, which is applied particularly to nonautonomous system, is to consider the time t as an additional dependent variable. Now we can choose a parameter τ as a new independent variable. The original problem therefore becomes one of finding q1 , · · · , qn with t as a function of an independent variable τ . Hence, we set the coordinate qi by adding t = qn+1 . The corresponding phase space must have 2n+2 dimensions, z = (p1 , p2 , · · · , pn , h, q1 , q2 , · · · , qn , t), here t and h being merely alternative notations for qn+1 and pn+1 . The new momentum h associated with the time t is, as its physical interpretation, the negative of the total energy. We call this new space the extended phase space. An advantage of adding another degree of freedom to the analysis is that the system now resembles an autonomous system [Arn89,Qin96,Gon96] with 2n + 2 degree freedom, because its Hamiltonian is not an explicit function of τ . In the extended phase space, (8.1) becomes 5 6 O −J2n+1 dz = J ∇ K(z), J = J2n+2 = , (8.2) dt J2n+1 O where K(z) = h + H(p1 , p2 , · · · , pn , q1 , q2 , · · · , qn , t), which we call the “extended Hamiltonian function”. We write (8.2) in another form d pi ∂H d qi ∂H =− , = , dτ ∂qi dτ ∂pi d pn+1 ∂H =− , dτ ∂qn+1 d qn+1 = 1. dτ

i = 1, 2, · · · ,

(8.3) (8.4) (8.5)

Equation (8.4) shows that our normalized parameter now becomes equal to qn+1 , which is the time t. The Equation (8.3) is the original canonical equation. The last Equation (8.5) gives the law according to which the negative of the total energy, pn+1 , changes with the time. The general form of the canonical Equation (8.2) has great theoretical advantages. It shows the role of conservative system in a new light. We notice that after adding the time t to the mechanical variables, every system becomes conservative. The extended Hamiltonian function k does not depend on variable τ explicitly and thus our system is a conservative system in the extended phase space. The method of generating function plays a central role in the construction of symplectic schemes. In [Fen86] a constructive

5.8 Symplectic Schemes for Nonautonomous System

243

general theory of generating function roughly reads as follows. Let a normal Darboux matrix be ⎤ 6 ⎡ 5 J −J Aα Bα ⎦ , B T = B, =⎣ 1 α= 1 Cα Dα (I + JB) (I − JB) 2

5 α−1 =

Aα

Bα

α

α

C

D

6

2

⎡ 1 (JBJ − J) ⎢ 2 =⎣ 1 (JBJ + J)

I

⎤ ⎥ ⎦.

I

2

Define a linear transformation in product space R2n+2 × R2n+2 by w  w

!

z z

=α

!

z z

,

! =α

−1

w  w

! ,

(8.6)

i.e., w  = J z − Jz,

1 2

1 2

w = (I + JB) z + (I − JB)z,

B T = B.

(8.7)

Let z → z = g(z, τ ) be the phase flow of the Hamiltonian system (8.2). It is a time-dependent canonical map. There exists, for a sufficiently small τ and in (some  = f (w, τ ) with neighborhood of) R2n+2 , a time-dependent gradient mapping w → w Jacobian fw (w, t) ∈ Sm(2n + 2) (i.e., symmetric everywhere) and a time-dependent generating function φ = φα,K (w, τ ), such that   ∂φ = −K Aα ∇φ(w) + B α w , ∂τ (8.8)

f (w, τ ) = ∇φα,K (w, τ ),

  Aα g(z, t) + Bα z ≡ (∇φ) Cα g(z, t) + Da z, t . On the other hand, for a given time-dependent scalar function ψ(w, t) : R2n+2 ×R → R, we can get a time-dependent canonical map g"(z, τ ). If ψ(w, τ ) approximates the generating function φα,K (w, τ ) of the Hamiltonian system (8.2), g"(z, t) approximates the phase flow g(z, t). For a sufficiently small s > 0 as the time-step, define ψ (m) =

m 

φ(k) (w) sk ,

k=1

where φ(1) (w) = −K(w), For k ≥ 0,

1 2

Aα = (JBJ − J).

(8.9)

244

5. The Generating Function Method

φ

(k+1)

k −1  1 (w) = k+1 m! m=1



·

2n  i1 ,···,im =1

Kzi1 ···zim (w)

   Aα ∇φ(j1 ) (w) i1 · · · Aα ∇φ(jm ) (w) im .



j1 + · · · + jm = k jl = 0

(8.10) Then ψ (m) (w, s) is the m-th approximation of φα,K (w, s), and the gradient mapping w −→ w  = f"(w, s) = ∇ψ (m) (w, s)

(8.11)

defines a canonical map z → z = g"(z, s) implicitly by equation Aα z + Bα z = (∇ψ (m) )(Cα z + Dα z, s).

(8.12)

An implicit canonical difference scheme z = z k −→ z = z k+1 = g"(z k , s),

(8.13)

for system (8.2) is obtained[Qin96] , and this scheme is of m-th order of accuracy. Let B = 0, φ(1) (w) = −K(w),

φ(2) = φ(4) = 0,

1 (∇K)T JKzz J∇K. 24

φ(3) =

We have a scheme of second order: J( z − z) = w  = ∇φ(2) (w, s) = −s∇K





z + z , 2

  k+1 z + zk z k+1 = z k + sJ∇K , 2  k+1  p + pk q k+1 + q k tk+1 + tk = pki − sHqi , , pk+1 , i 2

qik+1

=

qik

 + sHpi

hk+1 = hk − sHt



k+1

p

2

k

+p q , 2

k+1

2

k

+q t , 2

k+1

(8.14)



+ tk , 2



pk+1 + pk q k+1 + q k tk+1 + tk , , , 2 2 2

tk+1 = tk + s. This is the time-centered Euler scheme. Scheme of the fourth order:  k+1   k+1   k+1  z + zk z + zk z + zk = −s∇K J∇K J(z k+1 − z k ) = ∇φ(4) 2

2

2

  s3 + ∇z (∇K)T JKzz J∇K , 24 z k+1 = z k + sJ∇K



z k+1 + z k 2



−

  s3 J∇z (∇K)T JKzz J∇K , 24 (8.15)

5.8 Symplectic Schemes for Nonautonomous System

245

i.e., = pki − sHqi pk+1 i −



pk+1 + pk q k+1 + q k tk+1 + tk , , 2 2 2



s3 (Hpj pl qi Hqj Hql + 2Hpj pl Hqj qi Hql − 2Hqj pl qi Hpj Hql 24

−2Hqj pl Hpj qi Hql − 2Hqj pl Hpj Hql qi + 2Hqj ql Hpl qi Hpj + Hqj ql qi Hpl Hpj

qik+1

−2Hqj qi Hpj t − 2Hqj Hpj qi t + 2Hpj qi Hqj t + 2Hpj Hqj qi t + Hqi tt ),  k+1  p + pk q k+1 + q k tk+1 + tk = qik + sHpj , , 2

2

2

s3 + (Hpj pl pi Hqj Hql + 2Hpj pl Hqj pi Hql − 2Hqj pl pi Hpj Hql 24

−2Hqj pl Hpj pi Hql − 2Hqj pl Hpj Hql pi + 2Hqj pl Hpl qi Hpj + Hqj ql pi Hpl Hpj

hk+1

−2Hqj pi Hpj t − 2Hqj Hpj pi t + 2Hpj pi Hqj t + 2Hpj Hqj pi t + Hpi tt ),  k+1  p + pk q k+1 + q k tk+1 + tk = hk − sHt , , 2

−

2

2

3

s (Hpj pl t Hqj Hql + 2Hpj pl Hqj t Hql − 2Hqj pl t Hpj Hql 24

−2Hqj pl Hpj t Hql − 2Hqj pl Hpj Hql t + 2Hqj ql Hpl t Hql + Hqj ql t Hpl Hpj −2Hqj t Hpj t − 2Hqj Hpj tt + 2Hpj t Hqj t + 2Hpj Hqi t + Httt ), tk+1 = tk + s. (8.16) Let B=−

O I

I O

! ,

w=

p q

! .

We have φ(1) = −K(w), 1 2

φ(2) = − (Kqi Kpi )(w), 1 6

φ(3) = − (Kpj pl Kqj Kql + Kqj ql Kpj Kpl + Kqj pl Kpj Kql )(w), 1 2

φ(2) = − (Hqi Hpj + Ht )(w), 1 6

φ(3) = − (Hpj pl Hqi Hql + Hql Hqj pl Hpj + Hql Hpl t +Hqj ql Hpj Hpl + 2Hpj Hqj t + Htt ). Scheme of the first order

or (8.17)

246

5. The Generating Function Method

pk+1 = pki − sHqi (pk+1 , q k , tk ), i qik+1 = qik + sHpi (pk+1 , q k , tk ), hk+1 = hk − sHt (pk+1 , q k , tk ),

(8.18)

tk+1 = tk + s. Scheme of the second order pk+1 = pki − sHqi (pk+1 , q k , tk ) − i

s2 (Hqi t + Hqj qi Hpj + Hqj Hpj qi )(pk+1 , q k , tk ), 2

qik+1 = qik + sHpi (pk+1 , q k , tk ) +

s2 (Hpi t + Hqj pi Hpj + Hqj Hpj pi )(pk+1 , q k , tk ), 2

hk+1 = hk − sHt (pk+1 , q k , tk ) −

s2 (Htt + Hqj t Hpj + Hqj Hpj t )(pk+1 , q k , tk ), 2

tk+1 = tk + s. (8.19) Scheme of the third order = pki − sHqi (pk+1 , q k , tk ) − pk+1 i · (pk+1 , q k , tk ) −

s2 (Hqi t + Hqj qi Hpj + Hqj Hpj qi ) 2

s3 (Htt + 2Hpj Hqj t + Hql Hpl t + Hpj pl Hqj Hql 6

+Hql Hqj pl Hpj + Hqj ql Hpj Hpl )qi (pk+1 , q k , tk ), qik+1 = qik + sHpi (pk+1 , q k , tk ) + · (pk+1 , q k , tk ) +

s2 (Hpi t + Hqj pi Hpj + Hqj Hpj pi ) 2

s3 (Htt + 2Hpj Hqj t + Hql Hpl t + Hpj pl Hqj Hql 6

+Hql Hqj pl Hpj + Hqj ql Hpj Hpl )pi (pk+1 , q k , tk ), hk+1 = hk − sHt (pk+1 , q k , tk ) − · (pk+1 , q k , tk ) −

s2 (Htt + Hqj t Hpj + Hqj Hpj t ) 2

s3 (Htt + 2Hpj Hqj t + Hql Hpl t + Hpj pl Hqj Hql 6

+Hql Hqj pl Hpj + Hqj ql Hpj Hpl )t (pk+1 , q k , tk ), tk+1 = tk + s. (8.20)

Bibliography

[Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Berlin Heidelberg, Second edition, (1989). [CHMM78] A. Chorin, T. J. R. Huges, J. E. Marsden, and M. McCracken: Product formulas and numerical algorithms. Comm. Pure and Appl. Math., 31:205–256, (1978). [Coo87] G. J. Cooper: Stability of Runge–Kutta methods for trajectory problems. IMA J. Numer. Anal., 7:1–13, (1987). [Fen86] K. Feng: Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math., 4:279–289, (1986). [Fen93b] K. Feng: Symplectic, contact and volume preserving algorithms. In Z.C Shi and T. Ushijima, editors, Proc.1st China-Japan conf. on computation of differential equationsand dynamical systems, pages 1–28. World Scientific, Singapore, (1993). [FQ91] K. Feng and M.Z. Qin: Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study. Comput. Phys. Comm., 65:173–187, (1991). [FW91] K. Feng and D.L. Wang: A Note on Conservation Laws of Symplectic Difference Schemes for Hamiltonian Systems. J. Comput. Math., 9(3):229–237, (1991). [FW94] K. Feng and D.L. Wang: Dynamical systems and geometric construction of algorithms. In Z. C. Shi and C. C. Yang, editors, Computational Mathematics in China, Contemporary Mathematics of AMS Vol 163, pages 1–32. AMS, (1994). [FWQW89] K. Feng, H. M. Wu, M.Z. Qin, and D.L. Wang: Construction of canonical difference schemes for Hamiltonian formalism via generating functions. J. Comput. Math., 7:71–96, (1989). [Ge91] Z. Ge: Equivariant symplectic difference schemes and generating functions. Physica D, 49:376–386, (1991). [GF88] Z. Ge and K. Feng: On the Approximation of Linear Hamiltonian Systems. J. Comput. Math., 6(1):88–97, (1988). [Gon96] O. Gonzalez: Time integration and discrete Hamiltonian systems. J. Nonlinear. Sci., 6:449–467, (1996). [Hua44] L. K. Hua: On the theory of automorphic function of a matrix I, II. Amer. J. Math., 66:470–488, (1944). [LL99] L. D. Landau and E. M. Lifshitz: Mechanics, Volume I of Course of Theoretical Physics. Corp. Butterworth, Heinemann, New York, Third edition, (1999). [Qin96] M.Z. Qin: Symplectic difference schemes for nonautonomous Hamiltonian systemes. Acta Applicandae Mathematicae, 12(3):309–321, (1996). [Qin97a] M. Z. Qin: A symplectic schemes for the PDEs. AMS/IP studies in Advanced Mathemateics, 5:349–354, (1997). [QZ93] M. Z. Qin and W. J. Zhu: A note on stability of three stage difference schemes for ODE’s. Computers Math. Applic., 25:35–44, (1993). [Sie43] C.L. Siegel: Symplectic geometry. Amer. J. Math., 65:1–86, (1943). [Wan94] D. L. Wang: Some acpects of Hamiltonian systems and symplectic defference methods. Physica D, 73:1–16, (1994). [Wei72] A. Weinstein: The invariance of Poincar´ es generating function for canonical transformations. Inventiones Math., 16:202–213, (1972).

Chapter 6. The Calculus of Generating Functions and Formal Energy

In the previous chapter, we constructed the symplectic schemes of arbitrary order via generating function. However the construction of generating functions is dependent on the chosen coordinates. One would like to know under what circumstance will the construction of generating functions be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations.

6.1 Darboux Transformation Consider a cotangent bundle T ∗ Rn  R2n with natural symplectic structure[Fen98a] : 6 5 O In J2n = . (1.1) −In O Now we consider R4n and the product of cotangent bundles T ∗ R × T ∗ R  R4n with natural product symplectic structure: 5 6 −J2n O " J4n = . (1.2) O J2n Correspondingly, we consider the product space Rn × Rn  R2n . Its cotangent bundle, T ∗ (Rn × Rn ) = T ∗ R2n  R4n has a natural symplectic structure: 5 6 O I2n J4n = . (1.3) −I2n O Choose symplectic coordinates z = (p, q) on the symplectic manifold, then for symplectic transformation, g : T ∗ Rn → T ∗ Rn , we have 6 7 5 gz ∗ n , z∈T R , (1.4) gr (g) = z " 4n = (R4n , J"4n ). Note that on it is a Lagrangian submanifold of T ∗ Rn × T ∗ Rn in R 4n 4n R here is a standard symplectic structure (R , J4n ): A generating map K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

250

6. The Calculus of Generating Functions and Formal Energy

α : T ∗ Rn × T ∗ Rn −→ T ∗ (Rn × Rn ) maps the symplectic structure (1.2) to the standard one (1.3). In particular, α maps Lagrangian submanifolds in (R4n , J"4n ) to Lagrangian submanifolds Lg in (R4n , J4n ). Suppose that α satisfies the transversality condition of g Chapter 5, Equation (1.2), then 5 7 6 dφg (ω) ∗ 2n , (1.5) , ω∈T R Lg = ω φg is called generating function of g. We call this generating map α (linear case) or α∗ (nonlinear case) Darboux transformation, in other words, we have the following definition. Definition 1.1. A linear map 5

Aα

Bα

Cα

Dα

α=

6 ,

(1.6)

which acts as the followings: 5 6 5 6 5 6 5 6 z0 z0 Aα z0 + Bα z1 w0 4n −→ α = = ∈ R4n , R  z1 z1 Cα z0 + Dα z1 w1 is called a Darboux transformation, if α J4n α = J"4n .

(1.7)

Denote Eα = Cα + Dα ,

F α = A α + Bα ,

(1.8)

then, we have: Definition 1.2. Let α be a Darboux tramsformation. Then we define ) ( " J), Sp(J"4n , J4n ) = α ∈ GL(4n) | α J4n α = J"4n = Sp(J, ( ) Sp(J4n ) = β ∈ GL(4n) | β  J4n β = J4n = Sp(4n), ( ) Sp(J"4n ) = γ ∈ GL(4n) | γ  J"4n γ = J"4n = S"p (4n). Definition 1.3. A special case of Darboux transformation ⎡ ⎤ J2n −J2n ⎦ α0 = ⎣ 1 1 I2n I2n 2

is called Poincar´e transformation.

2

(1.9)

6.2 Normalization of Darboux Transformation

251

Remark 1.4. From the definition above, we know α0 ∈ Sp(J"4n , J4n ). Proposition 1.5. If α ∈ Sp(J"4n , J4n ), β ∈ Sp(4n), γ ∈ S"p (4n), then βαγ ∈ Sp(J"4n , J4n ). Proposition 1.6. ∀ α ∈ Sp(J"4n , J4n ), we have Sp(J"4n , J4n ) = Sp(4n)α0 = α0 S"p (4n), Sp(J"4n , J4n ) = Sp(4n)α = αS"p (4n). Proposition 1.7. Let α = 5 α−1 =

Aα Cα

Bα Dα

!

∈ Sp(J"4n , J4n ), then:

−J2n Cα

J2n Aα

J2n Dα

−J2n Bα

6

5 =

Aα−1

Bα−1

Cα−1

Dα−1

6 .

Hint: Using the first equation of (1.2) in Definition 1.2. Theorem 1.8. If α ∈ Sp(J"4n , J4n ) satisfies transversality condition |Cα + Dα | = 0, then for all symplectic diffeomorphism z → g(z), g ∼ I2n (near identity), gz ∈ Sp(2n), there exists a generating function: φα,g : R2n −→ R, such that Aα g(z) + Bα z = ∇φα,g (Cα g(z) + Dα z), i.e.,

(Aα g + Bα )(Cα g + Dα )−1 z = ∇φα,g (z).

6.2 Normalization of Darboux Transformation 1 4n(4n + 1) = 2 2 ∗ 8n +2n. Denote M ≡ {α ∈ M | |Eα | = 0} an open submanifold of M , dimM ∗ = dimM . Denote M  ≡ {α ∈ M | Eα = In , Fα = 0} ⊂ M ∗ ⊂ M . Denote M ≡ Sp(J˜4n , J4n ) a submanifold in GL(4n), dimM =

Definition 2.1. A Darboux transformation is called a normalized Darboux transformation[Fen98a] , if Eα = I2n , Fα = O2n . The following theorem answers the question on how to normalize a given Darboux transformation.

252

6. The Calculus of Generating Functions and Formal Energy

Theorem 2.2. ∀ α ∈ M ∗ , there exists 5 6 I2n P β1 = ∈ Sp(4n), O I2n ⎡ ⎤ T (T −1 ) O ⎦ ∈ Sp(4n), β2 = ⎣ O T

|T | = 0,

such that β2 β1 α ∈ M  . Proof. We need only to take P = −Fα Eα−1 = −(Aα +Bα )(Cα +Dα )−1 , T = Eα−1 , then 65 5 T 65 6 Aα B α Eα I −Fα Eα−1 O β2 β1 α = O I Cα Dα O Eα−1 ! Aβ2 β1 α Bβ2 β1 α = Cβ2 β1 α Dβ2 β1 α 5  6 Eα (Aα − Fα Eα−1 Cα ) Eα (Bα − Fα Eα−1 Dα ) . (2.1) = α1 = Eα−1 Cα Eα−1 Dα It’s easy to verify that Aβ2 β1 α + Bβ2 β1 α = O2n ,

Cβ2 β1 α + Dβ2 β1 α = I2n . 

The theorem is proved.

From now on we will assume α is a normalized Darboux transformation unless it is specified otherwise. Theorem 2.3. A Darboux transformation can be written in the standard form as ⎤ ⎡ −J2n J2n ⎦ , V ∈ sp(2n). α=⎣ 1 1 (I + V ) (I − V ) 2

2

Proof. It’s not difficult to show: 5 ∀ α1 ∈ M =⇒ ∃β ∈ Sp(4n),

Aβ

Bβ

Cβ

Dβ

β=

6 ,

such that α1 = β, where α0 is a Poincar´e transformation. After computation, we get ⎡ ⎤ 1 1 Aβ J2n + Bβ −Aβ J2n + Bβ 2 2 ⎢ ⎥ α1 = ⎣ ⎦. 1 1 Cβ J2n + Dβ −Cβ J2n + Dβ 2

2

6.2 Normalization of Darboux Transformation



Because α1 ∈ M , we have Dβ = I2n , Bβ = O, i.e., β = ! I2n O β ∈ Sp(4n), we have β = , Q ∈ Sm(2n). Thus: Q I2n ⎤ 5 6⎡ J −J2n 2n I2n O ⎦ ⎣ α1 = 1 1 Q I2n I2n I2n 2

⎡ =⎣ 1 ⎡

2

=⎣ 1 2

2

I2n + QJ2n

1 I2n − QJ2n 2

(I2n + V )

! . Since

⎦

⎤

−J2n

J2n

O I2n

⎤

−J2n

J2n

Aβ Cβ

253

1 (I2n − V ) 2

⎦,

where Q = Q, V = 2QJ. We shall write ⎡ αV = ⎣ 1 2

−J2n

J2n (I2n + V )

1 (I2n − V ) 2

Therefore, the theorem is completed. Corollary 2.4. Every α = αV ∈ M  ,

Aβ Cβ

Bβ Dβ

⎤ ⎦,

⎡ ⎢ αV−1 = ⎣

1 2

− (I2n − V ) I2n 1 (I2n + V ) 2

⎤ ⎥ ⎦.

I2n 

! ∈ M ∗ has a normalized Darboux form

V = (Cα + Dα )−1 (Cα − Dα ) ∈ sp(2n).

This result can be derived from (2.1). From the following theorem, we can show that the normalization condition is natural. Theorem 2.5. Let Gτ be a consistent difference scheme for equation z˙ = J −1 Hz , i.e., 1◦ Gτ (z)|τ =0 = z, ∀ z, H.  ∂Gτ (z)  = J −1 Mz , ∀ z, H. 2◦  ∂τ τ =0 iff the generating Darboux transformation is normalized with A = −J. Proof. We take symplectic difference scheme of first order via generating function of ! A B type α = , we have C D AGτ (z) + Bz = −τ Hz (CGτ (z) + Dz).

(2.2)

254

6. The Calculus of Generating Functions and Formal Energy

We first prove the “ only if ” part of the theorem. When taking τ = 0, we have AG0 (z) + Bz = (A + B)z = 0,

∀ z =⇒ A + B = O.

Differentiating (2.2) yields A Since

∂Gτ (z)  = −Hz ((C + D)z).  ∂τ τ =0  ∂Gτ (z)   = J −1 Hz (z),  ∂τ τ =0

we have

AJ −1 Hz (z) = −Hz ((C + D)z),

∀ H, z.

Take special form H(z) = z T b, and substitute it into above equation, we have AJ −1 b = −b,

∀ b,

which results in A = −J. On the other hand, since Hz (z) = Hz ((C + D)z), ∀ H, z take special form 1 2

H = z T z, and substitute it into the above equation, we get z = (C + D)z,

∀ z =⇒ C + D = I.

Now we prove the “ if ” part. Take A = −J,

A + B = O, then

C + D = I,

(2.3)

A(Gτ (z) − z) = −τ Hz (CGτ (z) + Dz),  A = −J, τ = 0 =⇒ Gτ (z)τ =0 = z.

On the other hand, we have % τ & & % τ ∂G (z)  ∂G (z)  A = −Hz ((C+D)z) =⇒ = J −1 Hz (z),   ∂τ ∂τ τ =0 τ =0

∀ z, H. 

Therefore, the theorem is completed.

Theorem 2.6. A normalized Darboux transformation with A = −J can be written in the standard form ⎤ ⎡ −J J ⎦ , ∀ V ∈ sp(2n). α=⎣ 1 1 (I − V ) (I + V ) 2

2

6.3 Transform Properties of Generator Maps and Generating Functions

255

6.3 Transform Properties of Generator Maps and Generating Functions Let

5

Aα

Bα

Cα

Dα

6 ∈ Sp(J"4n , J4n ),

α=

denote Eα = Cα + Dα , Fα = Aα + Bα . Let g ∈ Sp-diff. From now on, we always assume that transversality condition is satisfied, i.e., |Eα | = 0. ! (T −1 )T O ∈ Sp(4n), βT α ∈ Theorem 3.1. ∀ T ∈ GL(2n), let βT = O T Sp(J"4n , J4n ), we have: (3.1) φβT α,g ∼ = φα,g ◦ T −1 . Proof. Since ⎡ βT α = ⎣

T

T

(T −1 ) Aα

(T −1 ) Bα

T Cα

T Dα

⎤

5

⎦=

AβT α

BβT α

CβT α

DβT α

6 ,

we have Aα g(z) + Bα z = ∇φα,g ◦ (Cα g(z) + Dα z),

(3.2)

and T

T

(T −1 ) Aα g(z) + (T −1 ) Bα z = ∇φβT α,g ◦ (T Cα g(z) + T Dα z) ⇐⇒ Aα g(z) + Bα z = T  (∇φβT α,g ) ◦ T (Cα g(z) + Dα z) (3.3) = ∇(φβT α,g ◦ T )(Cα g(z) + Dα z). Comparing (3.2) with (3.3) for all z, we find: ∇φα,g (Cα g(z) + Dα z) = ∇(φβT α,g ◦ T )(Cα g(z) + Dα z). Thus we obtain or

φα,g ∼ = φβT α,g ◦ T φα,g ◦ T −1 ∼ = φβT α,g 

The theorem is proved. Theorem 3.2. ∀ S ∈ Sp(2n), define γS =

S O

O S

φαγS ,g ∼ = φα,S◦g◦S −1

!

" ∈ Sp(4n). Then we have (3.4)

256

6. The Calculus of Generating Functions and Formal Energy

Proof. Since: 5

Aα S

Bα S

Cα S

Dα S

6

5

Aα γS

Bα γS

Cα γS

Dα γS

=

αγS =

6 ,

we have Aα S ◦ g ◦ S −1 z + Bα z = ∇φα,S◦g◦S −1 (Cα S ◦ g ◦ S −1 z + Dα z). Since S is nonsingular, replacing z with S(z) results in Aα S ◦ g(z) + Bα Sz = ∇φα,S◦g◦S −1 (Cα S ◦ g(z) + Dα Sz),

∀ z.

(3.5)

∀ z.

(3.6)

On the other hand, (Aα S)g(z) + (Bα S)z = ∇φαγS ,g [(Cα S)g(z) + Dα Sz], Compare (3.5) with (3.6) and note that |Cα + Dα | = 0 ⇐⇒ |Cα S + Dα S| = 0 ⇐⇒ |Cα Sgz (z) + Dα S| = 0. Finally, we obtain: ∇φαγS,g = ∇φα,S◦g◦S −1 , i.e.,

φαγS,g ∼ = φα,S◦g◦S −1 . 

The proof can be obtained. Theorem 3.3. Take β =

I2n O

then:

!

P I2n

∈ Sp(4n), P ∈ Sm(2n), α ∈ Sp(J"4n , J4n ),

φβα,g ∼ = φα,g + ψp ,

(3.7)

1 2

where ψp = w P w ( function independent of g). Proof. Since: 5 βα =

I2n

P

O

I2n

6 5

Aα

Bα

Cα

Dα

6

5

Aα + P Cα

Bα + P Dα

Cα

Dα

= Eβα = Eα ,

6 ,

Fβα = Fα + P Eα ,

obviously, Aβα g(z) + Bβα z = ∇φβα,g (Cβα g(z) + Dβα z), (3.8) Aα g(z) + Bα z + (P Cα g(z) + P Dα z) = ∇φβα,g (Cα g(z) + Dα z). (3.9) On the other hand,

6.3 Transform Properties of Generator Maps and Generating Functions

∇ψP (Cα g(z) + Dα z) = P (Cα g(z) + Dα z).

257

(3.10)

Inserting (3.10) into (3.9), we obtain Aα g(z) + Bα z = ∇φβα,g (Cα g(z) + Dα z) − ∇ψP (Cα g(z) + Dα z) (3.11) = ∇(φβα,g − ψP )(Cα g(z) + Dα z). Compare (3.11) with Aα g(z) + Bα z = ∇φα,g (Cα g(z) + Dα z), we obtain

φβα,g − ψP ∼ = φα,g . 

Analogically, we have: O I2n

I2n Q

Theorem 3.4. If we take β =

! ∈ Sp(4n), Q ∈ Sm(2n), then

1 φα,g + (∇w φα,g (w)) Q(∇w φα,g (w)) ∼ = φβα,g (w + Q∇φα,g (w)). 2

(3.12)

Theorem 3.5. We have the following relation: φ⎡ ⎣

A C

B D

⎤ ⎦ ,g −1

∼ = −φ⎡ ⎣

−B D

⎤

−A C

.

(3.13)

⎦ ,g

Proof. Since Aα g −1 (z) + Bα z = ∇φα,g−1 (Cα g −1 (z) + Dα z), replacing z with g(z) yields Aα z + Bα g(z) = ∇φα,g−1 (Cα z + Dα g(z)).

(3.14)

Comparing (3.14) with −Bα g(z) − Aα z = ∇φ5 −B D

−A C

6

(Dα g(z) + Cα z), ,g



the proof is complete. Theorem 3.6. If φ⎡ ⎣

A C

B D

⎤ ⎦ ,g−1

∼ = −φ⎡ ⎣

A C

B D

⎤ ⎦ ,g −1

then A + B = O,

C = D.

,

∀ g,

258

6. The Calculus of Generating Functions and Formal Energy

Proof. By Theorem 3.5 and the uniqueness of Darboux transformation in Theorem 4.2, we have 5 6 5 6 A B −B −A =± . C D D C 

We only consider the case “+”, where we have A + B = O, C = D. 6 5 J −J , we have: Remark 3.7. For Poincar´e map α0 = 1 1 I I 2

2

φα0 ,g−1 ∼ = −φα0 ,g ,

∀ g ∈ Sp-diff.

t Theorem 3.8. Let gH be the phase flow of Hamiltonian system H(z). Then under t is an odd function w.r.t t, i.e., Poincar´e map α0 the generating function for gH

∀ w ∈ R2n ,

t (w, t) = −φα ,g t (w, −t), φα0 ,gH 0 H

t ∈ R.

t , we have Proof. By the properties of generating function for gH −t t −1 = (gH ) , gH t (w, −t) = φα ,(g t )−1 (w, t) = −φα ,g t (w, t). φα0 ,g−t (w, t) = φα0 ,gH 0 0 H H H

The theorem is proved. S O

Theorem 3.9. If S ∈ Sp(2n), α ∈ Sp(J"4n , J4n ), γ1 = αγ1 =

Aα S Cα S

Bα Dα



! O , then I

! .

Assume |Eαγ1 | = |Cα S + Dα | = 0, we have φα,S◦g ∼ = φαγ1 ,g , i.e.,

φ5 A C

Theorem 3.10. If 5 I γ2 = O

O

6

B D

,S◦g

(3.15)

∼ = φ5 AS CS

B D

6

. ,g

5

6 α ∈ Sp(J"4n , J4n ),

, S

Aα

Bα S

Cα

Dα S

6 ,

αγ2 =

assume |Bα + Dα S| = 0, we have φα,g◦S −1 ∼ = φαγ2 ,g , i.e.,

φ5 A C

B D

6 ,g◦S −1

∼ = φ5 A C

(3.16) BS DS

6

. ,g

6.3 Transform Properties of Generator Maps and Generating Functions

259

Proof. Since   Ag(S −1 z) + Bz = ∇φα,g◦S −1 Cg(S −1 z) + Dz ,

∀ z,

replacing z with Sz yields = ∇φα,g◦S −1 (Cg(z) + DSz)

Ag(z) + BSz

= ∇φ⎡

A C

⎣

φ⎡ ⎣

A C

BS DS

BS DS

∼ = φ⎡

⎤ ⎦ ,g

⎣

⎤

(Cg(z) + DSz),

⎦ ,g

A C

B D

⎤

.

⎦ ,g◦S −1



Therefore, the theorem is completed. The proof of Theorem 3.9 is similar. Theorem 3.11. If λI2n O

β= βα=

O I2n

λA λB C D

then we have

φ⎡ ⎣

! ∈ CSp(4n), !

α ∈ Sp(J"4n , J4n ),

∈ CSp(J"4n , J4n ),

λA λB C D

⎤ ⎦ ,g

λ = 0,

μ(βα) = λ,

∼ = λφ⎡ A ⎣ C

B D

⎤

.

(3.17)

⎦ ,g

Proof. Since α ∈ Sp(J"4n , J4n ) =⇒ Ag(z) + Bz = ∇φ⎡ ⎣

A C

B D

β ∈ CSp(J"4n , J4n ) =⇒ λAg(z) + λBz = ∇φ⎡ ⎣

L.H.S = λAg(z) + λBz = λ∇φ⎡ ⎣

R.H.S = ∇φ⎡ ⎣

then we have

λA C

λB D

⎤ ⎦ ,g

A C

B D

⎤ ⎦ ,g

(Cg(z) + Dz),

⎤

(Cg(z) + Dz),

⎦ ,g

λA λB C D

⎤ ⎦ ,g

(Cg(z) + Dz),

(Cg(z) + Dz),

260

6. The Calculus of Generating Functions and Formal Energy

∇φ⎡ ⎣

⎤

λA λB C D

(Cg(z) + Dz) = λ∇φ⎡

⎦ ,g

φ⎡ ⎣

A C

⎣

∼ = λφ⎡

⎤

λA λB C D

⎦ ,g

⎣

A C

B D

⎤

⎤

B D

(Cg(z) + Dz),

⎦ ,g

.

⎦ ,g



The theorem is proved. Theorem 3.12. Let 5 6 I2n O β= ∈ CSp(J4n ), λ = 0, α ∈ Sp(J"4n , J4n ), O λI2n 5 6 A B βα = ∈ CSp(J"4n , J4n ), μ(βα) = λ, λC λD then we have: φ⎡ ⎣

A λC

Proof. Since A λC

B λD

!

⎦ ,g

A λC

⎣

⎣

⎛ ⎜ ⎡ R.H.S = ⎜ ⎝φ ⎣

A C

B D

A λC ⎛

⎣

φ⎡ ⎣

A C

B D

⎤

A C

B D

⎤ ⎦ ,g

◦ λ−1 I2n .

(3.18)

⎤

B λD

⎤

(λCg(z) + λDz),

⎦ ,g

(Cg(z) + Dz),

⎦ ,g

⎞

B λD

⎜ ⎡ = λ−1 ∇ ⎜ ⎝φ

hence

⎣

∈ CSp(J"4n , J4n ),

Ag(z) + Bz = ∇φ⎡ L.H.S = ∇φ⎡

∼ = λφ⎡

⎤

B λD

A λC

⎤ ⎦ ,g

⎟ ⎟ ◦ λI2n (Cg(z) + Dz) ⎠ ⎞

B λD

∼ = λ−1 φ⎡

⎦ ,g

Therefore, the theorem is completed.

⎣

⎤ ⎦ ,g

A λC

⎟ ◦ λI2n ⎟ ⎠ (Cg(z) + Dz),

B λD

⎤ ⎦ ,g

◦ λI2n . 

6.4 Invariance of Generating Functions and Commutativity of Generator Maps

261

Before finishing this section, we will give two conclusive theorems which can include the contents of the seven theorems given before. They are easy to prove and the proofs are omitted here. Let ! a b , α ∈ CSp(J"4n , J4n ), β ∈ CSp(J4n ), β = c d obviously

β α ∈ CSp(J"4n , J4n ),

μ(βα) = λ(β)μ(α),

and then the following theorem. Theorem 3.13. For φβα,g , we have φβα,g (c∇w φα,g (w) + dw) ' 1   ∼ w (d b)w + (∇w φα,g (w)) = λ(β)φα,g (w) + 2 / 1    ·(c b)w (∇w φα,g (wλ)) (c a)(∇w φα,g (w)) . 2

(3.19)

(3.20)

We now formulate the other one. Let α ∈ CSp(J"4n , J4n ), γ ∈ CSp(J"4n ) ⇔ ! a b . γ  J"4n γ = ν(γ)J"4n ⇒ αγ ∈ CSp(J"4n , J4n ), μ(αγ) = μ(α)ν(γ), γ = c d We have the following theorem. Theorem 3.14. For φαγ,g , we have φαγ,g ∼ = φα,(ag+b)(cg+d)−1 .

6.4 Invariance of Generating Functions and Commutativity of Generator Maps First we present the uniqueness theorem of the linear fractional transformation. Theorem 4.1. Let 5

Aα

Bα

Cα

Dα

,

α= |Eα | = 0, If

5

6

Aα

Bα

Cα

Dα

α=

6 ∈ Sp(J"4n , J4n ),

|Eα | = 0.

(Aα M + Bα )(Cα M + Dα )−1 = (Aα M + Bα )(Cα M + Dα )−1 , ∀ M ∼ I2n ,

M ∈ Sp(2n),

then α = ±α.

(3.21)

262

6. The Calculus of Generating Functions and Formal Energy

Proof. Let

N0 = (Aα I + Bα )(Cα I + Dα )−1 = (Aα I + Bα )(Cα I + Dα )−1 .

Suppose β ∈ Sp(4n), first we prove that if (Aβ N + Bβ )(Cβ N + Dβ )−1 = N,

∀ N  N0 ,

N ∈ Sm(2n),

then β = ±I4n . Now we have two cases: 1◦ (Aβ N0 + Bβ )(Cβ N0 + Dβ )−1 = N0 ⇒ Aβ N0 + Bβ = N0 Cβ N0 + N0 Dβ . 2◦ Take N = N0 + εI ⇒ Aβ (N0 + εI) + Bβ = (N0 + εI)Cβ (N0 + εI) + (N0 + εI)Dβ . From 1◦ , 2◦ ⇒ εAβ = εN0 Cβ + εCβ N0 + εDβ + ε2 Cβ , ∀ ε, which results in Aβ − Dβ − N0 Cβ − Cβ N0 = εCβ = 0 =⇒ Cβ = 0, thus Aβ = Dβ . Aβ O

◦

From 1 , we have B =

Bβ Aβ

! , Bβ = Bβ . Therefore

−1 Aβ N A−1 β = N − Bβ Aβ . −1 Subtracting this formula by Aβ N0 A−1 β = N0 − Bβ Aβ yields

Aβ (N − N0 ) = (N − N0 )Aβ . Take N − N0 = εS, ∀ S ∈ Sm(2n) ⇒ Aβ S = SAβ , ∀ S ∈ Sm(2n) ⇒ Aβ = λI2n (This can be proved by mathematical induction). Then from 1◦ , Aβ N0 + Bβ = N0 Aβ ⇒ Bβ = 0, and ! Aβ O = λI4n ∈ Sp(4n) =⇒ λ = ±1. β= O Aβ Let β = αα−1 , then the fractional transformation of β preserves all symmetric N ∼ " J), α−1 ∈ Sp(J, J), " we have αα−1 ∈ Sp(J, J) = Sp(4n). N0 . Because α ∈ Sp(J, The theorem is proved.  We now present the uniqueness theorem for Darboux transformations. " J), then Theorem 4.2. Suppose α, α ∈ Sp(J, φα,g ∼ = φα,g ,

∀ g ∈ Sp-diff,

g ∼ I2n =⇒ α = ±α.

Proof. From the hypothesis, we have φα,g ∼ = φα,g =⇒ Hessian(φα,g ) = (φα,g )ww = (Aα g(z) + Bα )(Cα g(z) + Dα )−1 ,

6.4 Invariance of Generating Functions and Commutativity of Generator Maps

(φα,g )ww = (Aα g(z) + Bα )(Cα g(z) + Dα )−1 ,

263

∀ g(z) ∈ Sp(2n) ∼ I.

Then by uniqueness theorem of the linear fractional transformation α = ±α. From the above proof, we get Hessian (φα,g ) = Hessian (φ−α,g ),

∀ g ∈ I, α.

The generating function φα,g depends on Darboux transformation α, symplectic diffeomorphism g and coordinates. If we make a symplectic coordinate transformation w → S(z), then φ(S) ⇒ φ(S(z)), while the symplectic diffeomorphism g is represented in z coordinates as S −1 ◦ g ◦ S, i.e., φα,S −1 ◦g◦S = φα,g ◦ S For the invariance of generating function φg (S) under S, one would like to expect φα,S −1 ◦g◦S = φα,g ◦ S,

∀g ∼ I.

This is not true in general case. We shall study under what condition this is true for the normalized Darboux transformation αV . The following theorem answers this question.  Theorem 4.3. Let

⎡

α = αV = ⎣ 1 2

S ∈ Sp(2n), 5 γS =

(I + V ) 5 βS =

S

O

O

S

⎤

−J2n

J2n

1 (I − V ) 2

(S −1 )T

O

O

S

⎦,

∀ V ∈ sp(2n),

αV ∈ M  ,

6 ∈ Sp(J4n ),

6 ∈ Sp(J"4n ).

Then the following conditions are equivalent: 1◦ φαV ,S◦g◦S −1 = φα,g ◦ S −1 , ∀ g  I. 2◦ φαV γS ,g = φβS αV ,g , ∀ g  I. 3◦ αV γS = βS αV . 4◦ SV = V S. Proof. 1◦ ⇔ 2◦ from Theorems 3.1 and Theorem 3.2. 2◦ ⇒ 3◦ using the uniqueness theorem on Darboux transformation 4.2. For αV γS = ±βS αV , −1

since JS = S  J, sign “−” case is excluded. The rest of the proof is trivial. There is a deep connection between the symmetry of a symplectic difference scheme and the conservation of first integrals. Let F be the set of smooth functions defined on Rn . 

264

6. The Calculus of Generating Functions and Formal Energy

Theorem 4.4. If Hamiltonian function H is invariant under phase flow gF with Hamiltonian function F , then F is first integral of the system with Hamiltonian function H. Let H, F ∈ F, then t t = F ⇐⇒ {F, H} = 0 ⇐⇒ H ◦ gFt = H ⇐⇒ gH F ◦ gH t = gF−S ◦ gH ◦ gFS .

Theorem 4.5. Let F be a conservation law of Hamiltonian system, then phase flow t gH (or symplectic schemes φτH ) keeps phase flow gFt with F (or φτF ) invariant iff F ◦ gH = F + C. Let F ∈ F, g ∈ Sp-diff, then g = gF−t ◦ g ◦ gFt (or gFt = g −1 gFt (g(z)) ⇐⇒ F ◦ g = F + c. Proof. The “ if ” part of the proof is obvious. Since F ◦ g = F + c =⇒ ∇F = ∇F ◦ g =⇒ gFt = gFt ◦g = g −1 ◦ gFt ◦ g = g−1 gFt (g(z)) ⇐⇒ g = gF−t ◦ g ◦ gFt . On the other hand, take the derivative of both sides of the following equation w.r.t. t at t = 0, gFt (z) = g −1 gFt (g(z)), and notice that g∗ (z) ∈ Sp, g∗−1 J −1 = J −1 g∗T , we get J −1 ∇F (z) = g∗−1 (z)J −1 ∇F (g(z)) = J −1 g∗T (z)∇F (g(z)), then we have ∇F = ∇F ◦ g =⇒ F ◦ g = F + c. 

Therefore, the theorem is completed.

6.5 Formal Energy for Hamiltonian Algorithm Let F s be an analytic canonical transformation for s, i.e., 1◦ F s ∈ Sp-diff. 2◦ F 0 = id. 3◦ F s analytic if |s| is small enough. Then there exists a “formal” energy, i.e., a formal power series in s, hs (z) = h(s, z) =

∞  i=1

si hi (z)

6.5 Formal Energy for Hamiltonian Algorithm

265

with the following property: if hs (z) converges, the phase flow ght s is a canonical transformation with Hamiltonian function hs (z), which is considered as a timeindependent Hamiltonian with s as a parameter and satisfies “equivalence condition”  ght s t=s = F s . (5.1) Therefore hs (z) = hs (F s z), ∀z ∈ R2n , thus hs (z) is invariant under F s (for those s, z in the domain of convergence of hs (z)). The generating function with F s , the new Hamiltonian function and α, the Darboux transformation is ∞ 

φF s ,α (w) : ψ(s, w) =

sk ψ (k) (w).

(5.2)

k=1

Introduce formal power series hs (z) = h(s, w) =

∞ 

si hi (w).

k=1

Assuming it converges, we associate the phase flow with the generating function ∞ 

hs (z) −→ ψht s ,α (w) : χ(t, s, w) =

tk χ(k) (s, w),

k=1

χ

(1)

(s, w) = −h(s, w).

(5.3)

For k > 1, χ(k+1) (s, w) = −

k 

1 (k + 1)m! m=1

2n 



l1 ,···,lm =1 k1 +···+km =k

hwl1 ,···,wlm (s, w)

(k1 ) m) (s, w))l1 · · · (A1 χ(k (s, w))lm ·(A1 χw w

=

k 

1 (k + 1)m! m=1

 k1 +···+km =k

(k1 ) ·(A1 χw (s, w))l1

Let χ(k) (s, w) =

∞ 

m) · · · (A1 χ(k (s, w))lm . w

si χ(k,i) (w), then χ(t, s, w) =

i=0 ∞  i=0

si χ(k+1,i) (w) =

(1)

χwl1 ,···,wlm (s, w)

∞  ∞ 

(5.4)

tk si χ(k,i) (w). Then

k>1 i=0 ∞  i=0

·

si

k 

1 (k + 1)m! m=1

2n  l1 ,···,lm =1

 i0 + i1 + · · · + im = i k1 + · · · + km = k

(1,i )

1 ,i1 ) χwl1 0,···,wlm (w)(A1 χ(k (w))l1 · · · w

m ,im ) (w))lm . ·(A1 χ(k w

(5.5)

266

6. The Calculus of Generating Functions and Formal Energy

Thus χ(k+1,i) (w) =



2n 

i0 +i1 +···+im =i k1 +···+km =k

l1 ,···,lm =1

k 

1 (k + 1)m! m=1

(1,i )

χwl1 0,···,wlm (w)

(k1 ,i1 ) (km ,im ) ·(A1 χw (w))l1 · · · (A1 χw (w))lm .

Let χ(1) (s, w) =

∞ 

si χ(1,i) (w) = −h(s, w) = −

i=0 (k+1,i)

so the coefficient χ

∞ 

(5.6)

si hi (w),

i=0

can be obtained by recursion, χ(1,i) = −h(i) ,

i = 0, 1, 2, · · · . 

(5.7) 

Note that χ(k+1,i) is determined only by the values of χ(k ,i ) (k ≤ k, i ≤ i), χ(1,0)

χ(1,1)

.. .

χ(1,2)

.. .

χ(1,i)

···

.. .

χ(k,0)

χ(k,1)

χ(k,2)

χ(k+1,0)

χ(k+1,1)

χ(k+1,2)

χ(1,i+1)

.. .

.. .

χ(k,i)

···

· · · χ(k+1,i)

χ(k,i+1)

(5.8)

χ(k+1,i+1)

The condition (5.1) can be now reexpressed as χ(t, s, w)|t=s = χ(s, s, w) = ψ(s, w), i.e.,

∞ 

sk

i=0 k−1 

si χ(k,i) (w) =

i=0

k>1 ∞ 

∞ 

si



χ(k−j,j) (w) =

j=0

χ(k−j,j) (w) = ψ (k) ,

∞ 

sk ψ (k) (w),

k>1 ∞ 

si ψ (k) (w),

i=1

k = 2, 3, · · · ,

j=0

χ(1,0) = ψ (1) , k  i=0

So

χ(k+1−i,i) = ψ (k+1) ,

k = 2, 3, · · · .

(5.9)

6.5 Formal Energy for Hamiltonian Algorithm

−h(0)

−h(1)

−h(2)

···

−h(k−1)

−h(k)

ψ (1)

χ(1,0)

χ(1,1)

χ(1,2)

···

χ(1,k−1)

χ(1,k)

ψ (2)

χ(2,0)

χ(2,1)

χ(2,2)

···

χ(2,k−1)

.. .

.. .

.. .

ψ (k)

χ(k,0)

χ(k,1)

ψ (k+1)

χ(k+1,0)

and

267

(5.10)

χ(1,0) = ψ (1) , χ(2,0) + χ(1,1) = ψ (2) , χ(3,0) + χ(2,1) + χ(1,2) = ψ (3) ,

(5.11)

.. . χ(k+1,0) + χ(k,1) + · · · + χ(2,k−1) + χ(1,k) = ψ (k+1) . Now if ψ (1) , ψ (2) , · · · , ψ (k) , ψ (k+1) , · · · are known, then h(0) = −χ(1,0) , h(1) = −χ(1,1) , · · · , h(k−1) = −χ(1,k−1) , h(k) = −χ(1,k) , · · · can be determined. We get: h(0) = −ψ (1) , h(1) = −ψ (2) + χ(2,0) , h(2) = −ψ (3) + (χ(3,0) + χ(2,1) ), (5.12)

.. . h(k) = −ψ (k+1) + (χ(k+1,0) + χ(k,1) + · · · + χ(2,k−1) ), .. .

So h(0) , h(1) , h(2) , · · · can be recursively determined by ψ (1) , ψ (2) , · · ·. So we get the ∞  si h(i) (z), and in case of convergence, it satisfies formal power series hs = i=0

 ght s t=s = F s . We now give a special example to show how to calculate the formal energy. Let us consider normal Darboux transformation with

268

6. The Calculus of Generating Functions and Formal Energy

5 V = −E =

O

−I

−I

O

5

6 αV−1 =

,

where 1 A1 = (JV J − J) = 2

5

A1

B1

C1

D1

O

O

−I

O

6 ,

6 .

Suppose we just use the first term of the generating function of the generating map αV , i.e., we just consider the first order scheme F s ∼ ψ(s, w) = −sH(w) =

∞ 

sk ψ (k) .

i=1

Let us assume ψ (1) = −H(w). If ψ (2) = ψ (3) = · · · = 0, then χ(1,0) = ψ (1) = −H. We need to calculate χ(2,0) . Since 5 6 Hp (1,0) χz =− , Hq 5 6 5 6 65 O O O −Hp (1,0) = = , A1 χz −I O Hp −Hq 5 6 Hpp Hpq (1,0) χzz = − . Hqp Hqq By formula (5.6), we get χ(2,0) =

1 2 × 1! 1 2

1 =− 2

i0 +i1 =0 k1 =1 l1 =1

(1,0) 

= (χz

5

2n  2n 

2n 

0) 1 ,i1 ) χ(1,i (A1 χ(k )l1 zl z 1

(1,0)

) A1 χz

Hp Hq

6 5

0 Hp

6 1 2

= − Hq Hp .

From formula (5.10), we get χ(2,0) + χ(1,1) = ψ (2) = 0 =⇒ χ(1,1) = −χ(2,0) =

1  H Hp . 2 q

In order to obtain χ(1,2) , we first determine χ(3,0) and χ(2,1) , and for the latter we need to caculate

6.5 Formal Energy for Hamiltonian Algorithm

⎡ (1,1)

χz

⎤

n ∂  Hqj Hpj ⎥ ⎢ ∂p ⎢ ⎥

=

1⎢ ⎢ 2⎢

j=1

n ⎣ ∂ 

∂q

5

(1,1)

(2,0)

A1 χ z

Hqj Hpj

⎥ ⎥ ⎥ ⎦

j=1

Hpq Hp + Hpp Hq

1 = 2

A1 χz

269

6

, Hqq Hp + Hqp Hq 5 6 O O (1,1) = χz −I O 5 6 O 1 , =− 2 Hpq Hp + Hpp Hq (1,1)

= −A1 χz 5 1 = 2

6

O

. Hpq Hp + Hpp Hq

For k = 2, i = 0, we have χ(3,0) =



1 1 3 1!

2n 

χ(1,0) (A1 χz(k1 ,0) )l1 zl

i0 +i1 =0 k1 =2 l1 =1

2n 

+

2n 2n  



1



(k1 ,0) χ(1,0) )l1 (A1 χz(k2 ,0) )l2 zl ,zl (A1 χz

l1 ,l2 =1 i0 +i1 +i2 =0 k1 +k2 =2

=

1

2

1  (2,0) T 1 (2,0) (1,0) T (1,0) (1,0) χz A1 χz + A1 χz χzz A1 χz 3 6 1 6

= − (HqT Hpq Hp + HqT Hpp Hq + HpT Hqq Hp ). For k = 1, i = 1, we have χ(2,1) =

1 2

2n 

2n  2n 

i0 +i1 =1 k1 =1 l1 =1

0) 1 ,i1 ) χz(1,i (A1 χ(k )l1 z l 1

=

   (1,1) T 1  (1,0) T (1,1) (1,0) A1 χ z + χz A1 χz χz 2

=

1 1 (H T Hpp Hq + HpT Hqq Hp ) + HqT Hpq Hp . 4 q 2



270

6. The Calculus of Generating Functions and Formal Energy

From (5.11), we have χ(3,0) + χ(2,1) + χ(1,2) = ψ (3) = 0 =⇒ χ(1,2) = −(χ(3,0) + χ(2,1) ). For k = 1, i = 2,  1 1 χ(1,2) = − − (HqT Hpp Hq + HpT Hqq Hp ) − HqT Hpq Hp 6 6  1 1 + (HqT Hpp Hq + HpT Hqq Hp ) + HqT Hpq Hp 4

= −

2

1 (H T Hpp Hq + HpT Hqq Hp + 4HqT Hpq Hp ). 12 q

Finally, we get the formal power series of energy h(s, z) = −(χ(1,0) + sχ(1,1) + s2 χ(1,2) ) + O (s3 ) s 2

= H(z) − Hq Hp +

s2 (H  Hpp Hq + Hp Hqq Hp + 4Hq Hpq Hq ) + O (s3 ). 12 q

t , and Now, let H(z) be a time-independent Hamiltonian, let its phase flow be gH let its generating function be

t (w) = φ(t, w) = φgH

∞ 

tk φ(k) (w).

k=1

Then we have

φ(1) (w) = −H(w),

for k ≥ 1, φ(k+1) (w) =

2n   1 (1) φwl1 ···wlm (w) (k + 1)m! m=1 l1 ,···,lm =1 k1 +···+km =k     (k1 ) m) (w) l . (5.13) · A1 φw (w) l · · · A1 φ(k w k 

1

m

Theorem 5.1. [Fen98a] Let F s be a Sp-diff operator of order m for Hamiltonian H, i.e., φ(s, w) − ψ(s, w) = O(|s|m+1 ), and ⎧ ⎪ ψ (1) (w) = φ(1) (w) = −H(w), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ψ (2) (w) = φ(2) (w), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ then

.. . ψ (m) (w) = φ(m) (w),

6.5 Formal Energy for Hamiltonian Algorithm

271

h(0) (w) = H(w), h(1) (w) = h(2) (w) = · · · = h(m−1) (w) = 0, i.e., h(s, w) − H(w) = o (|s|m ) and

h(m) (w) = ψ (m+1) (w) − φ(m+1) (w). 



Proof. First we show that χ(k+1,i) depends only on derivatives of χ(k ,i ) (k  ≤ k, i ≤ i). The recursion for i = 0 is the same as the recursion of phase flow generat  ing function with Hamiltonian χ(1,0) (w). For i ≥ 1, χ(k+1,i) = 0, if χ(k ,i ) = 0 for all i , k  , such that 1 ≤ i ≤ i, 1 ≤ k  ≤ k. We have ψ (1) = χ(1,0) = χ(1,0) recursion

−−−−−→ χ(2,0) , χ(3,0) , χ(4,0) , · · · ψ (2) = χ(1,1) + χ(2,0) =⇒ χ(1,1) recursion

−−−−−→ χ(2,1) , χ(3,1) , χ(4,1) , · · · ψ (3) = χ(1,2) + χ(2,1) + χ(3,0) =⇒ χ(1,2) recursion

−−−−−→ χ(2,2) , χ(3,2) , χ(4,2) , · · · ψ (4) = χ(1,3) + χ(2,2) + χ(3,1) + χ(4,0) =⇒ χ(1,3) recursion

−−−−−→ χ(2,3) , χ(3,3) , χ(4,3) , · · · .. . ψ (k) = χ(1,k−1) + χ(2,k−2) + · · · + χ(k,0) =⇒ χ(1,k−1) recursion

−−−−−→ χ(2,k−1) , χ(3,k−1) , χ(4,k−1) , · · · So, χ(k,i) can be generated successively through (5.9), (5.6). Then h(s, w) =

∞ 

si χ(1,i) (w).

i=0

Using equation H = ψ (1) = φ(1) = χ(1,0) and (5.9), (5.14) ,we get χ(2,0) = φ(2) , χ(3,0) = φ(3) , · · · , χ(k,0) = φ(k) , · · · Using Equation (5.14), we get ψ (2) = φ(2) = χ(1,1) + φ(2) =⇒ χ(1,1) = 0. Applying Equations (5.9) and (5.14), we get

(5.14)

272

6. The Calculus of Generating Functions and Formal Energy

χ(2,1) = 0 =⇒ χ(3,1) = χ(4,1) = · · · = χ(k,1) = · · · = 0. Applying equation ψ (3) = φ(3) = χ(1,2) + χ(2,1) + χ(3,0) = χ(1,2) + 0 + φ(3) =⇒ χ(1,2) = 0, then χ(2,2) = χ(3,2) = χ(4,2) = · · · = χ(k,2) = · · · = 0. Finally ψ (m) = φ(m) = χ(1,m−1) + χ(2,m−2) + · · · + χ(m−1,1) + φ(m) =⇒ χ(1,m−1) = 0, then χ(2,m−1) = χ(3,m−1) = χ(4,m−1) = · · · = χ(k,m−1) = · · · = 0. Since χ(k,i) = 0, ∀ i = 1, 2, · · · , m − 1 and k = 1, 2, 3 · · ·, then the equation ψ (m+1) = χ(1,m) + χ(2,m−1) + · · · + χ(m,1) + χ(m+1,0) =⇒ χ(1,m) = ψ (m+1) − φ(m+1) , so we finally get h(s, z) =

∞ 

si χ(1,i) = H(z) + sm (ψ (m+1) − φ(m+1) ) + O (|s|m+1 ),

i=0

i.e., h(s, z) − H(z) = sm (ψ (m+1) (z) − φ(m+1) (z)) + O(|s|m+1 ). So in particular, if F s ∼ ψ(s, w) is given by the truncation of phase flow generating function, i.e., ψ (1) = φ(1) = H, ψ (2) = φ(2) , · · · , ψ (m) = φ(m) , ψ (m+1) = φ(m+1) = 0, then h(s, z) = H(z) − O(|s|m+1 ). Therefore, the theorem is completed.



6.6 Ge–Marsden Theorem

273

6.6 Ge–Marsden Theorem In this section, we describe the result of Ge–Marsden, which talks about nonexistence of symplectic schemes that preserving energy. Due to the importance of preserving energy for a numerical method, extensive effort has been made by many people in searching for energy-preserving symplectic scheme, yet none of them is successful. Ge Zhong, a former Ph.D. student of Prof. Feng, first proved in his thesis [Ge88] the non-existence of the energy-preserving symplectic schemes. The result was published later in [GM88] by himself and Marsden, and now is called Ge–Marsden theorem. Let H be such a Hamiltonian function, where in its neighborhood of some level surface (energy surface), there exists no other conservation law exception energy. In other words, given a function f defined in a neighborhood of energy surface H = c0 , if {f, H} = 0, then f = g(H), where g is a function on R1 . A symplectic scheme can be regarded as a one-parameter family of symplectic transformation φτ (τ ≥ 0). A well-posed difference scheme should satisfy the consistency condition which ensures φτ depends smoothly on parameter τ . Now suppose that we have a symplectic scheme which preserves the energy, i.e., H ◦ φτ = H, τ where φτ maps energy surface H = c to itself. We denote the mapping φτ , gH restricted on to the level surface H = c respectively as τ |H=c , fH

φτ |H=c .

Theorem 6.1 (G–M theorem). [Ge88] There exists a function τ = τ (c, t)defined on a neighborhoold of 0 ∈ R, such that   φτ (c,t) H=c = g t H=c . This means that if we can find a symplectic scheme preserving energy, we can solve the original Hamiltonian system equivalently by a reparametrization of time parameter in t phase flow gH . In general this is impossible. The proof of above Theorem 6.1 bases on the following Lemma 6.2. t t Lemma 6.2. Let gA , gA be solutions of following systems of ODE respectively. 1 2

dx = A1 (x, t), dt

dx = c(t)A1 (x, t), dt

where c(t) is function of t, then τ (t)

t = gA2 , gA 1

where τ (t) is the solution of following system: dτ = c(t), dt

τ (o) = 0.

274

6. The Calculus of Generating Functions and Formal Energy



Proof. omit. Now we give a proof of Theorem 6.1.

Proof. Let F (z, τ ) be a Hamiltonian function, whose phase flow is φτ . Then from H ◦ φτ = H and Theorem2.20 of Chapter 3 we get {F (·, τ ), H} = 0. According to the assumption, there exists a F1 such that:   F (z, τ ) = F1 H(z), τ , and the vector field generated by Hamiltonian function F (z, τ ) is J −1 F1 (H(z), τ )Hz , which is tangent to the energy surface H = c. Its restriction to the level surface H = c is   J −1 F1 (c, τ )Hz  . H=c

−1

It differs from the restriction of vector field J Hz to the level surface H = c only  by a constant F1 (c, τ ). By Lemma 6.2 the proof is completed. All symplectic transformations that keep H invariant compose a group S(H). S(H) is a rigidity under which S0 (H) is a contained connected support set of unit transformation of group S(H). S0 (H) induces the level surface H = c by the set of all transformation, denoted by S0 (H)|H=0 . Then S0 (H) is a curve  t  S0 (H) = {gH , t ∈ R}. H=c Note that the rigidity of S0 (H) exactly counteracts the existence of energy-preserving symplectc scheme.

Bibliography

[Fen98] K. Feng: The calculus of generating functions and the formal energy for Hamiltonian systems. J. Comput. Math., 16:481–498, (1998). [FQ87] K. Feng and M.Z. Qin: The symplectic methods for the computation of Hamiltonian equations. In Y. L. Zhu and B. Y. Guo, editors, Numerical Methods for Partial Differential Equations, Lecture Notes in Mathematics 1297, pages 1–37. Springer, Berlin, (1987). [FQ91a] K. Feng and M.Z. Qin: Hamiltonian Algorithms for Hamiltonian Dynamical Systems. Progr. Natur. Sci., 1(2):105–116, (1991). [FQ91b] K. Feng and M.Z. Qin: Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study. Comput. Phys. Comm., 65:173–187, (1991). [FQ03] K. Feng and M. Z. Qin: Symplectic Algorithms for Hamiltonian Systems. Zhejiang Press for Science and Technology, Hangzhou, in Chinese, First edition, (2003). [Ge88] Z. Ge: Symplectic geometry and its application in numerical analysis. PhD thesis, Computer Center, CAS, (1988). [Ge91] Z. Ge: Equivariant symplectic difference schemes and generating functions. Physica D, 49:376–386, (1991). [GM88] Z. Ge and J. E. Marsden: Lie–Poisson Hamilton–Jacobi theory and Lie–Poisson integrators. Physics Letters A, pages 134–139, (1988). [GW95] Z. Ge and D.L. Wang: On the invariance of generating functions for symplectic transformations. Diff. Geom. Appl., 5:59–69, (1995).

Chapter 7. Symplectic Runge–Kutta Methods

In this chapter we consider symplectic Runge–Kutta (R–K) method.

7.1 Multistage Symplectic Runge–Kutta Method Now we study Multistage Symplectic Runge–Kutta Method. A key feature of the R–K method is using the linear combination of the first-order derivatives of the numerical solution of differential equations to achieve the higher-order approximation.

7.1.1 Definition and Properties of Symplectic R–K Method Consider the following Hamiltonian system: dpi ∂H =− , dt ∂qi dqi ∂H = , dt ∂pi

i = 1, 2, · · · , n,

(1.1)

where H = H(p1 , · · · , pn , q1 , · · · , qn ) is a Hamiltonian function independent of t. For t-dependent Hamiltonian (e.g., nonautonomous system), we can introduce two new variables and transform the system into another one which has (1.1) form[Qin96,Gon96] . In order to facilitate the expression, we denote ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ z1 p1 Hz1 ⎢ .. ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎢ ⎢ pn ⎥ ⎢ zn ⎥ ⎥ ⎥=⎢ ⎥ , H z = ⎢ H zn ⎥ , z=⎢ ⎢ ⎢ q1 ⎥ ⎢ zn+1 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎦ ⎢ . ⎥ ⎢ . ⎥ ⎣ ⎣ .. ⎦ ⎣ .. ⎦ Hz2n qn z2n ! O In J = J2n = , J  = J −1 = −J, −In O where In is n × n identity matrix, J is a standard symplectic matrix. Using this notation, we can rewrite Equation (1.1) into K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

278

7. Symplectic Runge–Kutta Methods

dz = J −1 Hz , dt

(1.2)

or

dz = J −1 Hz = f (z). dt The s-stage R–K method for (1.3) has the following form: z k+1 = z k + h

s 

(1.3)

bi f (Yi ),

i=1

Yi = z k + h

s 

(1.4)

aij f (Yj ),

1 ≤ i ≤ s,

j=1

where h = tk+1 − tk (k ≥ 0), bi , aij (i, j = 1, 2, · · · , s) are real parameters. The properties of a R–K method (consistency, accuracy, and stability, etc.) are determined completely by these parameters. In scheme (1.4), if j ≥ i (1 ≤ i ≤ s), aij = 0, then all Yi i−1  Yi = z k + h aij f (Yj ) + aii f (Yi ), 1 ≤ i ≤ s, (1.5) j=1

can be computed in an explicit way from Y1 , Y2 , · · · , Yi−1 . Such scheme is therefore called explicit R–K scheme. In scheme (1.4), when j > i (1 ≤ i < s), aij = 0, and has certain aii = 0 in the diagonal line (1 ≤ i ≤ s), the scheme is called semiimplicit scheme. Each Yi may defined implicitly by a 2n-dimensional equation. The importance of semi-implicit methods is that the computation of Y1 , Y2 , · · · , Yi−1 can be carried out in sequence as s system of 2n algebraic equation rather than as one system of s × 2n equations. Sometimes we referred this scheme as diagonal implicit. If the method is neither explicit, nor diagonally implicit, and is just called implicit, then all Yi must be computed simultaneously. Explicit methods are much easier to apply than implicit ones. On the other hand implicit methods possess good stability properties. Butcher in[But87] proposed the so-called Butcher-array which provides a condensed representation of the R–K method (1.4),

c

A bT

where ci =

s 

c1

a11

···

a1s

c2 .. . cs

a21 .. . as1

··· ···

a2s .. . ass

b1

···

bs

(1.6)

aij (i = 1, 2, · · · , s). Thus, a s-stage R–K method is determined

j=1

completely by the mathmatical tableau (1.6). Therefore this kind of expression is often called the Butcher tableau (or form).

7.1 Multistage Symplectic Runge–Kutta Method

279

We regard a single-step difference scheme as a transition mapping from time tk to tk+1 . Definition 1.1. A symplectic R–K method is a R–K method whose transitional transformation of (1.4), i.e., Jacobian matrix

∂ z k+1 is symplectic. ∂z k

Definition 1.2. An s-stage R–K method is said to satisfy simplifying condition if B(p) : C(η) : D(ζ) :

s  i=1 s  j=1 s  i=1

bi ck−1 = i

1 , k

aij ck−1 = j

k = 1(1)p,

cki , k

bi ck−1 aij = i

bj (1 − ckj ) , k

k = 1(1)η, j = 1(1)s,

k = 1(1)ζ,

where A is s×s matrix, b and c are s×1 vectors of weights and abscissae, respectively. In 1964 Butcher proved the following fundamental theorem [But87] : Theorem 1.3. If the coefficients A, b, c of a R–K method satisfy B(p), C(η), D(ζ) (p ≤ η + ζ + 1, and p ≤ 2η + 2), then the R–K method is of order p[HNW93] . R–K method is based on high order quadrature rule. Thus one can derive a R–K method of order s for any set of distinct abscissas ci (i = 1, · · · , s). A high order can be obtained for the following special sets of abscissas: 1◦ Using shifted zeros of the Gauss–Legendre polynomial to obtain ci and condition C(s) of Definition 1.2 to obtain the Gauss–Legendre method. 2◦ Using zeros of Radau polynomial:  ds−1  s x (x − 1)s−1 (left Radau), (1) dxs−1  ds−1  s−1 (2) x (x − 1)s (right Radau), s−1 dx with condition D(s) of Definition 1.2 to obtain Radau I A method, or with condition C(s) to obtain Radau II A method. 3◦ Using zeros of Lobatto polynomial  ds−2  s−1 x (x − 1)s−1 s−2 dx with coefficients bi satisfying condition 1.2 B(2s − 2) to obtain (1) Lobatto III A if aij is determined by C(s); (2) Lobatto III B if D(s) is satisfied; (3) Lobatto III C if ai1 = b1 , ∀ i = 1, · · · , s, and the rest of aij is determined by C(s − 1). Radau I A:

280

7. Symplectic Runge–Kutta Methods

0

1 1

0

1 4

2 3

1 4

5 12

1 4

3 4

−

1 4

Radau II A:

1

1 3

5 12

1

3 4

1 4

3 4

1 4

−

1 12

1 1

Lobatto III A:

0

0

0

1

1 2

1 2

0

0

0

1 2

5 24

1 3

1

1 6

2 3

1 6

1 6

2 3

1 6

0

1 6

−

1 2

1 6

1 3

0

1

1 6

5 6

0

1 6

2 3

1 6

1 2

1 2

0 −

1 24

Lobatto III B:

0 1

1 2 1 2 1 2

Lobatto III C:

1 6

0

0 0 1 2

7.1 Multistage Symplectic Runge–Kutta Method

0 1

1 2

−

1 2

1 2

1 2

1 2

1 6

1 2

1 6

5 12

1

1 6

2 3

1 6

1 6

2 3

1 6

1 2

1 3

1 6

0

−

281

−

1 12

We present a table of these conditions for methods which are based on high order quadrature rule, see Table 1.1. Table 1.1.

The simplified conditions for s-stage method based on high order quadrature rule

method

simplified

Gauss–Legendre B(2s)

condition order of accuracy C(s)

D(s)

2s

Pad´e approx (s, s)

Radau I A

B(2s − 1) C(s − 1) D(s)

2s − 1

(s − 1, s)

Radau II A

B(2s − 1) C(s)

D(s − 1)

2s − 1

(s − 1, s)

Lobatto III A

B(2s − 2) C(s)

D(s − 2)

2s − 2

(s − 1, s − 1)

Lobatto III B

B(2s − 2) C(s − 2) D(s)

2s − 2

(s − 1, s − 1)

Lobatto III C

B(2s − 2) C(s − 1) D(s − 1)

2s − 2

(s − 2, s)

7.1.2 Symplectic Conditions for R–K Method In this subsection a sufficient condition for R–K method to be symplectic is given. Let B = diag[b1 , b2 , · · · , bs ] be a diagonal matrix, M = BA + A B − bb . The following condition was first proposed by Sanz-Serna during his visit to China[SS88] . Theorem 1.4. If M = 0, then an s-stage R–K method (1.4) is symplectic[SS88,Las88,Sur88] . Proof. Here we give our own proof[QZ92a] . To prove the scheme (1.4) is symplectic when M = 0, we only need to verify the Jacobian matrix is symplectic. From the scheme (1.4) we have  ∂z k+1 ∂Y = I + h bi Df (Yi ) ki , ∂z k ∂z s

(1.7)

i=1

 ∂Yi ∂Y =I +h aij Df (Yj ) kj , ∂z k ∂z s

j=1

where D f is the derivative of function f .

1 ≤ i ≤ s,

(1.8)

282

7. Symplectic Runge–Kutta Methods

Denote Di = D f (Yi ),

∂Yi = Xi (i = 1, 2, · · · , s), and let f = J −1 Hz , then ∂z k

JDi + Di J = 0, and %

∂z k+1 ∂z k

&

∂z k+1 J = J +h ∂z k

* + h

* s 

+

*

bi Di Xi

i=1 s 

(1.9)

J + hJ * J

h

i=1

= J +h

s 

+h

2

bi Di Xi

s 

+

bi Di Xi

i=1

bi [(Di Xi ) J + JDi Xi ]

i=1

* s 

+

i=1

+

bi Di Xi

s 

+ bi Di Xi

* J

i=1

s 

+ bi Di Xi

.

i=1

It follows from (1.8) (Di Xi ) JXi = (Di Xi ) J + h

s 

aij (Di Xi ) JDj Xj ,

j=1

(Xi ) JDi Xi = JDi Xi + h

s 

aij (Dj Xj ) JDi Xi .

j=1

Using Equation (1.9), we obtain %

∂z k+1 ∂z k

&

∂z k+1 J = J+ ∂z k

+h

* h

s 

+ bi Di Xi

* J

h

i=1

s 

s 

+ bi Di Xi

i=1

bi [Xi Di JXi + Xi JDi Xi ]

i=1

−h

s 

bi h

i=1

+h

s 

aij (Di Xi ) JDj Xj

j=1

s 



!

aij (Dj Xj ) JDi Xi

j=1

= J + h2

s s   (bi bj − bi aij − bj aji )(Di Xi ) JDj Xj . i=1 j=1

It is easy to see that if M = 0, then

7.1 Multistage Symplectic Runge–Kutta Method

%

∂ z k+1 ∂ zk

& J

283

∂ z k+1 = J, ∂ zk

i.e., the Jacobian matrix of transitional mapping

∂z k+1 is symplectic. ∂z k



Remark 1.5. If R–K method is non-reducible, then condition M = 0 is also necessary. From subsection 7.1.1 we know that a R–K method is determined completely by the coefficients ci , aij , bi (i, j = 1, · · · , s). Now we introduce the Gauss–Legendre method: let ci (i = 1, · · · , s) be zeros of shifted Legendre polynomial Qs (x), where the Legendre polynomials are defined as ds {(x2 − 1)s }, d xs   1 Qs (x) = Ps x − . 2

Ps (x) =

1

(1.10)

2s s!

(1.11)

Let this method satisfy simplified conditions B(s) and C(s). Solve equations

s 

bi ck−1 i

i=1

 1 1 = (1 ≤ k ≤ s) for bi (i = 1, · · · , s), and solve equations aij ck−1 = cki (1 ≤ j k k s

j=1

k ≤ s, 1 ≤ i ≤ s) for aij (i, j = 1, · · · , s). Then the scheme determined by bi and aij is the only R–K method that has achieved 2s-order of accuracy. We listed Butcher’s tableau for s ≤ 2 as follows s = 1: 1 2

1 2

(1.12)

1 s = 2:

√ 3− 3 6 √ 3+ 3 6

1 4

√ 3−2 3 12

√ 3+2 3 12

1 4

1 2

1 2

(1.13)

It is easy to see that s = 1 is exactly the case of the Euler centered scheme: % & 1 k k+1 k k+1 (z + z = z + hf ) . (1.14) z 2

284

7. Symplectic Runge–Kutta Methods

It is not difficult to verify that both schemes (1.12) and (1.13) satisfy the conditions M = 0, and hence are symplectic. Furthermore, we have the following conclusions: Theorem 1.6. An s-stage Gauss–Legendre method is a symplectic scheme with 2sorder of accuracy. Proof. Since the scheme satisfies conditions D(s), C(s), B(2s), i.e., s 

bi aij cl−1 = i

i=1

=

1 1 1 bj (1 − clj ) = bj − bj clj l l l s 

bi bj cl−1 − i

i=1

s 

bj aji cl−1 i ,

i=1

which results in s 

(bi aij + bj aji − bi bj )cl−1 = 0, i

l, j = 1, 2, · · · , s.

i=1

Since c1 , c2 , · · · , cs are not equal mutually, we obtain M = 0.



7.1.3 Diagonally Implicit Symplectic R–K Method In this subsection, we will give some diagonal symplectic R–K formulas. These schemes not only have advantages with regards to computational convenience and good stability, but also are symplectic. Let us consider a diagonally s-stage implicit R–K method that satisfies M = 0. Without loss of generality we assume that bi = 0 (i = 1, 2, · · · , s). Because of the condition M = 0, we have bi bj − bi aij − bj aji = 0,

i, j = 1, 2, · · · , s.

(1.15)

If bk = 0, then bi aik = 0 (i = 1, 2, · · · , s), the method is equivalent to a method with fewer stages. The following theorem is first proposed by the authors, sees the literature [QZ92a,SA91] . Theorem 1.7. If an s-stage diagonally implicit method satisfies M = 0, then we can write the method in the following form: c1

b1 2

c2

b1

b2 2

c3 .. .

b1 .. .

b2 .. .

cs

b1

b2

b3

···

bs 2

b1

b2

b3

···

bs

b3 2

(1.16)

.. .

7.1 Multistage Symplectic Runge–Kutta Method

where ci =

i 

bj−1 +

j=1

285

bi (i = 1, · · · , s, b0 = 0). 2

Proof. Since the scheme is diagonally implicit, aij = 0 (j > i); to satisfy M = 0, we have bi bj − bi aij − bj aji = 0 (i, j = 1, 2, · · · , s), which results in aij = bj ,

bi , 2

aii =

i = 1, · · · , s,

i > j. 

The theorem is proved.

Corollary 1.8. Explicit R–K method with any order does not satisfy condition M = 0. Remark 1.9. Tableau (1.16) Cooper[Coo87] has discussed the condition (1.15) and constructed a method of family (1.16) with s = 3 and order 3. Below we give diagonally implicit symplectic R–K methods for s ≤ 3: s = 1: 1 2

1 2

(1.17)

1 s=2: 1 4

1 4

0

3 4

1 2

1 4

1 2

1 2

(1.18)

s=3: 1 a 2

1 a 2

3 a 2

a

1 +a 2

1 a 2

(1.19)

a

a

1 −a 2

a

a

1 − 2a

where a = 1.351207, which is a real root of polynomial 6x3 − 12x2 + 6x − 1 [Coo87] . The above three schemes have accuracy o(Δt2 ), o(Δt2 ), o(Δt3 ) respectively.

286

7. Symplectic Runge–Kutta Methods

Corollary 1.10. If s = 3, and the elements in Butcher tableau are taken in symmetrical version (a11 = a33 ). 1 a 2

1 a 2

1 2

a

1 1− a 2

1 −a 2

(1.20)

a

1 − 2a

1 a 2

a

1 − 2a

a

Then this scheme has 4th-order accuracy. In Chapter 8 we will see that this is a typical example that using Euler centered scheme and multiplication extrapolation to achieve 4th order accuracy. Now we consider s = 4, Butcher tableau can be represented as follows: b1 2

b1 2

b1 +

b2 2

b1

b2 2

b 1 + b2 +

b3 2

b1

b2

b3 2

b1 + b 2 + b 3 +

b4 2

b1

b2

b3

b4 2

b1

b2

b3

b4

(1.21)

We expect this method to have 4th-order accuracy. According to Taylor expansion, the coefficients in the method must satisfy the system of equations: s  i=1 s  i=1 s 

bi = 1,

(1.22) 1 2

(1.23)

1 3

(1.24)

b i ci = , bi c2i = ,

i=1 s  i,j=1

1 6

bi aij cj = ,

(1.25)

7.1 Multistage Symplectic Runge–Kutta Method s 

1 4

bi c3i = ,

i=1 s  i,j=1 s 

(1.26) 1 8

(1.27)

1 , 12

(1.28)

bi ci aij cj = , bi aij c2j =

i,j=1 s 

287

bi aij ajk ck =

i,j,k=1

1 . 24

(1.29)

Now we have 8 equations with 4 unknowns. Luckily we find a set of solutions using computer, which is b1 = −2.70309412,

b2 = −0.53652708,

b3 = 2.37893931,

b4 = 1.8606818856.

Perhaps we can reduce the equations to the form of 4 equations with 4 unknowns and s  bi = 1, bi bj − bi aij = 0 (i, j = get the exact solution. For an example, using 1, 2, · · · , s), we have

s 

bi ai,j =

i=1,j=1

s  i=1

i=1

bi ci =

1 . So we can remove Equation 2

(1.23) from the system. In an implementation of this R–K method, we rewrite it in the following form: b1 h f (Y1 ), 2 b h Y2 = 2Y1 − z k + 2 f (Y2 ), 2 b h Y3 = 2Y2 − (2Y1 − z k ) + 3 f (Y3 ), 2 b h Y4 = 2Y3 − (2Y2 − 2Y1 + z k ) + 4 f (Y4 ), 2

Y1 = z k +

(1.30)

z k+1 = 2Y4 − (2Y3 − 2Y2 + 2Y1 − z k ). Corollary 1.11. This scheme (1.30) can be obtained by applying the implicit midpoint scheme over 4 steps of length b1 h, b2 h, b3 h, b4 h. It has 4-th order accuracy. Let

288

7. Symplectic Runge–Kutta Methods

1

z 4 = z 0 + b1 hf 2

1

3

2





z 4 = z 4 + b2 hf 

z 4 = z 4 + b3 hf 3

z 1 = z 4 + b4 hf



1

z 4 + z0 , 2



2

1



3

2



z4 + z4 , 2

(1.31)

z4 + z4 , 2 3



z1 + z 4 , 2

Rewrite it in the following form: 1

z 4 + z0 b = z 0 + 1 hf 2 2 2

%

1

1 z4 + z4 b = z 4 + 2 hf 2 2 3

2

2 z4 + z4 b = z 4 + 3 hf 2 2 3

3 z1 + z 4 b = z 4 + 4 hf 2 2

Let

1

z0 + z 4 = Y1 , 2 2

3

z4 + z4 = Y3 , 2

1

1

z 4 + z0 2

% % %

2

& , 1

z4 + z4 2 3

2

z4 + z4 2 3

z1 + z 4 2

& , &

(1.32) ,

& .

2

z4 + z4 = Y2 , 2 3

z 4 + z1 = Y4 , 2

then (1.32) becomes scheme (1.30). There are similar results for s ≤ 3. More detail can be seen later in Section 8.1. All schemes proposed in this section can be applied to general ODE’s as well. Exercise 1.12. Does there exist 5-stage diagonally implicit R–K method with 5thorder accuracy?

7.1.4 Rooted Tree Theory 1. High order derivatives and rooted tree theory The basic method to construct the numerical scheme for ordinary differential equations is Taylor expansion. If only a single (scalar) equation is considered, Taylor expansion can be used in studying the convergence, compatibility, and order conditions for R–K methods. However, if the system of differential equations are considered, Taylor expansion is intractable. Consider system of ODE’s: y˙ = f (y),

y(0) = η,

f : Rm → Rm ,

m > 1.

(1.33)

7.1 Multistage Symplectic Runge–Kutta Method

289

For brevity, let m = 2, and y = (1 y, 2 y) , f = (1 f, 2 f ) , introduce the following notations ∂if ∂ 2 (i f ) i , fj := j , i fjk := j ∂( y) ∂( y)∂(k y) we have 1 (1)

= 1 f,

2 (1)

= 2 f,

1 (2)

= 1 f1 (1 f ) + 1 f2 (2 f ),

2 (2)

= 2 f1 (1 f ) + 2 f2 (2 f ).

y y

y y

Using matrix and vector symbols, we have 6 5 1 f1 1 f2 f. y (2) = 2 f1 2 f2

(1.34)

(1.35)

The second order derivative can be expressed via Jacobian matrix. However, the third-order derivative y (3) , can no longer be expressed via matrix and vector symbol, not to mention the higher order derivative. This has motivated people to study the structure of the Taylor expansion of high order derivatives and search for a better symbol to simplify the Taylor expansion of high order derivatives. Then the rooted tree theory[But87,Lam91,HNW93,SSC94] (With the tree roots skill to express high order derivative) emerged. Take y(3) as an example:   1 (3) y = 1 f11 (1 f )2 + 1 f12 (1 f )(2 f ) + 1 f1 1 f1 (1 f ) + 1 f2 (2 f )   +1 f21 (2 f )(1 f ) + 1 f22 (2 f )2 + 1 f2 2 f1 (1 f ) + 2 f2 (2 f ) ,   2 (3) y = 2 f11 (1 f )2 + 2 f12 (1 f )(2 f ) + 2 f1 1 f1 (1 f ) + 1 f2 (2 f )   +2 f21 (2 f )(1 f ) + 2 f22 (2 f )2 + 2 f2 2 f1 (1 f ) + 2 f2 (2 f ) . (1.36) Definition 1.13. Let z, f (z) ∈ Rm , f (M ) (z) be the M -th Frechet derivatives of f . It is an operator on Rm × Rm × · · · × Rm (M times), and is linear in each operand, f (M ) (z)(K1 , K2 , · · · , KM ) m  m m  m   i = ··· fj1 j2 ···jM j1 K1j2 K2 · · · i=1 j1 =1 j2 =1

jM

KM · ei ,

(1.37)

jM =1

where z is the argument, K1 , K2 , · · · , KM operands, and Kt = i

and

1

Kt , 2 Kt , · · · , m Kt

T

∈ Rm ,

t = 1, 2, · · · , M,

M

fj1 j2 ···jM =

∂ i f (z) ∂(j1 z)∂(j2 z) · · · ∂(jM z)

ei = [0, 0, · · · , 0, 0123 1 , 0, · · · , 0]T ∈ Rm . i

(1.38)

290

7. Symplectic Runge–Kutta Methods

We have the following comments: (1) The value of f (M ) (z)(· · ·) is a vector in Rm . (2) Repeated subscripts are permitted in (1.37), so that all possible partial derivatives of order M are involved. Thus, if M = 3, m = 2, the following partial derivatives will appear: ∂ 3 (i f ) ; ∂(1 z)3

i

f111 =

i

f112 = i f121 = i f211 =

i

f122 = i f212 = i f221 =

∂ 3 (i f ) , ∂(1 z)∂(2 z)2

i

∂ 3 (i f ) , ∂(1 z)2 ∂(2 z)

f222 =

∂ 3 (i f ) , ∂(2 z)3

i = 1, 2.

(3) The argument z simply denotes the vector with respect to whose component we are performing the partial differentiations. (4) An M times Frechet derivatives has M operands. This is an important property to note. Take m = 2, we have Case M = 1, f (1) (z)(K1 ) =

2 2  

i

fj1 (j1 K1 )ei

i=1 j1 =1

5 = where i

f1 =

∂(i f ) , ∂(1 z)

i

1

f1 (1 K1 ) + 1 f2 (2 K1 )

2

f1 (1 K1 ) + 2 f2 (2 K1 )

f2 =

∂(i f ) , ∂(2 z)

6 ,

(1.39)

i = 1, 2.

Replace z with y, and K1 with f , (1.39) becomes 5 1 1 6 f1 ( f ) + 1 f2 (2 f ) f (1) (y)(f (y)) = = y (2) . 2 f1 (1 f ) + 2 f2 (2 f )

(1.40)

(1.40) can be briefly denoted as y (2) = f (1) (f ).

(1.41)

Case M = 2, f (2) (z)(K1 , K2 ) = 5 =

2  2 2  

i

fj1 j2 (j1 K1 )(j2 K2 )ei

i=1 j1 =1 j2 =1 1

1

1

f11 ( K1 )( K2 ) + 1 f12 (1 K1 )(2 K2 ) + 1 f21 (2 K1 )(1 K2 ) + 1 f22 (2 K1 )(2 K2 )

2

f11 (1 K1 )(1 K2 ) + 2 f12 (1 K1 )(2 K2 ) + 2 f21 (2 K1 )(1 K2 ) + 2 f22 (2 K1 )(2 K2 )

Use y to replace z, and let K1 = K2 = f, we obtain

6 .

7.1 Multistage Symplectic Runge–Kutta Method

5 f (2) (y)(f (y), f (y)) =

1

f11 (1 f )2 + 2(1 f12 )(1 f )(2 f ) + 1 f22 (2 f )2

2

f11 (1 f )2 + 2(2 f12 )(1 f )(2 f ) + 2 f22 (2 f )2

291

6 ,

(1.42)

which is only part of the right side of (1.36), but not all. The absent terms are    6 5 1 1 1 f1 f1 ( f ) + 1 f2 (2 f ) + 1 f2 2 f1 (1 f ) + 2 f2 (2 f ) (1.43)     . 2 f1 1 f1 (1 f ) + 1 f2 (2 f ) + 2 f2 2 f1 (1 f ) + 2 f2 (2 f ) Now if we replace the operand f (y) with f (1) (y)(f (y)) in (1.40), the result is exactly (1.43). Hence, shortening the notation as in (1.41), (1.36) can be written as y (3) = f (2) (f, f ) + f (1) (f (1) (f )).

(1.44)

Thus we have seen that y (2) is a single Frecht derivative of order 1, and that y (3) is a linear combination of Frecht derivatives of order 1 and 2. In general, y (p) turns out to be a linear combination of Frecht derivatives of order up to p − 1. The components in such linear combination are called elementary differentials. Definition 1.14. The elementary differential Fs : Rm → Rm of f , and their order are defined respectively by 1◦ f is only elementary differential of order 1, and 2◦ if Fs (s = 1, 2, · · · , M ) are elementary differential of order rs , then the Frecht derivative (1.45) F (M ) (F1 , F2 , · · · , FM ), is an elementary of order 1+

M 

rs .

(1.46)

s=1

Remark 1.15. We have following: 1◦ The elementary differential F1 , F2 , · · · , FM appearing as operands in (1.45) need not be distinct. 2◦ {F1 , F2 , · · · , FM } : f (M ) (F1 , F2 , · · · , FM ). (1.47) 3◦

The order of the elementary differential (1.47) is, by (1.46), the sum of the M  orders of the elementary differentials plus 1, i.e., 1 + rs , where 1 is “for the bracks=1

ets”. Order 1 has only one elementary differential, i.e., f . Order 2 has only one elementary differential, i.e., f (1) (f ) = {f }. Order 3 has two elementary differentials, i.e., M = 2 =⇒ operand f, f =⇒ elementary differential {f 2 }, M = 1 =⇒ operand f (1) (f ) = {f } =⇒ elementary differential {{f }} = {2 f }2 .

292

7. Symplectic Runge–Kutta Methods

Order 4 has four elementary differentials, i.e., M = 3 =⇒ operandf, f, f M = 2 =⇒ operand f, {f } operand {f 2 } M = 1 =⇒ operand {2 f }2

elementary differential {f 3 }, elementary differential {f {f }2 (≡ {2 f }f }), elementary differential {2 f 2 }2 , elementary differential {3 f }3 .

=⇒ =⇒ =⇒ =⇒

2. Labeled graph Let n be a positive integer. A labeled n-graph g is a pair {V, E} formed by a set V collection with card (V ) = n, and a set E of unordered pairs (v, w) as a collection of elements, of which v, w are point of the set V , and v = w. Therefore g may be empty. V and E elements are known as the vertices and edges respectively. Two vertices v, w is said to be the adjacent if (v, w) ∈ E. Fig. 1.1 shows labeled graph for n = 2, 3, 4. ◦ i

◦ j

◦ i

◦ j

n=2 l ◦

◦

n=3

k

j ◦ i

◦ j

◦ j

◦ k

◦ l i

◦ i Fig. 1.1.

◦ k

◦

k ◦

◦

◦

l

n=4

Labeled graph for n = 2, 3, 4

A graph can have many different types of label. However for the same graph, there exists a isomorphic mapping χ between two different labels. For an example, for one of the graphs in Fig. 1.1, depicted also in Fig. 1.2, we can take two types of label, as shown in Fig. 1.3. ◦ i Fig. 1.2.

◦ k

◦ l

Labeled of graph 4

◦ i Fig. 1.3.

◦ j

◦ j

◦ k

2 kind labeled of graph

◦ l

◦ m

◦ n

◦ p

◦ q

7.1 Multistage Symplectic Runge–Kutta Method

293

We take mapping χ to be χ : i −→ m,

j −→ n,

k −→ p,

l −→ q,

i.e., χ : V1 −→ V2 , then

V1 = {i, j, k, l},

V2 = {m, n, p, q},

(i, j) −→ (m, n),

χ:

(j, k) −→ (n, p), (k, l) −→ (p, q), i.e.,

E1 −→ E2 ,

χ:

E1 = {(i, j), (j, k), (k, l)}, E2 = {(m, n), (n, p), (p, q)}. Therefore χ : Lg1 → Lg2 , where Lg1 = {V1 , E1 }, Lg2 = {V2 , E2 } are two different labels in Fig. 1.3. In fact, if there exists a isomorphic mapping between two labels g1 , g2 , they can be regarded as two types of labels for the same tree. Therefore a graph is an equivalent class, which consists of a variety of different labeled graphs corresponding to different types of label. These labeled graphs are equivalent, i.e., there exists an isomorphic mapping between them. 3. Relationship between rooted tree and elementary differential Next we can see that there is a 1 to 1 correspondence between elementary and trees. (1) Let f be the unique elementary differential of order 1. Then f corresponds to the unique tree of order 1, which consists of a single vertex. (2) If the elementary differential Fs of order rs (s = 1, 2, · · · , M ) corresponds to trees ts of order rs (s = 1, 2, · · · , M ), then the elementary differential M M   rs corresponds to the tree of order 1 + rs , {F1 , F2 , · · · , FM } of order 1 + 1

obtained by grafting the M trees Fs (s = 1, 2, · · · , M ) onto a new root. Example 1.16. If F1 ∼ t1

◦ ◦

◦ ,

F2 ∼ t2 =

◦

,

F 3 ∼ t3 =

◦

s=1

◦ ◦ , then

◦ ◦ ◦ ◦◦ ◦ ◦ ◦ ◦ {F1 , F2 , F3 } ∼ ◦ ◦ . ◦ We need a notation to represent trees similar to the notation for elementary differential. All trees can be labeled with combination of the symbol of τ for the unique tree of order 1 (consisting of a single node) and the symbol [· · ·], meaning we have grafted the trees appearing between the brackets onto a new root. We shall denote n copies of [[[···[ ]···]]] tree t1 by tn1 , ktimes by [k , and ktimes by ]k . For example

294

7. Symplectic Runge–Kutta Methods

t1 = [τ ]= @

A

@

t2 = τ [τ ]

A

◦ ◦

, ◦

@

= [τ ]τ = τ [τ ]2 = @ A t3 = [t1 , t22 ] = [τ ] [τ [τ ]], ] [τ [τ ] ]

◦

◦

,

◦ ◦

◦ ◦ ◦ = [2 τ ] [τ [τ ]2 [τ [τ ]3 = ◦ . ◦ ◦ ◦ Definition 1.17. The order r(t), symmetry σ(t) and density (tree factorial) γ(t) are defined by ◦

◦

◦

r(τ ) = σ(τ ) = γ(τ ) = 1, and   r [tn1 1 tn2 2 · · ·] = 1 + n1 r(t1 ) + n2 r(t2 ) + · · · , (number of vertices)    n1  n σ(t2 ) 2 · · · , σ [tn1 1 tn2 2 · · ·] = n1 !n2 ! · · · σ(t1 )     n1  n γ(t2 ) 2 · · · . γ [tn1 1 tn2 2 · · ·] = r [tn1 1 tn2 2 · · ·] γ(t1 ) Let α(t) (tree multiplicity) be the number of essentially different ways of labeling the vertices of the tree t with the integers 1, 2, · · · , r(t) such that labels are monotone increase. Essentially different labeling is illustrated in the following examples: Example 1.18. t1 = [τ 3 ] =

◦ ◦

2 ◦ ◦ ◦ , its labeling trees are

◦ 1

3◦ 4 4 ◦ and ◦

◦ 1

2◦ 3 ◦,

are not regarded as essentially different labelings, hence α(t1 ) = 1. 4 ◦

◦ @ A ◦ Example 1.19. t2 = τ [τ ] =

◦ ◦

, its labeling trees are

3 ◦ and

2◦ ◦ 1

4 ◦ ◦3 3◦ ,

◦2 ◦ 1

◦2

4◦ ◦ 1

are regarded as essentially different labelings, and α(t2 ) = 3.

From above, we have a easy way of computing α(t), namely α(t) =

r(t) ! . σ(t)γ(t)

(1.48)

7.1 Multistage Symplectic Runge–Kutta Method

295

4. Order conditions for multi-stage R–K method Definition 1.20. The function F is defined on the set T of all trees by F (τ ) = f, F ([t1 , t2 , · · · , tM ]) = {F (t1 ), F (t2 ), · · · , F (tM )}.

(1.49)

The proof of the following two theorems was established by Butcher in 1987 [But87] . Theorem 1.21. Let y˙ = f (y), f : Rm → Rm , then y (q) =



α(t)F (t),

(1.50)

r(t)=q

where F (t) is defined by (1.49), and α(t) by (1.48). Below let us apply this theorem for p ≤ 4 to obtain y (q) : y (2) = {f }, y (3) = {f 2 } + {2 f }2 , y (4) = {f 3 } + 3{f {f }2 + {2 f 2 }2 + {3 f }3 . Let us define the right side of Equation (1.4) to be yn (h), which is then expanded as a Taylor series about h = 0, 1 y(xn+1 ) = y(xn ) + hy (1) (xn ) + h2 y (2) (xn ) + · · · , 2

(1.51)

  dq  y (h) = α(t)γ(t)φ(t)F (t).  n q dh h=0

(1.52)

where

r(t)=q

We first slightly modify the notation in Butcher array of a. Let as+1,i = bi (i = 1, 2, · · · , s), we get the following Table 1.2. Definition 1.22. 1◦ of all trees by:

For i = 1, 2, · · · , s, s + 1, define the function of φi on the set T

φi (τ ) =

s 

aij ,

j=1

φi ([t1 , t2 , · · · , tM ]) =

s  j=1

2◦

define φ(t) = φs+1 (t).

aij φj (t1 )φj (t2 ) · · · φj (tM ).

296

7. Symplectic Runge–Kutta Methods

Table 1.2.

Tree and elementary differential up to 4

tree

t

F (t)

◦

τ

f

◦

◦

[τ ] {f } = f  f ◦

◦

[τ 2 ] {f 2 } = f  (f, f )

2

1

j

i





aij cj

3

2

3

1

[[τ ]] {2 f }2 = f  f  f

◦ ◦

◦

3

1

6

1





aij c2j

4

6

4

1

aij



◦

ajk ck



[τ [τ ]] {f {f }2 = f  (f, f  f ) 4

1

8

3

◦ ◦ ◦

[[τ 2 ]] {2 f 2 }2 = f  f  (f, f ) 4

2

12

1



aij c3j





   ◦ [[τ ]] {3 f }3 = f f f f

4

1

24

1



aij cj





aij



 j

φi ([τ ]) =

as+1,j =



 j

aij φj (τ ) =

j

φ([τ ]) =

 j

φi ([[τ ]]) =

as+1,j cj =



 j





aij



 j

aij φj ([τ ]) =

as+1,j

 k

bi aij c2j

i,j

ajk



k

n

akn cn



bi aij ajk ck

i,j,k

∀ i = 1, 2, · · · , s,

bi ,

i

∀ i = 1, 2, · · · , s,

aij cj ,

j

j

φ([[τ ]]) =

bj =



ajk c2k

k



bi ci aij cj

i,j

Remark 1.23. Functions φi has representations on the set T  aij = ci , φi (τ ) = φ(τ ) =



ajk ck

k

j

j

bi c3i

i

j

◦

bi aij cj

i,j

k

j

◦



j

◦

bi c2i

i

 j

◦ [τ 3 ] {f 3 } = f  (f, f, f )

bi ci

i

j

◦ ◦

◦

1

◦

◦

◦

2

φ(t)  bi

j

◦

◦

r(t)σ(t)γ(t)α(t)φi (t), i = 1, · · · , s  1 1 1 1 aij (= ci )

b j cj = 



bi ci ,

i

aij

j

ajk ck =





∀ i = 1, 2, · · · , s,

ajk ck ,

k

bj ajk ck =

jk

Theorem 1.24. R–K method has order p, if

 ij

bi aij cj .

7.1 Multistage Symplectic Runge–Kutta Method

φ(t) =



bi φi =

i

1 , γ(t)

∀ r(t) ≤ p,

t ∈ T,

297

(1.53)

and does not hold for some tree of order p + 1. From Table 1.3, we then obtain the following number of orders trees (see Table 1.4). Table 1.3.

Number of trees up to order 10

order p

1

2

3

4

5

6

7

8

9

10

number of trees Tp

1

1

2

4

9

20

48

115

286

719

Table 1.4.

Number of conditions up to order 10

order p

1

2

3

4

5

6

7

8

number of conditions

1

2

4

8

17

37

85

200

9

10

486

1205

Number of order conditions for Multi-stage R–K up to order 10, can be seen in the following Table 1.4.

7.1.5 Simplified Conditions for Symplectic R–K Method There are four types of trees, which can be defined as follows[SA91] : (1) A labeled n-tree λτ is a labeled n-graph {V, E}, such that for any pair of distinct vertices v and w, there exists a unique path that joins v and w. (2) Two labeled n-trees {V1 , E1 }, {V2 , E2 } are said to be isomorphic, if a bijection of V1 onto V2 exists that transforms edges in E1 into edges in E2 , vertices V1 into V2 . n-trees τ is an equivalence class that consists of labeled n-trees isomorphic to it. Each of the labeled n-trees that represent τ is called a labeling of τ . (3) A rooted labeled n-tree ρλτ is a labeled n-tree, in which one of the vertices r, called the root, has been highlighted. The vertices adjacent to the root are called the sons of the root. The sons of the remaining vertices are defined in an obvious recursive way. In fact, when some point is defined as root, the tree becomes a directed graph, i.e., any edge (v, w) in set E has a direction to represent the relationship between father and son. Let T be a mapping from son to father. Since any point v has a path to the root, e.g. v = v0 , v1 , · · · , vm = r, r may be obtained through the sequential action of T on v. Therefore a direction can be defined from v to r, and the entire root also become oriented. (4) Two labeled n-trees {V1 , E1 , r1 }, {V2 , E2 , r2 } are said to be root isomorphic, if a bijection of V1 onto V2 exists that transforms edges in E1 onto E2 and maps

298

7. Symplectic Runge–Kutta Methods

r1 onto r2 . A rooted n-trees ρτ is an equivalence class that comprises of a the rooted labeled n-tree and all rooted labeled n-trees root-isomorphic to it. Fig. 1.4 is an example of rooted tree. Fig. 1.5 shows that there is only one unlabeled 3-tree for n = 3, τ31 , which represents three different labeled trees denoted by (A, B, C). Each labeled tree represents three rooted labeled trees denoted by lower case letter (a, b, · · ·). The 9 rooted labeled trees can be classified into two rooted trees ρτ31 , ρτ32 . The tree τ31 at the last row can be considered as the result of the identification of ρτ31 with ρτ32 . In general, trees can be considered to be equivalent classes of rooted trees, because a root isomorphism is an isomorphism. For each rooted tree ρτ , we denote α(ρτ ) as the number of the monotonic rooted labeled trees. The latter only allow so called monotonic rooted labelings where each vertex is labeled using an integer number (≤ n) smaller than all its sons.

j ◦

l ◦

◦k j ◦ R◦

k ◦

R◦

◦

l

s◦ j

◦+ + i

i +

i+

Fig. 1.4.

k ◦+

A rooted tree

Unless otherwise specified, it is assumed that the set of vertices of a labeled n graph is always {1, 2, · · · , n}. In order to clarify the above four types of trees, we use Fig. 1.5 to illustrate. τ31



(nolabeled 3-tree) (labeled 3-trees) 1

2

3

3

A

1

2

B

2

3

1 C

(rooted labeled 3-tree) (+)1-2-3 1-(+)2-3 1-2-(+)3 (+)3-1-2 3-(+)1-2 3-1-(+)2 (+)2-3-1 2-(+)3-1 2-3-(+)1 a c d e f g h i b ◦ ◦ ◦ ◦ (rooted 3-tree) ◦ ◦ ρτ32 ρτ31 (nolabeled 3-tree)

Fig. 1.5. 3-tree

◦

◦

◦

τ31

The 3-tree, labeled 3-trees (A) − (C), rooted labeled 3-tree (a) − (i), and rooted

7.1 Multistage Symplectic Runge–Kutta Method

Fig. 1.6.

+

τ11

•

ρτ21

• • +

τ21

••

Rooted n-tree (n=1,2) •

ρ τ31

• + •

ρτ41

ρτ43

Fig. 1.7.

•

ρτ11

• •

• • +

•

•

ρτ32

•

ρτ42

• + •

• τ31

• + • • • • +

τ41

•

• 1

•

•

• 2

• 3

•

•

• 4

•

• • • • +

ρτ44

τ42 •

Rooted n-tree (n=3,4)

p

o k

m

l

• •

• • + ρτI

n

• •

j

i

Fig. 1.8.

299

• •

• • + ρτJ

•

• •

•

• • + ρτ i

• •

• •

• •

•

• ρτj +

4 rooted tree in Lemma 1.25

Superfluous trees. Let τ be an n-tree and choose one of its labelings λτ . This labeling gives rise to n different rooted labeled trees ρλτ1 , · · · , ρλτn , where ρλτi has its root at the integer i (1 ≤ i ≤ n). If for each edge (i, j) in λτ , ρτi and ρτj represent different rooted trees; then τ is called non-superfluous. Consider the 3-tree τ31 in

300

7. Symplectic Runge–Kutta Methods

Fig.1.5. When choosing the labeled 3-tree A, we see that for the edge 1-2, choosing 1 as the root leads to ρτ31 , and choosing 2 as the root leads to ρτ32 . For the edge 2-3, choosing 2 as the root leads to ρτ32 , and choosing 2 as the root leads to ρτ31 . Therefore τ31 is non-superfluous. One the other hand the 4-tree with labeling is superfluous (see Fig. 1.6 and 1.7), since changing the root from 2 to the adjacent 3 does not result in different rooted trees. • • • • 1

2

3

4

In order to simplify the order conditions for symplectic R–K,we need some lemmas. Before introducing the lemmas, let us first look at Fig. 1.8: 4-rooted tree. Look at first rooted tree ρτi (i.e., root at i) and rooted tree ρτj . The root of the rooted trees ρτI , ρτJ in Fig. 1.8 is at vertex i and j, they are removed edge joining i and j in the top left-hand corner graph. Lemma 1.25. With the above notations 1◦ 1 1 1 1 + = · . γ(ρτi ) γ(ρτj ) γ(ρτI ) γ(ρτJ )

(1.54)

2◦ For the symplectic R–K method, weighted coefficients of elementary differential satisfy (1.55) φ(ρτi ) + φ(ρτj ) = φ(ρτI )φ(ρτJ ). 3◦

For order ≥ (r − 1), symplectic R–K method, φ(ρτi ) + φ(ρτj ) =

1 1 + . γ(ρτi ) γ(ρτj )

(1.56)

Therefore ρτi order conditions hold iff order conditions of ρτj hold. Proof. By the definition of γ, we have γ(ρτi ) = rγ(ρτJ )

γ(ρτI ) , r(ρτI )

(1.57)

γ(ρτj ) = rγ(ρτI )

γ(ρτJ ) , r(ρτJ )

(1.58)

where r(ρτI ) and r(ρτJ ) are the orders of ρτI and ρτJ . Then, insert r(ρτI ) in formula (1.57) and r(ρτJ ) in formula (1.58) into r(ρτI )+r(ρτJ ) = r to obtain (1.54). Rewrite the left side of formula (1.55) into  ;  ; φ(ρτi ) + φ(ρτj ) = bi aij bj aij + , (1.59) ij···

ij···

B

where represents a product of r − 2 factors akl . Equality (1.55) can be obtained using order condition (1.15) of the symplectic R–K method.  Example 1.26. See the simple examples below. From Fig. 1.8 we have

7.1 Multistage Symplectic Runge–Kutta Method

φ(ρτ v ) =



biv aiv iw aiv i1 aiv i2 aiw i3 aiw i4 ,

1 •

vw1···4

φ(ρτ w ) =



2 • • v

biw aiw iv aiw i3 aiw i4 aiv i1 aiv i2 .

301

•4 •3

• w

vw1···4

From Fig.1.8 we have φ(ρτv ) =



1 • biv aiv i1 aiv i2 ,

v12

φ(ρτw ) =



2 • • v •4

biw aiw i3 ai3 i4 .

w34

• 3 •w

Theorem 1.27. [SA91] Assume that a symplectic R–K method satisfies the order conditions for order ≥ (r − 1) with (r ≥ 2). Then, to ensure that the method to have order ≥ r, it is sufficient that, for each non-superfluous tree τ with r vertices, there is one rooted tree ρτ associated with τ for which φ(ρτ ) =

1 . γ(ρτ )

(1.60)

Proof. Choose first a non-superfluous tree τ . Assume that condition (1.60) is satisfied for a suitable rooted tree ρτi of τ . From the Lemma 1.25 we choose j as any of the vertices adjacent to i. By condition (1.56), the order condition (1.60) is also satisfied for ρτj . Since any two vertices of a tree can be joined through a chain of pairwise adjacent vertices, the iteration of this argument leads to the conclusion that the method satisfies the order conditions that arise from any rooted tree in τ . In the case of a superfluous tree τ , by definition, it is possible to choose adjacent vertices i, j, such that ρτi and ρτj are in fact the same rooted tree. Then condition (1.56) shows that (1.60) holds for the rooted tree ρτi . Therefore (1.60) holds for all rooted tree in τ .  Example 1.28. For r = 2, there is only one tree τ21 , this is a superfluous tree. Example 1.29. For r = 3, there is again only one tree τ31 . It has two rooted trees ρτ31 , ρτ32 . Hence the order conditions become 3 3   1 1 bi aij aik = , or bi aij ajk = . 3 6 i,j,k=1

i,j,k=1

Example 1.30. For r = 4, there is only one non-superfluous tree τ42 . We impose either the order conditions for ρτ43 or the order conditions for ρτ44 . We see that for symplectic R–K methods it is sufficient to obtain the order conditions only for non-superfluous trees rather than every rooted trees. The reduction in the number of order conditions is given in Table 1.5. Comparison order conditions between symplectic R–K (R–K–N) method.

302

7. Symplectic Runge–Kutta Methods

Table 1.5. Order 1 2 3 4 5 6 7 8

Order conditions between symplectic R–K (R–K–N) method R–K method 1 2 4 8 17 37 85 200

symp. R–K method 1 1 2 3 6 10 21 40

R–K–N method 1 2 4 7 13 23 43 79

symp. R–K–N method 1 2 4 6 10 15 25 39

Example 1.31. For diagonally symplectic R–K method, see tableau (1.16). If r = 3, according to Theorem 1.27 and Table 1.5, symplectic R–K method has only two conditions. One condition is for r = 1, b1 + b2 + b3 = 1.

(1.61)

another condition is for r = 3, which has only one non-superfluous tree with two rooted trees, ρτ31 , ρτ32 . Choose one of them r(ρτ31 )φ(ρτ31 ) = 1,

3 

bi c2i =

i=1

After simplifying, we obtain

b31 + b32 + b33 = 0.

1 , 3 (1.62)

Since we have two equations with three unknowns, one of which can be freely chosen, for example: √ 1 −32 √ , b2 = √ see (1.20). (1.63) b1 = b 3 = 2− 33 2− 33

7.2 Symplectic P–R–K Method In this section we study symplectic Partitioned–Runge–Kutta method(P–R–K method).

7.2.1 P–R–K Method In this subsection we focus on a class of special Hamiltonian system i.e., separable systems: H(p, q) = u(p) + v(q). (2.1) Its corresponding Hamiltonian equations are ⎧ dp ⎪ ⎨ = −vq (q) = f (q), dt

⎪ ⎩ d q = up (p) = g(p). dt

(2.2)

7.2 Symplectic P–R–K Method

303

Let us suppose that the component p of the first set of system (2.2) are integrated by an R–K method and the component q in second part of system are integrated with a different R–K method. The overall scheme is called a Partitioned– Runge–Kutta method, or shortly called P–R–K method. It can be specified by two Butcher tableaux: c1 a11 .. .. . . cs as1 b1

· · · a1s .. .. . . · · · ass ···

C1 A11 .. .. . . Cs As1 B1

bs

· · · A1s .. .. . . · · · Ass ···

Bs (2.3)

The application of (2.3) to the system (2.2) results in ⎧ s  ⎪ ⎪ n ⎪ P = p + h aij f (Qj ), ⎪ i ⎪ ⎪ ⎪ j=1 ⎪ ⎪ s ⎪  ⎪ ⎪ n ⎪ = q + h Aij g(Pj ), Q ⎪ i ⎨ j=1 i = 1, · · · , s, s  ⎪ ⎪ n+1 n ⎪ =p +h bi f (Qi ), ⎪ p ⎪ ⎪ ⎪ i=1 ⎪ ⎪ s ⎪  ⎪ ⎪ n+1 n ⎪ = q + h Bi g(Pi ). q ⎪ ⎩

(2.4)

i=1

These tableaux are coefficients of P–R–K method. Theorem 2.1. If coefficients of P–R–K (2.4) satisfies the following conditions: M = bi Aij + Bj aji − bi Bj = 0,

i, j = 1, · · · , s

(2.5)

then this P–R–K method is symplectic[AS93,Sur90,Sun93b,SSM92] . Proof. Let Ki = f (Qi ),

li = g(Pi ),

(2.6)

and d pn+1 ∧ d q n+1 − d pn ∧ d q n s    = h bi d Ki ∧ d Qi + Bi d Pi ∧ d li i=1

−h2

s 

(bi Aij + Bj aji − bi Bj )d Ki ∧ d lj .

i,j=1

Note that the first term on the right side of Equation (2.7) is

(2.7)

304

7. Symplectic Runge–Kutta Methods

bi d Ki ∧ d Qi + Bi d Pi ∧ d li =

s 

−vqq d Qi ∧ d Qj + upp d Pi ∧ d Pj = 0.

i,j=1

In order to satisfy the equality (2.5), it is sufficient to make bi Aij + Bj aji − bi Bj = 0. 

Therefore, the theorem is completed.

W -transformation (defined below) proposed by Hairer and Wanner in 1981 has the intention of simplifying the order condition C(·) and D(·), as well as their relationship. Through the W - transformation, it is easy to construct higher order R–K method[HW81] . Let us suppose polynomials pi (0 ≤ i ≤ (s − 1)), are orthogonal to the following inner product (p, q) =

s 

bi p(ci )q(ci ),

i=1

introducing matrix ⎡

p0 (c1 ) p1 (c1 ) · · · ps−1 (c1 )

⎢ ⎢ p0 (c2 ) p1 (c2 ) · · · ps−1 (c2 ) ⎢ W =⎢ ⎢ ··· ··· ··· ··· ⎣ p0 (cs ) p1 (cs ) · · · ps−1 (cs )

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

by the orthogonality of pi (i = 1, · · · , s − 1), we have W T BW = I. We take pk (x) as a standard shifted Legendre polynomial, defined by

pk (x) =

k  k  k + i   √ 2k + 1 (−1)k+i xi , i i

k = 0, 1, · · · , s − 1.

i=0

For an s-stage R–K method (A, b, c), let X = W −1 AW = W T BAW , then for the high order R–K method based on the high order quadrature formula their transformation matrix X is given by Table 2.1[Sun93a,Sun94,Sun95] .

7.2 Symplectic P–R–K Method

Table 2.1.

305

Matrix X form

method Gauss Lobatto III A Lobatto III B

Xs,s−1 ξs−1 ξs−1 u 0

Xs−1,s −ξs−1 0 −ξs−1 u

Lobatto III C

ξs−1 u

−ξs−1 u

Lobatto III S

ξs−1 uσ

−ξs−1 uσ

Radau I A

ξs−1

−ξs−1

Radau II A

ξs−1

−ξs−1

Radau I B Radau II B

ξs−1 ξs−1

−ξs−1 −ξs−1

For the Gauss–Legendre method ⎡ 1 −ξ1 ⎢ 2 ⎢ ξ1 0 ⎢ ⎢ ⎢ ξ2 XGL = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ O

Xs,s 0 0 0 u2 2(2s − 1) 0 1 4s − 2 1 4s − 2 0 0

symplectic yes no no no yes no no yes yes

⎤ −ξ2 .. .

..

.

..

..

.

..

.

..

.

O

..

.

. ξs−1

−ξs−1 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

1 (k = 0, 1, · · · , s − 1). 2 4k2 − 1

where ξk = √

Corollary 2.2. If coefficients of P–R–K method satisfy bi = Bi , ci = Ci , ci and bi = 0 (i = 1, · · · , s), and T

W T M W = X + X − e1 e T 1, then P–R–K method is symplectic. Corollary 2.3. (Sun)[Sun93b] Given a s-stage R–K method and its coefficients (A, b, c). If the coefficients ci , bi = 0 (i = 1, · · · , s) satisfy order conditions of R–K method B(p), C(η) and  D(ζ),  then the P–R–K method produced by the coefficient of a ¯ij = bj 1 − ji , bi , ci is symplectic with order r = min (p, 2η + 2, 2ζ + aij , a bi

2, η + ζ + 1). Corollary 2.4. Method of Radau IA has order 2s − 1, by Corollary 2.3, method of Radau IA and Radau I A is symplectic with order 2s − 1. Example 2.5. P–R–K method with first order accuracy Radau IA–I A 0

1

0

0 =⇒

1

1

1 2

1

306

7. Symplectic Runge–Kutta Methods

Example 2.6. P–R–K method with third order accuracy Radau I A – IA 0 2 3

1 4 1 4 1 4

1 4 5 12 3 4

−

0

0

0

2 3

1 3 1 4

1 3 3 4

1 8 7 24 1 4

=⇒

1 8 3 8 3 4

−

Corollary 2.7. Constructs symplectic P–R–K method Radau II A – Radau II A with the similar method. Example 2.8. P–R–K method with first order accuracy Radau II A – Radau II A 1

1

1 2

0

1

=⇒ 1

1

1

Example 2.9. P–R–K method with third order accuracy Radau II A – Radau II A 1 3

5 12 3 4 3 4

1

1 12 1 4 1 4

−

1 3

1 3

0

1

1

0

3 4

1 4

=⇒

3 8 7 8 3 4

1 24 1 8 1 4

−

Corollary 2.10. Using same method constructed symplectic P–R–K method Lobatto III C – III C with 2s − 2 order accuracy. Example 2.11. Symplectic P–R–K Lobatto III C–III C method with 2 order accuracy, its coefficients are 0 1

1 2 1 2 1 2

1 2 1 2 1 2

−

0

0

0

1

1

0

1 2

1 2

=⇒

1 4 3 4 1 2

1 4 1 4 1 2

−

Example 2.12. Symplectic P–R–K Lobatto III C – III C method, its coefficients are

7.2 Symplectic P–R–K Method

0 1 2

1

1 6 1 6 1 6 1 6

1 6 1 − 12 1 6 1 6

1 3 5 12 2 3 2 3

−

0

0

0

0

1 2

1 4

1 4

0

1

0

1

0

1 6

2 3

1 6

=⇒

1 12 5 24 1 12 1 6

1 6 4 12 5 6 2 3

−

307

1 12 1 − 24 1 12 1 6

Corollary 2.13. The s-stage P–R–K Lobatto III A–III B method is symplectic with 2s − 2 order accuracy. Example 2.14. Symplectic P–R–K Lobatto III A–III B method, its coefficients are 0 1

0

0

5 24 1 6 1 6

1 3 2 3 2 3

0

0

1 2 1 2

1 2 1 2

0 1 24 1 6 1 6

−

1 2 1 2 1 2

0 1

1 6 1 6 1 6 1 6

1 6 1 3 5 6 2 3

−

0 =⇒

0 1 2

0 0 =⇒ 0 1 6

1 4 1 2 1 2 1 12 2 16 1 6 1 6

0 1 4 1 2 1 12 1 3 2 12 2 3

−

0 1 48 1 12 1 6

−

With the help of symplectic conditions of P–R–K methods, we can construct symplectic R–K method. We have the following corollary: 1

Corollary 2.15. [Sun][Sun00] The s-stage R–K method with coefficients a∗ij = (aij + 2 Aij ), b∗i = bi = Bi and c∗i = ci are symplectic and at least satisfy B(p), C(ξ) and D(ξ), i.e., order r = min(p, 2ξ + 1),

where

ξ = min (η, ζ).

Example 2.16. If we take the coefficients in Example 2.14, we know that the right  of the table is a special case of 2-order accuracy of R–K methods of Lobatto III S i.e.,  1 σ = situation of literature [Chi97] , see Table 2.1 . 2

7.2.2 Symplified Order Conditions of Explicit Symplectic R–K Method The following s-stage scheme is well-known[ZQ95b]

308

7. Symplectic Runge–Kutta Methods

pi = pi−1 + ci hf (qi−1 ), qi = qi−1 + di hg(pi ),

i = 1, 2, · · · , s,

(2.8)

where f = −vq (q), g = up (p). We can regard f, g as a function of z = (p, q), for f (or g) with the p (or q) variables of coefficient 0, i.e., f (q, 0 · p) (or g(p, 0 · q). In order to facilitate the writing in a unified form, we make: p = ya , q = yb , f = fa , g = fb , ya,0 = p0 , yb,0 = q0 , and ya,1 = ps−1 , yb,1 = qs−1 , then Equation (2.8) is transformed into an s-stage P–R–K form: g1,a = ya,0 = p0 , g1,b = yb,0 = q0 , g2,a = ya,0 + c1 τ fa (q0 ) = ya,0 + c1 hfa (g1,b ) = p1 , g2,b = yb,0 + d1 τ fb (p1 ) = yb,0 + d1 hfb (g2,a ) = q1 , .. . s−1  gs,a = ya,0 + h cj fa (gj,b ) = ps−1 ,

(2.9)

j=1

gs,b = yb,0 + h

s−1 

dj fb (gj+1,a ) = qs−1 .

j=1

(2.9) is equivalent to i−1 

gi,a = ya,0 + h

cj fa (gj,b ),

j=1

gi,b = yb,0 + h

i−1 

dj fb (gj+1,a ),

i = 2, · · · , s,

j=1

ya,1 = ya,0 + h

s−1 

(2.10) cj fa (gj,b ),

j=1

yb,1 = yb,0 + h

s−1 

dj fb (gj+1,a ).

j=1

And (2.2) can be rewritten with new variables as 6 5 5 6 fa (yb ) y˙ a = . fb (ya ) y˙ b Let

(2.11)

a 1 = c1 ,

a2 = c2 ,

···,

as−1 = cs−1 ,

as = 0,

b1 = 0,

b2 = d1 ,

···,

bs−1 = ds−2 ,

bs = ds−1 ,

then schemes (2.10) now become

(2.12)

7.2 Symplectic P–R–K Method i−1 

gi,a = ya,0 + h

aj fa (gj,b ) = ya,0 + h

j=1

gi,b = yb,0 + h

i 

aj Rj,a ,

j=1

dj fb (gj,a ) = yb,0 +

j=1 s 

ya,1 = ya,0 + h yb,1 = yb,0 + h

i−1 

309

j=1 s 

i 

i = 2, · · · , s,

bj Rj,b ,

j=1

(2.13)

aj Rj,a , bj Rj,b .

j=1

Where Ri,a = fa (gi,b ),

Ri,b = fb (gi,a ).

(2.14)

Now, we just need to study the order conditions of scheme (2.13) when as = b1 = 0. Notice that as = b1 = 0 is necessary for (2.13) to be canonical and is also crucial for simplifying order conditions, as we will see later. A P -graph (denoted by P G) is a special graph which satisfies the following conditions: (i) its vertices are divided into two classes: “white” ◦ “black” •, sometimes instead “meagre” and “fat”. (ii) the two adjacent vertices of a P G cannot be of the same class. If we give the vertices of PG an arbitrary set of labels, we get a label P graph, and we say P -graph∈ P G. Two labeled P -graphs are said to be isomorphic labeled P -graphs if they are just two different labelings of the same P -graph. A simple path joins a pair of vertices v and w, v = w, and is a sequence of pairwise distinct vertices v = v0 , v1 , · · · , vn−1 = w, where vi = vi−1 , (vi−1 , vi ) ∈ E. Fig. 2.1 shows an example of a simple path of v and w for n = 4.

◦

Fig. 2.1.

◦

◦

A simple path of v and w black • and white ◦, for n=4

(1) The definition of a P -tree P τ , a labeled P -tree (denoted by λP τ ), a rooted labeled P -tree ρλP τ of the same order n are just as that of tree τ , labeled by tree λτ , rooted tree ρτ , and rooted labeled tree ρλτ , where the general graph is substituted by the P -graph. (2) We define the isomorphism of two labeled P -trees below. Generally tree’s isomorphism and the P -tree’s isomorphism are all in accordance with the type of labeling used. This has been described before. Here we give the precise definition for a P -tree. Two labeled P -trees {V1 , E1 } and {V2 , E2 } are called isomorphism, if the order of these tree is the same, and there exists a bijection mapping χ, from V1 to V2 and E1 to E2 satisfies

310

7. Symplectic Runge–Kutta Methods

K(χ(v1 )) = K(v1 ), where v1 ∈ V1 , and

 K(v) =

1, 0,

for v black, for v white.

A P -tree (n order) is the equivalent of such a class, it consists of a labeled P -tree and all of its isomorphism. We use [HNW93] the P -series and tree method to derive the order condition of Equation (2.13) below. We first introduce some definitions and notations. (3) Two rooted labeled P -trees with same order, {V1 , E1 , r1 } and {V2 , E2 , r2 } (where ri (i = 1, 2) denoted rooted label), are called rooted isomorphism if there exists a χ that satisfies the condition of (2), and χ(r1 ) = r2 holds. A rooted P -tree, denoted by ρP τ , is an equivalence class which contains a labeled P -tree and all of its isomorphic P -trees. We denote ρP τa (ρP τb ) as ρP τ with white (black) root, and ρλP τ as rooted labeled P -tree, which is obtained by adding label to ρP τ . Thus ρλP τ ∈ ρP τ . We denote by ρP τa (resp. ρP τb ) for a rooted P -tree ρP τ that has a white (resp. black) root. If we give the vertices of a rooted P -tree ρP τ such a set of labels so that the label of a father vertex is always smaller than that of its sons, we then get a monotonically labeled rooted P -tree M ρλP τ . We denote by α(ρP τ ) the number of possible different monotonic labelings of ρP τ when the labels are chosen from the set Aq = { the first q letters of i < j < k < l < · · ·}, where q is the order of ρP τ . The set of all rooted trees of order n with a white (resp. black) root is denoted by T Pna (resp. T Pnb ). Let us denote by λP τna (resp. λP τnb ) the set of all rooted labeled P -trees of order n with a white (resp. black) root vertex, and M λP τna (resp. M λP τnb ) the set of all monotonically labeled P -tree of order n with a white (resp. black) root vertex when the labels are chose from the set An . (4) The density γ(ρλτ ) of a rooted P -tree ρλτ is defined recursively as γ(ρλτ ) = r(ρλτ )γ(ρλτ 1 ) · · · γ(ρλτ m ), where r(ρλτ ) is order of ρλτ , and ρλτ 1 , · · · , ρλτ m are the sub-trees which arise when the root of ρλτ is moved from the tree. The density of rooted P -tree ρP τ is calculated by regarding them as general rooted tree neglecting the difference between with the black and white vertices . (5) Let ρP τ 1 , · · · , ρP τ m be rooted P -tree. We denote by ρP τ = a [ρP τ 1 , · · · , ρP τ m ] the unique rooted P -tree that arises when the roots of ρP τ 1 , · · · , ρP τ m are all attached to a white root vertex. Similarly denote it by b [ρP τ 1 , · · · , ρP τ m ] when the root of the P -tree is black. We say ρP τ 1 , · · · , ρP τ m are sub-trees of ρP τ . We further denote the rooted P -tree of order 1, which has a white (resp.black) root vertex by ta (resp.tb ). (6) (2.11) is defined recursively as: F (tb )(y) = fb (y), F (ta )(y) = fa (y), m   ∂ fw(ρP t) (y) F (ρP τ )(y) = F (ρP τ 1 )(y) · · · F (ρP τ m )(y) , (2.15) ∂yw(ρP τ 1 ) · · · ∂yw(ρP τ m )

7.2 Symplectic P–R–K Method

311

where y = (ya , yb ), and ρP τ = a [ρP τ 1 , · · · , ρP τ m ] or ρP τ = b [ρP τ 1 , · · · , ρP τ m ], and w(ρP τ ) is defined by: w(ρP τ ) =

 a, if ρP τ attached to a white root vertex, b,

if ρP τ attached to a black root vertex.

We see that F (ρP τ ) is independent of labeling. Here, and in the remainder of this book, in order to avoid sums and unnecessary indices, we assume that ya and yb are scalar quantities , and fa ,fb scalar functions. All subsequent formulas remain valid for vectors if the derivatives are interpreted as multi-linear mapping. Lemma 2.17. The derivatives of the exact solution of (2.11) satisfy:   (a) F (ρλP τ )(ya , yb ) = α(ρP τ )F (ρP τ )(ya , yb ), ya = (a) yb

ρP τ ∈T Pqa

ρλP τ ∈M λT Pqa



=

F (ρλP τ )(ya , yb ) =

ρλP τ ∈M λT Pqb



α(ρP τ )F (ρP τ )(ya , yb ),

ρP τ ∈T Pqb

(2.16) where q = 1, 2, · · ·. It is convenient to introduce two new rooted P -trees of order 0: ∅a and ∅b . The corresponding elementary differential are F (∅a ) = ya , F (∅b ) = yb . We further set T P a = ∅a ∪ T P1a ∪ T P2a ∪ · · · , T P b = ∅b ∪ T P1b ∪ T P2b ∪ · · · , λT P a = ∅a ∪ λT P1a ∪ λT P2a ∪ · · · , λT P b = ∅b ∪ λT P1b ∪ λT P2b ∪ · · · , M λT P a = ∅a ∪ M λT P1a ∪ M λT P2a ∪ · · · , M λT P b = ∅b ∪ M λT P1b ∪ M λT P2b ∪ · · · .

(2.17)

p-series: let c(∅a ), c(∅b ), c(ta ), c(tb ), · · · be real coefficients defined for all P trees, c : T P a ∪ T P b → R. The series p(c, y) = (pa (c, y), pb (c, y)) is defined as pa (c, y) =

 ρλP τ ∈M λT P a

=



hr(ρλP τ ) c(ρλP τ )F (ρλP τ )(y) r(ρλP τ )!

α(ρP τ )

ρP τ ∈T P a

pb (c, y)

=



ρλP τ ∈M λT P b

=



ρP τ ∈T P b

hr(ρλP τ ) c(ρλP τ )F (ρλP τ )(y), (ρλP τ )!

hr(ρλP τ ) c(ρλP τ )F (ρλP τ )(y) (ρλP τ )!

α(ρP τ )

hr(ρλP τ ) c(ρλP τ )F (ρλP τ )(y). r(ρλP τ ) !

(2.18)

312

7. Symplectic Runge–Kutta Methods

Notice that c is defined on T P a ∪T P b , and for any labelings of ρP τ (especially for monotonic labeling ρλP τ ), we have c(ρλP τ ) = c(ρP τ ). Lemma 2.17 states simply that the exact solution is a p-series, 

T   ya (t0 + h), yb (t0 + h) = p Y, ya (t0 ), yb (t0 ) ,

where Y (ρP τ ) = 1 for all rooted P -trees ρP τ . The following theorem is from the book[HNW93] . Theorem 2.18. Let c : T P a ∪ T P b → R, be a sequence of coefficients such that c(∅a ) = c(∅b ) = 1. Then  fa p(c, (ya , yb ))     = p cT , (ya , yb ) ,  h fb p(c, (ya , yb )) with cT (∅a ) = cT (∅b ) = 0, cT (ta ) = cT (tb ) = 1, cT (ρP τ ) = r(ρP τ )c(ρP τ 1 ) · · · c(ρP τ m ), ρP τ = a [ρP τ 1 , · · · , ρP τ m ], or ρP τ = b [ρP τ 1 , · · · , ρP τ m ]. Let

  ⎧ Ri,a = pa Ki , (ya,0 , yb,0 ) , ⎪ ⎪   ⎪ ⎨ R = p K , (y , y ) , i,b b i a,0 b,0   ⎪ = p , (y , yb,0 ) , G g i,a a i a,0 ⎪ ⎪   ⎩ gi,b = pb Gi , (ya,0 , yb,0 ) ,

i = 1, · · · , s,

(2.19)

where Ki (i = 1, · · · , s) : T P a ∪ T P b → R, Gi (i = 1, · · · , s) : T P a ∪ T P b → R are two sets of p-series. From (2.10), we have Gi (∅a ) = Gi (∅b ) = 1. Hence, From (2.14), we have  p Ki , (ya,0 , yb,0 )



  ! ! pa Ki , (ya,0 , yb,0 ) ! fa (gi,b ) Ri,a  =  =h = Ri,b fb (gi,a ) pb Ki , (ya,0 , yb,0 )       ! ! fa pb Gi , (ya,0 , yb,0 ) fa p Gi , (ya,0 , yb,0 )  = h      = h fb pa Gi , (ya,0 , yb,0 ) fb p Gi , (ya,0 , yb,0 )    = p Gi , (ya,0 , yb,0 ) .

Then from Theorem 2.18, we get Ki = Gi , But from (2.13), we have

∀ i = 1, · · · , s.

7.2 Symplectic P–R–K Method ⎡ 

 p Gi , (ya,0 , yb,0 )





= ⎡ ⎢ ⎢ ⎢ = ⎢ ⎢ ⎣

i−1 

⎢ ya,0 + h aj Rj,a pa Gi , (ya,0 , yb,0 ) ! ⎢ j=1 ⎢  =⎢  i ⎢  pb Gi , (ya,0 , yb,0 ) ⎣ yb,0 + h bj Rj,b ⎤j=1 i−1    ya,0 + h aj pa Kj , (ya,0 , yb,0 ) ⎥ ⎥ j=1 ⎥ . i  ⎥  ⎥  bj pb Kj , (ya,0 , yb,0 ) ⎦ yb,0 + h

313 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

j=1

Thus: Gi (ρP τa ) = Gi (ρP τb ) =

i−1  j=1 i 

aj Kj (ρP τa ),

∀ r(ρP τ ) ≥ 1.

(2.20)

bj Kj (ρP τb ),

j=1

From (2.13), we also have ya,1 = ya,0 + h

s  i=1

yb,1 = yb,0 + h

s 

  pa Ki , (ya,0 , yb,0 ) ,   pb Ki , (ya,0 , yb,0 ) .

(2.21)

i=1

Comparing the numerical solution obtained from (2.13) and the exact solution from (2.11), we get the order condition for scheme (2.13). Theorem 2.19. Scheme (2.13) is p-order accuracy iff its coefficients ai , bi satisfy: ⎧ s  ⎪ ⎪ ai Ki (ρP τa ) = 1, 1 ≤ r(ρP τa ) ≤ p, ⎪ ⎨ i=1 (2.22) s  ⎪ ⎪ ⎪ b K (ρP τ ) = 1, 1 ≤ r(ρP τ ) ≤ p, i i b b ⎩ i=1

where Ki (i = 1, · · · , s) are defined recursively by ⎧ Ki = Gi , ⎪ ⎪ ⎪ ⎪ Gi (∅a ) = Gi (∅b ) = 1, ⎪ ⎪ i−1 ⎪  ⎨ aj Kj (ρP τa ), Gi (ρP τa ) = r(ρP τa ), r(ρP τb ) ≥ 1. j=1 ⎪ ⎪ ⎪ i ⎪  ⎪ ⎪ ⎪ bj Kj (ρP τb ), ⎩ Gi (ρP τb ) =

(2.23)

j=1

From the first and second equations of (2.23) we know Ki (∅a ) = Ki (∅b ) = 0, Ki (ta ) = Ki (tb ) = 1, from the last two equations of (2.23) we can obtain Gi (ta ), Gi (tb ). Repeating this procedure, we can obtain Ki (ρP τa ), Ki (ρP τb ), by P tree order from low to high.

314

7. Symplectic Runge–Kutta Methods

Next we rewrite equations (2.23) into more intuitive forms. From (2.23), we have ⎧ * i + * i +   ⎪ ⎪ 1 m 1 ⎪ bj Kj (ρP τb ) · · · bj Kj (ρP τb ) , Ki (ρP τa ) = r(ρP τa ) ⎪ ⎪ ⎨ j=1 j=1 * i−1 + * i−1 + ⎪   ⎪ 1 m ⎪ 2 ⎪ Ki (ρP τb ) = r(ρP τb ) aj Kj (ρP τa ) · · · aj Kj (ρP τa ) , ⎪ ⎩ j=1

i = 2, 3, · · · , s,

j=1

(2.24)

where

ρP τa = a [ρP τb1 , · · · , ρP τbm1 ], ρP τb = b [ρP τa1 , · · · , ρP τam2 ].

(2.25)

We now define elementary weight φ(ρP τ ) for a rooted P -tree. Choose any labeling ρλP τ for ρP τ ; without loss of generality we choose a monotonic one with labels i < j < k < l < · · ·, where the rooted labeling is i. Then φ can be obtained recursively (Note the difference of solving for φ between the original tree and its subtree) ⎧ s−1    ⎪ ⎪ ⎪ ) = ai φ(ρP τb1 ), · · · , φ(ρP τbm1 ) , φ(ρP τ a ⎪ ⎪ ⎪ ⎨ i=1 s   r(ρP τa ), r(ρP τb ) ≥ 1,  ⎪ φ(ρP τ ) = bi φ(ρP τa1 ), · · · , φ(ρP τam2 ) , b ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎩ φ(∅a ) = φ(∅b ) = 1, (2.26) where ρP τa , ρP τb and (2.25) are the same. Here, notice that i is the root of ρP τa or ρP τb , and s is the label for an imaginary father vertex to the root i. The summation is always with respect to subscripts of the son’s vertex, from 1 adds to the father vertex or it reduces by 1. Now s is order of scheme (2.13). We are doing this only for ease of the recursive definition; otherwise vertex i has no father vertex and the summation superscript cannot be determined. Regarding the subtrees of ρP τa , ρP τb , the father vertex for their root labeling is i. It is not necessary to add an extra father vertex. So the weight for a p-tree as the original tree is different from the weight as another subset tree. By (2.26) we can see that the elementary weight of a tree φ is not related to its labeling as long as the imaginary father of maintaining root label is always the order of the scheme (2.13). Theorem 2.20.

[AS93,ZQ95b,SSC94]

φ(ρP τ ) =

1 , γ(ρP τ )

Order conditions in Theorem 2.19 are equivalent to ∀ ρP τ ∈ T P a ∪ T P b ,

r(ρP τ ) ≤ p.

(2.27)

Proof. We just need to prove ⎧ s−1  ⎪ ⎪ ⎪ aj Kj (ρP τa ), φ(ρP λτa )γ(ρλP τb ) = ⎪ ⎨ j=1

s  ⎪ ⎪ ⎪ φ(ρλP τ )γ(ρλP τ ) = bj Kj (ρλP τb ). ⎪ a b ⎩ j=1

(2.28)

7.2 Symplectic P–R–K Method

From (2.23), we have ⎛ ⎧ i  ⎪ ⎪ K (ρλP τ ) = r(ρλP τ ) a a ⎝ ⎪ i ⎨ ⎪ ⎪ ⎪ ⎩

⎛ Ki (ρλP τb ) = r(ρλP τb ) ⎝

⎞

j1 =1 i−1 

j1 =1

⎛

⎞

1 ajm Kjm (ρλP τa )⎠ 2 2

⎞

i 

1

bj1 Kj1 (ρλP τb )⎠ · · · ⎝

1

1

)⎠ , ⎞

i−1 

···⎝

m1

bjm Kjm (ρλP τb

jm1 =1

⎛

315

jm2 =1

m ajm Kjm (ρλP τa 2 )⎠ 2 2

,

(2.29)

where i = 2, 3, · · · , s, and ρλP τa = a [ρP τb1 , · · · , ρλP τbm1 ], ρλP τb = b [ρλP τa1 , · · · , ρλP τam2 ],

(2.30)

while j1 , · · · , jm1 and j1 , · · · , jm2 are the labels of the roots of ρλP τb1 , · · · , ρλP τbm1 and λP τa1 , · · · , ρλP τam2 respectively. Due to ⎧ s−1 * i + * + i    ⎪ m1 ⎪ 1 ⎪ a r(ρλP τ ) b K (ρλP τ ) · · · b K (ρλP τ ) , ⎪ i a j j j j b m m b 1 1 ⎪ 1 1 ⎪ ⎨ i=1 j1 =1 jm1 =1 R.S. of(2.28) ⇐⇒ * i−1 + * i−1 + s ⎪    ⎪ ⎪ 1 m ⎪ ⎪ bi r(ρλP τb ) aj1 Kj1 (ρλP τa ) · · · ajm Kjm (ρλP τa 2 ) , ⎪ 2 2 ⎩ i=1

L.S. of (2.28) ⇐⇒

j1 =1

jm2 =1

⎧ s−1     ⎪  ⎪ m m 1 1 ⎪ ⎪ ai r(ρλP τa ) φ(ρλP τb )γ(ρλP τb ) · · · φ(ρλP τb 1 )γ(ρλP τb 1 ) , ⎪ ⎨ i=1

s     ⎪  ⎪ 1 1 m m ⎪ ⎪ bi r(ρλP τb ) φ(ρλP τa )γ(ρλP τa ) · · · φ(ρλP τa 2 )γ(ρλP τa 2 ) , ⎪ ⎩ i=1

so we have to prove i 

φ(ρλP τbl )γ(ρλP τbl ) =

bjn Kjn (ρP τbl )

for

n = 1, · · · , m1 ,

jn =1

and

i−1 

φ(ρP τal )γ(ρP τal ) =

ajn Kjn (ρP τal )

n = 1, · · · , m2 .

for

jn =1

Continue this process and finally we see it is enough to prove f (l)−1

φ(ta )γ(ta ) =



f (l)−1

al Kl (ta ),

φ(tb )γ(tb ) =

l=1



bl Kl (tb ),

(2.31)

l=1

where l is the label of ta or tb and f (l) is the label of the father. Since f (l)−1

φ(ta )γ(ta ) =



f (l)−1

al · 1 =

l=1

φ(tb )γ(tb ) =

f (l)  l=1

bl · 1 =



f (l)−1

al · Kl (ta ) =

l=1 f (l) 

bl · Kl (tb ) =

l=1



al ,

l=1 f (l) 

bl ,

l=1

and Kl (ta ) = 1, The theorem is proved.

Kl (tb ) = 1. 

316

7. Symplectic Runge–Kutta Methods

Let P τ be a tree of order p (p ≥ 2) P -tree. Choose any label to obtain λP τ . Let v and w be two adjacent vertices. We consider four rooted P -tree . Denote ρP τ v (resp. ρP τ w ) as the rooted P -tree obtained by regarding the vertex v (resp.w) as the root of ρP τ . Denote ρP τv (resp.ρP τw ) the rooted P -tree, which arises when the edge (v, w) is deleted from P τ and has the root v (resp.w). Without loss of generality, let v be white, and w be black. Theorem 2.21. [AS93,ZQ95b,SSC94] With the above notations, we have: 1 1 1 + = . 1◦ v w γ(ρP τ )

2◦

γ(ρP τ )

γ(ρP τv )γ(ρP τw )

φ(ρP τ v ) + φ(ρP τ w ) = φ(ρP τv )φ(ρP τw ), when as = b1 = 0.

Proof. By the definition of γ, we have γ(ρP τv ) , r(ρP τv ) γ(ρP τw ) γ(ρP τ w ) = nγ(ρP τv ) . r(ρP τw )

γ(ρP τ v ) = nγ(ρP τw )

Due to r(ρP τv ) + r(ρP τw ) = n, therefore 1 1 r(ρP τv ) r(ρP τw ) + = + γ(ρP τ v ) γ(ρP τ w ) nγ(ρP τw )γ(ρP τv ) nγ(ρP τw )γ(ρP τv ) 1 = . γ(ρP τw )γ(ρP τv )

i.e., 1◦ . Also has φ(ρP τ v ) = φ(ρP τ w ) =

s−1 

aiv

iv iv  ;

iv  ;

resp.

1

iw  ;

iw ; ,

iv =1

1 iw =1

2

s 

iw i w −1 ;

iv ;

biw

aiv

2 iv =1

iw =1

where

biw

,

1

  is the product of all φ(ρP τ v ) resp. φ(ρP τ w ) , while

2

ρλP τa = a [ρP τb1 , · · · , ρλP τbm1 ],

ρλP τb = b [ρλP τa1 , · · · , ρλP τam2 ],

and iv , iw are labels of v and w respectively.

resp.

1

iv , (resp. iw ), therefore φ(ρP τv ) =

iv  ;

s−1  iv =1

aiv

iv ; , 1

φ(ρP τw ) =

iw  ;

varies only according to

2

s  iw =1

biw

iw ; 2

,

7.2 Symplectic P–R–K Method

317

then s−1 

φ(ρP τv )φ(ρP τw ) =

aiv

iv =1 s−1 

=

aiv

iv  s ;

1 iw =1 i iv v   ; 1

iv =1 s−1 

=

aiv

biw

2

biw

iw ;

biw

1 iw =1

iw ; 2

s 

+

2

iw =1

iv iv  ;

iv =1

iw ;

+

biw

2

iw =iv +1 iv ;

s−1 

iw  ;

s 

aiv

biw

1 iw =iv +1

iv =1

iw ;

.

2

After manipulation, we can get s−1  iv =1

aiv

iv ;

s 

biw

1 iw =iv +1

iw ;

=

2

=

s 

biw

iw i w −1 ;

aiv

iv ;

iw =2

2 iv =1

1

s 

iw i w −1 ;

iv ;

iw =1

biw

aiv

2 iv =1

,

b1 = 0.

1

From this, 2◦ holds.



Corollary 2.22. Let scheme (2.13) be at least (p − 1)-order, where p ≥ 2, then order conditions 1 φ(ρP τ v ) = γ(ρP τ v ) satisfies, iff φ(ρP τ w ) =

1 . γ(ρP τ w )

Proof. Because scheme (2.13) is at least (p − 1)-order, by Theorem 2.20, we know the following two relations hold: φ(ρP τ v ) =

1 , γ(ρP τv )

φ(ρP τ w ) =

1 . γ(ρP τw )

By Theorem 2.21, we have φ(ρP τ v ) + φ(ρP τ w ) =

1 1 + . γ(ρP τ v ) γ(ρP τ w )

The corollary is obviously established.



So far we draw the following conclusion for this section: Theorem 2.23. [AS93,ZQ95b,SSC94] Symplectic scheme (2.13) (as = b1 = 0) is of porder, if and only if any of P -tree pτ, r(pτ ) ≤ p there is a rooted P -tree ρP τ ∈ pτ , such that 1 φ(ρP τ ) = . γ(ρP τ )

318

7. Symplectic Runge–Kutta Methods

Proof. By Corollary 2.22, we know that any two kinds of rooted label method of the P τ lead to equivalent conditions. Therefore, we only need to take one of them to get the order conditions.  By Theorem 2.23, we simplify the order conditions. Originally, every rooted P tree has a corresponding order condition. Now every P -tree, no matter how different the root is chosen, has a corresponding order condition. For the case of 4-order, the number of order conditions reduces from 16 to 8, and the corresponding 8 P -trees are as follows: • •

•

• τb

τa

•

a[τb ]

b[τa , τa ] •

• •

• • b[a [τb , τb ]]

Fig. 2.2.

b[τb , τb ]

a[b [τa , τa ]]

• a[a [a [τb ]]]

8 P-trees

Finally, according to Theorem 2.23, we can simplify the order conditions for P–R–K method, which is given in Table 2.2. Calvo and Hairer[CH95] further reduce the number of independent condition in P–R–K method. See Table 2.3. For general Hamiltonian, the corresponding values are given by Table 2.4 which is obtained by Mirua[Mur97] . Table 2.2. case

Order conditions P–R–K method and Symplectic P–R–K method for separable

Order 1 2 3 4 5 6 7 8

P–R–K method 2 4 8 16 34 74 170 400

Symplectic P–R–K method 2 3 5 8 14 24 46 88

7.3 Symplectic R–K–N Method

Table 2.3.

Further reduction in Order conditions for P–R–K method in separable case

Order 1 2 3 4 5 6 7 8

Table 2.4.

319

P–R–K method 2 2 4 8 18 40 96 230

Symp. P–R–K method 1 1 2 3 6 10 22 42

expl. Symp. P–R–K method 1 1 2 3 6 9 18 30

Order conditions P–R–K method and Symplectic P–R–K method for general case

Order 1 2 3 4 5 6 7 8

P–R–K method 2 4 14 52 214 916 4116 18996

Symplectic P–R–K method 1 1 3 8 27 91 350 1376

7.3 Symplectic R–K–N Method Symplectic Runge–Kutta–Nystr¨om method is abbreviated as symplectic R–K–N method. The main purpose of this section is to develop and simplify the order conditions for R– K–N methods, while the simplified order conditions for canonical R–K–N methods, which are applied to special kind of ODE’s, are also obtained here. Then, using the simplified order condition , we construct some 5 stage, fifth-order symplectic R–K–N schemes.

7.3.1 Order Conditions for Symplectic R–K–N Method We consider a special kind of second order ODE’s: y¨J = f J (y 1 , y 2 , · · · , y n ),

J = 1, · · · , n,

(y 1 , · · · , yn ) ∈ Rn .

We can transform (3.1) into a system of first order ODE’s

(3.1)

320

7. Symplectic Runge–Kutta Methods

y˙ y¨

! =

y˙ f (y)

! ,

(3.2)

by adding another group of variables y˙ J (J = 1, · · · , n). Since canonical difference schemes are meaningful only to Hamiltonian systems, we assume that (3.2) can be written as ⎤ ⎡ ∂H(y, y) ˙ 5 6 y˙ ∂ y˙ ⎥ ⎢ (3.3) =⎣ ⎦, ∂H(y, y) ˙ y¨ − ∂y

where H(y, y) ˙ is scalar function that satisfies ⎧ ∂H(y, y) ˙ ⎪ = y, ˙ ⎨ ∂ y˙

⎪ ˙ ⎩ − ∂H(y, y) = f (y). ∂y

So H must be in the form H =

1 T ∂u y˙ y˙ − u(y), = f (y). Therefore only when 2 ∂y

f (g) is a gradient of some scalar function, its symplectic algorithm is meaningful. A general s-stage R–K–N method can be written as ⎧ s  ⎪ ⎪ ⎪ gi = y0 + ci hy˙ 0 + h2 aij f (gj ), i = 1, · · · , n, ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ s ⎨  y1 = y0 + hy˙ 0 + h2 bj f (gj ), (3.4) ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ s ⎪  ⎪ ⎪ ⎪ bj f (gj ). y˙ = y˙0 + h ⎪ ⎩ 1 j=1

The corresponding Butcher tableau is

c

A b b

c1 c2 .. . c3

a11 a21 .. . as1 b1 b1

··· ··· ··· ··· ··· ···

a1s a2s .. . ass bs bs

Theorem 3.1. If the coefficients of scheme (3.4) satisfy bj = bj (1 − cj ), 1 ≤ j ≤ s, bi aij − bj aji + bi bj − bi bj = 0, 1 ≤ i, j ≤ s, then scheme is symplectic[Sur90,Sur89,CS93,ZQ93] .

(3.5) (3.6)

7.3 Symplectic R–K–N Method

321

Proof. The proof of Theorem 3.1 can be find in [Sur89,OS92] . Here, we only point out that under conditions (3.5), (3.6) are equivalent to bi aij − bj aji + bi bj (cj − ci ) = 0,

1 ≤ i, j ≤ s.

(3.7) 

Therefore, the theorem is completed.

Similar to Section 7.1, we first introduce some necessary definitions and notations, and then derive order conditions. Some definitions of Section 7.1 can still be used. Here we only introduce some special definitions and notations. (1) S-graph. A S-graph, denoted as S-g, is a special P -graph where any two adjacent vertices belong to different categories: “ white (meagre)” or “ black (fat)”. The Labeled S-graph has a definition similar to the labeled P -graph. (2) S-tree. A S-tree, denoted as Sτ , has a definition similar to the P -tree of 7.2: replacing the P -graph in the definition of original P -tree with S-graph gives the definition of S-tree. The definition of labeled S- tree, λSτ , rooted S-tree, ρSτ , rooted labeled S-tree, ρλSτ , and isomorphic labeled S-trees, root-isomorphic labeled S-tree are defined using the same method as we have used to define P -trees, labeled P -trees, etc. We should point out that in this section, we just consider S-trees with “ black” root vertices. So when we refer to rooted S-tree, we mean that its S-tree has a “ black” root. Moreover, order r, density γ also has similar definition as mentioned in Section 7.1. But the elementary weight definition is completely different, which we will redefine subsequently. Definition 3.2. We define the elementary weight φ(ρλSτ ) corresponding to a rooted labeled S-tree. At first, for convenience, we assume ρλSτ is monotonically labeled. Later, we will see this is unnecessary. In the remainder of this section, without specification, the labels of the vertices are always j < k < l < m < · · ·. For a monotonic labeling, the label of the root is j. Then φ(ρλSτ ) is a sum over the labels of all fat vertices of ρλsτ , the general term of the sum is a product of 1◦ bj (j is a rooted vertex). 2◦ akl , if the fat vertex k is connected via a meagre son with another fat vertex l. 3◦ cm k , if the fat vertex k has m meagre end-vertices as its sons, where an endvertex is the vertex which has no son. We see that, for two different rooted labeled S-trees: ρλSτ 1 and ρλSτ 2 , we have φ(ρλSτ 1 ) = φ(ρλSτ 2 ). Thus, the choosing of the monotonic labeling is unnecessary. m j • •l • •l m k and , we have For example, for +k +j ρλSτ 1 ρλSτ 2   φ(ρλSτ 1 ) = bj cj ajm = bk akj ck = φ(ρλSτ 2 ) = φ(ρSτ ). j,m 1

j,k 2

Because ρλSτ and ρλSτ are rooted isomorphism, they belong to a rooted tree ρSτ : •

•

+ . Therefore, they form an equivalence class. The following theorem can be seen in the literature [HNW93] . We omit the proof here.

322

7. Symplectic Runge–Kutta Methods

Theorem 3.3. P-K-N method (3.4) is order of p iff: 1 , for rooted S-tree ρSτ, r(ρSτ ) ≤ p, (3.8) γ(ρSτ ) 1 , for rooted S-treeρλSτ, r(ρSτ ) ≤ (p − 1). (3.9) φ (ρSτ ) = γ(ρSτ )(r(ρSτ ) + 1)

φ(ρSτ ) =

The explanation for φ (ρSτ ) is similar to that of φ(ρSτ ), which only needs to substitute bj in φ(ρSτ ) (suppose j is the label of rooted tree, corresponding to a certain label choosing) by ¯bj . Because φ and φ is independent of the chosen label, (3.8) and (3.9) can take any of the labels to calculate. We find that (3.8) and (3.9) are not independent under symplectic conditions. Theorem 3.4. Under symplectic condition (3.5), order condition (3.8) implies condition (3.9)[ZQ95b] . Proof. Let ρSτ be ≤ p − 1 order S-tree, and let ρSu be such a rooted S-tree with r(ρSτ ) + 1 order that is obtained from ρSτ rooted tree by attaching a new branch with a meagre vertex to the root of τ . Therefore from definition of φ, we have  ;  ; bj cj bj φ(ρSu) = , φ(ρSτ ) = , j

j

where we assume that ρSτ and ρSu have monotonic labels j < k < l < · · ·. Then for ρSu, apart from the added root of the meagre leaf node, the remaining vertices ;have the same labeling as ρSτ , and is a sum for all non-fat root vertices, and is a product of aij and ci that are contained in ρSτ and ρSu. From the definition of φ, we have (r(ρSτ ) + 1)γ(ρSτ ) γ(ρSu) = , r(ρSτ ) therefore, φ (ρSτ ) =



bj

j

=

 j

=

bj

; ;

=



bj (1 − cj )

j

−



b j cj

;

;

= φ(ρSτ ) − φ(ρSu)

j

1 1 1 − = . γ(ρSτ ) γ(ρSu) (r(ρSτ ) + 1)γ(ρSτ )

(3.10)

Since the formula(3.8) is held for ≤ p-order S-tree and (3.9) is held only for ≤ p − 1-order S-tree, the order of the tree obtained by adding a leaf node to any ≤ p − 1-order tree on the root must be ≤ p and (3.8) must be satisfied. Therefore the final equal sign in (3.10) holds. Thus we reach the conclusion of this section.  Theorem 3.5. R–K–N method (3.4) is symplectic, and is of p-order iff:

7.3 Symplectic R–K–N Method

bi aij − bj aji + bi bj (cj − ci ) = 0, ¯bi = bi (1 − ci ), φ(ρSτ ) =

1 γ(ρSτ )

323

1 ≤ i, j ≤ s,

(3.11)

1 ≤ i ≤ s,

(3.12)

for rooted S-tree ρSτ, r(ρSτ ) ≤ p.

(3.13)

Note that the conditions we have given here are necessary and sufficient. However some conditions of (3.13) are still redundant, which means some conditions are mutually equivalent. We will see more details about this in Section 7.4.

7.3.2 The 3-Stage and 4-th order Symplectic R–K–N Method For convenience, we construct only explicit schemes here[QZ91] . Suppose the parameters aij of a R–K–N method to be a matrix A ⎡ ⎤ 0 0 ··· 0 0 ⎢ a21 0 · · · 0 0 ⎥ ⎢ ⎥ ⎢ a31 a32 · · · 0 0 ⎥ A=⎢ ⎥. ⎢ .. .. .. .. ⎥ ⎣ . . . . ⎦ as1

as2

· · · as,s−1

0

By the symmetry of Equations (3.11) in Theorem 3.5, we have bi aij − bj aji + bi bj (cj − ci ) = 0 ⇐⇒ bj aji − bi aij + bj bi (ci − cj ) = 0, hence Equations (3.11) can be simplified into bi aij − bj aji + bi bj (cj − ci ) = 0,

1 ≤ j < i ≤ s.

Since when j < i, aji = 0, the above formula can be written as bi aij + bi bj (cj − ci ) = 0,

1 ≤ j < i ≤ s.

For conditions of 3-stage of 3rd-order R–K–N method, we get equations for parameters : ⎧ b2 a21 + b2 b1 (c1 − c2 ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ b3 a31 + b3 b1 (c1 − c3 ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ b3 a32 + b3 b2 (c2 − c3 ) = 0, ⎪ ⎪ ⎪ ⎨ b1 + b2 + b3 = 1, (3.14) 1 ⎪ b c + b c + b c = , ⎪ 1 1 2 2 3 3 ⎪ 2 ⎪ ⎪ ⎪ 1 ⎪ 2 2 2 ⎪ b 1 c1 + b 2 c2 + b 3 c3 = , ⎪ ⎪ 3 ⎪ ⎪ ⎪ 1 ⎩ b2 a21 + b3 a31 + b3 a32 = , 6

and bi = bi (1 − ci ),

i = 1, 2, 3.

(3.15)

324

7. Symplectic Runge–Kutta Methods

Direct verification shows ⎡ 0 ⎢ 7 ⎢ A = ⎢ 36 ⎣ 0 b1 =

7 , 24

0 0

1 2 3 b2 = , 4

−

0

⎤

⎥ 0 ⎥ ⎥, ⎦ 0 b3 = −

1 , 24

c1 = c3 = 0,

c2 =

2 3

is a set of solutions for system (3.14). The number of order conditions becomes 7 for a scheme of order 4. In addition, if the scheme is of 3-stage, there are 3 equations for canonicity. Therefore we have a total of 10 equations with 9 unknown variables (systems (3.14) plus parameters ¯bi (i = 1, 2, 3)). So we first construct a 4-stage scheme of 4th order, which requires 13 equations with 14 variables: A 3-stage symplectic scheme needs to satisfy the following 6 conditions b2 a21 + b2 b1 (c1 − c2 ) = 0, b3 a31 + b3 b1 (c1 − c3 ) = 0, b3 a32 + b3 b2 (c2 − c3 ) = 0, b4 a41 + b4 b1 (c1 − c4 ) = 0, b4 a42 + b4 b2 (c2 − c4 ) = 0, b4 a43 + b4 b3 (c3 − c4 ) = 0, and equations that satisfy 4th-order conditions are as follows: b1 + b2 + b3 + b4 = 1, 1 2 1 2 2 2 2 b 1 c1 + b 2 c2 + b 3 c3 + b 4 c4 = , 3

b1 c1 + b2 c2 + b3 c3 + b4 c4 = , 1 6

b2 a21 + b3 a31 + b3 a32 + b4 a41 + b4 a42 + b4 a43 = , 1 4

b1 c31 + b2 c32 + b3 c33 + b4 c34 = , 1 8 1 b2 a21 c1 + b3 a31 c1 + b3 a32 c2 + b4 a41 c1 + b4 a42 c2 + b4 a43 c3 = . 24

b2 c2 a21 + b3 c3 a31 + b3 c3 a32 + b4 c4 a41 + b4 c4 a42 + b4 c4 a43 = ,

Set c4 = 0, we have b2 a21 + b2 b1 (c1 − c2 ) = 0, b3 a31 + b3 b1 (c1 − c3 ) = 0,

(3.16)

b3 a32 + b3 b2 (c2 − c3 ) = 0, b4 a41 + b4 b1 c1 = 0,

(3.18) (3.19)

(3.17)

7.3 Symplectic R–K–N Method

b4 a42 + b4 b2 c2 = 0, b4 a43 + b4 b3 c3 = 0, b1 + b2 + b3 + b4 = 1, 1 b1 c1 + b2 c2 + b3 c3 = , 2 1 b1 c21 + b2 c22 + b3 c23 = , 3

(3.20) (3.21) (3.22) (3.23) (3.24)

b2 a21 + b3 a31 + b3 a32 + b4 a41 + b4 a42 + b4 a43 = b1 c31 + b2 c32 + b3 c33 =

1 , 6

(3.25)

1 , 4

b2 c2 a21 + b3 c3 a31 + b3 c3 a32 =

(3.26) 1 , 8

(3.27)

b2 a21 c1 + b3 a31 c1 + b3 a32 c2 + b4 a41 c1 + b4 a42 c2 + b4 a43 c3 =

1 . (3.28) 24

We obtain a set of numerical solutions of (3.16) – (3.28): a21 = 0.2232896E − 01, a32 = 0.2886753, a42 = −0.1057342, b1 = −0.3867491E − 01, b3 = 0.5386746, c1 = 0.7886753, c3 = 0.7886752.

a31 = 0.2822977E − 08, a41 = 0.3053789E − 01, a43 = −0.4251137, b2 = 0.5000003, b4 = −0.9129767E − 07, c2 = 0.2113249,

Guessing from these numerical solutions, we obtain a31 = 0,

1 2

b2 = ,

b3 =

1 − b1 , 2

b4 = 0,

c2 = 1 − c1 .

c1 = c 3 ,

Inserting them into (3.16) – (3.28), we have (3.16) ⇐⇒ a21 + b1 (2c1 − 1) = 0, (3.17) ⇐⇒ 0 = 0,     1 1 1 − b1 a32 + − b1 (1 − 2c1 ) = 0, (3.18) ⇐⇒ 2

2

2

(3.19), (3.20), (3.21), (3.22), (3.23) ⇐⇒ 0 = 0, (3.24) ⇐⇒ (3.25) ⇐⇒ (3.26) ⇐⇒ (3.27) ⇐⇒ (3.28) ⇐⇒

325

1 2 1 1 c + (1 − c1 )2 = , 2 1 2  3 1 1 1 a21 + − b1 a32 = , 2 2 6 1 2 c1 − c1 + = 0, 6   1 1 1 (1 − c1 )a21 + c1 a32 − b1 = , 2 2 8   1 1 1 a21 c1 + − b1 a32 (1 − c1 ) = . 2 2 24

or

1 = 1,

326

7. Symplectic Runge–Kutta Methods

So we obtain a system of equations of the variables a21 , a32 , b1 , c1 a21 + b1 (2c1 − 1) = 0,     1 1 1 − b1 a32 + − b1 (1 − 2c1 ) = 0, 2

2

2

(3.29) (3.30)

1 2 1 1 c + (1 − c1 )2 = , (3.31) 2 1 2  3  1 1 1 a21 + − b1 a32 = , (3.32) 2 2  6  1 1 1 (1 − c1 )a21 + c1 a31 − b1 = , (3.33) 2 2 8   1 1 1 . (3.34) a21 c1 + − b1 a32 (1 − c1 ) = 2 2 24 √  √  1 3+ 3 3− 3 or c1 = . From (3.31), we have c21 −c1 + = 0, which leads to c1 = 6 6 6 1 Suppose b1 = , from (3.30) we obtain 2

a32 = so a32 =

1 (2c1 − 1), 2

(3.35)

√  √  3 3 or a32 = − . Under (3.35), Equation (3.32) becomes 6 6

1 a21 + 2

%

1 − b1 2

&

1 1 × (2c1 − 1) = . 2 6

Adding (3.36) and (3.29) together, we find

(3.36)

√ √ 2− 3  2 + 3 i.e., a21 = 2a21 or . 12 12 √  √  3−2 3 3+2 3 or . Therefore, we have reason to From (3.29), we obtain b1 = 12 12 speculate that √ √ √ √ 2− 3 3−2 3 3+ 3 3 a21 = , a32 = , b1 = , c1 = 12 6 12 6 and √ √ √ √ 2+ 3 3+2 3 3− 3 3 a21 = , a32 = − , b1 = , c1 = 12 6 12 6 are two sets of solutions of (3.29) – (3.34). Direct verification shows that they are indeed solutions of the equations of (3.16) – (3.28). Thus we obtain two sets of analytic solutions of the original equations of system (3.16)√– (3.28). √ 1 1 = − (2c1 − 1), 3 2

2− 3 3 , a31 = 0, a32 = , a41 , a42 , a43 arbitrary. 12 6 √ √ 3−2 3 1 3+2 3 , b2 = , b3 = , b4 = 0, b1 = 12 2 12 √ √ √ 3+ 3 3− 3 3+ 3 c1 = , c2 = , c3 = , c4 = 0. 6 6 6

Solution 1: a21 =

7.3 Symplectic R–K–N Method

327

√ √ 2+ 3 3 Solution 2: a21 = , a31 = 0, a32 = − , a41 , a42 , a32 arbitrary. 12 6 √ √ 3+2 3 1 3−2 3 b1 = , b2 = , b3 = , b4 = 0, 12 2 12 √ √ √ 3− 3 3+ 3 3− 3 c1 = , c2 = , c3 = , c4 = 0. 6 6 6

Since b4 = c4 = 0, and ¯b4 = b4 (1 − c4 ) = 0 in the two solutions, we obtain two 3-stage symplectic explicit R–K–N method of order 4. They are Scheme 1: ci

√ 3+ 3 6 √ 3− 3 6 √ 3+ 3 6

bi bi

aij 0

√ 2− 3 12

0 √ 5−3 3 24 √ 3−2 3 12

0

0

0

0

√

3 6 √ 3+ 3 12 1 2

0 √ 1+ 3 24 √ 3+2 3 12

Scheme 2: aij

ci

√ 3− 3 6 √ 3+ 3 6 √ 3− 3 6

bi bi

0

√ 2+ 3 12

0 √ 5+3 3 24 √ 3+2 3 12

0

0

0

0

√ 3 − 6 √ 3− 3 12 1 2

0 √ 1− 3 24 √ 3−2 3 12

Remark 3.6. We can obtain the required solutions easily by solving only the first 6 equations from the simplified order conditions of symplectic R–K–N.

7.3.3 Symplified Order Conditions for Symplectic R–K–N Method In subsection 7.3.1, we have a preliminary briefing of symplified order conditions for the symplectic R–K–N methods. In this section, we will simplify them further. The key is to make full use of symplectic conditions [ZQ95b] .

328

7. Symplectic Runge–Kutta Methods

Let Sτ be an S-tree of order n ≥ 3. It has at least two fat vertices. Let λSτ be a labeling. Let v and w be two fat vertices connected via a meagre vertex u. For order ≤ 2 S-tree, the root S-trees contains only one the first-order and one the secondorder. Therefore there are no such issues that the order conditions for the trees with the same order are related to each other. We consider six rooted S-trees. Let us denote ρSτ v (resp. ρSτ w ) as the rooted S-tree obtained by regarding the vertex v (resp. w) as the root of Sτ . Let us denote ρSτ vu (resp. ρSτ wu ) as the rooted S-tree with root v (resp. w) that arises when the edge (u, w)(resp. (v, u)) is deleted from sτ . Let us denote ρSτv and ρSτw as the rooted S-tree with root v and w respectively which arise when edges, (u, v) and (u, w) are deleted from Sτ . Fig. 3.1 shows the rooted trees of Theorem 3.7.

• u

v

+

w

• u

v

v ρSτ

Fig. 3.1.

vu

v

ρSτ v

Sτ

+

w

• u

• u ρSτ

w

+

ρSτ w

+ w

wu

• u

v

w +

ρSτv

ρSτw

+

Rooted S-trees

Theorem 3.7. With the above notations, we have: 1 1 1 1 − = − . γ(ρSτ v ) γ(ρSτ w ) γ(ρSτ vu )γ(ρSτw ) γ(ρSτ wu )γ(ρSτv )

1◦

And if the R–K–N method (3.4) satisfies (3.7), then, 2◦ φ(ρSτ v ) − φ(ρSτ w ) = φ(ρSτ vu )φ(ρSτw ) − φ(ρSτ wu )φ(ρSτv ). Proof. Let r(ρSτv ) = x,

r(ρSτw ) = y, ˙

n = r(Sτ ) = x + y + 1.

By definition of γ, we have γ(ρSτ v ) = n

; (x + 1)γ(ρSτw ),

γ(ρSτ w ) = n where

;; 1

2

1 ; (y + 1)γ(ρSτv ),

(3.37)

2

denotes the product of γ(τ1 )(γ(τ2 )) of the sub-trees, τi which arise

when v (resp. w) is chopped from ρSτv (resp. ρSτw ). Notice that γ is calculated as

7.3 Symplectic R–K–N Method

329

the general tree τ , with the difference between the black and white vertices neglected. Then from (3.37), we have 1 n 1 − = γ(ρSτ v ) γ(ρSτ w ) n

%

+ 1)γ(ρSτv ) − 1 (x + 1)γ(ρSτw ) (x + 1)γ(ρSτv )2 (y + 1)γ(ρSτw ) 1

2 (y

Because γ(ρSτ vu ) = (x + 1)

;

,

γ(ρSτ wu ) = (y + 1)

1

and γ(ρSτv ) = x

;

;

& .

(3.38)

,

2

,

γ(ρSτw ) = y

1

;

,

2

we have ⎛;

(y + 1)γ(ρSτv ) −

;

(x + 1)γ(ρSτw )

⎞

1 1 n⎜ 2 ⎟ 1 − = ⎝ ⎠ γ(ρSτ v ) γ(ρSτ w ) n γ(ρSτ vu )γ(ρSτ wu )γ(ρSτv )γ(ρSτw ) ;; 2 ⎛ ⎞ (x − y 2 + x − y) 1⎜ ⎟ 1 2 = ⎝ ⎠. n γ(ρSτ vu )γ(ρSτ wu )γ(ρSτv )γ(ρSτw )

However,

=

=

=

1 1 − γ(ρSτ vu )γ(ρSτw ) γ(ρSτ wu )γ(ρSτv )   n γ(ρSτ wu )γ(ρSτv ) − γ(ρSτ vu )γ(ρSτw )   n γ(ρSτ vu )γ(ρSτw )γ(ρSτ wu )γ(ρSτv ) * + ; ; ; ; n (y + 1) x− (x + 1) y 1 1 2  2  n γ(ρSτ vu )γ(ρSτ wu )γ(ρSτv )γ(ρSτw )   ;; (x + y + 1) x(y + 1) − (x + 1)y 1



2



n γ(ρSτ vu )γ(ρSτ wu )γ(ρSτv )γ(ρSτw ) ;; 2 (x − y 2 + x − y)

=

1 2  . n γ(ρSτ vu )γ(ρSτ wu )γ(ρSτv )γ(ρSτw )

Thus, we get 1◦ . By definition of φ, we have

(3.39) 

330

7. Symplectic Runge–Kutta Methods

⎧ v  ; ⎪ vu ⎪ φ(ρSτ ) = b c , ⎪ iv iv ⎪ ⎨

φ(ρSτv ) =



iv

biv

v ;

w  ; ⎪ ⎪ wu ⎪ ) = b c , φ(ρSτ ⎪ iw iw ⎩

φ(ρSτw ) =



iw

biw

w ;

,

iw

⎧ * v w+   ;; ⎪ ⎪ v ⎪ biv aiv iw φ(ρSτ ) = , ⎪ ⎪ ⎨ iv iv * v w+ ⎪   ;; ⎪ ⎪ ⎪ biw aiw iv , φ(ρSτ w ) = ⎪ ⎩

and

,

iv

iw

(3.40)

iw

v ; w  ;

denotes part of φ(ρSτ v )(resp. φ(ρSτ w ), which is the sum over black vertices of ρSτv (ρSτw ). From symplectic order condition (3.11), we have

where

φ(ρSτ v ) − φ(ρSτ w ) =



(biv aiv iw − biw aiw iv )

w ; v ;

iv ,iw

=



biv biw (civ − ciw )

v ; w ;

iv ,iw

=



biv civ

v  ;

iv

−



biw ciw

iw

biw

iw w ;

w ;

biv

v ;

iv

= φ(ρSτ vu )φ(ρSτw ) − φ(ρSτ wu )φ(ρSτv ). (3.41) Thus, the second part 2◦ of Theorem 3.7 is held. The following corollary is trivial. Corollary 3.8. Suppose that the R–K–N method (3.4) satisfying (3.7), has order at least n − 1, with n ≥ 3. If ρSτ v and ρSτ w are different rooted S-trees of order n, then the order condition is the same as given in Theorem 3.7. φ(ρSτ v ) =

1 γ(ρSτ v )

φ(ρSτ w ) =

1 γ(ρSτ w )

holds, iff

is satisfied. Proof. Because R–K–N method (3.4) is of at least n − 1-order, by Theorem 3.5, we have

7.3 Symplectic R–K–N Method

φ(ρSτ vu ) =

1 , γ(ρSτ vu )

φ(ρSτ wu ) =

1 , γ(ρSτv )

φ(ρSτw ) =

and φ(ρSτv ) =

331

1 , γ(ρSτ wu )

1 , γ(ρSτw ) 

similarly by Theorem 3.1, corollary is proved. So we have the conclusion of this subsection.

Theorem 3.9. [SSC94,ZQ95b] A R–K–N method (3.4) that satisfies symplectic conditions is order of p, iff for every S-tree Sτ there exists a rooted S-tree ρSτ v which arises when a black vertex v of Sτ is lifted as the root, such that φ(ρSτ ) =

1 . γ(ρSτ )

Proof. By Corollary 3.8 we know that any two different methods of choosing the corresponding root have equivalent order conditions. Hence the theorem is proved.  As an application of Theorem 3.9, we consider the explicit R–K–N method, i.e., aij = 0, for j > i (i, j = 1, 2, · · · , s), and the non-redundant case, i.e., bi = 0 (i = 1, 2, · · · , s), see [OS92] . Then, we have aij = bj (ci −cj ), for i ≥ j, (i, j = 1, 2, · · · , s). So we obtain the following corollary. Corollary 3.10. Non-redundant R–K–N method (3.4) is explicit symplectic and of order p, iff: 1◦ aij = bj (ci − cj ), 1 ≤ j < i ≤ s. 2◦ bj = bj (1 − cj ), 1 ≤ j ≤ s. 3◦ For every S-tree sτ , there exist a rooted S-tree ρSτ v , which arises when a black vertex v of sτ is lifted as the root, such that: φ(ρSτ ) =

1 . γ(ρSτ )

To obtain a 5-stage fifth order non-redundant symplectic explicit R–K–N method, the following equations are satisfied: aij = bj (ci − cj ),

1 ≤ j < i ≤ s,

(3.42)

bj = bj (1 − cj ),

1 ≤ j ≤ s,

(3.43)

and 5 

bj = 1,

(3.44)

j=1 5  j=1

1 2

b j cj = ,

(3.45)

332

7. Symplectic Runge–Kutta Methods 5 

1 3

bj c2j = ,

j=1 5 

1 6

bj ajl = ,

j,l=1 5 

(3.46)

(3.47)

1 4

bj c3j = ,

j=1 5 

(3.48) 1 8

bj cj ajm = ,

m,j=1 5 

(3.49)

1 5

bj c4j = ,

j=1 5 

(3.50)

bj c2j ajp =

j,p=1 5 

1 , 10

bj ajl ajp =

j,l,p=1 5 

bj cj ajl cl =

j,l=1

(3.51)

1 , 20

(3.52)

1 . 30

(3.53)

Replacing aij of system equations (3.44) – (3.53) by (3.42), we get a system of 10 equations for parameters bi , ci (i = 1, · · · , 5). Every order condition of system (3.44) − (3.53) corresponds to the S-trees of the same number in Fig. 3.2. l •k

•l j

+

j

+

1

•l j

+

+

3

2

•l

k •

k • j

+

4

•m

j

5 m•

p

m k•

•l +

6 Fig. 3.2.

j

k•

l •

m •

•p

k•

•l

l •m

k •

p •m

j + j + j 7 9 8 Rooted S-trees corresponding to order condition (3.44) – (3.53) +

l k•

• p j

+ 10

For the sake of convenience, we choose monotonic labelings for trees in Fig. 3.2. We obtain the Equations (3.46). In the following list we provide four sets of numerical solutions, whose laws are yet to be studied further.

7.4 Formal Energy for Symplectic R–K Method

1

2

3

4

i  bi

333

1

2

3

4

5

0.396826

−0.824374

0.204203

1.002182

0.221161

ci  bi

0.961729

0.866475

0.127049

0.754358

0.229296

0.221160

1.002182

0.204203

−0.824375

0.396827

ci  bi

0.770703

0.245641

0.872950

0.133524

0.038270

−1.670799

1.221431

0.088495

0.959970

0.400902

0.694313

0.637071

−0.020556

0.795861

0.301165

ci  bi

0.400902

0.959969

0.088495

1.221434

−1.670802

ci

0.698834

0.204138

1.020556

0.362928

0.305086

Remark 3.11. R–K, P–R–K, and R–K–N methods have corresponding order conditions. The order conditions for symplectic R–K, symplectic P–R–K, and symplectic R–K–N method can be simplified using symplectic conditions. The order conditions for order 1 to 8 have already been listed in Table 1.4. Calvo and Hairer[CH95] further reduce the number of independent condition in R–K–N method. See Table 3.1.

Table 3.1.

Order conditions R–K–N method and Symplectic R–K–N method for general case

Order 1 2 3 4 5 6 7 8

R–K–N method 1 1 2 3 6 10 20 36

Symplectic R–K–N method 1 1 2 2 4 5 10 14

7.4 Formal Energy for Symplectic R–K Method The energy H(z) of a Hamiltonian system is also an invariant of the system. However, under normal circumstances, no symplectic scheme can preserve all the original Hamilton energy [Fen98a] . On the other hand, any symplectic scheme preserves a formal Hamiltonian energy, which approaches the original Hamiltonian energy with the precision of numerical scheme. The calculation of formal energy can be done in many ways. First, we have obtained a complete method in theory to obtain the formal energy of a symplectic difference scheme constructed by generating function [Fen98a] . Yoshida [Yos90] uses Lie series of BCH Formula to determine the formal energy of separable Hamiltonian. What is insufficient is that the existing formal energy computational

334

7. Symplectic Runge–Kutta Methods

methods for symplectic R–K method mostly use Poincar´e lemma, and then use the quadrature method. Although theoretically primary function (total differential) does exist, obtaining the primary function through the integral is not that easy. Therefore, we attempt to calculate the formal energy of a symplectic R–K method in a easy way that does not need the integral and also does not need any differentiation.

7.4.1 Modified Equation Consider the numerical solution of ODEs z˙ = f (z),

z ∈ Rn .

(4.1)

The R–K method for Equation (4.1) is defined as follows: + * s  k i = f z0 + h aij kj ,

(4.2)

j=1

z1 = z0 + h

s 

bi ki .

(4.3)

i=1

Since the fundamental work of Butcher, numerical solution z1 can be written as (suppose f is sufficiently differentiable): * + s  hr(t)  α(t) γ(t) bi φi (t) F (t)(z0 ). (4.4) z1 = z0 + r(t) ! i=1 t∈T

Definition 4.1. [Hai94] Let t be a rooted tree. A partition of t into k subtrees {s1 , . . . , sk } is a set S, consisting of k − 1 branches of t such that the trees s1 , . . . , sk are obtained when the branches of S are removed from t. Such a partition is denoted by (t, S). We further denote α(t, S) as the number of possible monotonic labelings of t such that the vertices of the subtrees sj are labeled consecutively. Example 4.2. All partitions of t = [[τ ], [τ ]], t into k subtrees with the numbers α(t, S): k=1 • • •

• • 3

k=2 • • •

•. .. .. . ·•

• • \ • 2

• \• 1

k=3 •. • •. • •. .. .. .. .. .. .. . . . ·• • •· • •· \ / \• • •/ 1 2 1

•. • • .. .. . ·• • • \ / \•/ 1

•. .. .. . ·• \ \• 3

k=4 •. • •. .. .. .. .. . . ·• • •· \ / \•/ 2

k=5 •. •. .. .. .. .. . . ·• •· \ / \•/ 3

Suppose a numerical method can be expressed as a formal series z 1 = z0 +

 hr(t) α(t)a(t)F (t)(z0 ), r(t) ! t∈T

(4.5)

7.4 Formal Energy for Symplectic R–K Method

Table 4.1.

335

Relation between coefficients a(t) and b(t)(1)

t= •

t=

t=

t=

a(•) •

a

• • • • • •

a a

•

=

 • •

• •  • • •

 •

b(•)

=

b

=

b

=

a

 • + b ( • )2 • • •  3  •  b( • ) + b( • )3 + b 2 • • • •

  • • + 3b b ( • ) + b ( • )3 •

where a : T → R is an arbitrary function. Such a series is called a B-series. If function f (z) is only N -times continuously differentiable, then the series (4.5) has to be interpreted as a finite sum over t ∈ T with (r(t) ≤ N ). Theorem 4.3. [Hai94] Let a : T → R be an arbitrary mapping, and the right side of equation (4.1) f (z) is N -times continuously differentiable. The numerical solution given by (4.5) satisfies (4.6) z1 = z"(t0 + h) + O(hN +1 ). Here, z"(t) is the exact solution of the modified equation: 

" z˙ =

r(t)≤N

hr(t)−1 α(t)b(t)F (t)(" z ), r(t) !

(4.7)

where the coefficients b(t) can be defined recursively by: a(t) =

r(t)  α(t, S)  1  r(t) b(s1 ) · · · b(sk ). r(s1 ), · · · , r(sk ) k! α(t) k=1

(4.8)

(t,S)

The second sum in (4.8) is over all partitions of t into k subtrees {s1 , · · · , sk }. By (4.8), we can define relation between coefficients a(t) and b(t), See Table 4.1 to Table 4.4[LQ01] . According to Table 4.1 – Table 4.4, we can determine modified equation for R–K equation (up to 5 orders, it is clear that as long as the order continues to add 6, 7-order tree · · · equation can be modified to any order). Remark 4.4. If numerical method is symmetrical (or time-reversible), then when r(t) is even, b(t) = 0. Remark 4.5. If numerical method is p-order, in other words when r(t) ≤ p, a(t) = 1; then, if 2 ≤ r(t) ≤ p, b(•) = 1, b(t) = 0; if r(t) = p + 1, b(t) = a(t) − 1.

336

7. Symplectic Runge–Kutta Methods

Table 4.2.

t=

Relation between coefficients a(t) and b(t)(2)

•• • •

t=•

•

a

• • • • • • • •  • =b + 2b( • )b + 2b( • )2 b( •/ ) + b( • )4 • • • %

•

a

•

•

•

& •

•

% • &  • • 2 •  • 4 • + b( / )2 + b( • )b = b • • + b( • )b • • 3 3 • •

• 10 b( • )2 b( •/ ) + b( • )4 3 %• •& %• •&  • • •  • • • =b + 2b( • ) b a + 2b( • )b • • • • +

• t=

• •

•

• +4b( • )2 b( •/ ) + b(•)4 %•

• • t=• •

a

•

& • •

%• =b

•

& •

+ 4b( • ) b

•

•

 • • + 3b( / )2 • •

• +6b( • )2 b( •/ ) + b( • )4

Example 4.6. Centered Euler scheme zn+1 = zn + hf

 z + z n n+1 . 2

Modified equation can be defined1    h2 (2) h4 7 (4) " f (f, f ) − 2f (1) f (1) f + z˙ = f (" z) − f (f, f, f, f) 24

120 48

  3 1 1 + f (3) (f, f, f (1) f ) − f (2) f, f (2) (f, f ) − f (2) (f, f (1) f (1) f ) 4 4 2 3 4

7 (1) (3) 1 f f (f, f, f ) − f (1) f (2) (f, f (1) f ) 12 2

4

2

+ f (2) (f (1) f, f (1) f ) −

 1 3 + f (1) f (1) f (2) (f, f ) + f (1) f (1) f (1) f (1) f + O(h6 ).

Example 4.7. 2-stage Gauss–Legendre method [HNW93] : 1

b(t) by Section 7.5.

7.4 Formal Energy for Symplectic R–K Method

Table 4.3.

337

Relation between coefficients a(t) and b(t)(3)

   5  ••• • • 10 • • • • • • b( • )2 b( t = • • a • • = b • • + b( • )b( )+ ) • • 2 3 • • • • 5 + b ( • )3 b( •/ ) + b( • )5 2 % • • •& • • • % • • •&  • • 5 • • • 10 •  5 • • • b( • )2 b =b + b( • )b • + b( • )b t= • a + • 2 2 3 • • • • • +

 • • • 10 5 + 5b( • )3 b( / b( • )2 b • ) + b( • ) • 3

• % •& % •& %• & •  • • • • •  • • + 10b( • )2 b • • =b • + 5b( • ) b / t= • a + 10b( )b • • • • • • • • • • • • • +15b( • )b( •/ )2 + 10b( • )3 b( •/ ) + b( • )5

√ 1 3 1 − 2 6 4 √ √ 1 3 1 3 + + 2 6 4 6 1 2

√ 3 1 − 4 6 1 4 1 2

Modified equation is defined2 A 37h4 @ (4) h4 @ (3) " f (f, f, f, f) − 4f (1) f (3) (f, f, f ) − f (f, f, f (1) f ) z(t) ˙ = f (" z) − 8840 720   A −f (2) f, f (2) (f, f ) − 2f (1) (f (2) f, f (1) f ) + f (1) f (1) f (2) (f, f ) −

A 37h4 @ (2) (1) f (f f, f (1) f ) − 2f (2) (f, f (1) f (1) f ) + 2f (1) f (1) f (1) f (1) f 2880

+O(h6 ). Example 4.8. 2-order diagonal implicit R–K method: Modified equation: 2

b(t) by Section 7.5.

338

7. Symplectic Runge–Kutta Methods

Table 4.4.

Relation between coefficients a(t) and b(t)(4)

%• • %• • & % % •& % • •& • • • • • • • • ) + 52 b • )b • =b t= • a ) + 52 b( • )b + 5b(•/ )b • • • • • • •     • • • • •2 • • 2 2 10 • / + 20 b( • ) b b( • ) b + 5b( • )b( ) + 3 3 • • • • 3 5 / + 15 b( • ) b( ) + b( • ) 2 • • • • & % •& % • • %• •&  • • • %• •& • 5 • • • + 52 b( • )b • • + 56 b( • )b t= • a =b • + 3 b( • )b • • • • • • •  40 • 2 • • • + 9 b( • ) b • ) + 5b( • )b( / )2 + 53 b(•/ )b • • • • • 2 • 3 5 20 20 + 9 b( • ) b( ) + 3 b( • ) b( / •) • ) + b( % • % • & % • & • • &  • • • t = • • • a • • • = b • • • + 54 b( • )b + 54 b( • )b • • • • • •   •  • • • • • • 2 5 25 +6b + 53 b( • )b(•/ )2 •  b(•/ ) + 9 b( • ) b • • • 5 • + 15 + 59 b( • )2 b b( • )3 b( / 4 • ) + b( • ) • % • •& % • •& • • % • •& •  • • • =b • • + 58 b( • )b + 54 b(•/ )b t =• • a • • • • • • • % • &  • • •  • + 15 b( • )b • • + 10 b( • )2 b + 53 b( • )2 b 8 3 • • • • • + 54 b( • )b(•/ )2 + 35 b( • )3 b(•/ ) + b( • )5 8 • • • % • & % • & % •& • • •  5 • • • • 5 • + 2 b( / )b = b • • + 58 b( • )b t =• • a • • • + 4 b(•/ )b • • • • & • • •  • •  •2 2 15 10 • + 5b( • )b( / ) + 8 b( • )b • • + 3 b( • ) b • •  •  • • • 2 3 5 + 53 b( • ) b b( • ) b(•/ ) + b( • ) + 25 4 • & % & % •   • • • • • • • •  • • + 5b( • )b( / )2 t = • • a • • = b • • + 52 b( • )b • • + 53 b(•/ )b • • •    • • • • • • • • + 20 + 10 b( • )2 b b( • )2 b + 5b( • )3 b(•/ ) 9 9 • • +b( • )5

A h2 @ (2) 17h4 @ (4) " f (f, f ) − 2f (1) f (1) f − f (f, f, f, f) z(t) ˙ = f (" z) − 96

10240

A 13h @ (3) −4f (1) f (3) (f, f, f ) − f (f, f, f (1) f ) − f (2) (f, f (2) (f, f )) 4

2560

A h4 @ (2) (1) −2f (1) (f (2) f, f (1) f ) + f (1) f (1) f (2) (f, f ) − f (f f, f (1) f ) 2560 A −2f (2) (f, f (1) f (1) f ) + 2f (1) f (1) f (1) f (1) f + O(h6 ).

7.4 Formal Energy for Symplectic R–K Method

1 4 3 4

1 4 1 2 1 2

339

0 1 4 1 2

7.4.2 Formal Energy for Symplectic R–K Method Let T be a set of all rooted trees. On set T we define a relation ∼, as follows: 1◦ t ∼ t. 2◦ u ◦ v ∼ v ◦ u. 3◦ If u1 ◦ v1 ∼ u2 ◦ v2 , u2 ◦ v2 ∼ u3 ◦ v3 , then u1 ◦ v1 ∼ u3 ◦ v3 . Where u ◦ v and v ◦ u is defined by if u = [u1 , · · ·, um ], v = [v1 , · · · , vm ], then u ◦ v = [u1 , · · · , um , v],

v ◦ u = [v1 , · · · , vm , u].

Obviously “∼ ” expresses an equivalent relation. Using this equivalent relation to classify the root tree collection T , we can obtain a quotient space, denoted by T E. " which is obtained by filtering every equivalent Then we may construct another set T E, class of T E. The filtering rule is as follows: if t ∈ T E, then t is a rooted tree’s subset class. We sort the element of t according to σ(t), and choose t so that σ(t) is the biggest. In general, such t is not unique and we can choose any one of them. For each " we define a quasi elementary differential element in the T E   F ∗ (t) = f (m−1) F (t1 ), F (t2 ), · · · , F (tm ) . (4.9) The reason why we call (4.9) quasi elementary differential is because under normal circumstances, elementary differential has been defined as:   F (t) = f (m) F (t1 ), F (t2 ), · · · , F (tm ) , where f (m) (K1 , K2 , · · · , Km ) is m-order Frechet derivative. Here, we regard f (m−1) (K1 , K2 , · · · , Km ) as a formal definition. Obviously, when f is a differential of some function, f (m−1) (K1 , K2 , · · · , Km ) will become morder Frechet derivative of its primary function. For example, t = [τ, τ, τ, τ ], F ∗ [t] = f (3) (f, f, f, f). Let f = JHz , then F ∗ (t) becomes JH (4) (JHz , JHz , JHz , JHz ), which is obviously a fourth-order Frechet derivative. We use L to express this mapping, namely L : f (m−1) (K1 (f ), K2 (f ), · · · , Km (f )) −→ JH (m) (K1 (JHz ), K2 (JHz ), · · · , Km (JHz )). Obviously, L is a 1 to 1 mapping. From now on, we will always use L to express this mapping unless it is specified otherwise. We will no longer differentiate F ∗ (t) with L(F ∗ (t)). The operation of F ∗ (t) is always thought as operation of L(F ∗ (t)).

340

7. Symplectic Runge–Kutta Methods

Lemma 4.9. Let (A, b) be a symplectic R–K method, a(t) = γ(t)

s 

bj φj (t), then:

j=1

a(v ◦ u) a(u) a(v) a(u ◦ v) + = · , γ(u ◦ v) γ(v ◦ u) γ(u) γ(v) Proof.

[CS94]

u, v ∈ T P.

(4.10) 

.

" h) be the formal energy of symplectic R–K method (A, b), Lemma 4.10. Let H(z, then the corresponding modified equation is (possible differ by an arbitrary constant): " z˜. " z˙ = f"(" z) = J H

(4.11)

Conversely, if modified equation of symplectic R–K the method (A, b) are f"(" z ), then we have " z = −J f"(z). H (4.12) Proof. Note that both the modified equation and the formal energy can be obtained essentially via series expansion, Lemma 4.10 is obvious.  Lemma 4.11. Let (A, b) be a symplectic R–K method, a(t) = γ(t)

s 

bj φj (t), then:

j=1

b(v ◦ u) b(u ◦ v) + = 0, γ(u ◦ v) γ(v ◦ u)

u, v ∈ T P,

u = v,

(4.13)

where b(t) is determined recursively by (4.8). Proof. Using the method that proves Lemma 10 in literature[Hai94] and modifying it slightly will complete the proof. We leave out its detail here.  Remark 4.12. By (4.13) and relation α(t)γ(t)σ(t) = r(t) we obtain α(u ◦ v) b(u ◦ v) σ(v ◦ u) + = 0. α(v ◦ u) b(v ◦ u) σ(u ◦ v)

(4.14)

We now need another coefficient ν(t) related to rooted tree. Note first that the rooted trees u ◦ v and v ◦ u represent an identical unrooted tree with different root, i.e., selecting the other vertex of u ◦ v as root leads to tree v ◦ u. Thus, in an equivalent class, it can always transform one rooted tree to another by selecting a different root. Take t¯ ∈ T E, then t¯ is one equivalent class of rooted tree collection (t represents the element). Let u ∈ t¯, then u can be obtained by selecting some vertex of t as the root node. We denote ν(u) as number of vertices of t which may be selected. Example 4.13. t = [τ, τ, [τ ]], then t¯ = {t1 , t2 , t3 , t4 }, where • • • • • t2 = • • , t3 = • t1 = • • • , , • • • • We have

• • • . t4 = • •

7.4 Formal Energy for Symplectic R–K Method

ν(t1 ) = 1,

ν(t2 ) = 1,

ν(t3 ) = 1,

341

ν(t4 ) = 2.

Then, the following relation for ν(t) is hold ν(u ◦ v) · σ(u ◦ v) = ν(v ◦ u) · σ(v ◦ u).

(4.15)

Lemma 4.14. Let rooted tree u = v, then F ∗ (u ◦ v) = −F ∗ (v ◦ u).

(4.16)

Proof. F ∗ (u ◦ v) = JJ −1 F (u)JJ −1 F (v). Let α = J −1 F (u), β = J −1 F (v), where F (u), F (v) are elementary differentials. If u = [u1 , u2 , · · · , um ], then   F (u) = f (m) F (u1 ), F (u2 ), · · · , F (um ) , where f = JHz , and F (v) is similar to F (u). By properties of elementary differential (multilinear mapping), α,β are 2n-dimensional vectors, and F ∗ (u ◦ v) = J(α, Jβ) = JαT Jβ = −Jβ T Jα = −F ∗ (v ◦ u). 

Therefore, the lemma is completed. " then Lemma 4.15. Let t∗ ∈ T E, α(t∗ )b(t∗ )∇F ∗ (t∗ ) =



α(t)b(t)F (t).

(4.17)

t∈t¯∗

Proof. First the right side of (4.17) is uniquely determined, and on the left side the selection of t∗ may not necessarily be unique. Therefore, it is required to prove that (4.17) the left side of the formula is independent of selection of t∗ . We explain it as follows: given t∗1 ∈ t∗ , such that σ(t∗ ) = σ(t∗1 ), then there exists a series of u ◦ v, such that t∗ = u1 ◦ v1 ∼v1 ◦ u1 = u2 ◦ v2 ∼v2 ◦ u2 = · · · ∼um ◦ vm = t∗1 .

(4.18)

Let m = 2, i.e., t∗ = u1 ◦ v1 , t∗1 = v1 ◦ u1 , then by (4.14) α(t∗ )b(t∗ ) = −α(t∗1 )b(∗1 ), and by Lemma 4.14 F ∗ (u ◦ v) = −F ∗ (v ◦ u) =⇒ ∇F ∗ (u ◦ v) = −∇F ∗ (v ◦ u), therefore

α(t∗ )b(t∗ )∇F ∗ (t∗ ) = α(t∗1 )b(t∗1 )∇F ∗ (t∗1 ). ∗1

(4.19)

For m > 2, t must be a node of (4.18), (4.19) is also held. Next, ∇F ∗ (t∗ ) must be a linear combination of basic differentials in the same class, i.e.,

342

7. Symplectic Runge–Kutta Methods

∇F ∗ (t∗ ) =



β(t)F (t).

t∈t¯∗

Consider a special case: •k l t∗ = •

m •

•j

,

•i

F ∗ (t∗ ) = f (2) (f, f, f (1) f ). The differentiation of F ∗ (t∗ ) can be seen as a following process: first differentiate w.r.t. the root node one time, obtain f (3) (f, f, f (1) f ), and then differentiate w.r.t. each vertex one time, i.e., add 1 to each vertex superscript and then move it in front of its father along with its substance, continue moving until it reaches the front of root node. In this process, according to (4.4), every move accompanies a change in sign. Using the above example, differentiating w.r.t. the point i, j, k, l, m respectively, we obtain m

−→ −f (1) f (2) (f, f (1) f ),

l −→ −f (1) f (2) (f, f (1) f ),

j

−→ −f (2) (f, f (2) (f, f )),

k −→ f (1) f (1) f (2) (f, f ).

Thus, we get: ∇F ∗ (t∗ ) = f (3) (f, f, f (1) f ) − 2f (1) f (2) (f, f (1) f ) −f (2) (f, f (2) (f, f )) + f (1) f (1) f (2) (f, f ). It is easy to see, β(t) = ±

ν(t) , ν(t∗ )

where “ ± ” is selected using the following rule: if d(·) expresses the distance between the vertex to the root node, i.e., the least number of vertices passed from this vertex to the root node along the connection between vertices (including initial point and root node), then sign (β(t)) = (−1)d(t) . Using the above example d(i) = 0,

d(m) = d(l) = d(j) = 1,

d(k) = 2.

By sign (b(u ◦ v)) = −sign (b(v ◦ u)), we have (−1)d(t) = therefore: ∇F ∗ (t∗ ) =

sign (b(t)) , sign (b(t∗ ))

 α(t)b(t) F (t). α(t∗ )b(t∗ ) ¯∗

t∈t

7.4 Formal Energy for Symplectic R–K Method

Thus, we get α(t∗ )b(t∗ )∇F ∗ (t∗ ) =



343

α(t)b(t)F (t).

t∈t¯∗



Therefore, the lemma is completed. With the above results, we describe the main result of this section[Hai94] .

Theorem 4.16. Given a R–K method (A, b), A = (aij )s×s , b = (b1 , b2 , · · · , bs ) , its formal energy is " h) = −J H(z,

 hρ(t)−1 α(t)b(t)F ∗ (t), ρ(t)!

" t ∈ T E,

(4.20)

ρ(t)≤N

where b(t) is determined by a(t) (According to Table 4.1 to Table 4.4), i.e., a(t) = γ(t)

s 

bj φj (t).

j=1

Proof. Let the modified equation be " z˙ =

 hr(t)−1 α(t)b(t)F (t)(" z ), r(t) !

t∈T P

then " z˙ =

 hr(t∗ )−1  α(t)b(t)F (t)(" z) r(t∗ ) ! ¯∗

" t∗ ∈T E

=

t∈t

 hr(t∗ )−1 α(t∗ )b(t∗ )∇F ∗ (t∗ ). r(t∗ ) !

" t∗ ∈T E

By Lemma 4.10 " z = −J H

 hr(t∗ )−1 α(t∗ )b(t∗ )∇F ∗ (t∗ ), r(t∗ )!

" t∗ ∈T E

which leads to (differ by an arbitrary constant) " h) = −J H(z,

 hr(t∗ )−1 α(t∗ )b(t∗ )F ∗ (t∗ ). r(t∗ ) !

" t∗ ∈T E

The theorem is proved.



Remark 4.17. Literature [Tan94] pointed out that each item of series expansion of formal energy of symplectic R–K scheme has 1 to 1 corresponding relationship with unrooted trees collection. This theorem specifically indicates this 1 to 1 correspondences.

344

7. Symplectic Runge–Kutta Methods

Finally we sum up the method to construct the formal energy of  a symplectic R–K method: given a symplectic R–K method (A, b, c), let a(t) = γ(t) bj φj (t). Then j

according to Table 4.1 to Table 4.4 identify the corresponding b(t) to each rooted " Using (4.20), we can directly write the formal energy. Without loss of tree in T E. generality, in practice we can choose  7 • • • • • • • • • • "= •, • • • , • • , ··· . TE , , • •, • • • • • If we know the order of method or the method are time reversible (symmetrical), then according to Remark 4.4 and Remark 4.5 many calculations can be left out. Example 4.18. Centered Euler scheme zn+1 = zn + hf

 z + z n n+1 . 2

Its formal energy[Tan94] : 2 4 " h) = H(z) + h Jf (2) (f, f ) − 7h Jf (3) (f, f, f, f) H(z,

24

−

5760

4

h h4 Jf (2) (f, f, f (1) f ) − Jf (2) (f (1) f, f (1) f ) + O(h6 ). 480 160

Example 4.19. Gauss–Legendre method √ 1 3 − 2 6 √ 1 3 + 2 6

√ 3 1 − 4 6 1 4 1 2

1 4

√ 1 3 + 4 6 1 2

Its formal energy: 4 4 " h) = H(z) + 37h Jf (3) (f, f, f, f) + h Jf (2) (f, f, f (1) f ) H(z,

8840

720

4

+

37h Jf (1) (f (1) f, f (1) f ) + O(h6 ). 2880

Example 4.20. Diagonal implicit R–K method 1 4 3 4

1 4 1 2 1 2

0 1 4 1 2

7.5 Definition of a(t) and b(t)

345

Its formal energy 2 4 " h) = H(z) + h Jf (1) (f, f ) + 17h Jf (3) (f, f, f, f) H(z,

96

10240

4

+

13h h4 Jf (2) (f, f, f (1) f ) + Jf (1) (f (1) f, f (1) f ) + O(h6 ). 2560 2560

7.5 Definition of a(t) and b(t) We consider following schemes.

7.5.1 Centered Euler Scheme 1 2

1 b( • ) = 1

 • b •/ = 0

a

  1 b • • =− • 4

b

 • • • 1 = • 2

 •  a • • =1 •

b

a

 • • 3 • = 2 •

 • a • • =3 •

  5 a •• •• = 16 •  • • 15 a •• = 16 •

a( • ) = 1

• a(•/ ) = 1

  3 a • • = • 4

a

•  3 • = 2 •

•  1 • = 2 •

 • • • =0 •

 •  b • • =0 •

b

 • • • =0 •

 • b • • =0 •

 •  5 a ••• = 8 •

  7 b •• •• = 48 •

 •  1 b ••• = 24 •

•  •  15 a • • = 8 •

 • • 1 b •• =− 16 •

•  •  3 b • • =− 8 •

346

7. Symplectic Runge–Kutta Methods

• a •

•

• 5 • = 4

• • • 5 a • = 2 • • • 15 • = 2 • •

a

 • • • 5 • = 4 •

• b •

a

•

 • • • 7 • =− 12 •

• 1 • =

b

4

• • • 1 b • =− 6 •

• •  15 a • = • 4 •

• •  1 • = • 4 •

b

• • 3 • = 2 • •

b

7.5.2 Gauss–Legendre Method √ 1 3 − 2 6 √ 1 3 + 2 6

1 4

√ 1 3 − 4 6

√ 1 3 + 4 6

1 4

1 2

1 2

  b /• = 0 • •  b • =0 •

a( • ) = 1

a( /•) = 1 •

b( • ) = 1

a( • • ) = 1 •

• a( •) = 1 •

  b • • =0 •

••• )=1 a( •

 • • a • =1 •

• • • b =0 •

b

 •  a • • =1 •

 • a • • =1 •

 •  b • • =0 •

 • b • • =0 •

• a •

 •  35 a ••• = 36 •

• b •

 •  1 b ••• =− 36 •

• a •

•

•

• 35 • = 72

• 35 • = 72

• b •

•

•

 • • • =0 •

• 37 • = − 72 • 37 • =− 72

7.6 Multistep Symplectic Method

347

7.5.3 Diagonal Implicit R–K Method 1 4 3 4

a( • ) = 1 15 a( • • ) = • 16 1 ••• a( )= • 2  •  3 a • • = 2 •  • • 205 a • • = 256 • • • 65 a • • = 64 •

1 4 1 2 1 2

a( /•) = 1 • • 9 •) = a 8 •  • • a • =1 •  • a • • =3 •  •  115 a ••• = 128 •

0 1 4 1 2

b( • ) = 1   1 b • • =− • 16 • • • =0 b •  •  b • • =0 •  • • 51 b • • = − 256 • • • 1 b • • =− 64 •

  b /• = 0 • •  1 b • = 8 •  • • b • =0 •  • b • • =0 •  •  13 b ••• =− 128 •

7.6 Multistep Symplectic Method We present in this section Multistep method for Hamiltonian system.

7.6.1 Linear Multistep Method Consider the autonomous ODEs on Rn dz = a(z), dt

(6.1)

where z = (z1 , · · · , zn ) and a(z) = (a1 (z), · · · , an (z)) is a smooth vector field on Rn . For Equations (6.1) we define a linear m step method (LMM) in standard form by m m   αj zj = τ βj Qj , (6.2) j=0

j=0

where αj and βj are constants subject to the conditions αm = 1,

|α0 | + |β0 | = 0.

348

7. Symplectic Runge–Kutta Methods

If m = 1, we call (6.2) a single step method. Otherwise, we call it a multi-step method. The linearity means that the right hand of (6.2) linearly depends on the value of a(z) on integral points. For compatibility of (6.2) with Equation (6.1), it must be of at least order one and thus satisfies (1) α1 + α2 + · · · + αm = 0; m  (2) β0 + β1 + · · · + βm = jαj = 0. j=0

LMM method (6.2) has two characteristic polynomials ξ(λ) =

m 

αi λi ,

σ(λ) =

i=0

m 

βi λi .

(6.3)

i=0

Equation (6.2) can be written as ξ(E) = τ a(σ(E)yn ).

(6.4)

In next subsection, we propose a new definition for symplectic multi-step methods. This new definition differs from the old ones given for the single step method. It is defined directly on M which corresponds to the m step scheme defined on M , while the old definitions are given by defining a corresponding one step method on M × M × · · · × M = M m with a set of new variables. The new definition introduces a step transition operator g : M → M . Under our new definition, the leap-frog method is symplectic only for linear Hamiltonian systems. The transition operator g will be constructed via continued fractions and rational approximation.

7.6.2 Symplectic LMM for Linear Hamiltonian Systems First we consider a linear Hamiltonian system dz = az, dt

(6.5)

where a is an infinitesimal 2n × 2n symplectic matrix a ∈ sp(2n). Its phase flow is z(t) = exp (ta)z0 . The LMM for (6.5) is αm zm + · · · + α1 z1 + α0 z0 = τ a(βm zm + · · · + β1 z1 + β0 z0 ).

(6.6)

Our goal is to find a matrix g, i.e., a linear transformation g : R2n → R2n which can satisfy (6.6)   αm g m (z0 ) + · · · + α1 g(z0 ) + α0 z0 = τ a βm g m (z0 ) + · · · + β1 g(z0 ) + β0 z0 . (6.7) Such a map g exists for sufficiently small τ and can be represented by continued fractions and rational approximations. We call this transformation step transition operator[Fen98b] . Definition 6.1. If g is a symplectic transformation, then its corresponding LMM (6.6) is symplectic (we simply call the method SLMM).

7.6 Multistep Symplectic Method

349

From (6.7), we have τa =

α0 I + α1 g 1 + · · · + αm g m . β0 I + β1 g 1 + · · · + βm g m

(6.8)

The characteristic equation for LMM is ξ(λ) = τ μσ(λ),

(6.9)

where μ is the eigenvalue of the infinitesimal symplectic matrix a and λ is the eigenvalue of g. Let ξ(λ) ψ(λ) = , (6.10) σ(λ) then (6.9) can be written as τ μ = ψ(λ).

(6.11)

λ = φ(τ μ).

(6.12)

Its inverse function is To study the symplecticity of the LMM, one only needs to study the properties of functions φ and ψ. We will see that if φ is of the exponential form or ψ is of logarithmic form, the corresponding LMM is symplectic. We first study the properties of the exponential functions and logarithmic functions. Explike and loglike functions First we describe the properties of exponential functions: (1) exp (x)|x=0 = 1. d

(2) exp (x)|x=0 = 1. dx (3) exp (x + y) = exp (x) · exp (y). If we substitute y with −x, we have exp (x) exp (−x) = 1.

(6.13)

Definition 6.2. If a function φ(x) satisfies φ(0) = 1, φ (0) = 1 and φ(x)φ(−x) = 1, we call this function an explike function. It is well known that the inverse function of an exponential function is a logarithmic function x → log (x). It has the following properties: (1) log x|x=1 = 0; (2)

d log x|x=1 = 1; dx

(3)

log xy = log x + log y.

If we take y =

1 , we get x

log x + log

1 = 0. x

(6.14)

350

7. Symplectic Runge–Kutta Methods

Definition 6.3. If a function ψ satisfies ψ(1) = 0, ψ  (1) = 1, and % & 1 ψ(x) + ψ = 0, x

(6.15)

we call it a loglike function. Obviously, polynomials can not be explike functions or loglike functions, so we try to find explike and loglike functions in the form of rational functions. Theorem 6.4. [Fen98b] LMM is symplectic for linear Hamiltonian systems iff its step transition operator g = φ(τ a) is explike, i.e., φ(μ)·φ(−μ) = 1, φ(0) = 1, φ (0) = 1. Theorem 6.5.

[Fen98b]

LMM is symplectic for linear Hamiltonian systems iff ψ(λ) =   = 0, ψ(1) = 0, ψ (1) = 1.

ξ(λ) 1 is a loglike function, i.e., ψ(λ) + ψ σ(λ) λ

1

Proof. From Theorem 6.4, we have φ(μ)φ(−μ) = 1, so λ = φ(μ), = φ(−μ). The λ     1 1 inverse function of φ satisfies ψ(λ) = μ, ψ = −μ, i.e., ψ(λ) + ψ = 0, λ λ  condition (1), (2). ψ(1) = 0, ψ (1) = 1 follows from consistency   1 , let ψ(λ) = μ, then its inverse function is λ

On the other side, if ψ(λ) = −ψ

φ(μ) = λ and φ(−μ) =

1 , we then have φ(μ) · φ(−μ) = 1. λ



Theorem 6.6. If ξ(λ) is a antisymmetric polynomial, σ(λ) is a symmetric one, then ψ(λ) =

ξ(λ) , satisfies σ(λ)

  ψ(1) = 0,

ψ

1 λ

+ ψ(λ) = 0.

Proof. Since  

1 ξ(λ) = λ ξ λ m

=

 

1 σ(λ) = λm σ λ

ψ(λ) =

ξ(λ) , σ(λ)

=

m 

am−i λ = − i

i=0 m 

ψ

1 λ

βm−i λi =

ψ(1) =

ξ(1) = 0. σ(1)

1 λ

m 

βi λi = σ(λ),

i=1

% & % & 1 1 λm ξ λ λ −ξ(λ) % & = , = % & = σ(λ) 1 1 σ λm σ λ λ ξ

  we obtain ψ(λ) + ψ

ai λi = −ξ(λ),

i=1

i=0

 

m 

= 0. Now ξ(1) =

m  k=0

αk = 0, σ(1) =

m 

βk = 0, then

k=0



7.6 Multistep Symplectic Method

351

Corollary 6.7. If above generating   polynomial is consistent with ODE (6.1), then 1 ˙ ψ(λ) is loglike function, i.e., ψ + ψ(λ) = 0, ψ(1) = 0, ψ(1) = 1. λ

˙ − ξξ ˙ ˙ ξσ ξ(1) Proof. ψ  (1) = = = 1. This condition is just consistence condition.  2 σ σ(1)

Theorem 6.8. Let ψ(λ) =

ξ(λ) irreducible loglike function, then ξ(λ) is an antiσ(λ)

symmetric polynomial while σ(λ) is a symmetric one. Proof. We write formally ξ(λ) = αm λm + αm−1 λm−1 + · · · + α1 λ + α0 , σ(λ) = βm λm + βm−1 λm−1 + · · · + β1 λ + β0 . If deg ξ(λ) = p < m, set ai = 0 for i > p; if deg σ(λ) = q < m, set βi = 0 for i > q. ψ(1) = 0 ⇒ ξ(1) = 0, since otherwise, if ξ(1) = 0, then ψ(1) =

ξ(1) = 0. σ(1)

Now ξ(1) = 0 ⇔ σ(1) = 0, since otherwise ξ(1) = σ(1) ⇒ ξ(λ), σ(λ) would have common factor. So we have ξ(1) = σ(1) =

m  k=0 m  k=0

αk = βk =

p  k=0 q 

αk = 0, βk = 0.

k=0

If m = deg ξ = p, then am = ap = 0. If m = deg σ = q, then βm = βp = 0.   ψ

1 λ

% & % & 1 1 λm ξ λ λ ξ(λ) % & = . = % & = σ(λ) 1 1 m σ λ σ λ λ ξ

  Since ψ(λ) + ψ

1 λ

= 0, we have

ξ(λ) ξ(λ) =− ⇐⇒ ξ(λ)σ(λ) = −ξ(λ)σ(λ) σ(λ) σ(λ)

=⇒ ξ(λ)|ξ(λ)σ(λ) and σ(λ)|σ(λ)ξ(λ). Since ξ(λ), σ(λ) have no common factor, then ξ(λ)|ξ(λ), σ(λ)|σ(λ). If m = deg ξ(λ) ⇒ deg ξ ≤ deg ξ ⇒ ∃ c, ξ(λ) = cξ(λ) =⇒ σ(λ) = −cσ(λ). Since αm = 0 ⇒ αm λm + αm−1 λm−1 + · · · + α0 = c(αm + · · · + α0 wn ) ⇒ αm = cα0 , α0 = cαm ⇔ αm = c2 αm , therefore c2 = 1, c = ±1. Suppose c = +1, then

352

7. Symplectic Runge–Kutta Methods

σ(λ) = −¯ σ (λ), and

m 

βk = σ(1), then−¯ σ (1) = σ(1) ⇔ σ(1) = 0, which leads to

k=0

a contradiction with the assumption σ(1) = 0. Therefore c = −1, i.e., ˜ ξ(λ) = −ξ(λ), σ(λ) = σ ˜ (λ),

αj = −αm−j , βj = βm−j ,

j = 0, 1, · · · , m, j = 0, 1, · · · , m. 

The theorem is proved.

The proof for the case m = deg σ(λ) proceeds in exactly the same manner as above.

7.6.3 Rational Approximations to Exp and Log Function 1. Leap-frog scheme We first study a simple example: z2 = z0 + 2τ az1 .

(6.16)

Let z1 = cz0 , then z0 = c−1 z1 , insert this equation into (6.16),we get % & 1 1 1 z2 z2 = 2τ az1 + z1 = 2τ a + z1 = , z = z1 = d1 z1 , 1 2 c c d1 2τ a + c ⎛ ⎞ ⎜ z3 = z1 + 2τ az2 = ⎝2τ a +

⎟ z = d 2 z2 , 1⎠ 2 2τ a + c 1

1

z2 = 2τ a +

⎞

⎛ ⎜ ⎜ 1 ⎜ z4 = ⎜2τ a + 1 ⎜ 2τ a + ⎝ 1 2τ a + c

⎟ ⎟ ⎟ ⎟ = d4 z3 , ⎟ ⎠

z3 ,

1 2τ a +

1 c

···.

Where dk can be written in the form of continued fractions 1 1 1 , 2τ a + 2τ a + · · · + 2τ a + · · ·  lim dk = g = τ a + 1 + (τ a)2 .

dk = 2τ a + k→∞

(6.17) (6.18)

We assume the transition operator of Leap-frog to be g, from (6.16) we have g 2 − 1 = 2τ ag,

 2 now we  have g = τ a ± 1 + (τ a) . Here only sign + is meaningful, thus g = 2 τ a + 1 + (τ a) which is just the limit of continued fraction (6.17). It is easy to verify that g is explike, i.e., g(μ)g(−μ) = 1. So the Leap-frog scheme is symplectic for linear Hamiltonian systems according to our new definition.

7.6 Multistep Symplectic Method

2.

Exponential function exp(z) = 1 +

∞  zk . k!

353

(6.19)

k=1

We have Lagrange’s continued function z −z z −z 1 + 2 + · · · + 2n − 1 + 2 +· · · a 1 a2 a2n−1 a2n = b0 + , b1 + b2 + · · · + b2n−1 + b2n +· · ·

exp (z) = 1 +

(6.20)

where a1 = z, a2 = −z, · · · , a2n−1 = z, a2n = −z, b0 = 1, b1 = 1, b2 = 2, · · · , b2n−1 = 2n − 1, b2n = 2,

n ≥ 1, n ≥ 1,

and Euler’s contract expansion z2 z2 2z 2 − z + 6 + · · · + 2(2n − 1) + · · · A1 A2 An = B0 + , B1 + B2 + · · · + Bn + · · ·

exp (z) = 1 +

(6.21)

where A1 = 2z, A2 = z 2 , · · · , An = z 2 , B0 = 1, B1 = 2 − z, B2 = 6, · · · , Bn = 2(2n − 1),

n ≥ 2, n ≥ 2.

We have P0 p0 p1 1+z P1 2+z p2 = = 1, = = = , , Q0 q0 q1 1 q2 Q1 2−z 6 + 4z + z 2 P2 12 + 6z + z 2 p3 p4 = = = + ···. , q3 6 − 2z q4 Q2 12 − 6z + z 2

(6.22)

In general p2n−1 (z) is a polynomial of degree n, q2n−1 is a polynomial of degree n−1, so

p2n−1 is not explike. While p2n = Pn (x), q2n = Qn (x) are both polynomials of q2n−1

degree n and from the recursions P0 = 1,

P1 = 2 + z,

Pn = z 2 Pn−2 + 2(2n − 1)Pn−1 ,

Q0 = 1,

Q1 = 2 − z,

Qn = z 2 Qn−2 + 2(2n − 1)Qn−1 .

It’s easy to see that for n = 0, 1, · · ·, Qn (z) = Pn (−z),

Pn (0) > 0.

So the rational function φn (z) = is explike and

Pn (z) Pn (z) = Qn (z) Pn (−z)

(6.23)

354

7. Symplectic Runge–Kutta Methods

φn (z) − exp(z) = o(|z|2n+1 ), where P0 = 1,

P1 = 2+z,

Pn (z) = z 2 Pn−2 (z)+2(2n−1)Pn−1 (z),

n ≥ 2. (6.24)

This is just the diagonal Pad´e approximation. 3.

Logarithmic function log w =

∞  (w − 1)k , kwk

(6.25)

k=1

we have the Lagrange’s continued fraction (n − 1)(w − 1) n(w − 1) w − 1 w − 1 w − 1 2(w − 1) + ··· + + + ··· 1 + 2 + 3 + 2 2n − 1 2 a 1 a2 a3 a4 a2n−1 a2n = , (6.26) b1 + b2 + b3 + b4 + · · · + b2n−1 + b2n + · · ·

log w =

where a1 = w − 1, a2 = w − 1, b0 = 0, b1 = 1, b2 = 2,

and

a3 = w − 1, a4 = 2(w − 1), b3 = 3, b4 = 2, · · · ,

a2n−1 = (n − 1)(w − 1), b2n−1 = 2n − 1,

a2n = n(w − 1), b2n = 2,

···,

n ≥ 2, n ≥ 2,

and the Euler’s contracted expansion 

2

2(w − 1) 2(w − 1) 2 × 2(w − 1) log w = w + 1 – 6(w + 1) – 2.5(w + 1) A1 A2 A3 An = , B1 + B2 + B3 + · · · + Bn + · · ·



2 2(n − 1)(w − 1)

– · · · – 2(2n − 1)(w + 1) – · · · (6.27)

where A1 = 2(w − 1), A2 = −2(w − 1), · · · , An = −(2(n − 1)(w − 1))2 , B0 = 0, B1 = w + 1, B2 = 6(w + 1), · · · , Bn = 2(2n − 1)(w + 1),

n ≥ 3, n ≥ 2.

The following can be obtained by recursion P0 2(w − 1) p0 p1 p2 P1 = = 0, = w − 1, = = , Q0 q0 q1 q2 Q1 w+1 3(w2 − 1) w 2 + 4w − 5 P2 p3 p4 = = = 2 , . q3 4w + 2 q4 Q2 w + 4w + 1

(6.28)

In general p2n−1 (w) − log (w) = O (|w − 1|2n ), q2n−1 (w)

The rational function

p2n (w) − log (w) = O (|w − 1|2n+1 ). q2n (w)

p2n−1 (w) approximates log w only by odd order 2n − 1, it does q2n−1 (w)

not reach the even order 2n, and is not loglike. However

7.6 Multistep Symplectic Method

Rn = ψn (w) =

355

p2n (w) Pn (w) = q2n (w) Qn (w)

is a loglike function. In fact, by recursion, it’s easy to see that % & 1 , Pn (w) = −wn Pn w % & 1 Qn (w) = w n Qn , w

(6.29)

and ∀ n, Qn (1) = 0.We also have P1 (w) = 2(w − 1), Q1 (w) = w + 1,

P0 = 0, Q0 = 1,

P2 (w) = 3(w2 − 1), Q2 (w) = w 2 + 4w + 1,

and for n ≥ 3, Pn (w) = −(2(n − 1)(w − 1)2 Pn−2 (w) + 2(2n − 1)(w − 1)Pn−2 (w)), Qn (w) = −((2n − 1)(w − 1)2 Qn−2 (w) + 2(2n − 1)(w − 1)Qn−2 (w)).

(6.30)

3(λ2 − 1)

So we see R1 (λ) is just the Euler midpoint rule and R2 (λ) = 2 is just the λ + 4λ + 1 Simpson scheme. Conclusion: The odd truncation of the continued fraction of the Lagrange’s approximation to exp(x) and log (x) is neither explike nor loglike, while the even truncation is explike and loglike. The truncation of the continued fraction obtained from Euler’s contracted expansion is explike and loglike. 4.

Obreschkoff formula Another rational approximation to a given function is the Obreschkoff formula[Obr40] : Rm,n (x) =

n  k=0

=

Ckn Ckm+n

1 (m + n)!

k!

-

(x0 − x)k f (k) (x) −

m 

Ckm (x k C k! k=0 m+n

− x0 )k f (k) (x0 )

x

(x − t)m (x0 − t)n f m+n+1 (t)dt.

(6.31)

x0

. (1) Take f (x) = ex , x0 = 0, we obtain Pad´e approximation exp(x) = Rm,n (x). If m = n, we obtain Pad´e approximation Rm,m (x). . (2) Take f (x) = log(x), x0 = 1, we obtain log(x) = Rm,n (x). If m = n, we obtain loglike function Rm (x), Rm (λ) =

m 1  Ckm (λ − 1)k (λm−k + (−1)k−1 λm ), k λm C k k=1 2m

i.e., Rm (λ) + Rm We have

% & 1 = 0. λ

356

7. Symplectic Runge–Kutta Methods

Rm (λ) − log (λ) = O (|λ|2m+1 ), λ2 − 1 , 2λ 1 (−λ4 + 8λ3 − 8λ + 1), R2 = 12λ2 1 (λ6 − 9λ5 + 45λ4 − 45λ2 + 9λ − 1), R3 = 60λ3

R1 =

··· where R1 (λ) is just the leap-frog scheme. 5.

Nonexistence of SLMM for Nonlinear Hamiltonian Systems (Tang Theorem) For nonlinear Hamiltonian systems, there exists no symplectic LMM. When equation (6.1) is nonlinear, how to define a symplectic LMM? The answer is to find the step-transition operator g : Rn → Rn , let z = g 0 (z), z1 = g(x), z2 = g(g(z)) = g ◦ g(z) = g 2 (z), .. .

(6.32)

zn = g(g(· · · (g(z)) · · ·)) = g ◦ g ◦ · · · ◦ g ◦ (z) = g n (z), we get from (6.2) k  i=0

αi g i (z) = τ

n 

βi f ◦ g i (z).

(6.33)

i=0

It’s easy to prove that if LMM (6.33) is consistent with Equation (6.1), then for smooth f and sufficiently small step-size τ , the operator g defined by (6.32) exists and it can be represented as a power series in τ and with first term equal to identity. Consider a case where Equation (6.1) is a Hamiltonian system, i.e., a(z) = J∇H(z), we have the following definition. Definition 6.9. LMM is symplectic if the transition operator g defined by (6.32) is symplectic for all H(z) and all step-size τ , i.e., g∗T Jg∗ (z) = J.

(6.34)

This definition is a completely different criterion that can include the symplectic condition for one-step methods in the usual sense. But Tang in[Tan93a] has proved that nonlinear multistep method can satisfy such a strict criterion. Numerical experiments of Li[Fen92b] show that the explicit 3-level centered method (Leap-frog method) 1 is symplectic for linear Hamiltonian systems H = (p2 + 4q 2 ) (see Fig. 0.2 in 2 introduction of this book) but is non-symplectic for nonlinear Hamiltonian systems 1 2 H = (p2 + q 2 ) + q4 (see Fig. 0.3 (a,b,c) in introduction of this book). 2

3

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Chapter 8. Composition Scheme

In this chapter, we consider a class of reversible schemes also called symmetrical schemes. In algebraic language, it is not other, just like self-adjoint schemes . Here, we only deal with one-step reversible schemes. We will introduce the concept of adjoint methods and some of their properties. We will show that there is a self-adjoint scheme of even order in every method. Using the self-adjoint schemes with lower order, we can construct higher order schemes by “composing” a method, and this constructing process can be continued to obtain arbitrary even order schemes. The composing method presented here can be used to in both non-symplectic and symplectic schemes. In [Yos90] , H.Yoshida proposed a new method to get multistage higher order explicit symplectic schemes for separable Hamiltonian systems by composing lower order ones. However, in [QZ92,Wru96,Suz92] , we found that this method can also be applied to non-symplectic schemes for both Hamiltonian and non-Hamiltonian systems.

8.1 Construction of Fourth Order with 3-Stage Scheme In this section, we construct a 3-stage difference scheme of fourth order by the method of composing 2nd order schemes symmetrically.

8.1.1 For Single Equation We know that trapezoid scheme Zk+1 = Zk +

 h f (Zk ) + f (Zk+1 ) 2

(1.1)

with h the step length, is order 2 for ODEs, Z˙ = f (Z). We expect that the 3-stage method of the form   Z1 = Z0 + c1 h f (Z0 ) + f (Z1 ) ,   Z2 = Z1 + c2 h f (Z1 ) + f (Z2 ) ,   Z3 = Z2 + c3 h f (Z2 ) + f (Z3 )

(1.2)

(1.3)

K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

366

8. Composition Scheme

would be of order 4 (i.e., Z3 − Z(t + h) = O(h5 )), where Z0 = Z(t). Z(t + h) is the exact solution at t + h and Z3 the numerical one when the parameters c1 , c2 , and c3 are chosen properly. We will use the method of Taylor expansion to deal with the simple case when there is only one ordinary differential equation (ODE). When we deal with the case of systems of ODEs, the Taylor expansions become very complex, although they surely can be applied and the same conclusion as in the former case can be derived. We introduce another method[HNW93] , known as “trees and elementary differentials” to deal with the latter case. In fact, the essence of the two methods is the same; they are just two different ways of expression. In this section, without specific statements, the values of all functions and their derivatives are calculated at Z0 , and we consider only the terms up to o(h4 ) in the following calculations, while the higher order terms of h are omitted, f = f (Z0 ),

f  = f  (Z0 ), . . . , etc.

First, we calculate the Taylor expansion of the exact solution. Since ⎧ Z˙ = f, ⎪ ⎪ ⎪ ⎪ ⎨ Z¨ = f  · Z˙ = f  f, 2 ⎪ Z (3) = f  f 2 + f  f, ⎪ ⎪ ⎪  ⎩ (4) Z = f  f 3 + 4f f  f 2 + f  3 f,

(1.4)

we have, with Z0 = Z(t), h4 h2  h3 2 3 f f + (f  f 2 +f  f )+ (f  f 3 +4f  f  f 2 +f  f )+O(h5 ). 2! 3! 4! (1.5) Now, we turn to the Taylor expansion of the numerical solution. We can rewrite (1.3) as

Z(t+h) = Z0 +hf +

Z3 = Z0 + c1 h(f (Z0 ) + f (Z1 )) + c2 h(f (Z1 ) +f (Z2 )) + c3 h(f (Z2 ) + f (Z3 )) = Z0 + h(c1 f (Z0 ) + (c1 + c2 )f (Z1 ) +(c2 + c3 )f (Z2 ) + c3 f (Z3 )).

(1.6)

We use the same technique to expand the Taylor expansions of f (Z2 ), f (Z3 ). Since Z2 − Z0 = c1 h(f (Z1 ) + f (Z0 )) + c2 h(f (Z2 ) + f (Z1 )) = c1 hf (Z0 ) + (c1 + c2 )hf (Z1 ) − c2 hf (Z2 ).

(1.7)

We need to calculate f (Z1 ), f (Z2 ), f (Z3 ) Taylor expansion up to the terms of order 3 of h f (Zi ) = f (Z0 ) + f  (Zi − Z0 ) +

f  f  (Zi − Z0 )2 + (Zi − Z0 )3 + O(h4 ). (1.8) 2! 3!

8.1 Construction of Fourth Order with 3-Stage Scheme

367

Since Z1 = Z0 + c1 h(f (Z1 ) + f (Z0 )) by (1.8), we have

  f (Z1 ) = f (Z0 ) + f  c1 hf (Z0 ) + c1 hf (Z1 ) 2   + f2 ! c1 hf (Z0 ) + c1 hf (Z1 ) 3   + f3 ! c1 hf (Z0 ) + c1 hf (Z1 ) + O(h4 ).

(1.9)

Inserting the Taylor expansion of f (Z1 ) into right side of (1.9), we get

   f (Z1 ) = f (Z0 ) + c1 hf  f (Z0 ) + f (Z0 ) + c1 hf  f (Z0 ) + f (Z1 )   2    + f2 ! (c1 h)2 f (Z0 ) + f (Z0 ) + f2 ! (c1 h)2 f (Z0 ) + f (Z1 ) 2 3    + f3 ! (c1 h)3 f (Z0 ) + f (Z0 ) + O(h4 ) +c1 hf  f (Z0 ) + f (Z1 )    = f (Z0 ) + c1 hf  2f (Z0 ) + c1 hf  f (Z0 ) + f (Z0 ) + c1 2hf  f (Z0 )    2    + (c1 h)2 f2 ! 2f (Z0 ) + c1 hf  f (Z0 ) +(c1 h)2 f2 ! f (Z0 ) + f (Z0 ) 2 3   + (c1 h)3 f3 ! 2f (Z0 ) + O(h4 ) +f (Z0 )     2 = f (Z0 ) + c1 h 2f  f (Z0 ) + (c1 h)2 2f  f (Z0 ) + 2f  f 2 (Z0 )   3 +(c1 h)3 2f  f (Z0 ) + 6f  f  f 2 (Z0 ) + 43 f  f 3 (Z0 ) + O(h4 ).

(1.10) Similarly, developing f (Z2 ) and f (Z3 ), since

    Z2 − Z0 = c1 h f (Z1 ) + f (Z0 ) + c2 h f (Z2 ) + f (Z1 ) = c1 hf (Z0 ) + (c1 + c2 )hf (Z1 ) + c2 hf (Z2 ),

therefore  f    f (Z2 ) = f (Z0 ) + hf  c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + c2 f (Z2 ) + h2 c1 f (Z0 ) 2! 2 f   h3 c1 f (Z0 ) + (c1 + c2 )f (Z1 ) +(c1 + c2 )f (Z1 ) + c2 f (Z2 ) + 3! 3 +c2 f (Z2 ) + O(h4 ) %   = f (Z0 ) + hf  c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + c2 f (Z0 ) + hf  c1 f (Z0 )    +(c1 + c2 )f (Z1 ) + c2 f (Z0 ) + hf  c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + c2 f (Z0 ) 2 & f  % f   + h2 c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + c2 f (Z0 ) + h2 c1 f (Z0 ) 2! 2!   +(c1 + c2 )f (Z1 ) + c2 f (Z0 ) + hf  c1 f (Z0 ) + (c1 + c2 )f (Z1 ) &2 f   3 +c2 f (Z2 ) + h3 c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + c2 f (Z0 ) + O(h4 ). 3!

(1.11)

368

8. Composition Scheme

Similarly, inserting Taylor expansion of f (Z1 ) into (1.11), we get   f (Z2 ) = f (Z0 ) + hf  c1 f (Z0 ) + (c1 + c2 ) f (Z0 ) + (c1 h)2f  f (Z0 )    2 +(c1 h)2 2f  f (Z0 ) + 2f  f 2 (Z0 ) + c2 f (Z0 )     +hf  c1 f (Z0 ) + (c1 + c2 ) f (Z0 ) + (c1 h)2f  f (Z0 ) + c2 f (Z0 )  f   h2 (c1 + c2 )2 4f 2 (Z0 ) +hf (c1 + c2 )2f (Z0 ) + 2!   f  2  + h c1 f (Z0 ) + (c1 + c2 ) f (Z0 ) + (c1 h)2f  f (Z0 )) + c2 f (Z0 ) 2!  2 f  + h3 (c1 + c2 )3 8f 3 (Z0 ) +hf  (c1 + c2 ) f (Z0 ) + f (Z0 ) 3! +O(h4 )      2 = f (Z0 ) + h 2(c1 + c2 )f f (Z0 ) + h2 (c1 + c2 )2 2f  f (Z0 )   3 +2f  f 2 (Z0 ) + h3 (c1 + c2 )(c21 + c1 c2 + c22 )2f  f (Z0 )   + (c1 + c2 )2c21 + 2c2 (c1 + c2 )2 + 4(c1 + c2 )3 f  f  f 2 (Z0 )  4 + (c1 + c2 )3 f  f 3 (Z0 ) + O(h4 ). 3

(1.12)

Using the above identical method, we have   f (Z3 ) = f (Z0 ) + hf  c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z3 ) 2 f   c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z3 ) +h2 2! 3   3f +h c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z3 ) 3! +O(h4 )   = f (Z0 ) + hf  c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z0 )   +hf  c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z0 )   +hf  (c1 + c3 )f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) 2  f   +h2 c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z0 ) 2!    2f c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z0 ) +h 2! 3   +hf c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z0 ) 3 f   (c1 + c3 )f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) +h3 3! +O(h4 ).

(1.13) Inserting the Taylor expansion of f (Z1 ) and f (Z2 ) into (1.13)    2 f (Z3 ) = f (Z0 ) + h 2(c1 + c2 + c3 )f  f (Z0 ) + h2 (c1 + c2 + c3 )2 2f  f (Z0 )

8.1 Construction of Fourth Order with 3-Stage Scheme

369

  +(c1 + c2 + c3 )2 2f  f 2 (Z0 ) + h3 c1 + c2 )c21 + (c2 + c3 )(c1 + c2 )2  3  +c3 (c1 + c2 + c3 )2 2f  f (Z0 ) + (c1 + c2 )c21 + (c2 + c3 )(c1 + c2 )2  +c3 (c1 + c2 + c3 )2 + 2(c1 + c2 + c3 )3 2f  f  f 2 f (Z0 )  4 (1.14) + (c2 + c2 + c3 )3 f  f 3 (Z0 ) + O(h4 ). 3 w

w

Let c1 = c3 = 1 , c2 = 0 , take into account (1.5) and (1.6), and compare the 2 2 Taylor expansion of the exact solution (1.5) with the above one. In order to get fourth order accuracy schemes (1.3), we need to solve the following equations for coefficients c1 , c2 , c3 : hf  : c1 + (c1 + c2 ) + (c2 + c3 ) + c3 = 1 =⇒ 2w1 + w0 = 1, (1.15) h f f : (c1 + c2 )2c1 + (c2 + c3 )2(c1 + c2 ) + c3 2(c1 + c2 + c3 ) 2 

1 2

= ,

(1.16)

2

h3 f  f 2 , h3 f  f : (c1 + c2 )2c21 + (c2 + c3 )2(c1 + c2 )2 + c3 2(c1 + c2 + c3 )2 1 6

= ,

(1.17) 4 3

4 3

4 3

h4 f  f 3 : (c1 + c2 ) c31 + (c2 + c3 ) (c1 + c2 )3 + c3 (c1 + c2 + c3 )3 =

1 , 24

(1.18)

3

h4 f  f : (c1 + c2 )2c31 + (c1 + c2 )2(c21 + c22 + c1 c2 )(c2 + c3 )   +c3 2 (c1 + c2 )c21 + (c2 + c3 )(c1 + c2 )2 + c3 (c1 + c2 + c3 )2 = 4   2

h f ff

1 , 24

: (c1 +

(1.19)



c2 )6c31

3

2c21 (c1

+ (c2 + c3 ) 4(c1 + c2 ) + + c2 )  2 2 +2c2 (c1 + c2 ) + c3 2(c1 (c1 + c2 ) + (c2 + c3 )(c1 + c2 )2  1 +c3 (c1 + c2 + c3 )2 + 2(c1 + c2 + c3 )3 = . (1.20) 24

1

1

When 2w1 + w0 = 1 holds, the Equation (1.16) becomes = , i.e., identity, and 2 2 the Equations (1.17) – (1.20) become the same, i.e., 6w13 − 12w12 + 6w1 − 1 = 0. Thus, we get the conditions for the difference scheme (1.3) to be of order 4:  2w1 + w0 = 1, 6w13 − 12w12 + 6w1 − 1 = 0. Thus we get,

1

w0 =

−2 3 1

2 − 23

,

w1 =

1 1

2 − 23

.

(1.21)

370

8. Composition Scheme

Now, scheme (1.3) becomes ⎧   1 ⎪ Z1 = Z0 + ⎪ 1 h f (Z0 ) + f (Z1 ) , ⎪ ⎪ 2(2 − 2 3 ) ⎪ ⎪ 1 ⎨   −2 3 Z2 = Z1 + 1 h f (Z1 ) + f (Z2 ) , 2(2 − 2 3 ) ⎪ ⎪ ⎪ ⎪   1 ⎪ ⎪ ⎩ Z3 = Z2 + 1 h f (Z2 ) + f (Z3 ) .

(1.22)

2(2 − 2 3 )

8.1.2 For System of Equations We use the “method of tree and elementary differentials” [HNW93] given in Chapter 7. We first rewrite the scheme (1.3) in the R–K methods: ⎧   ⎪ ⎨ Z1 = Z0 + hc1 f (Z0 ) + c1 f (Z1 ) ,  Z2 = Z0 + h c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + c2 f (Z2 ) , ⎪   ⎩ Z3 = Z0 + h c1 f (Z0 ) + (c1 + c2 )f (Z1 ) + (c2 + c3 )f (Z2 ) + c3 f (Z3 ) . (1.23) Obviously, the above equation is equivalent to following ⎧ g1 = Z0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ g2 = Z0 + c1 hf (g1 ) + c1 hf (g2 ), g3 = Z0 + c1 hf (g1 ) + (c1 + c2 )hf (g2 ) + c2 hf (g3 ), ⎪ ⎪ ⎪ g4 = Z0 + c1 hf (g1 ) + (c1 + c2 )hf (g2 ) + (c2 + c3 )hf (g3 ) + c3 hf (g4 ), ⎪ ⎪ ⎪   ⎩ Z = Z0 + h c1 f (g1 ) + (c1 + c2 )f (g2 ) + (c2 + c3 )f (g3 ) + c3 f (g4 ) , (1.24) where g2 = Z1 , g3 = Z2 , g4 = Z3 , and Z = Z3 . Thus, the Butcher tableau A

C

bT takes the following form: 0

0

0

0

0

2c1

c1

c1

0

0

2(c1 + c2 )

c1

c1 + c2

c2

0

2(c1 + c2 + c3 )

c1

c1 + c2

c2 + c3

c3

c1

c1 + c2

c2 + c3

c3

From the previous chapter, we have the order condition for R–K method as follows: Theorem 1.1. R–K method

8.1 Construction of Fourth Order with 3-Stage Scheme

giJ = Z0J + Z1J = Z0J +

s  j=1 s 

371

aij f J (gj1 , · · · , gjh ), bj hf J (gj1 , · · · , gjh )

j=1

is order of p, iff

s 

bj φj (ρt) =

j=1

1 γ(ρt)

for all rooted tree ρt, have r(ρt) ≤ p, where Z0 = (Z01 , · · · , Z0n )T , f J = (f 1 , f 2 , · · · , f n )T .

Since rooted tree ρt of Theorem 1.1 is defined in Chapter 7, definitions of φj (ρt) are as follows:  ajk a · · · , φj (ρt) = k,l,···

where ρt is a labelled tree with root j, the sum over the r(ρt) − 1 remaining indices k, l, · · ·. The summand is a product of r(ρt) − 1 a’s, where all fathers stand two by two with their sons as indices. ⎧ 4 4  4  C ⎪ 1 ⎪ ⎪ b = 1, b ajk = , j j ⎪ ⎪ 2 ⎪ j=1 ⎪ j=1 k=1 ⎪ ⎪ 4 4 4 4 ⎪ ⎪     ⎪ 1 1 ⎪ ⎪ bj ajk ajl = , bj ajk akl = , ⎪ ⎨ 3 6 j=1 j=1 k,l=1 k,l=1 (1.25) 4 4 4 4     ⎪ ⎪ 1 1 ⎪ bj ajk ajl ajm = , bj ajk akl ajm = , ⎪ ⎪ ⎪ 4 8 ⎪ j=1 j=1 k,l,m=1 k,l,m=1 ⎪ ⎪ ⎪ 4 4 4 4 ⎪     ⎪ 1 1 ⎪ ⎪ bj ajk akl akm = , bj ajk akl alm = . ⎪ ⎩ 12 24 j=1

k,l,m=1

j=1

k,l,m=1

From the previous chapter on simplifying condition of symplectic R–K, we know that the system of Equation (1.25) only exists in 3 independent conditions (equation). In the above equation, last 4 conditions have only one independent condition. By symmetrically choosing c1 = c3 , this condition is satisfied automatically. Taking w w c1 = c3 = 1 , c2 = 0 , by the first two conditions of Equation (1.25), we obtain the 2 2 same equation 2w1 + w0 = 1. (1.26) Substituting the relation (1.26) in the order of conditions (1.25), we get 2w13 + w03 = 0.

(1.27)

These Equations (1.26) and (1.27) are just the same as in Subsection 8.1.1 for single equation.

372

8. Composition Scheme

From the literature[Fen85] , we know that the centered Euler scheme is symplectic, but trapezoidal scheme (1.1) is right non-symplectic as a result of nonlinear transformation [Dah75,QZZ95,Fen92] , the scheme (1.1) can transform the Euler center point form, therefore the trapezoidal form is nonstandard symplectic, just as discussed in Section 4.3 of Chapter 4. It is the same with the trapezoidal form, the centered Euler scheme may also be used to construct the higher order scheme. Because ⎧   Z0 + Z1 ⎪ = Z + d hf , Z ⎪ 1 0 1 ⎪ 2 ⎪ ⎪ ⎨   Z + Z2 , Z2 = Z1 + d2 hf 1 2 ⎪ ⎪ ⎪   ⎪ ⎪ ⎩ Z3 = Z2 + d3 hf Z2 + Z3 , 2

it is equally the same in R–K method with in the following Butcher tableau: d1 2

d1 2

0

0

d1 +

d2 2

d1

d2 2

0

d1 + d2 +

d3 2

d1

d2

d3 2

d1

d2

d3

Using the same method, we can prove that , when d1 = d3 = 1

−2 3 1

2 − 23

1 1

2 − 23

, d2 =

, the above scheme is fourth order, and the coefficient is entirely the same as in

trapezoidal method.

8.2 Adjoint Method and Self-Adjoint Method Here, we will introduce the concept of adjoint scheme and self-adjoint scheme. These two kinds of schemes are the foundation that construct higher order scheme in the future. First, we see several higher order schemes as the example, and seek common character which may supply method for constructing higher order scheme; In the Section 4.4 of Chapter 4, we discussed an explicit scheme for separable Hamiltonian system. We know  n+1 = pn − τ Vq (q n ), p (2.1) q n+1 = q n + τ Up (pn+1 ) (where τ is step size, pn , qn are numerical solution in step n) is of order 1. We shall compose this scheme (2.1) to a 2nd order scheme choosing a suitable coefficient of τ ,

8.2 Adjoint Method and Self-Adjoint Method

373

⎧ n+ 1 τ p 2 = pn − Vq (q n ), ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨ q n+ 12 = q n + τ Up (pn+ 12 ), 1 1 τ ⎪ pn+1 = pn+ 2 − Vq (q n+ 2 ), ⎪ ⎪ 2 ⎪ ⎪ ⎩ n+1 1 = q n+ 2 . q This scheme is equal to the following: ⎧ n+ 1 p 2 = pn − ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨ q n+ 2 = q n + 1

⎪ q n+1 = q n+ 2 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ pn+1 = pn+ 2

τ Vq (q n ), 2 1 τ Up (pn+ 2 ), 2 1 τ + Up (pn+ 2 ), 2 τ − Vq (q n+1 ). 2

(2.2)

This 2nd order scheme can also be defined as a self-adjoint scheme, see also [Yos90] . Ruth[Rut83] , using scheme (2.1), constructed a 3rd order scheme via composition method, ⎧ k k q1 = q k + d1 τ Up (p1 ), ⎪ ⎨ p1 = p − c1 τ Vq (q ), p2 = p1 − c2 τ Vq (q1 ), q2 = q1 + d2 τ Up (p2 ), (2.3) ⎪ ⎩ k+1 k+1 k+1 p = p2 − c3 τ Vq (q2 ), q = q2 + d3 τ Up (p ). 7

4

1

2

2

When c1 = , c2 = , c3 = − , d1 = , d2 = − , d3 = 1, this scheme is 24 3 24 3 3 3rd order. We may construct multistage schemes, in order to achieve the higher order precision. In literature[QZ92,Fen86,Fen91,FR90,Rut83] , we may see the following 4th order form: ⎧ p1 = pk − c1 τ Vq (qk ), q1 = q k + d1 τ Up (p1 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p2 = p1 − c2 τ Vq (q1 ), q2 = q1 + d2 τ Up (p2 ), (2.4) ⎪ q3 = q2 + d3 τ Up (p3 ), ⎪ ⎪ p3 = p2 − c3 τ Vq (q2 ), ⎪ ⎪ ⎩ k+1 p = p3 − c4 τ Vq (q3 ), q k+1 = q3 + d4 τ Up (pk+1 ), where ⎧ 1 1 ⎪ c1 = 0, c2 = c4 = − (2 + α), c3 = (1 + 2α), ⎪ ⎪ 3 3 ⎪ ⎪ ⎨ 1 d1 = d4 = (2 + α), 6 D ⎪ ⎪ √ ⎪ 1 ⎪ 3 3 1 ⎪ d , ⎩ 2 = d3 = (1 − α), α = 2 + 6

or

2

⎧ 1 1 ⎪ c1 = c4 = (2 + α), c2 = c3 = (1 − α), ⎪ ⎪ 6 6 ⎪ ⎪ ⎨ 1 d1 = d3 = (1 + 2α), 3 D ⎪ ⎪ √ ⎪ 1 ⎪ 3 3 1 ⎪ = − (1 + 2α), d = 0, α = 2 + . d ⎩ 2 4 3 2

374

8. Composition Scheme

The above examples in the 3-stage fourth-order scheme give us an understanding. We can construct the higher order scheme through the low-order scheme, and this method is not limited the symplictic type. Because the usual structure uses the Taylor expansion, the comparison of coefficient of structure in the higher order form will be cumbersome, therefore in this section and next, we will use the Lie series method. This method is already used widely , for example Stanly. Stenberg, Alex, J. Dragt and F. Neri used the Lie series to study the differential equation, see also the literature [Ste84,DF76,DF83,Ner87] . Using the Lie series to study our question is convenient as there is no need to extract the Lie series item truly to which form it corresponds, but must use its form only. We will see this later. We know that each scheme is always formal and is expressed as follows: yn+1 = S(τ )yn ,

(2.5)

where τ is the step size. S(τ ) is called the integrator, but yn+1 and yn are numerical solutions of equation in steps n + 1 and n . Definition 2.1. An integrator S ∗ (τ ) is called the adjoint integrator S(τ ), if S ∗ (−τ )S(τ ) = I, ∗

S(τ )S (−τ ) = I.

(2.6) (2.7)

That means yn+1 = S(τ )yn , yn = S ∗ (−τ )yn+1 , or yn+1 = S ∗ (−τ )yn , yn = S(τ )yn+1 . In fact, (2.6) – (2.7) equations are equivalent to S(−τ )S ∗ (τ ) = I, S ∗ (τ )S(−τ ) = I.

(2.8) (2.9)



In order to prove this, set τ = −τ , then (2.6) – (2.7) becomes ⎧   ⎨ S ∗ (τ )S(−τ ) = I, 



⎩ S(−τ )S ∗ (τ ) = I. τ expresses the length of arbitrary step; therefore, the above equations are formula of (2.8), (2.9). Further, we would like to point out that the two conditions (2.6) and (2.7) are the same. Since form S ∗ (−τ )S(τ ) = I, we get S ∗ (−τ ) = S −1 (τ ), where S −1 (τ ) is the inverse of the integrator S(τ ). Here, we always assume S(τ ) is invertible. So, we have S(τ )S ∗ (−τ ) = S(τ )S −1 (τ ) = I. The formula may result in (2.7) by (2.6), and vice versa. But note the difference between S ∗ (τ ) and S −1 (τ ), that is, S ∗ (−τ ) = S −1 (τ ). Here, S ∗ (τ ) and S(τ ) is the same push-forward mapping, but S −1 (τ ) is the pull-back mapping.

8.2 Adjoint Method and Self-Adjoint Method

375

For a convenient deduction in Section 8.3, we give another definition of a selfadjoint method[HNW93] here, and show that it is equivalent to definition adjoint (2.1). We rewrite (2.5) as follows: yn+1 = yn + τ φ(x, yn , τ ).

(2.10)

yn , yn+1 is numerical solution for the equation y  = f (x) in the n and n + 1 step, and φ is increment function which the form (2.5) corresponds. Definition 2.2. Scheme yn+1 = yn + τ φ∗ (x, yn , τ ) is an adjoint scheme (2.10), if it satisfies: B = A − τ φ(x + τ, A, −τ ), A = B + τ φ∗ (x, B, τ ).

(2.11) (2.12)

Theorem 2.3. Definition 2.1 and Definition 2.2 are equivalent. Proof. Since (2.8) – (2.9) and (2.6) – (2.7) are equivalent, (2.6) and (2.9) are also equivalent. It is enough to prove that (2.9) is equivalent to (2.11) – (2.12). Let  ∗ S (τ )yn = yn + τ φ∗ (x, yn , τ ), S(τ )yn = yn + τ φ(x, yn , τ ), first prove that (2.9) → (2.11) – (2.12). Let  A = yn+1 , B = A − τ φ(x + τ, A, −τ ), prove that (2.12), due to   S ∗ (τ )S(−τ )yn+1 = S ∗ (τ ) yn+1 − τ φ(x + τ, yn+1 , −τ ) = yn+1 − τ φ(x + τ, yn+1 , −τ )   +τ φ∗ x, yn+1 − τ φ(x + τ, yn+1 , −τ ), τ = yn+1 − τ φ(x + τ, yn+1 , −τ ) + τ φ∗ (x, B, τ ) = B + τ φ∗ (x, B, τ ) = IA. By the last equality, also because S ∗ (τ )S(−τ ) = I, we have B + τ φ∗ (x, B, τ ) = A, this is the formula (2.12). Now, we will prove : (2.11) – (2.12) ⇒ (2.9). Let ' A = yn+1 , B = A − τ φ(x + τ, A, −τ ).

(2.13)

376

8. Composition Scheme

Since

  S ∗ (τ )S(−τ )yn+1 = S ∗ (τ ) yn+1 − τ φ(x + τ, yn+1 , −τ ) = A − τ φ(x + τ, A, −τ ) + τ φ∗ (x, B, τ ),

from (2.12), we have S ∗ (τ )S(−τ )A = B + τ φ∗ (x, B, τ ) = A = IA, i.e., S ∗ (τ )S(−τ ) = I.



Definition 2.4. An integrator S(τ ) is self-adjoint, if S ∗ (τ ) = S(τ ),

i.e.,

S(−τ )S(τ ) = I.

In[Yos90] , the integrator with the property S(−τ )S(τ ) = I. H. Yoshida called this operator as reversible. We see that time reversibility and self-adjointness are the same. The time reversible (i.e., self-adjoint) integrator plays an important role in this chapter due to its special property.         τ τ τ τ Theorem 2.5. For every integrator S(τ ), S ∗ S or S S∗ is self2

adjoint integrator[QZ92,Str68] . Proof. We must prove that S ∗

  τ 2

2

2

2

  τ 2

=S

∗



S ∗ (τ )S(τ )

is self-adjoint, i.e.,

= S ∗ (τ )S(τ ).

By Definition 2.1,we have S ∗ (τ )S(−τ ) = I, then 

∗

S ∗ (τ )S(τ )

Because we also have

−1 S ∗ (−τ )S(−τ )  −1 = S −1 (−τ ) S ∗ (−τ )  −1 = S ∗ (τ ) S ∗ (−τ ) .

S ∗ (−τ )S(τ ) = I,

i.e., therefore



=



−1

S ∗ (−τ )



∗

S ∗ (τ )S(τ )

Therefore, the theorem is completed.

= S(τ ),

= S ∗ (τ )S(τ ). 

Theorem 2.6. If S1 (τ ) and S2 (τ ) are self-adjoint integrators, then symmetrical composition S1 (τ )S2 (τ )S1 (τ ) is self-adjoint[QZ92] .

8.3 Construction of Higher Order Schemes

377

Proof. Consider S1 (τ ) and S2 (τ ) are self-adjoint integrators, then ∗  −1  = S1 (−τ )S2 (−τ )S1 (−τ ) S1 (τ )S2 (τ )S1 (τ ) = S1 (−τ )−1 S2 (−τ )−1 S1 (−τ )−1 = S1∗ (τ )S2∗ (τ )S1∗ (τ ) = S1 (τ )S2 (τ )S1 (τ ). 

The theorem is proved.

We pointed out that generally, a combination of self-adjoint operators is not necessarily from the self-adjoint. One simple example is ∗  S1 (τ )S2 (τ ) = S2 (τ )S1 (τ ) = S1 (τ )S2 (τ ), where S1 (τ ) and S2 (τ ) are self-adjoint operators, but they not commutative. We will construct a higher order form in the below section.

8.3 Construction of Higher Order Schemes We will first give constructed method for the higher difference scheme, and will further prove that the Gauss–Legendre method is a self-adjoint method. We have given some example for structured higher order schemes. In this section, we will introduce first-order differential operators, Lie series and some of their properties, all these are the basis of further deduction. Denote: f = (f1 , f2 , · · · , fn )T g = (g1 , g2 , · · · , gn )T % &T d d D= ,···, , d y1

d yn

where f1 , f2 , · · · , fn and g1 , g2 , · · · , gn are scalar function. Let Lf = f T D =

n  i=1

fi

∂ ∂yi

(3.1)

be a first-order differential operator. The action of Lf on a scalar function ϕ is, * n +  ∂ Lf ϕ = fi ϕ = f T Dϕ(y). ∂y i i=1 It is linear and satisfies the Leibniz formula, i.e., for two scalar functions ϕ1 and ϕ2 , (1) (2)

Lf (λ1 ϕ1 + λ2 ϕ2 ) = λ1 Lf ϕ1 + λ2 Lf ϕ2 , Lf (ϕ1 ϕ2 ) = ϕ1 Lf ϕ2 + ϕ2 Lf ϕ1 .

∀ λ1 , λ2 ∈ R.

(3.2) (3.3)

378

8. Composition Scheme

Definition 3.1. The commutator of two first-order differential operators Lf and Lg is defined by [Lf , Lg ] = Lf Lg − Lg Lf . (3.4) The commutator of the two first-order differential operators is still a first differential operator, since Lf Lg ϕ = f T Dg T Dϕ =

n 

fj

n n n   ∂  ∂ ∂gi ∂ ∂2ϕ gi ϕ= fj ϕ+ f j gi , ∂yj i=1 ∂yi ∂yj ∂yi ∂yj ∂yi i,j=1 i,j=1

gj

n n n   ∂  ∂ ∂fi ∂ ∂2ϕ fi ϕ= gj ϕ+ gj fi , ∂yj i=1 ∂yi ∂yj ∂yi ∂yj ∂yi i,j=1 i,j=1

j=1

Lg Lf ϕ = g T Df T Dϕ =

n  j=1

therefore, (Lf Lg − Lg Lf )ϕ =

n %  i,j=1

∂gi ∂fi − gj fj ∂yj ∂yj

&

∂ ϕ, ∂yi

this means [Lf , Lg ] = Lc ,

c = [c1 , c2 , · · · , cn ],

ci =

n % 

fj

j=1

∂gi ∂fi − gj ∂yj ∂yj

& ,

and Lc will still be the first-order differential operator. It is very easy to prove the following properties of bracket

[λ1 Lf1

[Lf , Lf ] = 0, + λ2 Lf2 , Lg ] = λ1 [Lf1 , Lg ] + λ2 [Lf2 , Lg ],

∀ λ1 , λ2 ∈ R.

(3.5) (3.6)

The commutator also satisfies the Jacobi identity, i.e., if Lf , Lg , Lh are three firstorder differential operators, then A @ A @ A @ (3.7) [Lf , Lg ], Lh + [Lg , Lh ], Lf + [Lh , Lf ], Lg = 0. (3.7) is very easy to prove, the detailed proof process is seen [Arn89] . From the above, we know that first-order differential operator forms a Lie algebra. Definition 3.2. A Lie series is an exponential of a first-order linear differential operator ∞ n n  t Lf etLf = . (3.8) n! n=0 The action of a Lie series a scalar function ϕ(y) is given by: etLf ϕ =

∞ k k  t Lf k=0

k!

ϕ(y) =

∞  k tk  T f (y)D ϕ(y) k!

k=0

= ϕ(y) + tf T (y)(D(y)) +

  t2 T f (y)D f T (y)Dϕ(y) + · · · . 2

(3.9)

8.3 Construction of Higher Order Schemes

379

Taylor expansion gives us an one elementary example of a Lie series et[1,1,···,1]D ϕ(y)

* n +k ∞ k   d t = ϕ(y) k ! i=1 dyi k=0   = ϕ y + t(1, 1, · · · , 1)T .

(3.10)

We have seen several properties of Lie series, which are similar to those of [Ste84] . Let, f = (f1 (y), f2 (y), · · · , fn (y))T , g = (g1 (y), g2 (y), · · · , gn (y))T , and etf

T

D

T  T T T g = etf D g1 , etf D g2 , · · · , etf D gn ,

then, we have the following: Property 3.3. The Lie series has the compositionality etLf g(y) = g(etLf y).

(3.11)

Proof. It is enough to prove etLf gm (y) = gm (etLf y). Since e

tLf

y=

∞ k k  t Lf k=0

∞  tk y= k! k! k=0

*

n 

∂ fi ∂y i i=1

+k y,

considering j component (1 ≤ j ≤ n), in etLf y , we have etLf yj =

∞  n  k=0

then gm (e

i=1

 tk ∂ k Lk−1 fj , yj = yj + ∂yi k! f ∞

fi

k=1

+

*

tLf

∞ k ∞   t k−1 tk k−1 Lf f1 (y), · · · , yn + L y) = gm y1 + fn (y) k! k! f k=1 k=1 * n +k ∞  ∂ tk  fi (y) gm (y) = etLf gm (y). = k! ∂yi k=0

i=1



The proof can be obtained. Property 3.4. Product preservation property etLf (pq) = (etLf p)(etLf q),

(3.12)

380

8. Composition Scheme

where p(y), q(y) are scalar functions. Proof. By (3.2) – (3.3) and (3.8), (3.12) can be obtained by direct computation.



Property 3.5. Baker-Campbell-Hausdroff formula (simply called BCH formula): All first differential operators constituted a Lie algebra. Therefore, we have the following BCH formula: 2 3 4 (3.13) etLf etLg = et(Lf +Lg )+t w2 +t w3 +t w4 +··· , where

1 2

w2 = [Lf , Lg ], w3 =

A A 1@ 1@ [Lf , Lg ], Lf + [Lf , Lg ], Lg , 12 12

w4 =

@ @ AA 1@ Lf , Lg Lg , Lf ] , 24

···. Property 3.6. If the differential equation property is, y(t) = etf

T

D

y,

y = y(0),

then y(t) ˙ = f (y(t)). Proof. Since yi (t) = etf

T

(y)D

yi (0), then

T T d yi (t) = etf (y)D f T (y)Dyi (0) = etf D fi (y). dt

From Property 3.3, we have   T d yi (t) = fi (etf D y) = fi y1 (t), y2 (t), · · · , yn (t) = fi (y(t)). dt 

The proof can be obtained.

From Property 3.6, we know that equation y˙ = f (y) can express the solution y(t) = etLf y(0), Section 8.2 has discussed that the integral S(τ ) can also be represented in this form. If S(τ ) has the group property about τ , it will be the phase flow of dy

= f (y). However, in our problem, there is just one parameter autonomous ODE dτ family S(τ ) without group property. So, there just exists a formal vector field f τ (y), which defines only the formal autonomous system dy = f τ (y). dt

Its formal phase flow concerning two parameters τ, t, can be expressed by etLf τ . Take the diagonal phase flow

8.3 Construction of Higher Order Schemes

381

etLf τ |t=τ = eτ Lf τ . This is just S(τ ) Lie series expression. See the next chapter to know more about the formal vector field and the formal phase flow. Since f τ (y) is a formal vector field, it is a formal power series in τ . Thus, the exponential representation of S(τ ) will the following form: S(τ ) = exp (τ A + τ 2 B + τ 3 C + τ 4 D + τ 5 E + · · ·), and series

τ A + τ 2B + τ 3C + τ 4D + τ 5E + · · ·

may not be convergence, where A, B, C, D, E, · · · are first-order differential operators. Therefore, we have: Theorem 3.7. Every integrator S(τ ) has a formal Lie expression [QZ92] . We use Theorem 3.7 to derive an important conclusion. Theorem 3.8. Every self-adjoint integrator has an even order of accuracy[QZ92] . Proof. Let S(τ ) be a self-adjoint integrator. Expand S(τ ) in the exponential form S(τ ) = exp (τ w1 + τ 2 w2 + τ 3 w3 + · · ·). Suppose S(τ ) is of order n, then S(τ )y(0) = eτ Lf y(0) + O(τ n+1 ), when the ODE to be solved is y˙ = f (y). Since eτ Lf + o(τ n+1 ) = eτ Lf +O(τ then

S(τ ) = eτ Lf +O(τ

n+1

)

n+1

)

,

.

We must show that n is an even number. This means that we have to prove w2 = w4 = w6 = w8 = · · · = 0. Since

S(−τ ) = exp (−τ w1 + τ 2 w2 − τ 3 w3 + · · ·),

and using the BCH formula, we get S(τ )S(−τ ) = exp (2τ 2 w2 + O(τ 3 )).

(3.14)

Since S(τ ) is self-adjoint, i.e., S(τ )S(−τ ) = I, So (3.14) means w2 = 0, and (3.14) becomes   S(τ )S(−τ ) = exp 2τ 4 w4 + O(τ 5 ) . This leads to w4 = 0. Continuing this process, we have

382

8. Composition Scheme

w2 = w4 = w6 = · · · = w2k = · · · = 0. Thus S(τ ) becomes S(τ ) = exp (τ w1 + τ 3 w3 + τ 5 w5 + · · ·). Therefore, if S(τ ) is of order n, then n must be an even number. Since S(τ ) is at least of order 1, and if n  2, we have w1 = Lf , because S(τ ) Lie series expression is unique.  Now, we will provide a corollary on the construction of higher order schemes. Corollary 3.9. Let S(τ ) be a self-adjoint integrator with order 2n, if c1 , c2 satisfies 2c2n+1 + c2n+1 = 0, 1 2

2c1 + c2 = 1,

then composition integrator S(c1 τ ) S(c2 τ ) S(c1 τ ) is of order 2n + 2, solving the above equations, we get [QZ92] : √ 2n+1 1 2 √ √ . c1 = , c = 2 2 − 2n+1 2 2 − 2n+1 2 Proof. From Theorem 2.6, we know S(c1 τ )S(c2 τ )S(c1 τ ) is a self-adjoint operator and Theorem 3.8 shows it to be even order. Since S(τ ) is of order 2n, the expansions in exponential form of S(c1 τ ), S(c2 τ ) are   S(c1 τ ) = exp c1 τ w1 + τ 2n+1 c2n+1 w2n+1 + O(τ 2n+3 ) , 1   S(c2 τ ) = exp c2 τ w1 + τ 2n+1 c2n+1 w2n+1 + O(τ 2n+3 ) . 2 Again, by BCH formula, we get S(c1 τ )S(c2 τ )S(c1 τ )   = exp (2c1 + c2 )τ w1 + (2c2n+1 + c2n+1 )τ 2n+1 w2n+1 + O(τ 2n+3 ) 1 2   = exp τ w1 + O(τ 2n+3 ) . The proof can be obtained.



H. Yoshida in [Yos90] obtained the same result for symplectic explicit integrator used to solve separable systems. The result can be applied to non-Hamiltonian systems and non-symplectic integrators. In this chapter, we will extend these results to solve general autonomous system’s form. Some examples of adjoint scheme and its construction are given below. A concrete method to construct an adjoint for any given scheme is also given. This method can be referred in literature [HNW93] . If the numerical solution is yτ , then any given scheme may be expressed as: yτ (x + τ ) = yτ (x) + τ φ(x, yτ (x), τ ),

(3.15)

8.3 Construction of Higher Order Schemes

383

where φ is increment function corresponding to the scheme and τ is step size. By substituting −τ instead of τ in (3.15), we get   y−τ (x − τ ) = y−τ (x) − τ φ x, y−τ (x), −τ , and x + τ instead of x, we get y−τ (x) = y−τ (x + τ ) − τ φ(x + τ, y−τ (x + τ ), −τ ).

(3.16)

For sufficiently τ , Equation (3.16), for y−τ (x + τ ) possesses a unique solution (by the implicit function theorem) and expresses in the following form:   y−τ (x + τ ) = y−τ (x) + τ φ∗ x, y−τ (x), τ , (3.17) and (3.17) is just the adjoint scheme for (3.15). y−τ is the adjoint scheme of numerical solution. φ∗ is the increment function corresponding to the adjoint scheme (the above process equals to: first solve S(−τ ), then solve S −1 (−τ )). In fact; let y−τ (x + τ ) = A, y−τ (x) = B, from (3.16) and (3.17), we have  B = A − τ φ(x + τ, A, −τ ), A = B + τ φ∗ (x, B, τ ), as in Equations (2.11) and (2.12) in Definition 2.2. Next, we would like to consider self adjoint conditions for R–K method. Since most one-step multistage methods can be written in R–K form, we now turn to the R–K methods to get some results which may be useful. The general s-stage R–K method is in the form R–K . ⎧ s    ⎪ ⎪ ⎪ ki = f x0 + ci τ, y0 + τ aij kj , ⎪ ⎨ j=1 (3.18) s ⎪  ⎪ ⎪ ⎪ bi ki , ⎩ y1 = y 0 + τ i=1

where y0 is numerical solution in x0 , y1 is numerical value in x0 + τ , then ci =

s 

aij

(3.19)

j=1

may be expressed in Butcher tableau: c1

a11

a12

···

a1s

c2 .. .

a21 .. .

a22 .. .

··· .. .

a2s .. .

cs

as1

as2

···

ass

b1

b2

···

bs

[HNW93]

The proof for the following Lemma 3.10 see . It can be proved by Definition 2.2 directly. Since we have proved the equivalence between Definition 2.1 and 2.2.

384

8. Composition Scheme

Lemma 3.10. Every R–K method has an adjoint method, whose coefficients a∗ij , b∗j , c∗j (i, j = 1, · · · , s) can be written as follows: ⎧ ∗ ∗ ⎪ ⎨ ci = 1 − cs+1−i , a∗ij = bs+1−j − as+1−i,s+1−j , (3.20) ⎪ ⎩ ∗ bj = bs+1−j . Lemma 3.11. If as−i+1,s−j+1 + aij = bs−j+1 = bj , then the corresponding R–K method (3.18) is self-adjoint. Concentrating on semi-explicit symplectic R–K method, we have: Theorem 3.12. The semi-explicit symplectic R–K method for autonomous systems is self-adjoint if its Butcher tableau is of the form[QZ92] . Table 3.1.

Butcher tableau in theorem 3.12 b1 2 b2 b1 2 .. .. . .

..

.

b1

b2

···

b1

b2

···

b2 2 b2

b1

b2

···

b2

b1 2 b1

Proof. We know that the Butcher tableau of semi-explicit symplectic R–K method must be of the form b1 2

b1 .. .

b2 2

.. .

..

b1

b2

···

bs−1 2

b1

b2

···

bs−1

bs 2

b1

b2

···

bs−1

bs

.

Tab. 3.1is obvious from Lemma 3.11. By Lemma 3.11, we know that possesses such form of Table 3.1 is evident. For non self-adjoint symplectic integrator S(τ ), S ∗ (τ )S(τ ) is self-adjoint, and is symplectic. In order to prove that it is symplectic, it is enough to prove S ∗ (τ ) is symplectic. If S ∗ (τ )S(−τ ) = I, then S ∗ (τ ) = S −1 (−τ ). As S(τ ) is symplectic, S(−τ )

8.3 Construction of Higher Order Schemes

385

and S −1 (−τ ) are symplectic integrators, and therefore S ∗ (τ ) is also symplectic. The two Lemmas given below can be seen in paper[HNW93] . Lemma 3.13 is derived from Theorem 1.24 of Chapter 7 and Theorem 1.1 of Chapter 8.  Lemma 3.13. If in an implicit s-stage R–K method, all ci (i = 1, · · · , s) are different and at least of order s, then it is a collocation method iff it is satisfies: s 

aij cq−1 = j

j=1

cqi , q

i = 1, · · · , s,

q = 1, · · · , s.

(3.21)

Lemma 3.14. Based on the symmetrical distribution, collocation algorithm is selfadjoint. As the Legendre polynomial is the orthogonal, coefficient of Gauss–Legendre method ci (i = 1, · · · , s) takes the root of Legendre polynomial Ps (2c − 1) in which the root is not mutually same. Moreover, the coefficient of Gauss–Legendre method aij (i, j = 1, · · · , s) satisfies formula (3.21) , and the Gauss–Legendre method is of the order 2s; therefore, it is the collocation method. For details, see[HNW93] . We have: Theorem 3.15. Gauss–Legendre methods are self-adjoint. Proof. Since Gauss–Legendre method is collocation method, we only need to prove ci = 1−cs+1−i , and ci are symmetrical distributions, where c1 , c2 , · · · , cs are the root of Legendre polynomial Ps (2c − 1) (lower index denotes order of polynomial). If the root of Ps (w) are w1 , w2 , · · · , ws , then ci =

1 wi − , 2 2

i = 1, · · · , s,

i.e., ci = 1 − cs+1−i and wi + ws+1−i = 0 are equivalent. Legendre polynomial can be constructed by  q0 (w) = 1, q−1 (w) = 0, 2 qi−1 (w), qi+1 (w) = (w − δi+1 )qi (w) − γi+1

where δi+1 = 2 = γi+1

⎧ ⎨ 0,

(wqi , qi ) , (qi , qi )

(qi , qi ) , ⎩ (qi−1 , qi−1 )

and

(qi , qj ) = 1

i = 0, 1, · · · ,

i ≥ 0, for

i = 0,

for

i ≥ 1,

1 −1

qi (w)qj (w) d w.

We obtain q1 = w, q2 = w2 − , assuming q2n (w) is an even function and q2n−1 (w) 3 is an odd function. We proceed by induction on n, for n = 1, this has established.

386

8. Composition Scheme

Suppose q2n is an even function, and q2n−1 is an odd function. Prove that n + 1 is also true. Since - 1 odd d w even|1−1 (wq2n , q2n ) = −1 = = 0, δ2n+1 = (q2n , q2n ) (q2n , q2n ) (q2n , q2n ) 2 then q2n+1 = wq2n − γ2n+1 q2n−1 (w) = odd function−odd function = odd function. - 1 odd dw (wq2n+1 , q2n+1 ) −1 = = 0, δ2n+2 = (q2n+1 , q2n+1 ) (q2n+1 , q2n+1 ) 2 q2n (w) = is an even function. We have proved this then q2n+2 = wq2n+1 − γ2n+2 conclusion for n + 1. From this, the P2n (w) root may be written in the following sequence: −w1 , −w2 , · · · , −wn , wn , · · · , w2 , w1 .

But the root of p2n+1 (w) has the following form: −w1 , −w2 , · · · , −wn , 0, wn , · · · , w2 , w1 , where wi > 0, wi > wi+1 (i = 1, · · · , n), therefore wi + wn+1−i = 0. Even though we have the direct proof of Theorem 3.15, the computation is tedious. As a result of Gauss–Legendre method coefficient aij , bj satisfies the following equation: s  j=1 s 

aij cq−1 = j

cqi , q 1 q

bj cjq−1 = ,

j=1

i = 1, · · · , s,

q = 1, · · · , s,

(3.22)

q = 1, · · · , s.

(3.23)

Using the linear algebra knowledge, we have aij =

cki k ϕ , ; kj (cj − cl )

s (−1)n+k  k=1

s 2

n= ,

j=l

bj =

1 k ϕ , kj (cj − cl )

s (−1)n+k 

;

k=1

i, j, l = 1, · · · , s,

j=l

where

⎧ ⎪ ⎨ ϕkj = ⎪ ⎩

 {t1 ,t2 ,···,ts−k }⊂{1,2,···,j−1,j+1,···,s}

ϕsj = 1.

ct1 ct2 · · · cts−k ,

k < s,

8.3 Construction of Higher Order Schemes

387

The direct calculation may result in cki - ci k ; ϕkj = lj (t)d t, (cj − cl ) 0

s (−1)n+k  k=1

j = 1, · · · , s,

(3.24)

j = 1, · · · , s,

(3.25)

j=l

1 - 1 k ; ϕkj = lj (t)d t, (cj − cl ) 0

s (−1)n+k  k=1

j=l

B k=j

where lj = B

k=j

(t − ck )

(cj − ck )

, when ci = 1 − cs+1−i , we have li (t) = ls+1−i(1−t) , then as+1−i,s+1−j + aij = bs+1−j = bj , 

from (3.24) and (3.25), it is easy to prove.

Below given is an example to construct a self-adjoint scheme using a given scheme .

[QZ92]

Example 3.16. It is well know that scheme (2.1) is of the first order. From the above method, the adjoint scheme will be  q n+1 = q n + τ Up (pn ), (3.26) pn+1 = pn − τ Vq (qn+1 ). Composition scheme (2.2) from (2.1) and (3.26) is of order 2. In order to maintain τ the transient step size, original τ will shrink and will become , because the present 2     τ ∗ τ S . If it is self-adjoint, the transient length of step size is mainscheme is S 2 2 tained as τ . Example 3.17. It is easy to prove that scheme y 1 = y0 +

A τ@ f (y1 ) + f (y0 ) 2

is self-adjoint, and will be of order 2. Symmetrical composition scheme (1.22) is 3stage of 4th order and also self-adjoint. Example 3.18. The explicit 4th order symplectic scheme (2.4) can be composed by S2 (x1 τ ) S2 (x2 τ ) S2 (x1 τ ) and developed as follows: SV (c1 τ )SU (d1 τ )SV (c2 τ )SU (d2 τ )SV (c3 τ )SU (d3 τ )SV (c4 τ )SU (d4 τ ). 0 12 3 0 12 30 12 3 S2 (x1 τ )

noting Corollary 3.9, we get:

S2 (x2 τ )

S2 (x1 τ )

388

8. Composition Scheme x1 1 √ = 0.6756035, = 2 2− 33 1 √ = 1.35120719, d 1 = d 3 = x1 = 2(2 − 3 2) √ −32 √ = x2 = −1.7024142, d2 = d4 = 0, 2− 32 √ x + x2 1− 32 √ = −0.1756036. c2 = c3 = 1 = 2 2(2 − 3 2)

c1 = c4 =

Example 3.19. By literature[FQ91] , we know that one class of symplectic scheme for equation

dy = J∇H is dt

%

& 1 1 (I + JB)y k+1 + (I − JB)y k , 2 2 6 5 6 5 O −In In O , J= , I= O In In O

y k+1 = y k + τ J(∇H)

B T = B, (3.27)

this scheme is of order 1, if B = O; if B = O, then the scheme will be of order 2. In scheme (3.27), if B = O, we can prove it is self-adjoint. When B = O, if integrator of scheme (3.27) is S(τ, H, B), then adjoint integrator of scheme (3.27) will be S(τ, H, −B) = S ∗ (τ, H, B).     τ τ and S ∗ is self-adjoint with second order, Composition integrator from S 2 2 concrete scheme is ⎧   τ 1 1 ⎪ ⎨ y1 = y k + J(∇H) (I − JB)y1 + (I + JB)y k , 2 2 2   ⎪ ⎩ y k+1 = y1 + τ J(∇H) 1 (I + JB)y k+1 + 1 (I − JB)y1 . 2

2

2

8.4 Stability Analysis for Composition Scheme In this paragraph, will discuss the stability of the three-stage scheme which was constructed in Section 8.1   1 1 yn+ 13 = yn + 1 τ f f (yn ) + f (yn+ ) , 3

[QZ93]

, fourth order

2(2 − 2 3 )

1

yn+ 23 = yn+ 13 + yn+1 = yn+ 23 +

−2 3 1 3

2(2 − 2 ) 1 1 3

2(2 − 2 )

  τ f (yn+ 13 ) + f (yn+ 23 ) ,

  τ f (yn+ 23 ) + f (yn+1 ) .

(4.1)

8.4 Stability Analysis for Composition Scheme

389

We will prove that although the trapezoid method is A-stable, scheme (4.1) is not A-stable. Fortunately , the unstable region is very small, as Fig. 4.2 (enlaged figure is Fig.4.1) shows, and scheme (4.1) is still useful for solving stiff systems. Judging from the size and location of the unstable region of scheme (4.1), we know it is safe for systems which have eigenvalues not very adjacent to the real axis, while some other methods which have unstable regions near the imaginary axis, such as Gear’s are safe for systems which have eigenvalues not very adjacent to the imaginary axis. 0.020

S

0.012 0.004 –0.004 –0.012

–0.020 –0.596 –0.579 –0.563 –0.603 –0.587 –0.571

Fig. 4.1.

Closed curve S which contains all zero point of scheme (4.1) 1.0 0.6 0.2 –2.0 –1.6

–1.2

–0.8

–0.4

0.0 –0.2 –0.6 –1.0

Fig. 4.2.

Stability region size and position of (4.1)

Just the same as in scheme (4.1), the Euler midpoint rule can also be used to construct a scheme: ⎧ % & yn + yn+ 1 1 ⎪ 3 ⎪ , ⎪ ⎪ yn+ 13 = yn + 2 − 2 13 τ f 2 ⎪ ⎪ ⎪ ⎪ & % 1 ⎪ ⎨ yn+ 1 + yn+ 2 −2 3 3 3 , τ f yn+ 23 = yn+ 13 + 1 (4.2) 2 2 − 23 ⎪ ⎪ ⎪ & % ⎪ ⎪ yn+ 2 + yn+1 1 ⎪ 3 ⎪ yn+1 = yn+ 23 + . ⎪ 1 τf ⎪ 2 ⎩ 2 − 23

390

8. Composition Scheme

Scheme (4.2) is symplectic, but scheme (4.1) is non-symplectic. We now study the stability of scheme (4.1). Note that scheme (4.1) is not A- stable, whereas the trapezoid method is. To show this, we apply scheme (4.1) to test equation

which yields

λ ∈ C,

y(0) = y0 ,

y˙ = λy,

Re λ < 0,

 ⎧ τ yn+ 13 = yn + c1 λyn + λyn+ 13 , ⎪ ⎪ 2 ⎪ ⎪  ⎨ τ yn+ 23 = yn+ 13 + c2 λyn+ 13 + λyn+ 23 , 2 ⎪ ⎪  τ ⎪ ⎪ 2 = y + c y λyn+ 23 + λyn+1 , ⎩ n+1 1 n+ 3

(4.3)

(4.4)

2

i.e.,

⎧ c1 λτ ⎪ 1+ ⎪ ⎪ 2 y , ⎪ ⎪ yn+ 13 = ⎪ c1 λτ n ⎪ ⎪ 1− ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ c2 λτ ⎪ ⎨ 1+ 2 y 1, 2 yn+ 3 = c2 λτ n+ 3 ⎪ ⎪ 1 − ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ λτ c 1 ⎪ ⎪ 1+ ⎪ ⎪ 2 y 2, ⎪ y = n+1 ⎪ ⎪ c1 λτ n+ 3 ⎩ 1−

where c1 =

1 1

2 − 23

, c2 =

−2

1 3 1

2 − 23

yn+1 =

. Let

(4.5)

2

λτ = z, z ∈ C, we have 2

(1 + c1 z)(1 + c2 z)(1 + c1 z) yn . (1 − c1 z)(1 − c2 z)(1 − c1 z)

(4.6)

Definition 4.1. The stable region R of scheme (4.1) is: &% &% & ⎧  % ⎫ c1 λτ c2 λτ c1 λτ   ⎪ ⎪ 1+ 1+ ⎨  1+  ⎬ 2 2 2   &% &% &  < 1, Re (λτ ) < 0 , R = λτ ∈ C   % c2 λτ c1 λτ  c1 λτ  ⎪ ⎪ ⎩ ⎭ 1− 1−  1− 2

i.e.,

2

2

 7    (1 + c1 z)(1 + c2 z)(1 + c1 z)  z ∈ C   < 1, Re z < 0 .  (1 − c1 z)(1 − c2 z)(1 − c1 z)

 R=

Obviously, when z →

1 (< 0), we have c2

   (1 + c1 z)(1 + c2 z)(1 + c1 z)     (1 − c1 z)(1 − c2 z)(1 − c1 z)  −→ ∞.

(4.7)

8.4 Stability Analysis for Composition Scheme

391

1

This means schemes (4.1) cannot be stable in the adjacent region . Thus, we obtain c2 the following theorem: Theorem 4.2. Scheme (4.1) is not A-stable. Since scheme (4.1) is not A-stable, we will figure out the stable region of it. To do this, we will first study the roots of the following equation:    (1 + c1 z)(1 + c2 z)(1 + c1 z)    (4.8)  (1 − c1 z)(1 − c2 z)(1 − c1 z)  = 1. Once the roots of (4.8) are known, it is not difficult to get the stable region of (4.1). Note Equation (4.8) is equivalent to (1 + c1 z)(1 + c2 z)(1 + c1 z) = eiθ , (1 − c1 z)(1 − c2 z)(1 − c1 z)

0 ≤ θ < 2π.

(4.9)

From (4.9), we get the following polynomial: c21 c2 (1 + eiθ )z 3 + (2c1 c2 + c21 )(1 − eiθ )z 2 +(2c1 + c2 )(1 + eiθ )z + (1 − eiθ ) = 0,

0 ≤ θ < 2π.

(4.10)

Since 2c1 + c2 = 1, and a = c21 c2 , b = 2c1 c2 + c21 , then (4.10) becomes: a(1 + eiθ )z 3 + b(1 − eiθ )z 2 + (1 + eiθ )z + (1 − eiθ ) = 0,

0 ≤ θ < 2π. (4.11)

Consider the roots of (4.11) in two cases. Case 4.3. 1 + eiθ = 0 (i.e., 0 ≤ θ < 2π, θ = π). By computing the roots of polynomial (4.11), we get z1 = x + yi,

z2 = −x + yi,

z3 = wi,

x, y, w ∈ C

(4.12)

when θ is chosen as a definite value. z1 , z2 , z3 should satisfy (z − z1 )(z − z2 )(z − z3 ) = a(1 + eiθ )z 3 + b(1 − eiθ )z 2 + (1 + eiθ )z +(1 − eiθ ) = 0 ⎧   ⎪ b 1 − eiθ ⎪ z1 + z2 + z3 = − , ⎪ ⎪ ⎪ a 1 + eiθ ⎪ ⎪ ⎪ ⎨ 1 z1 z2 + z1 z3 + z2 z3 = , ⇐⇒ a ⎪ ⎪ ⎪   ⎪ ⎪ 1 1 − eiθ ⎪ ⎪ z z = − . z ⎪ ⎩ 1 2 3 a 1 + eiθ

Since

(4.13)

392

8. Composition Scheme

sin θ (1 − cos θ) − i sin θ 1 − eiθ =− i, = i θ (1 + cos θ) + i sin θ 1 + cos θ 1+e then

⎧ b sin θ ⎪ ⎪ i = ip1 , z1 + z2 + z3 = ⎪ ⎪ a 1 + cos θ ⎪ ⎪ ⎨ 1 z1 z2 + z1 z3 + z2 z3 = = p2 , a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z1 z2 z3 = 1 sin θ i = ip3 ,

(4.14)

a 1 + cos θ

where p1 , p2 , p3 are real numbers. From (4.12) and (4.14), we have equations of x, y, w as the following: 2y + w = p1 , 2

2

x + y + 2yw = −p2 , x2 w + y 2 w = −p3 .

(4.15) (4.16) (4.17)

Now, we will prove that Equations system (4.15)–(4.17) exists as a set of real solution. In fact, from (4.16) and (4.17), we get: p2 w + 2yw 2 = p3 .

(4.18)

w3 − p1 w2 − p2 w + p3 = 0.

(4.19)

From (4.15) and (4.18), we have

Since p1 , p2 , p3 are all real, (4.19) is a polynomial with real coefficient and has one real root and two conjugate complex roots. Using the real root w from (4.19), we can get a real value of y from (4.15) , and from (4.16) and (4.17), we have x2 = real, then x is real or pure imaginary. If x is pure imaginary, then from (4.12), z1 , z2 , z3 are all pure imaginaries, so all the the roots of (4.11) are on the imaginary axis. This means that if we assume    (1 + c1 z)(1 + c2 z)(1 + c1 z)   , V (z) =  (1 − c1 z)(1 − c2 z)(1 − c2 z)  then, V (z) > 1 or V (z) < 1 for Re (z) < 0. For the same reason that scheme (4.1) cannot be A-stable, V (z) > 1 for Re (z) < 0 is possible, but we have V (−0.5) < 1, 1

and V (z) is continuous except at . Thus, x is impossible to be pure imaginary, so c2 it must be real. Since polynomial (4.11) only has three roots, we will get the same results of z1 , z2 , z3 if we use other value of w, real or complex. Case 4.4. 1 + eiθ = 0 (i.e., θ = π). 1 When θ = π, (4.11) becomes z = − > 0, then it has two real roots ± b 2

Eventually, we get the following:

D 1 − . b

8.4 Stability Analysis for Composition Scheme

393

Theorem 4.5. The three roots of polynomial (4.11) are in the form : z1 = −x + yi,

z2 = x + yi,

z3 = wi,

where x, y, w are all real. Theorem 4.5 tells us that there is a root of (4.10) on the imaginary, and that the two other roots are located symmetrically with respect to the imaginary axis. Thus, there is only one root on the left open semi-plane. Computation shows that these roots form a closed curve S (when θ changes from 0 to 2π), as in Fig. 4.1. From (4.15) – (4.17), we get the equation for S:  2 x − 3y 2 + 2p1 y + p2 = 0, 0 ≤ θ < 2π, θ = π. (4.20) 2yx2 + 2y 3 − p1 x2 − p1 y 2 − p3 = 0, D

and x=± where z = −x + iy, p1 =

1 − , b

y = 0,

for θ = π,

(4.21)

b sin θ 1 sin θ , p2 = , p3 = . a(1 + cos θ) a a(1 + cos θ)

Since in S, V (z) > 1, and when z → ∞, V (Z) → 1, we can conclude the stability of scheme (4.1). Theorem 4.6. The stable region R of scheme (4.1) is [QZ93] : R = {z ∈ C | z outside S and Re z < 0},

i.e.,

( ) z R = {λτ ∈ C | λτ outside S ∗ and Re (λτ ) < 0}, where S ∗ = z ∈ C | ∈ S . 2

Scheme (4.1) is not A-stable, the outside region of it is infinite, and the unstable region is very small. The unstable region is blackened in Fig. 4.2, it is a little “disk” about −1.18 on the real axis. For every definite λ, we can choose some special step length τ , such that λτ will not be in S ∗ , while the step-length τ need not be very small for λ which has a big modulus. Because of linear cases, scheme (4.2) is equivalent to (4.1). So, scheme (4.2) has exactly the same stable region as (4.1). Thus, we conclude that scheme (4.1) and (4.2) are still useful for solving stiff problems, which we wanted to show in example. Following are some numerical tests for stability of scheme (4.1). Example 4.7. Numerical test for orders of schemes (4.1) and (4.2). To test the order scheme (4.1) and (4.2), we apply them to the following Hamiltonian system: ⎧ ⎪ d p = − ∂ H = −w 2 q − q 3 , ⎪ ⎨ dt ∂q (4.22) ⎪ ⎪ ⎩ d q = ∂ H = p, dt

∂p

394

8. Composition Scheme

1 1  where the Hamiltonian H = p2 + w2 q 2 + q 4 , and compare the numerical solu2 2 tions with trapezoid method and centered Euler scheme. For convenience, the numerical solution of p and q can be denoted as 1◦ (4.1) by T 4p, T 4q. 2◦ (4.2) by E4p, E4q. 3◦ trapezoid scheme by T 2p, T 2q. 4◦ centered Euler scheme by E2p, E2q. Respectively, we use double precision in all computations. We can see the following explicit scheme: ⎧ τ p 1 = pn − Vq (qn ), ⎪ ⎪ ⎪ n+ 2 2 ⎪ ⎪ ⎪ ⎨ qn+ 1 = qn + τ Up (pn+ 1 ), 2 2 2 (4.23) τ ⎪ ⎪ ⎪ qn+1 = qn+ 12 + 2 Up (pn+ 12 ), ⎪ ⎪ ⎪ τ ⎩ p = p 1 − V (q ), n+1

n+ 2

2

q

n+1

x1 1 √ = 0.6756035, = 2 2− 33 1 √ = 1.35120719, d 1 = d 3 = x1 = 2(2 − 3 2) √ −32 √ = x2 = −1.7024142, d4 = 0, d2 = 2− 32 √ x + x2 1− 32 √ = −0.1756036. c2 = c3 = 1 = 2 2(2 − 3 2)

c1 = c4 =

A separable system with H = V (q) + U (p) is self-adjoint, so it can be used to construct fourth-order scheme to get (4.1) and (4.2). From Sections 8.2 and 8.3, 1 the simplified fourth-order scheme can be written taking c1 = 1 = c3 , c2 = 2 − 23

1

−2 3 1

2 − 23

, x1 = x3 = 1.35120719, x2 = −1.7024142. For details see Example 3.18. ⎧ τ pn+ 14 = pn − x1 Vq (qn ), ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 1 = qn + x1 τ Up (p q ⎪ n+ n+ 14 ), ⎪ 3 ⎪ ⎪ ⎪ x + x ⎪ 2 ⎪ τ Vq (qn+ 13 ), pn+ 12 = pn+ 14 − 1 ⎪ ⎪ 2 ⎪ ⎨ qn+ 23 = qn+ 13 + x2 τ Up (pn+ 12 ), ⎪ ⎪ x2 + x3 ⎪ ⎪ τ Vq (qn+ 23 ), ⎪ ⎪ pn+ 34 = pn+ 12 − 2 ⎪ ⎪ ⎪ ⎪ qn+1 = q 2 + x3 τ Up (p 3 ), ⎪ n+ 3 n+ 4 ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎩ pn+1 = pn+ 3 − x3 Vq (qn+1 ), 4

(4.24)

2

where pn+ 14 , pn+ 12 , pn+ 34 and qn+ 13 , qn+ 23 denote the numerical solution of different stages at every step. Scheme (4.24) has been proved by H. Yoshida to be a fourthorder scheme in [Yos90] . We can apply scheme (4.24) to Equation (4.22) and compare

8.4 Stability Analysis for Composition Scheme

Table 4.1.

395

Numerical comparison between several schemes

Step N

N = 10

N = 20

N = 100

N = 500

N = 1000

Numerical solution and exact solution EXp = −1.131 156 917 000000 EXq = −0.021 512 660 000000 T 4p − EXp = 0.000 014 55000 T 4q − EXq = −0.000 003 728 E4p − EXp = 0.000 068 24300 E4q − EXq = −0.000 029 687 T 2p − EXp = 0.000 641 21600 T 2q − EXq = 0.003 917 96400 E2p − EXp = 0.000 025 85700 E2q − EXq = 0.004 206 14700 EXp = −0.578 997 162 000000 EXq = −0.479 477 967 00000 T 4p − EXp = 0.000 004 11500 T 4q − EXq = −0.000 002 660 E4p − EXp = −0.000 116 088 E4q − EXq = −0.000 029 838 T 2p − EXp = −0.014 158 525 T 2q − EXq = −0.003 977 057 E2p − EXp = −0.015 197 562 E2q − EXq = −0.004 255 307 EXp = −1.083 692 040 00000 EXq = 0.163 258 193 0000000 T 4p − EXp = −0.000 104 873 T 4q − EXq = −0.000 195 865 E4p − EXp = 0.000 145 7860 E4q − EXq = 0.000 131 2730 T 2p − EXp = 0.024 490 7400 T 2q − EXq = 0.036 283 1300 E2p − EXp = 0.027 254 9000 E2q − EXq = 0.039 223 1760 EXp = −1.089 537 517 000000 EXq = −0.153 288 801 000000 T 4p − EXp = 0.000 560 51300 T 4q − EXq = −0.001 139 354 E4p − EXp = −0.000 250 063 E4q − EXq = 0.000 559 4940 T 2p − EXp = −0.040 591 714 T 2q − EXq = 0.188 655 9980 E2p − EXp = −0.037 488 191 E2q − EXq = 0.204 743 2350 EXp = −0.966 531 326 000000 EXq = −0.293 028 275 000000 T 4p − EXp = 0.002 470 90100 T 4q − EXq = 0.002 014 58300 E4p − EXp = −0.001 281 080 E4q − EXq = −0.000 988 873 T 2p − EXp = −0.603 588 331 T 2q − EXq = −0.233 974 665 E2p − EXp = −0.668 484 708 E2q − EXq = −0.243 402 518

the results with that of schemes we mentioned above. We denote EXp and EXq as the exact solution of p and q for system (4.22), and present our results when taking w = 2, τ = 0.1, p0 = 0.5, q0 = 0.5 in Table 4.1. From Table 4.1, we can see that T 4p, T 4q and E4p, E4q are more approximate to EXp, EXq than T 2p, T 2q and E2p, E2q. Thus, we conclude that scheme (4.1) and (4.2) have a higher order than trapezoid method and centered Euler scheme. Table 4.1 also shows that although the trapezoid scheme (4.1) is non-symplectic, it can be used to solve a Hamiltonian system to get satisfactory results than the centered Euler scheme, by nonlinear transformation; the latter can be obtained from the former, see Section 8.1. Example 4.8. Numerical test for stability of schemes (4.1) and (4.2). To consider the unstable case, we take λ = −11.8, τ = 0.1, and initial value y0 = 1.0 in the test equation, so λτ falls into the unstable region. While the exact solution decreases quickly, the numerical solution obtained by scheme (4.1) grows to infinity as shown in Table 4.2. Example 4.9. For the stable case, we consider a linear stiff system  y˙1 = −501y1 + 500y2 , y˙2 = 500y1 − 501y2 , which has eigenvalues λ1 = −1001, λ2 = −1. The exact solution is

(4.25)

396

8. Composition Scheme

Table 4.2.

Stability test

Step number

Numerical and exact solution

Step1

0.576776990×101

0.307278738

Step10

0.407404568×108

0.000007504

Step50

0.112235299×1039

0.000000000

Step100

0.816583328×1075

0.000000000

Table 4.3. Step N N = 10

N = 30

N = 50

N = 100

N = 200

Test for stiff system Numerical solution and exact solution EXY 1 = 0.998364638 EXY 2 = 0.991660285 T 4Y 1 = 0.998453117 T 4Y 2 = 0.991571619 T 4Y 1 − EXY 1 = 0.000088478 T 4Y 2 − EXY 2 = −0.000088666 EXY 1 = 0.985112102 EXY 2 = 0.985111801 T 4Y 1 = 0.985111988 T 4Y 2 = 0.985111662 T 4Y 1 − EXY 1 = −0.000000114 T 4Y 2 − EXY 2 = −0.000000138 EXY 1 = 0.975309908 EXY 2 = 0.975309908 T 4Y 1 = 0.975309788 T 4Y 2 = 0.975309788 T 4Y 1 − EXY 1 = −0.000000120 T 4Y 2 − EXY 2 = −0.000000120 EXY 1 = 0.006571583 EXY 2 = 0.006571583 T 4Y 1 = 0.006571770 T 4Y 2 = 0.006571771 T 4Y 1 − EXY 1 = −0.000000186 T 4Y 2 − EXY 2 = −0.000000188 EXY 1 = 0.000000298 EXY 2 = 0.000000298 T 4Y 1 = 0.000000298 T 4Y 2 = 0.000000298 T 4Y 1 − EXY 1 = −0.000000000 T 4Y 2 − EXY 2 = −0.000000000

⎧     ⎨ y1 (t) = f1 (y1 , y2 ) = 0.5 y1 (0) − y2 (0) e−1001t + 0.5 y1 (0) + y2 (0) e−t ,     ⎩ y2 (t) = f2 (y1 , y2 ) = −0.5 y1 (0) − y2 (0) e−1001t + 0.5 y1 (0) + y2 (0) e−t , (4.26) where y1 (0), y2 (0) denote the initial value. Since system (4.25) is linear, schemes (4.1) and (4.2) are equivalent in this case. We present a numerical solution using scheme (4.1) here. In Table 4.3, we denote the numerical solution of y1 and y2 using (4.1) by T 4Y 1, T 4Y 2, and the exact solution of y1 and y2 by EXY 1 and EXY 2. We also assume τ = 0.1, y1 (0) = 1.5, y2 (0) = 0.5, in the Table 4.3, while τ = 0.0005 in the first 50 steps, and τ = 0.1 in the remaining steps.

8.5 Application of Composition Schemes to PDE When solving partial differential equations (PDEs), there are several methods such as spectral methods and finite difference methods which can be used to achieve highorder accuracy in the space direction, while it is difficult to obtain high-order accuracy in time direction. So it is obvious that the overall accuracy is often influenced strongly by the relatively unsatisfactory approximation in the time direction. Though the self-adjoint schemes (also called symmetrical schemes or reversible schemes) are well known, such as the composed Strang scheme [Str68] which is of order 2, the advantage of these schemes which can be used to construct higher order schemes is long

8.5 Application of Composition Schemes to PDE

397

neglected. In this section, we use scheme (4.1) to solve two kinds of PDEs in order to show that the technique introduced in previous section can be used to overcome the deficiency in the time direction, since theoretically, we can construct arbitrary even order schemes in the time direction[ZQ93b] . Let us first consider the following one-dimensional first-order wave equation 

ut + ux = 0, u(x, 0) = f (x),

0 ≤ x ≤ 2π,

(5.1)

with periodic boundary conditions u(0, t) = u(2π, t). Since collocation, Galerkin, and tau methods are identical in the absence of essential boundary conditions, we will analyze the Fourier collocation or pseudospectral method. Let us introduce the collocation points xn = 2πn/2N (n = 0, · · · , 2N − 1), and let u = (u0 , · · · , u2N −1 ), where un = u(xn , t). The collocation equation that approximates (5.1) is as follows: ∂u = C −1 DCu, ∂t

(5.2)

where C and D are 2N × 2N matrices whose entries are ckl = √ dkl

1

A @ exp (k − N )xl ,

2N = −k ∗ δkl ,

(5.3) (5.4)

where k ∗ = k − N (1 ≤ k ≤ 2N − 1), and k∗ = 0, if k = 0. For the process of the discretization, see also literature [GO77] , we leave out the proof in this, but directly quote. To solve this, let us consider Equation (5.1) with initial value f (x) = sin x, and compare the numerical solution with the exact solution u(x, t) = sin (x − t), we use scheme (4.1) and trapezoid scheme (crank-Nicolson) to solve Equation (5.2) (N = 5). All u values are calculated in the collocation points taking the time step size τ = 0.1 and 0.01, and respectively calculating 100 steps with the double precision. ORD.4 and ORD.2, represent results that use (4.1) and the trapezoidal form obtained numerical solution respectively. ERR.4 and ERR.2 represent error between numerical solution ORD.4 and ORD.2 and the exact solution, where the collocation point is n. We list u(x, t) in each step with values 0, 5, 9 as collocation points. The exact solution is denoted by EX. From Table 5.1 and Table 5.2 we can see that the solution of the 4th order scheme is more precise than the solution of the 2nd order scheme, when τ = 0.1 precise 2, when τ = 0.01 precise 4.

398

8. Composition Scheme

Table 5.1. Step N

Comparison between numerical and exact solution when τ =0.1 n

EX

ORD.4

ORD.2

0 −0.099833416647 −0.099832763924 −0.099750623437 N =1

N = 10

5

0.099833416647

0.099832763924

9 −0.665615704994

−0.66561545443

ERR.4

ERR.2

0.000000652723

0.000082793209

0.099750623438 −0.000000652723 −0.000082793209 −0.665553604585

0.000000489551

0.000062100409

0 −0.841470984808 −0.841467440655 −0.841021115481

0.000003544153

0.000449869327

5

0.841470984808

0.841021115481 −0.000003544153 −0.000449869327

0.841467440655

9 −0.998346054152 −0.998346431587 −0.998393545150 −0.000000377435 −0.000047490998 0

0.544021110889

0.537020563223 −0.000055042829 −0.007000547666

0.543966068061

N = 100 5 −0.544021110889 −0.543966068061 −0.537020563223 9

Table 5.2. Step N N =1

N = 10

0.933316194418

0.000055042829

0.007000547666

0.930296266090 −0.000023553213 −0.003019928328

0.933292641025

Comparison between numerical and exact solution when τ = 0.01 n

EX

ORD.4

ORD.2

0

0.009999833340 −0.099998333280 −0.009999750000

5

0.009999833340

ERR.4 0.000000000007

ERR.2 0.000000083334

0.009999750000 −0.000000000007 −0.000000083334

0.009999833280

9 −0.595845898383 −0.595845898378 −0.595845831454

0.000000000005

0 −0.099833416647 −0.099833416582 −0.099832587427

0.000000000065

0.000000829220

5

0.099832587427

0.000000000042

−0.000000829220

9 −0.665615704994 −0.665615704952 −0.665615083044

0.000000000003

0.000000621950

0 −0.841470984808 −0.841470984547 −0.841466481987

0.000000000261 −0.000004502821

N = 100 5

0.099833416647

0.841470984808

0.099833416582

0.000000066929

0.841466481987 −0.000000000267 −0.000004502871

0.841470984547

9 −0.998346054152 −0.998346054304 −0.998346533230 −0.000000000152 −0.000000479078

Similarly, in 2nd order PDE, the result of the 4th order scheme is more precise when compared to the result of the 2nd order scheme in 2 - 4 precision. Let us take the second order heat conductivity equation ⎧ ∂u(x, t) ∂ 2 u(x, t) ⎪ = , ⎪ ⎨ 2 ∂t

⎪ ⎪ ⎩

0 < x < π,

∂x

u(0, t) = u(π, t) = 0,

t > 0,

u(x, 0) = f (x),

0 ≤ x ≤ π.

t ≥ 0, (5.5)

By applying Fourier sine approach in Equation (5.5), we get uN (x, t) =

N 

an (t) sin nx,

(5.6)

n=1

and

⎧ d an 2 ⎪ ⎪ ⎨ d t = −n an , - π 2 ⎪ (0) = f (x) sin nx d x. a ⎪ n ⎩ π

0

(5.7)

8.5 Application of Composition Schemes to PDE

Table 5.3. Step N N =1

N = 10

N = 50

399

Comparison between numerical and exact solution when τ =0.1 n

EX

ORD.4

ORD.2

ERR.4

ERR.2

1

0.531850090044

0.5318500444815

0.531805704455

0.0000003547710

−0.000044385589

2

0.860551522611

0.8605520966420

0.860479705219

0.0000005740310

−0.000071817391

3

0.860551522611

0.8605520966420

0.860479705219

0.0000005740310

−0.000071817391

4

0.531850090044

0.5318504448150

0.531805704455

0.0000003547710

−0.000443855890

1

0.216234110142

0.2162355525360

0.216053719560

0.0000001442394

−0.000180390582

2

0.349814139737

0.3498764735800

0.349582261644

0.0000023338430

−0.000291878093

3

0.349814139737

0.3498764735800

0.349582269644

0.0000123338430

−0.000291878093

4

0.216234110142

0.2162355522536

0.216053719560

0.0000014423940

−0.000180390582

1

0.003960465877

0.0039605979700

0.003943973573

0.0000001320940

−0.000164923040

2

0.006408168400

0.0064083821320

0.006381483292

0.0000002137320

−0.000026685108

3

0.006408168400

0.0064083821320

0.006381483292

0.0000002137320

−0.000026685108

4

0.003960465877

0.0039605979700

0.003943973573

0.0000001320940

−0.000164923040

The initial value of an can be represented in another form. Let xj = 1, · · · , N ) be collocation points, from collocation equation N 

an sin

n=1

πjn = u(xj ), N +1

j = 1, · · · , N,

πj (j = N +1

(5.8)

we get explicit solution an =

N πjn 2  , u(xj ) sin N +1 N +1

n = 1, · · · , N.

j=1

Since u(x, 0) = f (x), we get: an (0) =

N 2  πjn f (xj ) sin , N +1 N +1

n = 1, · · · , N.

(5.9)

j=1

The exact solution for Equation (5.5) with boundary condition f (x) = sin x is e−t sin x. In Table 5.3 and Table 5.4, all symbols carry the same significance. We take N = 4 for computation. To solve the semi-discrete spectral approximations ut = LN u,

(5.10)

ut = Lu,

(5.11)

of the differential equation where L denotes the spacial operator, we often use the Crank–Nicolson scheme, backward Euler scheme, and leap-frog scheme. However, we know the backward and forward Euler schemes are not self-adjoint, nor the leap-frog scheme. But the first two schemes are adjoint to each other and the composition is the Crank–Nicolson scheme

400

8. Composition Scheme

Table 5.4. Step N N =1

N = 100

Comparison between numerical and exact solution when τ =0.01 n

EX

ORD.4

ORD.2

ERR.4

ERR.2

1

0.581936691312

0.581936691316

0.581936642817

−0.000000000004

−0.000000048495

2

0.941593345844

0.941593345850

0.941593267377

0.000000000006

−0.000000078467

3

0.941593345844

0.941593345850

0.941593267377

0.000000000006

−0.000000078467

4

0.581936693120

0.581936691316

0.581936642817

−0.000000000004

−0.000000048495

1

0.216234110142

0.216234110285

0.216232308172

0.0000000001430

−0.000001801970

2

0.349874139137

0.349874139969

0.349811224088

0.0000000002310

−0.000002915049

3

0.349874139137

0.349874139969

0.349811224088

0.0000000000231

−0.000002915049

4

0.276234110142

0.216234110285

0.216232308172

0.0000000001430

−0.000001801970

un+1 − un = Δt

 1 LN un+1 + LN un . 2

(5.12)

which is self-adjoint and of order 2. We can construct a fourth-order scheme by composition   1 Δt LN un + LN un+1/3 , un+1/3 = un + 1/3 2(2 − 2

  2 Δt LN un+1/3 + LN un+2/3 , 2(2 − 21/3 )   1 + Δt LN un+2/3 + LN un+1 . 1/3 2(2 − 2 )

un+2/3 = un+1/3 − un+1 = un+2/3

)

1/3

(5.13)

Finally, we can point out that scheme (5.13) is unstable for some special step size of t. Since the diameter of the unstable region is very small, we can always avoid taking those step-size Δt which make λΔt (λ denotes the eigenvalue of the system to be solved) fall into the unstable region. Fig. 5.1 shows the solution of the heat equation when we use scheme (5.13) to solve the (5.11) We take Δt = 0.0097 and N = 24. We can conclude that while the Crank–Nicolson remains stable, the scheme (5.13) does not, and solution tends to overflow. For a Detailed numerical test about this problem, see[ZQ93b] .

Fig. 5.1. Stability comparison between schemes of Crank–Nicolson (L), (5.13) (M) and exact solution (R) of the heat equation

8.6 H-Stability of Hamiltonian System

401

8.6 H-Stability of Hamiltonian System We know that Hamiltonian system always appears in space of even dimension. A more important fact is that there is no asymptotically stable linear Hamiltonian system. They are either Liapunov stable or unstable, so are the linear symplectic algorithms. Therefore, the usual stability concepts in numerical methods for ODEs are not suitable to symplectic algorithms for Hamiltonian systems, for example, A-stability and A(α) π π and A-stability stability, α ≤ . Hence, usual A(α) stability is useless for α ≤ 2 2 needs to be modified. Here, we introduce a new test system and a new concept-Hstability (Hamiltonian stability) for symplectic algorithms and discuss the H-stability of symplectic invariant algorithms and the H-stability intervals of explicit symplectic algorithms. For the linear Hamiltonian system dz = Lz, dt

L = JA ∈ sp(2n),

H = (z, Az), AT = A,

(6.1)

a linear symplectic algorithm t z k+1 = gH (z k ) = G(s, A)z k ,

k0

(6.2)

∀k > 0,

(6.3)

is stable, if ∃ C > 0, such that z k  = Gk (s, A)z 0   Cz 0 ,

where •  is a well-defined norm, such as Euclidean norm. Evidently, it is equivalent to Gk (s) bounded, or the eigenvalues of G(s) are in the unit disk and its elementary divisors corresponding to the eigenvalues on the unit circle are linear. Since G(s) is symplectic, then (6.4) G−1 (s) = J −1 G(s)T J. Hence, if λ is an eigenvalue of G(s), so is λ−1 , and they have the same elementary divisors. Therefore, the eigenvalue with the module less than 1 is always accompanied with the eigenvalue with the module great than 1. This implies that the linear symplectic method (6.1) cannot be asymptotically stable. We have: Theorem 6.1. Linear symplectic method (6.1) is stable iff the eigenvalues of G(s) are unimodular and their elementary divisors are linear [Wan94] . Here, we introduce the test Hamiltonian system dz = αJz, dt with H(z) = H(p, q) =

α ∈ R,

α α T z z = (p2 + q 2 ), 2 2

(6.5)

A = αI.

402

8. Composition Scheme

Definition 6.2. A symplectic difference method is H-stable at μ = αs, if it is stable for the test Hamiltonian system (6.2) with the given μ, such μ is called a stable point. The maximum interval in which every point is stable and which contains the original point is called the H-stability interval of the method. A symplectic difference method is H-stable if its H-stability interval is the whole real axis. In this case, its numerical solutions are bounded for (6.2) with α ∈ R. Remark 6.3. It is reasonable to choose (6.5) as the model equation because any linear Hamiltonian system may turn into the standard form 1 αi (p2i + qi2 ). 2 n

H(p, q) =

i=1

Test Equations (6.2) and (6.1) become z k+1 = G(μ)z k ,

(6.6)

where G(μ) is 2 × 2 symplectic matrix. If 5 G(μ) =

a1

a2

a3

a4

6 ,

then det G(μ) = a1 a4 − a2 a3 = 1. Its characteristic polynomial is    a1 − λ a2   |G(μ) − λI| =   = λ2 − (a1 + a4 )λ + 1.  a3 a4 − λ  So, its eigenvalues are E a1 + a4 λ± = ± 2

%

a1 + a4 2

&2 − 1.

(6.7)

Lemma 6.4. Scheme (6.6) is stable at μ = 0, iff 

a1 + a4 2

2

< 1,

i.e., −1 <

a1 + a4 < 1. 2

(6.8)

Example 6.5. Applying the centered Euler scheme to the test system (6.5), it becomes 1 2

z + z), z = z + μJ(

μ = αs,

 −1   1 1 z = I + μJ I − μJ z, 2

where

2

8.6 H-Stability of Hamiltonian System

⎡

−μ

μ

1 1 − μ2 4

⎣ 1 1 + μ2 4

G(μ) =

therefore %

⎢

1

1 4

1 − μ2

a1 + a4 2

&2

1 ⎞ 4 ⎠ < 1, =⎝ 1 1 + μ2 4

⎛

1 − μ2

403

⎤ ⎥ ⎦,

(6.9)

∀μ = 0.

By Lemma 6.4, we know that the centered Euler scheme to all μ = 0 is stable, certainly it is also stable for μ = 0, therefore, the centered Euler scheme is H-stable. For the stability region of certain explicit scheme, see the literature [Wan94,QZ90] . In Section 8.2, we have constructed schemes of difference from 1st order to 4th order. We will now discusses its stability by applying these schemes to the model Equation (6.5), we get z k+1 = Gi (μ)z k ,

μ = αs,

i = 1, 2, 3, 4

Gi is the step transition equation. * + 1 −μ , G1 (μ) = μ 1 − μ2 ⎛

1

1 − μ2 2 ⎜ G2 (μ) = ⎝   1 μ 1 − μ2 4

⎛

1

−μ 1 2

1 − μ2 1

1 − μ2 + μ4 2 72 ⎜ G3 (μ) = ⎝   1 2 1 μ 1 − μ + μ4 6

* G4 (μ) =

a1

a2

a3

a4

1 2

a1 = 1 − μ2 +

72

⎞ ⎟ ⎠,  1 −μ 1 − μ2 + 6

7 μ4 1728

 ⎞

1 5 7 μ6 1 − μ2 + μ4 − 2 72 1728

⎟ ⎠,

+ ,

1 4 1 μ + (1 + β)2 μ6 , 24 144

  1 1 a2 = −μ 1 − μ2 − (2 + β)(1 + 2β)2 μ4 , 6

216

  1 1 1 (2 + β)(1 − β)μ4 + (2 + β)(1 + β 2 )μ6 , a3 = μ 1 − μ2 − 6

1 2

a4 = 1 − μ2 +

216

1 4 1 μ + (1 + β)2 μ6 . 24 144

864

404

8. Composition Scheme

Theorem 6.6. From the explicit scheme above, the H-stability intervals are (−2, 2), (−2, 2), (−2.507, 2.507) and (−1.573, 1.573). Proof. Proof of this theorem can be found in paper of Daoliu Wang[Wan94] and paper  of Mengzhao Qin and Meiqing Zhang[QZ90] .

Bibliography

[Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Berlin, Second edition, (1989). [Dah75] G. Dahlquist: Error analysis for a class of methods for stiff nonlinear initial value problems. In G.A. Watson, editor, Lecture Notes in Mathematics, Vol. 506, Numerical Analysis, Dundee, pages 60–74. Springer, Berlin, (1975). [DF76] A. J. Dragt and J. M. Finn: Lie series and invariant functions for analytic symplectic maps. J. of Math. Phys., 17:2215–2227, (1976). [DF83] A.J. Dragt and E. Forest: Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods. J. of Math. Phys., 24(12):2734–2744, (1983). [Fen85] K. Feng: On difference schemes and symplectic geometry. In K. Feng, editor, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, pages 42–58. Science Press, Beijing, (1985). [Fen86] K. Feng: Difference schemes for Hamiltonian formalism and symplectic geometry. J. Comput. Math., 4:279–289, (1986). [Fen91] K. Feng: The Hamiltonian Way for Computing Hamiltonian Dynamics. In R. Spigler, editor, Applied and industrial Mathmatics, pages 17–35. Kluwer, The Netherlands, (1991). [Fen92] K. Feng: Formal power series and numerical methods for differential equations. In T. Chan and Z.C. Shi, editors, International conf. on scientific computation, pages 28–35. World Scientific, Singapore, (1992). [For92] E. Forest: Sixth-order Lie group integrators. J. of Comp. Phys., 99:209–213, 1992. [For06] E. Forest. Geometric integration for particle accelerators. J. Phys. A: Math. Gen., 39:5321–5377, (2006). [FQ91] K. Feng and M.Z. Qin: Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study. Comput. Phys. Comm., 65:173–187, (1991). [FR90] E. Forest and R. D. Ruth: Fourth-order symplectic integration. Physica D, 43:105–117, (1990). [GO77] D. Gottlib and A. Orsag: Numerical Analysis of Spectral Methods, Theory and Application. SIAM, Philadelphia, (1977). [HNW93] E. Hairer, S. P. Nørsett, and G. Wanner: Solving Ordinary Differential Equations I, Nonstiff Problems. Springer-Verlag, Berlin, Second revised edition, (1993). [McL95a] R. I. McLachlan: Comment on “ Poisson schemes for Hamiltonian systems on Poisson manifolds”. Computers Math. Applic., 29:1, (1995). [McL95b] R. I. McLachlan: Composition methods in the presence of small parameters. BIT, 35:258–268, (1995). [McL95c] R. I. McLachlan: On the numerical integration of ODE’s by symmetric composition methods. SIAM J. Numer. Anal., 16:151–168, (1995). [McL95d] R. I. McLachlan: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput., 16:151–168, (1995). [MSS99] A. Murua and J. M. Sanz-Serna: Order conditions for numerical integrators obtained by composing simpler integrators. Phil. Trans. Royal Soc. A, 357:1079–1100, (1999).

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[MSSS97] A. Murua, J. M. Sanz-Serna, and R. D. Skeel: Order conditions for numerical integrators obtained by composing simpler methods. Technical Report 1997/7, Departemento de Matem´atica Aplicada y Computati´on, Universidad de Valladolid, Spain, (1997). [Mur97] A. Murua: On order conditions for partitioned symplectic methods. SIAM J. Numer. Anal., 34:2204–2211, (1997). [Mur99] A. Murua: Formal series and numerical integrators, part I: Systems of ODEs and symplectic integrators. Appl. Numer. Math., 29:221–251, (1999). [Mur06] A. Murua: The Hopf algebra of rooted trees, free Lie argebra,and Lie series. Foundations of Computational Mathematics, 6(4):387–426, (2006). [Ner87] F. Neri: Lie algebras and canonical integration. University of Maryland Tech. report, (1987). [QWZ91] M. Z. Qin, D. L. Wang, and M. Q. Zhang: Explicit symplectic difference schemes for separable Hamiltonian systems. J. Comput. Math., 9(3):211–221, (1991). [QZ90] M. Z. Qin and M. Q. Zhang: Multi-stage symplectic schemes of two kinds of Hamiltonian systems for wave equations. Computers Math. Applic., 19:51–62, (1990). [QZ90a] M. Z. Qin and M. Q. Zhang: Explicit Runge–Kutta–like schemes to solve certain quantum operator equations of motion. J. Stat. Phys., 60(5/6):839–843, (1990). [QZ92] M. Z. Qin and W. J. Zhu: Construction of higher order symplectic schemes by composition. Computing, 47:309–321, (1992). [QZ93] M. Z. Qin and W. J. Zhu: A note on stability of three stage difference schemes for ODEs. Computers Math. Applic., 25:35–44, (1993). [QZZ95] M. Z. Qin, W. J. Zhu, and M. Q. Zhang: Construction of symplectic of a three stage difference scheme for ODEs. J. Comput. Math., 13:206–210, (1995). [Rut83] R. Ruth: A canonical integration technique. IEEE Trans. Nucl. Sci., 30:26–69, (1983). [Ste84] S. Steinberg: Lie series and nonlinear ordinary equations. J. of Math. Anal. and Appl., 101:39–63, (1984). [Str68] G. Strang: On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 5:506–517, (1968). [Suz77] M. Suzuki: On the convergence of exponential operators the zassenhuas formula, BCH formula and systematic approximants. Communications in Mathematical Physics, 57:193–200, (1977). [Suz90] M. Suzuki: Fractal decomposition of exponential operators with applications to manybody theories and Monte Carlo simulations. Physics Letters A, 146:319–323, (1990). [Suz92] M. Suzuki: General theory of higher-order decomposition of exponential operators and symplectic integrators. Physics Letters A, 165:387–395, (1992). [Wan94] D. L. Wang: Some acpects of Hamiltonian systems and symplectic defference methods. Physica D, 73:1–16, (1994). [Wru96] O. Wrubel: Qin-Kompositionen mit Lie-Reihen. Diplomarbeit Uni Karlsruhe (TH), (1996). [Yos90] H. Yoshida: Construction of higher order symplectic integrators. Physics Letters A, 150:262–268, (1990). [ZQ93] W. Zhu and M. Qin: Applicatin of higer order self-adjoint schemes of PDEs. Computers Math. Applic., 26(3):15–26, 1993.

Chapter 9. Formal Power Series and B-Series

We study vector fields, their associated dynamical systems and phase flows together with their algorithmic approximations in RN from the formal power series approach [Fen93a,Fen92] .

9.1 Notation Our considerations will be local in both space and time, all related objects are C∞ smooth. We use coordinate description and matrix notation, the coordinate vectors in RN and vector functions a : RN → RN are denoted by column matrices. The identity vector function 1N is given by 1N (x) = x. For vector function a = (a1 , · · · , aN )T : RN → RN ,   ∂ ai = Jacobian matrix a, a∗ : = ∂ xj

a∗ : =



ai

∂ = linear differential operator of first order associated to a, ∂xi

the association a → a∗ is linear, a∗ operates on scalar functions φ : RN → R a∗ φ =



ai

∂φ , ∂xi

and on vector functions b : RN → RN as a∗ b = a∗ (b1 , · · · , bN )T = (a∗ b1 , · · · , a∗ bN )T = b∗ a,

a∗ 1N = a.

Multiple applications of linear differential operators are naturally defined, such as a∗ b∗ , (a∗ b∗ )c∗ , a∗ (b∗ c∗ ), etc. The operations are multilinear, associative but noncommutative; thus, powers can be defined as a∗k = a∗ a∗ · · · a∗ (k time),

ak := a∗k 1N ,

the identity operator I operates on scalar and vector functions φ and b as Iφ = φ, Ib = b. K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

408

9. Formal Power Series and B-Series

We identify all vector functions a : RN → RN as vector fields. All vector fields in RN form a (∞-dimensional) real Lie bracket under Lie bracket [a, b] := a∗ b − b∗ a = b∗ a − a∗ b = (b∗ a∗ − a∗ b∗ )1N . The Lie algebra VN is associated with the (∞-dimensional) local Lie group DN of near-identity diffeomorphisms—or simply near-1 maps—of RN . Consider the dynamical system in RN dx = a(x), dt

(1.1)

defined by a vector field a. It possesses a phase flow eta = et , which is a one-parameter (in t) group of near-1 maps of RN , e0 = 1N ,

et+s = et ◦ es ,

and generates the solution by x(0) → eta x(0) = x(t). The phase flow is expressible as a convergent power series in t: eta = 1N +

∞ 

tk e k ,

k=1

e 0 = 1N ,

ek =

1 ∗ 1 1 a ek−1 = (a∗ )k 1N = ak . k k! k!

We define Exp ta∗ := I +

∞  1 (ta)∗k , k!

I is the identity operator.

k=1

This is an operator power series operating on scalar and vector functions, and defined by ∗

exp ta := (Exp ta )1N

∞ ∞ k   1 t k ∗ k = 1N + (ta ) 1N = 1N + a , k! k! k=1

then

eta = (Exp ta∗ )1N = exp ta,

for scalar function

(1.2)

k=1

(1.3)

φ ◦ eta = φ ◦ exp ta = (Exp ta∗ )φ,

for vector function b ◦ eta = b ◦ exp ta = (Exp ta∗ )b = (Exp ta∗ )b∗ 1N . Each numerical algorithm solving the system (1.1) possesses the step transition map fas which is one-parameter (in step-size s) family (in general not a one-parameter group in s) of near-1 maps on RN , expressible as a convergent power series in s

9.2 Near-0 and Near-1 Formal Power Series

fas = 1N +

∞ 

sk fk ,

409

(1.4)

k=1

the coefficients can be determined recursively form the defining difference equation. The transition generates the numerical solution x(0) → (fas )N x(0) ≈ x(N s) by iterations with step-size s chosen fixed in general. The main problem is to construct and analyze the algorithmic approximations  fas ≈ eta t=s = esa in a proper way. For this purpose, we propose a unified framework based on the apparatus of formal power series, Lie algebra of vector fields, and the corresponding Lie group of diffeomorphisms [Lie88,Olv93] .

9.2 Near-0 and Near-1 Formal Power Series ∞ 

Among the formal power series

sk ak , ak : RN → RN , we pick out two special

0

classes. The first class consists of those with a0 = 0, called near-0 formal vector fields and the second class consists of those with a0 = 1N , called near-1 formal maps (diffeomorphisms). ∞  s sk ak form a (∞-dim.) real Lie algebra All near-0 formal vector fields a = 1

FVN under the Lie bracket 5∞ 6 ∞ ∞     s s k k [a , b ] = s ak , s bk := sk [ai , bj ]. k=1

k=1

k=2

i+j=k

The associated near-0 formal differential operators and their products are * s

(a )∗ : = * as∗ :=

∞ 

+ k

s ak

k=1 ∞ 

as∗ bs∗ : =

k=2

∗

+∗

sk ak

k=1 ∞ 

:=

∞ 

sk ak∗ ,

k=1

:=

∞ 

sk a∗k ,

k=1

sk



a∗i b∗j ,

(as∗ )2 : = as∗ as∗ , etc.

i+j=k

For any vector function a = (a1 , · · · , aN )T : RN → RN and any near-1 formal ∞  map g s = 1 + sk gk , we define the composition, k=1

410

9. Formal Power Series and B-Series

(a ◦ g s )(x) = a(gs (x)) = a(x) +

∞ 

sk (a ◦ g)k (x),

k=1

(a ◦ g)k =

k 



m=1 k1 +···+km =k

1 (D m a)(gk1 , · · · , gkm ), m!

where Dm a = (Dm a1 , · · · , Dm aN )T , N 

Dm ai (v1 , · · · , vm ) =

j1 ,···,jm

∂ m ai v1j · · · vmjm , ∂xj1 · · · ∂xjm 1 =1

is the usual m-th differential multi-linear form for m tangent vectors vi = (vi1 , · · ·, viN )T (i = 1, · · · , m) at point x ∈ RN , which is invariant under permutation of vectors. Using the identities, (D 1 a)(b) = b∗ a, (D2 a)(b, c) = (c∗ b∗ − (c∗ b)∗ )a, (D3 a)(b, b, b) = (b∗3 + 2b3∗ − 3b∗ b2∗ )a. We get in particular (a ◦ g)1 = g1∗ a, 1 2

(a ◦ g)2 = g2∗ a + (g1∗2 − g12∗ )a, (a ◦ g)3 = g3∗ a + ((g2∗ g1∗ − (g2∗ g1 )∗ )a +

1 ∗3 (g + 2g13∗ − 3g1∗ g12∗ )a. 3! 1

For any two near-1 formal maps s

f = 1N +

∞ 

k

s fk ,

s

g = 1N +

k=1

∞ 

sk gk ,

k=1

the composition f ◦ g is defined in a term by term way: s

s

(f s ◦ g s )(x) = f s (g s (x)) = 1N (g s (x)) +

∞ 

sk fk (g s (x))

k=1

=: 1N (x) + (f ◦ g)1 = f1 + g1 ,

∞ 

sk (f ◦ g)k (x),

k=1

(f ◦ g)k = fk + gk + δ(f1 , · · · , fk−1 ; g1 , · · · , gk−1 ), δ(f1 , · · · , fk−1 ; g1 , · · · , gk−1 ) =

k−1 

i 



i=1 m=1 i1 +···+im

k ≥ 2,

1 (D m fk−i )(gi1 , · · · , gim ). m! =i

9.2 Near-0 and Near-1 Formal Power Series

411

In particular we get, (f ◦ g)2 = f2 + g2 + g1∗ f1 , 1 2

(f ◦ g)3 = f3 + g3 + g1∗ f2 + g2∗ f1 + (g1∗2 − g12∗ )f1 , 1 2

(f ◦ g)4 = f4 + g4 + g1∗ f3 + g2∗ f2 + g3∗ f1 + (g1∗2 − g12∗ )f2 +(g2∗ g1∗ − (g2∗ g1 )∗ )f1 +

1 ∗3 (g + 2g13∗ − 3g1∗ g12∗ )f1 . 3! 1

Under this composition rule, all near-1 formal maps f s = 1N +

∞ 

sk fk

k=1

form a (∞-dim) formal Lie group FDN . In group FDN , inverse elements, square roots, rational powers, etc., always exist, and their coefficients can always be determined recursively by the defining composition relations. For example, the inverse ∞  sk hk = hs is defined by (f s ◦ hs ) = 1N , hence (f s )−1 := 1 + k=1

f1 + h1 = 0,

fk + hk + δ(f1 , · · · , fk−1 ; h1 , · · · , hk−1 ) = 0,

k ≥ 2.

In particular, h1 = −f1 ,

h2 = −f2 + f12 , 1 2

1 2

h3 = −f3 + f1∗ f2 + (f2∗ − f12∗ )f1 − f13 + f12∗ f1 . There is an obvious one-one correspondence between the Lie algebra FVN and the Lie group FDN , established simply by +1N and −1N . However, the more significant one-one correspondence between them is given by exp and its inverse log. exp : FVN → FDN , ∞ ∞   1 s∗ m as = (a ) 1N sk ak =⇒ exp as =: 1N + m! m=1 k=1

=: 1N +

∞ 

sk fk = f s .

(2.1)

k=1

Note that s∗ m

(a )

=

*

∞  k1 =1

+ sk1 a∗k1

* ···

∞  km =1

+ skm a∗km

=

∞  k1 ,···,km =1

sk1 +···+km a∗k1 · · · a∗km ,

412

9. Formal Power Series and B-Series

so we get easily fk =

k  1 m! m=1

f k = ak +



a∗k1 · · · a∗km 1N ,

k ≥ 1, f1 = a1 ,

k1 +···+km =k

k  1 m! m=2

 k1 +···+km =k

1 a∗k1 · · · a∗km 1N , k ≥ 2, f2 = a2 + a21 . (2.2) 2

Note that (2.2) provides a 2-way recursion formula from a1 , · · · , ak to f1 , · · · , fk and vice versa. Therefore, exp maps FVN one-one onto FDN and its inverse, i.e., log is defined by the same (2.2): log = (exp)−1 : FDN −→ FVN , log exp as = as ,

exp log f s = f s .

In particular, a1 = f1 ,

1 2

1 2

a2 = f2 − a21 ,

a3 = f3 − (a∗1 a2 + a∗2 a1 ) −

1 2

a4 = f4 − (a∗1 a3 + a22 + a∗3 a1 ) − ak = fk −

k−1  m=2

1 m!



1 3 a , 3! 1

1 ∗ ∗ 1 (a a a2 + a∗1 a∗2 a1 + a∗2 a∗1 a1 ) − a41 , 3! 1 1 4!

a∗k1 · · · a∗km 1N −

k1 +···+km =k

1 k a , k! 1

k ≥ 3.

An equivalent construction of log f s = as is ∞  (−1)m−1 s hm , log f = m m=1 s

where hs1 = f s − 1N ,

hsm = hsm−1 ◦ f s − hsm−1 .

It is easy to compute hs1 =

∞ 

sk fk =

hs2 =

sk1 (1N ◦ f )k1 ,

k1 =1

k=1 ∞ 

∞ 

sk1 +k2 ((1N ◦ f )k1 ◦ f )k2 ,

k1 ,k2 =1

hs3 =

∞  k1 ,k2 ,k3 =1

···.

sk1 +k2 +k3 (((1N ◦ f )k1 ◦ f )k2 ◦ f )k3 ,

(2.3)

9.2 Near-0 and Near-1 Formal Power Series

Substituting in (2.3) and equating with

∞ 

413

sk ak , we get

k=1

ak =

k  (−1)m−1 m m=1



(· · · ((1N ◦ f )k1 ◦ f )k2 · · · ◦ f )km .

(2.4)

k1 +···+km =k

It is easy to verify log exp as = as for this log, so this is precisely the inverse of exp, thus agreeing with the previous one. We use the above construction (2.4) to establish the formal Baker–Campbell– Hausdorff formula[Bak05,Hau06] . For arbitrary near-1 formal maps f s , g s , log (f s ◦ g s ) = log f s + log g s +

∞ 

dk (log f s , log g s ),

k=1

where log f s = as , log g s = bs , then[Dyn46] dk (as , bs ) =

k 1  (−1)m−1 k m m=1

 p1 +q1 +···+pm +qm =k pi +qi ≥1,pi ≥0,qi ≥0

[(as )p1 (bs )q1 · · · (as )pm (bs )qm ] , p1 !q1 ! · · · pm !qm !

where (x)p = xx · · · x (p times),

[x1 x2 x3 · · · xn ] = [[· · · [[x1 , x2 ], x3 ], · · ·], xn ].

In particular, d1 =

1 s s [a , b ], 2

d2 =

Let log (f s ◦ g s ) = cs =

∞ 

 1 s s s [a b b ] + [bs as as ] , 12

d3 = −

1 s s s s [a b b a ]. 24

sk ck , then

k=1

c1 = a1 + b1 ,

1 2

c2 = a2 + b2 + [a1 b1 ], 1 2

c3 = a3 + b3 + ([a1 b2 ] + [a2 b1 ]) + c4 = a4 + b4 +

1 ([a1 b1 b1 ] + [b1 a1 a1 ]), 12

1 ([a1 b3 ] + [a2 b2 ] + [a3 b1 ]) 12

+

1 ([a1 b1 b2 ] + [a1 b2 b1 ] + [a2 b1 b1 ] + [b1 a1 a2 ] + [b1 a2 a1 ] + [b2 a1 a1 ]) 12

−

1 [a1 b1 b1 a1 ], 24

etc.

Note that the classical BCH formula is restricted to the composition of two oneparameter groups, where log f s = sa1 and log g s = sb1 . The log transform reduces matters at the Lie group level to those at the easier level of Lie algebra. All properties of near-1 formal maps have their logarithmic interpretations.

414

9. Formal Power Series and B-Series

Proposition 2.1. 1◦ 2◦ 3◦

[Fen93a,Fen92,Fen93b]

We list some of them, let log f s = as =

∞ C

sk ak :

k=1

5◦ 6◦

f s is a phase flow, i.e., f s+t = f s ◦ f t ⇔ log f s = sa1 . f s is revertible, i.e., f s ◦ f −s = 1N ⇔ log f s is odd in s. f s raised to real μ-th power (f s )μ ⇔ log (f s )μ = μ log f s . In particular, √ 1 log (f s )−1 = − log f s , log f s = log f s . 2 f s scaled to f αs ⇔ log (f αs ) = (log f )αs . In particular, log (f −s ) = (log f )−s . f s − g s = O(sp+1 ) ⇔ log f s − log g s = O(sp+1 ). f s ◦g s = g s ◦f s ⇔ [log f s , log g s ] = 0 ⇔ log (f s ◦g s ) = log f s +log g s .

7◦

(f s ◦g s ) = hs ⇔ log hs = log (f s ◦g s ) = log f s +log g s +

4◦

∞ C

dk (log f s , log g s ).

k=1

8◦ f s symplectic ⇔ all ak are Hamiltonian fields (see Chapter 5). 9◦ f s contact ⇔ all ak are contact fields (see Chapter 11). 10◦ f s volume-preserving ⇔ all ak are source-free fields (see Chapter 10). The log transform has important bearing on dynamical systems with Lie algebra structure. The structure-preserving property of maps f s at the Lie group (G ⊂ Dm ) level can be characterized through their logarithms at the associated Lie algebra (L ⊂ Vm ) level.

9.3 Algorithmic Approximations to Phase Flows 9.3.1 Approximations of Phase Flows and Numerical Method We return to the main problem of approximation to the phase flow for dynamical system

dx = a(x). dt

fas = f s = 1N +

∞  k=1

sk fk ≈ esa = 1N +

∞ 

sk ek ,

k=1

ak ek = . k!

If fk = ek (1 ≤ k ≤ p), we say fas is accurate to order ≥ p, if moreover, fp+1 = ep+1 , we say it is accurate to  order p. sk ak . Note that the first p + 1 equations in (2.2) completely Let log f s = as = determine a1 , a2 , · · ·, ap+1 and f1 , f2 , · · ·, fp+1 each other. It is then easy to establish fk = ek , 1 ≤ k ≤ p; fp+1 = ep+1 ⇐⇒ a = a1 = e1 ; ak = 0, 1 < k ≤ p; ap+1 = fp+1 − ep+1 = 0. So, the orders of approximation for fas ≈ esa and for log fas − sa are the same. Moreover, note that we have a formal field

(3.1)

9.3 Algorithmic Approximations to Phase Flows

s−1 log f s = s−1 as = a +

∞ 

415

sk+1 ak+1 = a + O(sp ),

k=1

which is the original field a up to a near-0 perturbation and defines a formal dynamical system ∞  dx = (s−1 log f s )(x) = a(x) + sk+1 ak+1 (x) dt

k=1

having a formal phase flow (in two parameters t and s with group property in t) ets−1 as = exp ts−1 as whose diagonal formal flow ets−1 as |t=s is exactly f s . This means that any compatible algorithm fas of order p gives perturbed solution of a right equation with field a; however, it gives the right solution of a perturbed equation with field s−1 log fas = a + O(sp ). There could be many methods with the same formal order of accuracy but with quite different qualitative behavior. The problem is to choose among them those leading to allowable perturbations in the equation. For systems with geometric structure, the 8◦ , 9◦ , 10◦ of Proposition 2.1 provide guidelines for a proper choice. The structure-preservation requirement for the algorithms precludes all unallowable perturbations alien to the pertinent type of dynamics. Take, for example, Hamiltonian systems. A transition map fas for Hamiltonian field a is symplectic if and only if all fields ak are Hamiltonian, i.e., the induced perturbations in the equation are Hamiltonian. So symplectic algorithms are clean, inherently free from all kinds of perturbations alien to Hamiltonian dynamics (such as artificial dissipation inherent in the vast majority of conventional methods), this accounts for their superior performance. The situations are the same for contact and volume-preserving algorithms . The Proposition 2.1 profound impact on later developed called “Backward error series” work, “Modified equation” and “Modified integrator”[Hai94,CHV05,CHV07] .

9.3.2 Typical Algorithm and Step Transition Map Finally we give, as an illustration, four simplest methods together with step transition maps and their logarithms. 1 2

esa = 1N + sa + s2 a2 + (1)

1 3 3 s a + O(s4 ). 3!

Explicit Euler method (E): x1 − x0 = sa(x0 ), f s − 1N = sa, fEs = 1N + sa, log fEs = sa −

non-revertible, order = 1. (2) Implicit Euler method (I):

s2 2 a + O(s3 ), 2

416

9. Formal Power Series and B-Series

x1 − x0 = sa(x1 ), f s − 1N = sa ◦ f s , fIs = (1N − sa)−1 = (fE−s )−1 = 1 + sa + s2 a2 + O(s3 ), log fIs = sa +

s2 2 a + O(s3 ), 2

non-revertible, order = 1. (3) Trapezoidal method (T ): s 2 s f s − 1N = (a ◦ f s + a), 2     s s s −1 s fTs = 1N − a ◦ 1N + a = fI2 ◦ fE2 2 2

x1 − x0 = (a(x1 ) + a(x0 )),

s

s

= (fE2 )−1 ◦ fCs ◦ fE2 = 1N + sa +

s2 2 s3 a + a3 + O(s4 ), 2 4

s3 3 a + O(s5 ), 12

log fTs = sa +

revertible, order = 2, symplectic for linear Hamiltonian but non-symplectic for nonlinear Hamiltonian systems, where fCs denoting following centered Euler scheme. (4) Centered Euler method (C):   1 x1 − x0 = sa (x1 + x0 ) , 2   1 s (f + 1N ) , f s − 1N = sa ◦ 2

2-stage version recommended for implementation: s 2

x1 = 2¯ x − x0 ,

x ¯ = x + a(¯ x), s

s

x ¯ = fI2 (x0 ), s

fCs = 2fI2 − 1N

x1 = 2fI2 (x0 ) − 1N (x0 ),     s s s s −1 = 1N + a ◦ 1N − a = fE2 ◦ fI2 2

2

= 1N + sa + log fCs = sa + s3

2

3

s 2 s a + (a∗ a2 + a3 ) + O(s4 ), 2 8





1 1 a∗ a2 − a3 + O(s5 ), 8 24

revertible, order = 2, unconditionally symplectic with preservation of all quadratic invariants for Hamiltonian systems. Note the similarities and delicate differences between C and T : Both can be coms s posed by a implicit and a explicit stages but in opposite orderings. Moreover, they 2

2

9.4 Related B-Series Works

417

are conjugate to each other. C is far less known than T , it becomes prominent only after the recent development of symplectic algorithms [Fen85] . In crucial aspects, C is superior. Remark 3.1. The above log fCs is not others but just formal vector fields for centered Euler scheme or present called backward error analysis s2 1 f¯ = f + (f  f  f − f  (f, f )). 12

2

9.4 Related B-Series Works Consider the numerical solution of ODEs z˙ = f (z),

z ∈ Rn .

(4.1)

B-series methods: B-series were introduced by Harier and Wanner[HW74] . The Taylor series of exact solution of (4.1) with initial value z(0) = z can be written as z(h) = z + hf (z) +

 h2  h3   f (f (z), f (z)) + f  (z)f  (z)f (z) + · · · . f (z)f (z) + 2! 3!

(4.2) B-series methods are numerical integrators zn+1 = Φh (zn ) whose Taylor series have the same structure with real coefficients a(τ ): Φh (z) = z + ha( )f (z) + h2 a( )f  (z)f (z) + h3  +a( )f  (z)f  (z)f (z) + · · · ,

 a( ) 2!

f  (f (z), f (z)) (4.3)

where coefficients a(τ ) are defined for all rooted trees and characterize the integrator. Every numerical integrator (including R–K method) can be expanded into a Bseries as introduced and studied in[HW74] . Definition 4.1 (rooted tree and forest). The set of rooted tree T and forest F are defined recursively by , only one vertex belong to T ; 1◦ The tree 2◦ If τ1 , · · · , τn are n tree of τ , the forest u = τ1 , · · · , τn is the commutative product of τ1 , · · · , τn ; 3◦ If u is a forest of F, then u = |τ | is a tree of T . Let T = { , , , · · ·} be the set of rooted trees and let ∅ be the empty tree. For τ1 , · · · , τn ∈ T , we denote by τ = [τ1 , · · · , τn ] the tree obtained by grafting the roots of τ1 , · · · , τn to a new vertex which becomes the root of τ . Elementary differentials Ff (τ ) are defined by induction as   Ff ( )(z) = f (z), Ff (τ )(z) = f (m) (z) Ff (τ1 ), · · · , Ff (τm )(z) . (4.4) For real coefficients a(∅) and a(τ ), τ ∈ T a B-series is a series of the form

418

9. Formal Power Series and B-Series

B(f, a, z) = a(∅)Id +

 h|τ | a(τ ) Ff (τ )(z) σ(τ )

(4.5)

τ ∈T

= a(∅)Id + ha( )f + h2 a( )f  f + h3 a( )/2f  (f, f ) + · · · , (4.6) where Id stands for the identity; Id(z) = z and the scalars a(τ ) are the known normalization coefficients[BSS96] . Now, we give following examples: Example 4.2. The Taylor series of the exact solution of (4.1) can be written as a B1 series z(h) = B(f, e)(z0 ) with coefficients a(τ ) = e(τ ) = ,∀τ ∈ T. γ(τ )

Example 4.3. The coefficient B-series for explicit Euler scheme a(τ ) = 0, ∀ τ ∈ T except a( ) = 1. Example 4.4. The coefficient B-series for implicit Euler scheme a(τ ) = 1, ∀ τ ∈ T .  |τ |−1 1 Example 4.5. The coefficient B-series for centered Euler scheme a(τ ) = , 2 ∀τ ∈ T. Example 4.6. The coefficient B-series for trapezoidal scheme a( ) = 1, a( ) = 1 , a( 2

1 2

1 4

) = , a( ) = , · · · .

Example 4.7. The coefficient B-series for R–K method (A, b, c), a(τ ) = bT φ(τ ), ∀ τ ∈ T. Partitions and skeletons: A partition pτ of a tree τ is obtained by cutting some of the edges [CHV07] . The resulting list of trees is denoted by P (pτ ). Eventually, the set of all partitions pτ of τ is denoted by P (pτ ). Now, given a partition pτ , the corresponding skeleton χ(pτ ), as introduced in [CHV07] , is the tree obtained by contracting each tree of P (pτ ) to a single vertex and by re-establishing the cut edges (see Tables 4.1 – 4.25). We observe that a tree τ ∈ T has exactly 2|τ |−1 partitions pτ and that different partitions may lead to the same P (pτ ). An admissible partition is a partition with at most one cut along any part from the root to any terminal vertex. We denote AP τ as the set of admissible partition of τ and by convention, we suppose that ∅ ∈ AP τ . We denote #(pτ ) as number of subtrees. We denote this distinguished tree by R(pτ )(or rp ). We denote P ∗ (pτ ) = P (pτ ) \ R(pτ ) as the list of forest that do not contain the root of τ . We distinguish rp as the tree vp (or P (pτ )) whose root coincides with the root of τ . This tree is usually referred to as a subtree of τ and we denoted by vp∗ (or P ∗ (pτ )) the forest obtained by removing rp from vp . The above definition can be seen in Tables 4.1 – 4.25.

9.4.1 The Composition Laws The following theorem result on the composition of B-series was obtained by[HW74] . Now we formulate this theorem in the form [CHV07] :

9.4 Related B-Series Works

419

Theorem 4.8. Let a, b : T ∪{∅} → R be two mappings, with a(∅) = 1. Then B-series B(f, a)(z) inserted into B(f, b)(·) is still a B-series   B(f, b) B(f, a)(z) = B(f, a · b)(z),

(4.7)

and a · b : T ∪ {∅} → R is defined by

(a · b) = b(∅) = b(∅),

∀ τ ∈ T,

(a · b) =



b(rp )a(vp∗ ),

(4.8)

p∈AP(τ )

where a is extended to F, as follows:

∀u = τ1 · · · τn ∈ F,

a(u) =

n ;

a(τi ).

i=1

Table 4.1.

The partitions of a tree of order 2 with associated skeleton and forest •

p

τ

· •·

•

' τ

•

•

χ(pτ )

•

/  •

P (p )

•

R(pτ )

•

•  ∗

τ

P (p )

∅

·•

••



• 



•

#(pτ )

1

2

pτ ∈ AP τ

yes

yes



(4.9)

420

9. Formal Power Series and B-Series

Table 4.2.

The partitions of a tree of order 3 with associated skeleton and forest •

• p

τ

•

'• τ



•• •

τ

P (p )

∅

· · ·•·

• • 

•••

•

•

• 

•





•



••

#(pτ )

1

2

2

3

pτ ∈ AP τ

yes

yes

yes

yes

Table 4.3.

·•

•

••

• 

•·

• 

• 

·•

•

• 

•

R(p )

· •·

•

•/ 

• τ

•

•

•

P (p )

• ·• •

•

χ(pτ )

∗

•· ·



The partitions of a tree of order 3 with associated skeleton and forest • •

pτ

• ·· •·

• • ·· •·

•

•· ·· • ·· •·

•

• •

•

τ

χ(p )

•

•

•

•

•

'• /  •   • •  • • ••• • • •

P (pτ )

• • R(pτ )  ∗

τ

P (p ) #(pτ ) p ∈ AP τ

τ

• ∅

•

• 

'

/ •

•

•

• 

•





••

1

2

2

3

yes

yes

yes

no



9.4 Related B-Series Works

Table 4.4.

The partitions of a tree of order 4 with associated skeleton and forest •

• •

• pτ

•·

•

χ(p )

•

R(p ) 

•· ·• · · ·•·

•

• ·· · • ·• · •· •

•

• ·· •· ·• · ·•

• ·· •· ··• · · ·•· •

•

•

•

•



•   •

• 2 yes

•

•

•

•

• •

•

•

•

•

•

•

•

•

•

•  

• 2 yes

'•

•

•· · •·

•

•

• ·· •· •

•

•

χ(pτ )

P ∗ (pτ )

•

•

  • •

• 2 yes

• •

 

• 3 yes

•• 3 no

•

•  

•• 3 yes



•••



4 no

The partitions of a tree of order 4 with associated skeleton and forest

pτ

R(pτ )

•

1 yes

•

P (p )



∅

τ

τ

•

•

•

•

τ

Table 4.5.

·• · •·

•

• •

•

P (p ) #(pτ ) τ p ∈ AP τ

•

·· •

• ·· ·• •

/' / ' / ' / ' • / ' • / ' /'  • • • • • • • • • • • •• •• •• •••• • • • • • • • • • •

∗

•

•

•

τ

P (pτ )

421

•

•· • ·· • •

• •

• •

•· ·• ·· ·· • • •

• •

•

• ·· •·· · •· • • •

•· • ·· •· · •· • • •

•· ·• ·· ·· •· · •· • • • •

•/ '

/ ' / ' / /' • / ' • / '  • • •  • • •• •• •• •••• • • • • • • • • • • • • • • • • • • • • • • • • • • • /' / /     '    '  • • • •  ∅ •, •, • • • , • •,•, • • • • • •

#(pτ )

1

2

2

2

3

3

3

4

pτ ∈ AP τ

yes

yes

yes

yes

yes

no

no

no

422

9. Formal Power Series and B-Series

Table 4.6.

The partitions of a tree of order 4 with associated skeleton and forest • •

pτ

χ(p ) ' P (pτ )

•

P ∗ (pτ ) #(pτ ) τ p ∈ AP τ

• ·· •· •

•

•· ·· • ·· · • •

•· ·· • •·· ·•

•

•

•

•

•

•

•

•

•

•

•

•· ·· • ·· · •· ··• • • • •

• ·· •·· ··• •

•

•

/ ' • / ' • / ' /' /' /' /  • • • • • • • • • •• •• •• •••• • • • • • • • • •

• 

p

•

•

•

• R(pτ )

•· ·· • • •

•

•· ··•

•

•

τ

Table 4.7.

•

•

•

• •

•

∅



' •

•

• /

•



•   •

•

• 2 yes

1 yes

•

•

  •

2 yes

•

•

2 yes

•

 • •

•• 3 no

•

•

 • •

3 no

 •

3 no

•••



4 no

The partitions of a tree of order 5 with associated skeleton and forest • • • •· • • • •· • • • ·• •· •· • •· • ·• • •· ·• •· •· ·• ·· ·· · · ·· · · ·· ·· · · ·· ·· ·· • • • •· • • •· •

τ

χ(p )

P (pτ )

τ

R(p )

•

•

τ

• •

•

•

• •

•

•

•

•

•

•

•

•

• • • •

' • /' /' /' /' / ' / ' / ' / • • • • • • • • • • • •• •• •• •••• • • • • • • • • • • • • • • '

/

•

•

•

• '

•

•

• /

'

•

•

• /

• •

/ '

'

• •

/'

•

• /'

/'

/

∅

•

•

•

• •

• •

• •

•••

#(pτ )

1

2

2

2

3

3

3

4

pτ ∈ AP τ

yes

yes

yes

yes

yes

yes

yes

yes

P ∗ (pτ )

9.4 Related B-Series Works

Table 4.8.

The partitions of a tree of order 5 with associated skeleton and forest •

p

•

τ

χ(pτ ) '• •

τ

P (p )

• •

τ

R(p )

 P ∗ (pτ ) #(pτ ) τ p ∈ AP τ

Table 4.9. forest

• • • • •

/'

•

•

χ(p )

•

•

• • /'

• • •

•

• • /'

• • •



•

•

•

• ·· •·· ··• •

•

•

•

•

•/'

• /'

• /'

• /'

•/

•

•

•

•

•

•

•

• • •

• 2 yes

2 yes



•

•

' / • •



• • 2 yes

• ·· •· •

•

• ·· •· • •· ··• •

•

• •• •

• •• •

• •• •

•

•

•

• ·· •·· ·· • • • •

•

• • •

•

•

•

•

• ·· •· • ·· •· •

• •· ·· • ·· · • •

•

•

•

•

•

• •

'• •

P ∗ (pτ ) #(pτ ) τ p ∈ AP τ

•

•

•

•

•

•

•

•

•

' •/   ' •/ ' •/ •• •• •• • • • • 2 3 3 3 yes no no no

Continuous partitions of the above tree of order 5 with associated skeleton and

τ

R(p )

• •· ·· • • •

•

1 yes

pτ

τ

• •· ··•

• ·· •· • • •

•

• •

∅

•

•

• •· ·· • •· ··•

P (pτ )

423

•

• /'• • /'• •

•• •

'• • •• 3 no

•

• ·· •·· ·· • ·· · • •

•

•• •

•

3 no

•

•

• •

• •

• •

• ·· •· • ·· •·· ··• • •

• •

• ·· •·· ·· • ·· · •· ··• • • • • •

' / • /' •/ ' •/ ' • / •• •• • ••• ••• ••• ••• • • • •

•

/' • •

•

•

• ·· •·· ·· • •· ··•

/'

••

•

•

• •· ·· • ·· · •· ··•

• •

/' • • 3 no

• /

•

•

•

•

•

' •/ ' •/ ' • /   •• •• •• ••• •••• • • • 4 no

4 no

4 no

4 no

5 no

424

9. Formal Power Series and B-Series

Table 4.10.

p

The partitions of a tree of order 5 with associated skeleton and forest •

τ

• •

• • • • • • • • • • • • • • • •· • • ·· • • ·· • • ·• •· ···· • • ··· ··· • • ·····• ·· · · · ·• • • • •· •· • •

τ

P (p )

τ

•

•

χ(pτ )

•

•

•

•

•

•

•

•

•

•

•

•

•

•

• •

 • •   •   •   •   •   • •  • •  • • • • • • • • • • • • •• •• •• • • • • • • • • • • • • •

• • •

R(p ) 

∅

P ∗ (pτ )

•

• •



•



•

• • 

•



•

• • 

•



•

• • 



•

•

•



•

•

• 

•

•

•  

• •

 



•

•

•

•

••

••

••

#(pτ )

1

2

2

2

2

3

3

3

pτ ∈ AP τ

yes

yes

yes

yes

yes

yes

yes

yes

Table 4.11. forest

p

τ

p

τ

Continuous partitions of the above tree of order 5 with associated skeleton and

• • • • • • • • • • • • • • • • • ·· ·• •· ·· • •· • •· ·· ·· • • ·· ·· ·• •· ·· ·• •· ·· ·• •· ·· ·· ·• · ·· ··· · ·· ·· · ·· · ·· · · · ··· · · ··· · ··· · ·· • • • • • • • • •

P (pτ )

•

•

•

•  • ••

•

•

• •  • •• •

•

•

•

R(pτ ) 

• • • • • • • • • • • • • • • • • •

•

•

•  

• • •

•

•

•  

• •

 

• •



• •



• 

•   ••••

••

••

••

•••

•••

•••

•••

#(pτ )

3

3

3

4

4

4

4

5

pτ ∈ AP τ

yes

yes

yes

yes

yes

yes

yes

yes

P ∗ (pτ )

•

' • / ' • / ' • / ' • /  • • • • • • ••• ••• ••• ••• • • • • •

•  • •• •

•

9.4 Related B-Series Works

Table 4.12.

pτ

χ(pτ )

P (pτ )

R(pτ )

The partitions of a tree of order 5 with associated skeleton and forest • • • • • • • • · • • •·· • • • • • • • • •·· • • • · • ·· ·· · · · · ·• ·• • •· • •· •· •· · ·· · ·· · •· • • • • •· • •· • • • • • • • • • • • • • • • • • • • • • • • • '• • •/ ' /' /' /  • •  • •  • •  •  • • • • • • •• •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 

P ∗ (pτ ) 3 τ p ∈ AP τ

Table 4.13. forest

pτ

χ(pτ )

P (pτ )

  ' • /  • • • • 1 2 2 yes yes yes ∅



 • 2 yes



 '• • • 2 yes

•

3 no

/' • • •

•

3 no

/' • • •

/ • •

4 no

Continuous partitions of the above tree of order 5 with associated skeleton and • •·· • • • · • • •·· · • • •·· · • • •·· · • • • ·• • •·· • • •·· · • ·· · · ·· · · ·· · ·· · · ·· · ·· · ·· ·· · •· · •· · •· ·• ·• · •· •· •· · ·· · · · ·· ·· •· • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • / / / / / / / ' ' ' ' ' ' • • • '   • • • • ••• ••• ••• ••••• ••• ••• ••• ••• • • • • • • • •

•

•

R(pτ )

•

• •

 P ∗ (pτ ) #(pτ ) pτ ∈ AP τ

425

•

• •





•

•

• 



• •

• •

• •

•••

3 yes

3 yes

3 yes

4 yes

' • •• 4 no

•

•

•

/' • /' • /   •• •• •••• • • • 4 no

4 no

5 no

426

9. Formal Power Series and B-Series

Table 4.14.

pτ

χ(pτ )

P (pτ )

R(pτ )

The partitions of a tree of order 5 with associated skeleton and forest

•

•

∅

P (pτ )

R(pτ )

P ∗ (pτ ) #(pτ ) τ p ∈ AP τ

• •· • ··• •

•

•

•

•

• ·· •·

•

•



1 yes

•

•

•

•· • ·· •

• ·· •·· • ··• • •

•

•

•

•· ·• ·· ·· • •

• ·· •· • ·• · •· •

•



•

•



•

•/' • / / ' • /    ' • • • • • • • • • • • 2 2 2 3 3 3 yes yes yes no yes no

'•

2 yes

Continuous partitions of the above tree of order 5 with associated skeleton and • ·· •· •· • ·· • •

• χ(pτ )

• ·· •· • • •

• • • • • • • ' • ' • • / ' ' / ' / ' / / / / ' / ' • • • • • • •• •• • • • •• • • • • • • • • • • •• •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

P ∗ (pτ ) #(pτ ) pτ ∈ AP τ

pτ

•

•

•



Table 4.15. forest

•

•

• •· • ···• · •· • • •

• •· •· ··• ·· • •

• •

• ·· •·· • ···• · •· • •

• ·· •·· •· ··• ·· •

• ·· •· •· ·• ·· ·· •

• •· •· ···• ·· ·· •

• ·· •·· •· ···• ·· ·· •

•

•

•

•

• •

•

•

•

•

•

•

•

•

•

•

•

• ·· •·

' • / ' • • /' • • /' /' /' /' /  • • • • •• • • ••••• • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • •    ' • / ' • /   ' • / ' • /  •••• •• •• • • ••• ••• • • • • • • 3 yes

3 no

3 yes

4 no

4 no

4 no

4 no

5 no

9.4 Related B-Series Works

Table 4.16.

427

The partitions of a tree of order 5 with associated skeleton and forest • •

pτ

• •

• •

•

•· · •·

•· · •·

•

• •

• •

•

• • ·· •·

• • •· • ·· •

•· · •·

•· · •·

• • •· ·• ·· ·· •

•

•

•

• •

•· · •· ·· · •

• •

χ(pτ )

P (pτ )

R(pτ ) P ∗ (pτ ) #(pτ ) τ p ∈ AP τ

Table 4.17. forest

•

•

•

•

•

•

•

•

• •

• •

•/ ' •/ ' • / ' • /' • / ' /' • • / ' • / • • • • • • •• • • • •• • •• • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •           '• • /   • •   • • ∅ • • • • •• • • 1 2 2 2 2 3 3 3 yes yes yes yes yes yes yes no

Continuous partitions of the above tree of order 5 with associated skeleton and

pτ

• •

χ(pτ )

• '

• •· ·· •

•· · •·

•· · •·

• • ·· •·

•

•

•

•

•

•

•· • · •·· ·• ·· ·· • • •

• •

• •· · •· ·•· ·· ·· • •

•· · •·

•· · •· ·· · • •

•

•

•

• •

•

•· · •·· ·· • •

•· · •·

•

• •

•· •· · · •·· ·•· ·· ·· • • •

• • •

• /'  • • / ' • • /' • /' • /' • /' • /  •• • ••• ••• ••• ••• ••••• • • •• •• • • • • • • •

τ

•

R(p )  P ∗ (pτ ) #(pτ ) τ p ∈ AP τ

•

'• •

•· • · •·· • ·· •

P (pτ )

•

•

• •

•

3 yes

3 yes

•

•

•

• •

•

•

/ / '       ' • / ' • / • ' • ••• • • • • • • • • • • • • ••• • • • • • • 3 no

4 no

4 no

4 no

4 no

5 no

428

9. Formal Power Series and B-Series

Table 4.18.

pτ

The partitions of a tree of order 5 with associated skeleton and forest • • • • • • •· •· ·· ·· • • • • • • • • • • • • • · • • · ·• • • • •· • •· · • •· ·· ·· · · · ·· ·· ·· ·· · ·· • • • • •· • • • • •

χ(pτ )

P (pτ )

R(pτ ) P ∗ (pτ )

•

• •

•

•

•

•

•

•

•

• •

•

•

• •

• /' ' • /' • /' /' ' • / /' •• /  •  • • •• • • • • • • •• • •• • • • • • •• •• • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • /  ' • /      '       • • • • ∅ • • • • • • •

#(pτ )

1

2

2

2

2

3

3

3

pτ ∈ AP τ

yes

yes

yes

yes

yes

no

yes

yes

Table 4.19. forest

pτ

Continuous partitions of the above tree of order 5 with associated skeleton and •· •· • • · · • •· ·• •· • ·• • •· ·• •· •· ·• ·· ·· · ·· ·· ·· ··· • •· • •

•· •· •· •· · · · · • •· ··• •· • ··• •· •· ·• •· •· ··• · · ·· · ·· · ·· · · ·· · · · • • • •· •

•

• •

χ(pτ )

• ••

•

•

/' • • ••

•



•

• •

• •

• •

•· • · ·•

• • • • • • • • •

/' /' /' / /' '  •• / • • • •  • • • • • • • • • • • • • • • • • • • • • • • • ••

•

•

R(pτ )

• •

' P (pτ )

•

• • • •

• •



• •

•

•

• •

• •

•

•

 ' • / ' • /       ••• ••• ••• •••• • • •• •

• •

• •

#(pτ )

3

3

3

4

4

4

4

5

pτ ∈ AP τ

yes

yes

yes

yes

no

no

yes

no

P ∗ (pτ )

9.4 Related B-Series Works

Table 4.20.

429

The partitions of a tree of order 5 with associated skeleton and forest •

•

•

•

• •

•

pτ

•

•

χ(pτ )

• •· • • • • • •· ·• • • •· • ·· · · · · • ·•· •· • ·• • ·• •· • •· ·• •· ·• · ·· ·· ·· ·· · • • • • • • • •

•

•

•

•

• •

•

•

•

•

•

•

•

•

•

• /'

•' ' • / • • •• • • •

P (pτ )

•

•

•

•

•

•

R(pτ )  P ∗ (pτ )

•

•

•

•/ • /' •/ ' / '• /' /' • • • • • • • • • • •• • •• •• •• •• • • • • •

• 

∅

•

•

• •

•





 '



•

•

•

•

• •

•

• •

• / 

•

•

•

• 

•



•

' • • • •

/ •  •

•  • •

#(pτ )

1

2

2

2

2

3

3

3

p ∈ AP τ

yes

yes

yes

yes

yes

yes

yes

yes

τ

Table 4.21. forest

Continuous partitions of the above tree of order 5 with associated skeleton and •· • • ·• • ·• •· ·• •· • •· ·• •· ·• • ·• ·· ·· ·· ·· ·· ·· • •·· • ··•·· • •·· • ··• • • • • • • • · · · · · · · · · · · ·· ··· · · ·· ·· ·· ·· ·· ·· • • •· •· •· • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ' / ••  ••    •  •  •  •  •• • ••••• ••• ••• ••• ••• • • •• •• • • • • • •

pτ

χ(pτ )

P (pτ )

• • R(pτ ) ∗

•  • •

τ

P (p ) τ

#(p ) p ∈ AP τ

τ

•

•

•

•  • •

 •

•

•

• •    • •• • • •••

•

•

•

 • ••

•  •

•••

•    ••••

3

3

3

4

4

4

4

5

no

no

yes

no

no

no

no

no

430

9. Formal Power Series and B-Series

Table 4.22.

The partitions of a tree of order 5 with associated skeleton and forest •

• • •

χ(pτ )

•

• •

•

pτ

•

• ·· · • •

•

• •· ·· •

•· • ·· • •

•

•

• • • ' • / ' • •/ ' /' •• • • • • • • •• • • • • • • • • •

P (pτ )

R(pτ )

P ∗ (pτ ) #(pτ ) pτ ∈ AP τ

Table 4.23. forest

P (pτ )

•· • ·· • • ·· •·

•

•· ·• ·· ·· • • • •

• •

•· • ·· •· ·· • • • • •

•

•

•

• ·• · •·· ·· • • •

•

• ·• · •· • ·· •· •

•

•· ·• ·· ·· •· ·· • •

•· ·• ·· ·· • • ·· •· • • • •

•· • ·· •· ·· • ·· · • • • • •

• ·• · •·· ·· • ·· •· • • • •

•· ·• ·· ·· •· ·· • ·· · • • • • • •

• • • •

•

•

•

•

• • • •/ ' /  '  •  •  •  •   •• • • • ••••• ••• ••• ••• ••• •• •• • •• • • • • • •

R(pτ ) ' P ∗ (pτ ) #(pτ ) τ p ∈ AP τ

•

•· ·· • ·· · • • • •

Continuous partitions of the above tree of order 5 with associated skeleton and

• χ(p )

• •

•

/' • / ' /' • /' / •• • • • • •• • •• • •• • • • • • • • • • • • • • • • • • • /  /     '  ' • /   ' • •/ ' • • • • • • • • •• • ∅ • • • • 1 2 2 2 2 3 3 3 yes yes yes yes yes no yes no

pτ

τ

• • •

•

• ·· •· • •

•

•

•

• •

•

• •

•/  •  '  •  •  •   •/  • •• •• •• ••• ••• • • • • • • • • • • • 3 3 3 4 4 4 4 5 no no no no no no no no

9.4 Related B-Series Works

Table 4.24.

The partitions of a tree of order 5 with associated skeleton and forest • •

• •

pτ

'

τ

R(p )

P ∗ (pτ ) #(pτ ) pτ ∈ AP τ

Table 4.25. forest

•

R(pτ ) P ∗ (pτ ) #(pτ ) pτ ∈ AP τ

•

•

•· ·· •

• •

• •

•

• •

• ·· •·

•

•· ·· • ·· •·

• •

•

•

•

• •

• •· ·• ·· ·· •

• •

/ / ' • /' • ' • • / '• • / ' • / • •• • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • /           ' • • • • • • • ∅ • • 1 2 2 2 2 3 yes yes yes yes yes no

•· · •

• •· · •· •

• •

•

•

 • ••

• ' • / •• • • •

•

• •

 • ' • / • • • • • 3 3 yes no

Continuous partitions of the above tree of order 5 with associated skeleton and

• ·· •· · · •·

•

•

•· ·· • •· · •·

•

P (pτ )

• •

•

χ(pτ )

•· • ·· • •

•

• pτ

• •· · •·

•

χ(pτ )

•

•

• •

P (pτ )

431

•· · •· ·• ·· • •

•· · •· ··• ·· ·· • •

•

•

•

•

• •

•

• •

•· ·· • ·· •· · · •· •

•

•

•

• •

•

•· · •· ·• ·· •· · •·

•· ·• ·· ·· •· · •·

•· · •· ··• ·· ·· •· · •·

•

•

•

• • •

• • •

• • •

•

• •/ /'  •      • • • • • • ••• •• ••••• • • • • • • • • • • •• • • • • • • • • • • • • • • • • •              • •  • • • • • •• ••• •• • • •• •• •• • • • • •• • 3 3 3 4 4 4 4 5 no no yes no no no no no  • ••

'

432

9. Formal Power Series and B-Series

9.4.2 Substitution Law In[CHV07,CHV05] ,they introduce a new composition law on B-series, denoted by ∗ and called law of substitution, obtained as a result of the substitution of vector field g(z) = 1 B(f, b)(z) with b(∅) = 0, into another B-series B(g, a)(z). Chartier, Hairer, and h

Vilmart give the following theorem:

Theorem 4.9. For b(∅) = 0, the vector field h−1 B(f, b) gives a B-series B(h−1 B(f, b), a) = B(f, b ∗ a).

(4.10)

We have (b ∗ a)(∅) = a(∅) and for all τ ∈ T ,  (b ∗ a)(τ ) = a(χp )b(vp ),

(4.11)

p∈P(τ )

where b is extended to F as follows: ∀ u = τ1 · · · τn ∈ F,

a(u) =

n ;

a (τi ).

(4.12)

i=1

Remark 4.10. The composition law for the trees of order ≤ 5 is listed in Example 4.22. Remark 4.11. The Substitution law for the (backward error) trees of order ≤ 5 is listed in Example 4.24. Remark 4.12. The Substitution law for the trees of order ≤ 5 is listed in Example 4.23. Modified integrators (called generating function method or preprocessed vector field integrators): Let Ψf,h is the exact h-flow for Equation (4.1) which is a B-series with 1 coefficient e(τ ) = . Consequently, the coefficient ˘b(τ ) of the modified differenγ(τ )

tial equation for Φf,h = B(f, a) is obtained from (˘b ∗ a)(τ ) = e(τ ),

∀ τ ∈ T.

(4.13)

Backward error analysis (called formal vector field, modified equation or postprocessed vector field): The modified differential equation of a method Ψf,h = B(f, e) is obtained by putting Φf,h equal to the exact flow. Its coefficient b(τ ) is therefore obtained from (b ∗ e)(τ ) = a(τ ), ∀ τ ∈ T. (4.14) Remark 4.13. Substituting the expression

1 given in (4.13) into (4.14) gives ˘b∗b∗a = γ

a. Therefore, ˘b and b(τ ) are inverse elements for substitution law ∗ ˘b(τ ) ∗ b(τ ) = b(τ ) ∗ ˘b(τ ) = δ• (τ ).

(4.15)

9.4 Related B-Series Works

433

Proposition 4.14. Using formulae (4.13) and (4.11) in Example 4.23, we easily obtain modified centered Euler scheme of sixth order first find in[CHV07] :  A h2 @ (2) h4 3 (4) z˙ = f (z) + f (f, f ) − 2f (1) f (1) f + f (f, f, f, f ) 24 120 48 1 (3) 1 (2) (1) (2) − f (f, f, f f ) + f (f, f (f, f )) − f (2) (f, f (1) f (1) f ) 4 4 2 (2) (1) 3 1 (1) + f (f f, f f ) − f (1) f (3) (f, f, f ) + f (1) f (2) (f, f (1) f ) 4 12  2 1 (1) (1) (2) (1) (1) (1) (1) − f f f (f, f ) + f f f f f + O(h6 ). 4

Proof. First, we must point out ˘b(τ ) = 0, ∀ |τ | = even. We calculate coefficient ˘b( as follows ˘b(

)

) + 2a( )˘b( )2˘b( ) + a( )˘b( )2˘b( )

+a( )˘b( )2˘b( ) + a(

)˘b( )5 = e(

).

Note the formula in Example 4.23, coefficient a(τ ) in Example 4.5, and γ(τ ) in Table 4.26, we have ˘b(

)=

1 1 1 1 + − = . 40 24 16 240



The proof of others is left to the reader.

Proposition 4.15. In 2001, the author first obtained modified equation for centered Euler scheme given in Example 4.6 of Chapter 7. Using formulas (4.14) and (4.23) in Example 4.24, we can obtain this formula again. Proof. First, we must point out b(τ ) = 0, ∀ |τ | = even. We calculate coefficient b( ) as follows b(

1 ) + 6e( )b( )2b( ) + ( γ

) = a(

).

Note the formula in Example 4.22, coefficient a(τ ) in Example 4.5, and γ(τ ) in Table 1 1 1 7 4.26, we have b( )= + − = . The proof of others is left to the reader. 16 6 5 240  Remark 4.16. After calculating the coefficients ˘b(τ ) and b(τ ), we list following in Table 4.26. Remark 4.17. Can directly test following equation ˘b(τ ) ∗ b(τ ) = δ• (τ ),

(4.16)

via ∗ operation formula (4.11). Remark 4.18. Relating the previous laws to two Hopf algebra introduced, respectively, by Connes and Kremer[CK98] and by Calaque, Ebrahimi-Fard, and Manchon[CEFM08] , we can see these papers[Bro00],[CHV08]] .

434

9. Formal Power Series and B-Series Coefficients σ(τ ), γ(τ ), ˘b(τ ), and b(τ ) for trees of order  5

Table 4.26.

∅

τ σ(τ )

1

1

2

1

6

1

2

1

γ(τ )

1

2

3

6

4

8

12

24

˘b(τ )

0

1

0

1/12

−1/12

0

0

0

0

b(τ )

0

1

0

−1/12

1/12

0

0

0

0

σ(τ )

24

2

2

6

1

1

2

1

2

γ(τ )

5

10

20

20

40

30

60

120

15

˘b(τ )

1/80

−1/80

1/240

b(τ )

7/240

τ

−1/240 1/120 1/240

1/80

−1/120 −1/240 1/120

−7/240 −1/240

−1/80

1/240

1/80

1/240 −1/240

9.4.3 The Logarithmic Map The coefficient ω(τ ) can be interpreted as the coefficient of the modified field obtained by backward error analysis, for the explicit Euler method z1 = z0 + hf (z0 ), corresponding to a = δ∅ + δ• . They can be computed by formula (4.11) or (4.22). Murua in[Mur06] gives the following formula log (a) = (a − b∅ ) ∗ ω.

(4.17)

Properties of logarithmic map has been discussed in Proposition 2.1. Using formula of Example 4.24, determined ω(τ )(= b(τ )) recursively , because a(τ ) = 0 ∀τ ∈ T except a( ) = 1. For example: from 14 formula of Example 4.24, we have 1 1 1 1 1 1 1 1 1 1 1 1 1 ) + ( )(− )( ) + (− )( ) + (− ) + ( ) + 2( )( ) 12 2 2 6 2 2 3 2 4 6 6 6 4 1 1 1 1 1 1 1 1 1 +( )( ) + 2( )( ) + ( )(− ) + 3( )(− ) + = 0, 3 4 3 3 24 2 8 2 30

b(

1 2

) + (−

then we get ω(

) = b(

)=

1 . 20

The test of others ω(τ ) is left the reader. We give following Table 4.27(compare with [Mur06],[CHV05][CHV08],[CEFM08] ).

9.4 Related B-Series Works

435

Coefficient ω(τ ) for trees of order  5

Table 4.27.

τ

∅

ω(τ )

0

1

−1/2

1/6

1/3

0

−1/12

−1/6

−1/4

–1/30

–1/60

1/30

1/30

1/10

1/20

3/20

1/5

1/60

τ

ω(τ )

Definition 4.19. (Lie derivative of B-series) Let b(τ ) with b(∅) = 0 and a(τ ) be the coefficient of two B-series and let z(t) be a formal solution of the differential equation hz(t) ˙ = B(b, z(t)). The Lie derivatives of the function B(a, z(t)) with respect to the vector field B(b, z(t)) is again B-series h

d B(a, z(t)) = B(∂b a, z(t)). dt

(4.18)

Its coefficients are given by ∂b a(∅) = 0 and for |τ | ≥ 1 by ∂b a(τ ) =



a(θ)b(τ \ θ),



∂b a(τ ) =

a(χp )b(vp ).

(4.19)

pτ ∈P(τ )

θ∈SP (τ )

Exercise 4.20. [HLW02] Prove that the coefficient of modified differential equation are recursively defined by b(∅) = 0, b(·) = 1 and b(τ ) = a(τ ) −

|τ |  1 j=2

j!

∂bj−1 b(τ ),

(4.20)

where ∂bj−1 b(τ ) is the (j − 1)-th iterative of the Lie-derivative ∂b . Proposition 4.21. The above-mentioned formula (4.20) is just formula b ∗

1 = γ(τ )

a(τ ) namely |τ |  1 j=1

j!

∂bj−1 b(τ ) = b(τ ) ∗

1 . γ(τ )

(4.21)

Proof. Note that formula (4.23) in Example 4.24 and Tables 4.1 – 4.25, can obtain this results directly. 

436

9. Formal Power Series and B-Series

For Example: from 4th formula of Example 4.24, we have e( )b( ) + 2e( )b( )b( ) + e( )b( )3 = a( ), 1 6

b( ) + b( )b( ) + b( )3 = a( ), 1 6

b( ) = a( ) − b( )b( ) − b( )3 . Example 4.22. The composition laws for the trees of order ≤ 5 are a · b( ) = b(∅) · a( ) + b( ) a · b( ) = b(∅) · a( ) + b( ) · a( ) + b( ) a · b( ) = b(∅) · a( ) + b( ) · a( )2 + 2b( ) · a( ) + b( ) a · b( ) = b(∅) · a( ) + b( ) · a( ) + b( ) · a( ) + b( ) a · b(

) = b(∅) · a( +b(

a · b(

) + b( ) · a( )3 + 3b( ) · a( )2 + 3b( ) · a( )

)

) = b(∅) · a(

) + b( ) · a( )a( ) + b( ) · a( ) + b( ) · a( )2

+b( ) · a( ) + b( ) · a( ) + b( a · b(

) = b(∅) · a(

)

) + b( ) · a( ) + b( ) · a( )2 + 2b( ) · a( ) + b(

)

a · b( ) = b(∅) · a( ) + b( ) · a( ) + b( ) · a( ) + b( ) · a( ) + b( ) ) = b(∅) · a( ) + b( ) · a( )4 + 4b( ) · a( )3 + 6b( ) · a( )2 a · b( +4b( ) · a( ) + b( ) a · b( ) = b(∅) · a( ) + b( ) · a( )3 + b( ) · a( ) · a( )2 +2b( ) · a( ) · a( ) + b( ) · a( ) + b( ) · a( )2 +2b( ) · a( )2 + 2b( ) · a( ) + b( ) · a( ) + b( a · b(

) = b(∅) · a(

) + 2b( ) · a( ) · a( ) + b( ) · a( ) + b( ) · a( )2

+2b( ) · a( ) + 2b( a · b(

) = b(∅) · a( +b( ) · a(

a · b(

) = b(∅) · a( +b(

a · b(

) · a( ) + b(

)

) + b( ) · a( )3 + 3b( ) · a( )2 + 3b( ) + b(

) · a( )

)

) + b( ) · a( ) · a( ) + b( ) · a( )2 + b( ) · a( )

) · a( ) + b( ) · a( ) + b( ) · a(

) = b(∅) · a(

)

2

) + b(

)

) + b( ) · a( ) · a( ) + b( ) · a( ) · a( ) + b( ) · a( )

+b( ) · a( ) + b( ) · a( )2 + b( ) · a( ) + b(

) · a( ) + b(

)

9.4 Related B-Series Works

a · b( ) = b(∅) · a( ) + b( ) · a( ) + b( ) · a(

437

) + b( ) · a( )2

+2b( ) · a( ) + b( ) a · b( ) = b(∅) · a( ) + b( ) · a( ) + b( ) · a( ) + b( ) · a( ) +b( ) · a( ) + b( ) a · b( ) = b(∅) · a( ) + 2b( ) · a( )2 + b( ) · a( ) · a( ) + b( ) · a( )2 +b(

) · a( ) + 2b(

) · a( ) + b( ) · a( ) + b( )

Example 4.23. The substitution law ∗ defined in for the trees of order ≤ 5. b ∗ a( ) = a( )b( ) b ∗ a( ) = a( )b( ) + a( )b( )2 b ∗ a( ) = a( )b( ) + 2a( )b( )b( ) + a( )b( )3 b ∗ a( ) = a( )b( ) + 2a( )b( )b( ) + a( )b( )3 b ∗ a( ) = a( )b( ) + 3a( )b( )b( ) + 3a( )b( )2 b( ) + a( b ∗ a(

) = a( )b(

) + a( )b( )b( ) + a( )b( )2 + a( )b( )b( )

+2a( )b( )2 b( ) + a( )b( )2 b( ) + a( b ∗ a(

) = a( )b(

)b( )4

)b( )4

) + a( )b( )b( ) + 2a( )b( )a( ) + a( )b( )2 b( )

+2a( )b( )2 b( ) + a(

)b( )4

b ∗ a( ) = a( )b( ) + 2a( )b( )b( ) + a( )b( )2 + 3a( )b( )2 b( ) +a( )b( )4 b ∗ a( b ∗ a(

) = a( )b( ) + 4a( )b( )b( ) + 6a( )b( )2 b( ) )b( )5 +4a( )b( )3 b( ) + a( ) = a( )b( ) + a( )b( )b( ) + 2a( )b( )b( ) + a( )b( )b( ) +a( )b( )2 b( ) + 2a( )b( )2 b( ) + a( )b( )2 b( ) +2a( )b( )b( )2 + 2a( )b( )3 b( ) + 2a( )b( )3 b( ) +a(

b ∗ a(

) = a( )b(

)b( )5 ) + 2a( )b( )b(

) + 2a( )b( )b( )

2

+a( )b( ) b( ) + 2a( )b( )2 b( ) + 3a( )b( )b( )2 +4a(

)b( )3 b( ) + a(

)b( )5

438

9. Formal Power Series and B-Series

b ∗ a(

) = a( )b(

) + a( )b( )b(

) + 3a( )b( )b(

+3a( )b( )2 b( ) + 3a( )b( )2 b( ) + a( )b( )3 b( ) + a(

+3a( b ∗ a(

) = a( )b(

) + a( )b( )b(

) )b( )3 b( )

)b( )5 ) + a( )b( )b( ) + a( )b( )b(

)

+a( )b( )b( ) + 2a( )b( )2 b( ) + a( )b( )b( )2 +a( )b( )b( )2 + a( )b( )2 b( ) + a( )b( )2 b( )

b ∗ a(

+a(

)b( )3 b( ) + a( )b( )3 b( ) + 2a(

+a(

)b( )5

) = a( )b(

) + a( )b( )b(

)b( )3 b( )

) + a( )b( )b( ) + a( )b( )b( )

+a( )b( )b( ) + a( )b( )2 b( ) + 2a( )b( )b( )2 +a( )b( )b( )2 + 2a( )b( )2 b( ) + a( )b( )3 b( ) +3a(

)b( )3 b( ) + a(

b ∗ a( ) = a( )b( ) + a( )b( )b(

)b( )5 ) + 2a( )b( )b( ) + a( )b( )b( )

+2a( )b( )b( )2 + 2a( )b( )2 b( ) +a( )b( )2 b( ) +a( )b( )2 b( ) + 2a(

)b( )3 b( ) + 2a( )b( )3 b( )

+a( )b( )5 b ∗ a( ) = a( )b( ) + 2a( )b( )b( ) + 2a( )b( )b( ) + 3a( )b( )2 b( ) +3a( )b( )b( )2 + 4a( )b( )3 b( ) + a( )b( )5 b ∗ a( ) = a( )b( ) + 2a( )b( )b(

) + a( )b( )b( ) + a( )b( )b(

)

+2a( )b( )2 b( ) + 2a( )b( )2 b( ) + 2a( )b( )b( )2 +a( +a(

)b( )3 b( ) + 2a(

)b( )3 b( )

)b( )3 b( ) + a( )b( )5 (4.22)

9.4 Related B-Series Works

439

Example 4.24. The substitution law ∗ defined in for the trees of order ≤ 5 b ∗ e( ) = e( )b( ) b ∗ e( ) = e( )b( ) + e( )b( )2 b ∗ e( ) = e( )b( ) + 2e( )b( )b( ) + e( )b( )3 b ∗ e( ) = e( )b( ) + 2e( )b( )b( ) + e( )b( )3 b ∗ e( b ∗ e(

) = e( )b( ) = e( )b(

) + 3e( )b( )b( ) + 3e( )b( )2 b( ) + e(

) + e( )b( )b( ) + e( )b( )2 + e( )b( )b( )

+2e( )b( )2 b( ) + e( )b( )2 b( ) + e( b ∗ e(

) = e( )b(

)b( )4

)b( )4

) + e( )b( )b( ) + 2e( )b( )e( ) + e( )b( )2 b( )

+2e( )b( )2 b( ) + e(

)b( )4

b ∗ e( ) = e( )b( ) + 2e( )b( )b( ) + e( )b( )2 + 3e( )b( )2 b( ) +e( )b( )4 (4.23) b ∗ e(

) = e( )b( +4e(

b ∗ e(

) = e( )b(

) + 4e( )b( )b( )b( )3 b( ) + e(

b ∗ e(

) + 2e(

+2e(

)b( )b( )2 + 2e(

+e(

)b( )5

) = e( )b(

+e( b ∗ e(

) = e( )b(

)

)b( )2 b(

) + e(

) + e( )b( )b(

)b( )3 b( ) + 2e(

)b( )3 b( )

) + 2e( )b( )b( ) + e( )b( )b( )2 + 4e(

) + e( )b( )b( ) + 3e(

)b( )3 b( ) + e(

) + 3e( )b( )b( )b( )2 b( ) + e( )b( )5

)

)b( )2 b( )

)b( )2 b(

)b( )3 b( )

)b( )5

+3e( )b( )2 b( +3e(

) + 2e( )b( )b(

) + 2e( )b( )b(

+2e( )b( )2 b( ) + 3e(

)b( )2 b(

)b( )5

) + e( )b( )b(

+e( )b( )2 b(

) + 6e(

) )b( )3 b( )

)

440

b ∗ e(

9. Formal Power Series and B-Series

) = e( )b(

) + e( )b( )b( ) + e( )b( )b(

+e( )b( )b( ) + 2e( )b( )2 b( ) + e(

)b( )b( )2

+e( )b( )b( )2 + e( )b( )2 b(

)b( )2 b( )

) = e( )b(

) + e( )b( )b(

) + e( )b( )b(

+e( )b( )b( ) + e( )b( )2 b( +e( +3e(

) + e(

)b( )3 b( ) + e( )b( )3 b( ) + 2e(

+e( b ∗ e(

) + e( )b( )b(

)b( )b( )2 + 2e(

)b( )3 b( ) + e( ) + e( )b( )b( )

) + 2e( )b( )b( )2

)b( )5 ) + 2e( )b( )b( ) + e( )b( )b(

+2e( )b( )b( )2 + 2e( )b( )2 b( ) + e( )b( )2 b( +e(

)b( )5

)b( )2 b( ) + e( )b( )3 b( )

)b( )3 b( ) + e(

b ∗ e( ) = e( )b( ) + e( )b( )b(

)

)b( )2 b( ) + 2e(

)

)

)b( )3 b( ) + 2e( )b( )3 b( ) + e( )b( )5

b ∗ e( ) = e( )b( ) + 2e( )b( )b( ) + 2e( )b( )b( ) + 3e( )b( )2 b( ) +3e( )b( )b( )2 + 4e( )b( )3 b( ) + +e( )b( )5 b ∗ e( ) = e( )b( ) + 2e( )b( )b( +2e( +e(

)b( )2 b(

) + 2e(

)b( )3 b( ) + 2e(

) + e( )b( )b(

) + e( )b( )b(

)

)b( )2 b( ) + 2e( )b( )b( )2 )b( )3 b( ) + e(

)b( )3 b( ) + e( )b( )5

Bibliography

[Bak05] H. F. Baker: Alternants and continuous groups. Proc. London Math. Soc., 3:24–47, (1905). [Bro00] Ch. Brouder: Runge–Kutta methods and renormalization. Euro. Phys. J. C, 12:521– 534, (2000). [BSS96] J. C. Butcher and J. M. Sanz-Serna: The number of conditions for a Runge–Kutta method to have effective order p. Appl. Numer. Math., 22:103–111, (1996). [CEFM08] D. Calaque, K. Ebrahimi-Fard, and D. Manchon: Two Hopf algebra of trees interacting. arXiv: 0806.2238 v 2, (2008). [CHV05] P. Chartier, E. Hairer, and G. Vilmart: A substitution law for B-series vector fields. Technical Report 5498, INRIA, (2005). [CHV07] P. Chartier, E. Hairer, and G. Vilmart: Numerical integration based on modified differential equations. Math. Comp., 76(260):1941–1953, (2007). [CHV08] P. Chartier, E. Hairer, and G Vilmart: Composing and substituting S-series and Bseries of integrators and vector fields. Preprint, www.irisa.fr/ipso/fichiers/algebraic.pdf, (2008). [CK98] A. Connes and D. Kreimer: Hopf algebra, renormazation and noncommutative geometry. Communications in Mathematical Physics, 199:203–242, (1998). [Dyn46] E. B. Dynkin: Normed Lie algebra and analytic groups, volume 1. Amer. Math. Soc. (translation), (1946). [Fen85] K. Feng: On difference schemes and symplectic geometry. In K. Feng, editor, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, pages 42–58. Science Press, Beijing, (1985). [Fen92] K. Feng: Formal power series and numerical methods for differential equations. In T. Chan and Z. C. Shi, editors, International conf. on scientific computation, pages 28–35. World Scientific, Singapore, (1992). [Fen93a] K. Feng: Formal dynamical systems and numerical algorithms. In K. Feng and Z. C. Shi, editors, International conf. on computation of differential equations and dynamical systems, pages 1–10. World Scientific, Singapore, (1993). [Fen93b] K. Feng: Symplectic, contact and volume preserving algorithms. In Z.C. Shi and T. Ushijima, editors, Proc.1st China-Japan conf. on computation of differential equations and dynamical systems, pages 1–28. World Scientific, Singapore, (1993). [Hai94] E. Hairer: Backward analysis of numerical integrators and symplectic methods. Annals of Numer. Math., 1:107–132, (1994). [Hau06] F. Hausdorff: Die symbolische exponentialformel in der gruppentheorie. Berichte der Sachsischen Akad. der Wissensch., 58:19–48, (1906). [HLW02] E. Hairer, Ch. Lubich, and G. Wanner: Geometric Numerical Integration. Number 31 in Springer Series in Computational Mathematics. Springer-Verlag, Berlin, (2002). [HW74] E. Hairer and G. Wanner: On the Butcher group and general multivalue methods. Computing, 13:1–15, (1974). [Lie88] S. Lie; Zur theorie der transformationsgruppen. Christiania, Gesammelte Abh., Christ. Forh. Aar., 13, (1988).

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[Mur06] A. Murua: The Hopf algebra of rooted trees, free Lie algebra, and Lie series. Foundations of Computational Mathematics, 6(4):387–426, (2006). [Olv93] P. J. Olver: Applications of Lie Groups to Differential Equations. GTM 107. SpringerVerlag, Berlin, Second edition, (1993). [Ote91] J. A. Oteo: The Baker–Campbell–Hausdorff formula and nested commutator identities. J. of Math. Phys., 32(2):419–424, (1991). [OW00] B. Owren and B. Welfert: The Newton iteration on Lie groups. BIT, 40(1):121–145, (2000). [Owr06] B. Owren: Order conditions for commutator-free Lie group methods. J. Phys. A: Math. Gen., 39:5585–5599, (2006). [Rei99] S. Reich: Backward error analysis for numerical integrators. SIAM J. Numer. Anal., 36:475–491, (1999). [SS96] J. M. Sanz-Serna: Backward Error Analysis for Symplectic Integrators. In J. E. Mardsen, G. W. Patrick, and W. F. Shadwick, editors, Integration Algorithms and Classical Mechanics, pages 193–206. American Mathematical Society, New York, (1996). [SS97] J. M. Sanz-Serna: Geometric integration. In The State of the Art in Numerical Analysis (York, 1996), volume 63 of Inst. Math. Appl. Conf. Ser. New Ser., pages 121–143, Oxford Univ. Press, New York, (1997).

Chapter 10. Volume-Preserving Methods for Source-Free Systems

Source-free dynamical systems is an important system in recent mechanics and physics. It has a abroad application. Therefore, designing a proper numerical method for this system is significant. It is well known that phase flow of source-free system is a volume-preserving transformation. Therefore, the transient operator of the numerical method that we design should be volume-preserving. We call this algorithm the volume-preserving algorithm.

10.1 Liouville’s Theorem Let x = (x1 , x2 , · · · , xN )T , and f (x) = (f1 (x), f2 (x), · · · , fN (x))T : RN → RN , then the dynamical system dx = f (x) dt

is source-free (i.e., divergence-free), when

(1.1)

N  ∂fi i=1

∂xi

= 0 (i.e., div f (x) = 0). The flow

of a source-free system is volume-preserving, i.e., det (etf (x))∗ = 1,

∀ x, t,

here etf denotes the flow of system (1.1) and (etf (x))∗ the Jacobian of etf at x. Thus, volume-preserving schemes are required for computing the numerical solution of   ∂x

n+1 = 1, we call this scheme volume-preserving, where xn denotes (1.1). If det ∂ xn the numerical solution at step n. We know that the phase flow of Hamiltonian system preserves phase volume invariable. The source-free system is more general than the Hamiltonian system, we must prove that the phase flow preserving phase volume is invariable, considering the dynamic system (1.1), its phase flow is

g t (x) = x + f (x)t + o(t2 ).

(1.2)

Let D(0) be a region in x space and V (0) is its volume, then V (t) = volume of D(t),

D(t) = g t D(0).

K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

444

10. Volume-Preserving Methods for Source-Free Systems

Theorem 1.1 (Liouville’s Theorem). If div f = 0, then g t preserving the volume is invariable, V (t) = V (0). Proof. First proof



d V (t)  =  d t t=0

div f d x,

(1.3)

D(0)

for any t, using the formula for changing variables in a multiple integral gives ∂ gt x det d x. V (t) = ∂x

D(0)

Calculating

∂ gt x by formula (1.2), we find ∂x ∂ gtx ∂f =E+ t + O(t2 ), ∂x ∂x

but

as t → 0.

det (E + At) = 1 + t tr A + O(t2 ),

where tr A =

n 

t → 0,

aii . Therefore

i=1

V (t) = 

[1 + t div f + O(t2 )]d x, = div f d x.

(1.4)

D(0)

d V (t)   d t t=0

D(0)

Then Equation (1.3) is proved. Now we take t = t0 is no worse than t = 0, therefore  d V (t)  = div f d x,  dt

and if div f = 0,

t=t0

D(t0 )

d V (t) = 0. This completes the proof. dt

In particular, for Hamiltonian equation     ∂ ∂H ∂ ∂H − + = 0, div f = ∂p

∂q

∂q

∂p

Liouville’s theorem is proved specially.

10.2 Volume-Preserving Schemes 10.2.1 Conditions for Centered Euler Method to be Volume Preserving Let us consider centered Euler scheme



10.2 Volume-Preserving Schemes

 xn+1 = xn + τ f

xn+1 + xn 2

445

 ,

(2.1)

where τ is the step size in t. We then have    ∂ xn+1 xn+1 + xn 1 ∂ xn+1 1 = IN + τ Df + IN , ∂ xn

IN ∂ xn+1 = ∂ xn IN

2

2 ∂ xn

2

τ + Df (x∗ ) 2 . τ − Df (x∗ ) 2





∂f x + xn ∂ xn+1 = ≡ B = (bij ), x∗ = n+1 . The condition det ∂x 2 ∂xn τ |IN + Df (x∗ )| 2 1, now requires = 1. Let P (λ) = |Df (x∗ ) − λIn | be the characterτ |IN − Df (x∗ )| 2

Here, Df = fx =

istic matrix of Df (x∗ ). Since

 % &     % & τ 2     ∗ τ 2 Df (x ) + IN   IN + Df (x∗ ) P 2    τ 2 τ N   =  % &, &  = (−1) %    2 2 IN − τ Df (x∗ )  τ  P − Df (x∗ ) − IN   −  2 τ  2  τ

we then get the condition for scheme (2.1) to be volume-preserving[QZ93] , i.e., P (λ) = (−1)N P (−λ). Let us consider some particular cases of N to show that scheme (2.1) is not always volume preserving. Case 2.1. In this case, we have P (λ) = λ2 + (b11 + b22 )λ + b11 b22 − b12 b21 .

(2.2)

N  ∂fi = 0, i.e., tr B = 0, then P (λ) = λ2 + b11 b22 − b12 b21 , and ∂x i i=1

Since

P (−λ) = P (λ). Thus, the scheme (2.1) is always volume-preserving for source-free systems of dim.2. Case 2.2. Here P (λ) = −λ3 + (b11 + b22 + b33 )λ2 − cλ + |B| = −λ3 − cλ + |B|, where

  b11  c=  b21

  b12   b22 + b22   b32

  b23   b11 + b33   b31

 b13  . b33 

(2.3)

446

10. Volume-Preserving Methods for Source-Free Systems

The volume-preserving condition for Euler method is now |B| = 0. For example, (ABC flow) when system (1.1) takes the form dx = cy − bz, dt dy = az − cx, dt dz = bx − ay, dt

a, b, c ∈ R,

we have |B| = 0. For this dynamical system, centered Euler method is volumepreserving. Lemma 2.3. Let P (λ) be the characteristic polynomial of matrix AN ×N , then   P (λ) = |A − λIN | = (−1)N λN − P1 λN −1 + P2 λN −2 + · · · + (−1)N PN , (2.4) where P1 =

N 

aii = tr A,

i

  N  a    ii aij  P2 =  ,  aji ajj  i<j   aii aij aik  N    a P3 =  ji ajj ajk i<j
    ,   

(2.5)

··· PN = |A|. Using Lemma 2.3, we can discuss the case N = 4. Case 2.4. At this time, P (λ) = λ4 − P1 λ3 + P2 λ2 − P3 λ + |B|. Since P1 = tr (B) = 0, then P (−λ) = (−1)4 P (λ) requires P3 = 0. It must be pointed out that, when N increases, more increasing number of conditions is required for system (2.1) to be volume-preserving, and it seems impossible to satisfy all these condition. But fortunately, for the special case when system (1.1) is Hamiltonian, i.e., 6 5 O −Ik , N = 2k. f = J∇H, J = Ik O Scheme (2.1) is volume-preserving. This is because the Hamiltonian system is sourcefree and Df is an infinitesimal symplectic matrix, we have the following Lemma.

10.2 Volume-Preserving Schemes

447

Lemma 2.5. Let M be an infinitesimal symplectic matrix, if λ is an eigenvalue of M , ¯ −λ. ¯ so are −λ, λ, From Lemma 2.5, we know that P (−λ) = (−λ)2k P (λ) is valid when system (1.1) is Hamiltonian, so Euler method is volume-preserving for Hamiltonian systems. In fact, the method is even symplectic for Hamiltonian systems, that is to say it also preserve the symplectic structure of Hamiltonian systems which is a much stronger property than volume-preserving.

10.2.2 Separable Systems and Volume-Preserving Explicit Methods In this section, we consider a special kind of source-free systems called separable systems. System (1.1) is separable if d xi = fi (x1 , x2 , · · · , xi−1 , xi+1 , · · · , xN ), dt

i = 1, 2, · · · , N.

(2.6)

We can divide the above system into N source-free systems: ⎧ d x1 ⎪ = f1 (x2 , · · · , xN ), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ d x2 ⎪ ⎨ = 0, dt

.. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ d x N ⎩ = 0.

(2.7)

dt

⎧ d x1 ⎪ = 0, ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ d x2 ⎪ ⎨ = f2 (x1 , x3 , · · · , xN ), dt

.. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ d x N ⎩ = 0.

(2.8)

dt

··· ⎧ d x1 ⎪ ⎪ = 0, ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ .. . ⎪ d xN −1 ⎪ ⎪ = 0, ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ d xN = fN (x1 , · · · , xN −1 ). dt

(2.9)

448

10. Volume-Preserving Methods for Source-Free Systems

The first order explicit Euler method can be applied to them to get the exact solutions of them, i.e., the phase flows of them. Using the composition method[QZ92] , we can construct first order explicit Euler volume-preserving scheme for system (2.6). The adjoint of this scheme is obtained from the implicit Euler method and is also explicit. Composing these two schemes, we get a reversible explicit. This process can be expressed by formal power series as shown below. From Chapter 9, we know the flow of (1.1) can be represented by power of series. eτf = 1N +

∞ 

ek,f : RN −→ RN ,

τ k ek,f ,

ek,f =

k=1

where f ∗ denotes the first order differential operator , f ∗ =

N  i=1

1 ∗k f 1N , k!

fi

∂ , f ∗2 = f ∗ × ∂xi

f ∗ , f ∗ 3 = f ∗ × f ∗ × f ∗ , · · · , 1N is the identity vector function, 1N (x) = x. For simplicity, we just write out eτA · eτB = eτcτ , (2.10) the first several terms are cτ = A + B +

τ [A, B] + o(τ 2 ), 2

where [A, B] = A∗ B − B∗ A is the Lie bracket of A and B, A∗ , B∗ denotes the Jacobian matrix of A and B . We now rewrite system of Equations (2.7) – (2.9) in compact form as dx = ai (x), dt

ai = (0, · · · , 0, fi , 0, · · · , 0)T ,

i = 1, 2, · · · , N.

(2.11)

These integrable systems have flow eτai = 1N +

inf 

τ k ek,ai ,

i = 1, 2, · · · , N.

(2.12)

k=1 ∗k Since we have a∗N i 1N (x) = ai x = 0, when k ≥ 2, then

eτai (x) = x +

inf 

τ k ek,ai (x) = x +

k=1

inf  τk k=1

k!

a∗i k 1N (x) = x + τ ai (x).

(2.13)

Using the formula (2.10), we find eτaN × eτaN −1 × · · · × eτa2 × eτa1 = eτf +o(τ ) .

(2.14)

This means the concatenation eτaN × eτaN −1 × · · · × eτa1 approximates the flow eτf to the first order of τ . Because the equations in the system (2.11) are all source-free, their flows are all volume-preserving and the concatenation of them remains volume-preserving, so

10.3 Source-Free System

449

  det (eτaN × eτaN −1 × · · · × eτa1 )(x) ∗  = det eτaN (xN −1 ))∗ × det (eτaN −1 (xN −2 ))∗ × · · · × det (eτa1 (x0 ))∗ = 1, where x0 = x, x1 = eτa1 (x0 ), · · · , xN −1 = eτaN −1 (xN −2 ), xN = eτaN (xN −1 ). Thus, from system (2.6) we get volume-preserving scheme of first order. This is an explicit scheme since eτai (i = 1, · · · , N ) are flows of integrable systems which can be written as (2.13). From [QZ92] , we know the concatenation eτaN ×eτaN −1 ×· · ·×eτa1 with its adjoint eτa1 × eτa2 × · · · × eτaN produces a reversible scheme τ

τ

τ

τ

ea2N × ea2N −1 × · · · × eτa1 × · · · × ea2N −1 × ea2N of second order, but is still explicit. We can use theory of composition[QZ92] to construct symplectic scheme of arbitrary order.

10.3 Source-Free System Source-free dynamical systems on the Euclidean space Rn are defined by source-free (or divergence-free) vector fields a : Rn → Rn , div a(x) =

n  ∂ ai (x) i=1

∂ xi

= 0,

∀x ∈ Rn ,

(3.1)

through equations dx = x˙ = a(x), dt here and hereafter, we use the coordinate description and matrix notation x = (x1 , · · · , xn )T ,

a(x) = (a1 (x), · · · , an (x))T ,

(3.2)

(3.3)

where T denotes the transpose of a matrix. In this subsection, we mainly analyze and construct numerical algorithms proper for source-free systems. Such systems constitute one of the most important classical cases of dynamical systems preserving certain geometric structure and arise in many physical problems such as particle tracking in incompressible fluids and toroidal magnetic surface-generation in stellarators. Because of the difficulty and even impossibility of solving equations by quadrature, the numerical methods certainly play an important role in understanding the dynamic behavior of a system and in solving physical and engineering problems. On the other hand, the problem of whether a numerical algorithm is proper for a system is closely related to the problem of whether the algorithmic approximation to the corresponding phase flow approximates perfectly in some sense and even strictly preserve the structure of the system itself if the system has such structure. It has been evidenced with some typical examples in the Hamiltonian case that “nonproper” algorithms will result in essentially wrong approximations

450

10. Volume-Preserving Methods for Source-Free Systems

to the solutions of systems and “proper” algorithms may generate remarkably right ones. But how does one evaluate a numerical algorithm to be proper for source-free systems? It is well known that intrinsic to all source-free systems there is a volume form of the phase space Rn , say α = dx1 ∧ dx2 ∧ · · · ∧ dxn

(3.4)

such that the evolution of dynamics preserves this form. In other words, the phase flow eta , of source-free system (3.2), satisfies the volume-preserving condition (eta )∗ α = α,

(3.5)

or equivalently, det

∂ eta (x) = 1, ∂x

∀ x ∈ Rn , t ∈ R.

(3.6)

In addition to this, eta satisfies the group property in t, e0a = identity,

et+s = eta ◦ esa . a

(3.7)

In fact, (3.5) and (3.7) completely describe the properties of the most general source-free dynamical systems. This fact suggests that a proper algorithmic approximation gas to phase flow esa for source-free vector field a : Rn → Rn should satisfy these two requirements. However, the group property (3.7) is too stringent in general for algorithmic approximations because only the phase flows satisfy it. Instead of it, a weaker requirement, i.e., ga0 = identity,

gas ◦ ga−s = identity,

(3.8)

is reasonable and practicable for all vector fields a : Rn → Rn . We call such algorithmic revertible approximations, that means gas always generate coincident forward and backward orbits. As for the volume-preserving property (3.5), it characterizes the geometric structure —volume-preserving structure—of source-free systems. Our aim here is just to construct difference schemes preserving this structure, which we call volumepreserving schemes, in sense that the algorithmic approximations to the phase flows satisfy (3.5) for the most general source-free systems.

10.4 Obstruction to Analytic Methods We note that for n = 2, source-free vector fields = Hamiltonian fields, and areapreserving maps = symplectic maps, so the problem for area-preserving algorithms has been solved in principle. But for n ≥ 3, the problem is new, since all the conventional methods plus even the symplectic methods are generally not volume-preserving, even for linear source-free systems. As an illustration, see example and Lemma of Feng and Shang[FS95] .

10.4 Obstruction to Analytic Methods

451

Example 4.1. Solve on R3 dx = a(x) = Ax, dt

tr A = 0,

(4.1)

by the Euler centered method, we get algorithmic approximation Gs to esa = exp (sA) with  s −1  s  Gs = I − A I+ A . (4.2) 2 2 Simple calculations show that in 3-dimensions, if tr A = 0, then det Gs = 1 ⇔ det A = 0, which is exceptional. A more general conclusion in linear case is Lemma 4.2. Let sl(n) denote the set of all n × n real matrices with trace equal to zero and SL(n) the set of all n × n real matrices with determinant equal to one. Then for any real analytic function φ(z) defined in a neighborhood of z = 0 in C satisfying the conditions: 1◦ φ(0) = 1; ˙ = 1. 2◦ φ(0) We know that φ(sl(n)) ⊂ SL(n) for some n ≥ 3 if and only if φ(z) = exp (z). Proof. “If part” is a known conclusion, for the “only if part” it suffices to show it for n = 3. For this, we consider matrices of the diagonal form ⎡ ⎤ s 0 0 ⎢ ⎥ ⎥ ∈ sl(3), s, t ∈ R. 0 D(s, t) = ⎢ (4.3) ⎣ 0 t ⎦ 0 0

−(s + t)

Since φ is analytic in a neighborhood of the origin in C, we have ⎡ ⎤ φ(s) 0 0 ⎢ ⎥ ⎥ , s, t  0. φ(t) 0 φ(D(s, t)) = ⎢ ⎣ 0 ⎦ 0

0

(4.4)

φ(−(s + t))

By assumption, det φ(D(s, t)) = 1, for s, t  0. So s, t  0,

φ(s)φ(t)φ(−(s + t)) = 1,

(4.5)

together with the condition φ(0) = 1, we have φ(s)φ(−s) = 1,

s  0.

(4.6)

Multiplying the both sides of Equation (4.5) by φ(s + t) and using (4.6), we get φ(s)φ(t) = φ(s + t),

s, t  0.

(4.7)

This, together with the conditions 1◦ and 2◦ of the lemma, implies φ(z) = exp (z), which completes the proof.



452

10. Volume-Preserving Methods for Source-Free Systems

Lemma 4.2 says that there are no consistent analytic approximations to the exponential function sending sl(n) into SL(n) at the same time other than the exponential itself. This shows that it is impossible to construct volume-preserving algorithms analytically depending on source-free vector fields. Thus we have: Theorem 4.3 (Feng-Shang). All the conventional methods including the well-known Runge–Kutta methods, linear multistep methods and Euler methods (explicit, implicit and centered) are non-volume-preserving. The above lemma tell us we cannot construct volume-preserving scheme for all source-free system. But we can split class sl(n) to subclass and perhaps in subclass, there exists volume-preserving scheme. In Subsection 10.2.1, we get some condition for centered Euler scheme to be volume-preserving scheme. It is the best elucidation. Consequently, to construct volume-preserving algorithms for source-free systems, we must break through the conventional model and explore new ways.

10.5 Decompositions of Source-Free Vector Fields In R2 , every source-free field a = (a1 , a2 )T corresponds to a stream function or 2-dimensional Hamiltonian ψ, unique up to a constant: a1 = −

∂ψ , ∂ x2

a2 =

∂ψ . ∂ x1

(5.1)

In R3 , every source-free field a = (a1 , a2 , a3 )T corresponds to a vector potential b = (b1 , b2 , b3 )T , unique up to a gradient: ∂ b3 ∂ b2 − , ∂ x2 ∂ x3 ∂ b2 ∂ b1 a3 = − , ∂ x1 ∂ x2

a1 =

a = curl b, a2 =

∂ b1 ∂ b3 − , ∂ x3 ∂ x1

then we get source-free decomposition ⎡ ⎤ ⎡ ∂b 0 ⎡ ⎤ − 2 a1 ⎢ ⎥ ⎢ ∂ x3 ⎢ ∂ b1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ a 0 a=⎢ + ⎢ ⎥ 2 ⎣ ⎦ ⎢ ∂ x3 ⎥ ⎢ ⎢ ⎣ ∂b ⎦ ⎣ ∂b 2 a3 − 1 ∂ x2

∂ x1

⎤ ⎡

∂ b3 ∂ x2

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂ b3 ⎥+⎢ − ⎥ ⎢ ∂ x1 ⎦ ⎣ 0

(5.2)

⎤ ⎥ ⎥ ⎥ ⎥ = a(1) +a(2) +a(3) . (5.3) ⎥ ⎦

As a generalization of cases n = 2, 3, on Rn , we have[FS95] : Lemma 5.1. To every source-free field a = (a1 , a2 , · · · , an )T , there corresponds a skew symmetric tensor field of order 2, b = (bik )1≤i,k≤n , bik = −bki , so that ai =

n  ∂ bik k=1

∂ xk

,

i = 1, 2, · · · , n.

(5.4)

10.5 Decompositions of Source-Free Vector Fields

453

Proof. With the given a = (a1 , · · · , an )T , we define the 1-form on Rn α=

n 

ai (x)d xi .

(5.5)

i=1

Since a is source-free, we have δα = −

n  ∂ ai i=1

∂ xi

= −div a = 0,

where δ is codifferential operator. The above equation means that α is δ-closed. By Poincar´e’s lemma, there exists a 2-form, say β, so that α = δβ.

(5.6)

But for the 2-form β, there exists a skew symmetric tensor of order 2, b = (bik )1≤i.k≤n , bik = −bki , so that n  bik d xi ∧ d xk . (5.7) β= i,k=1

Take (5.7) codifferential, * n + n   ∂ bik δβ = d xi , ∂ xk i=1

(5.8)

k=1

and from Equations (5.5) and (5.6), we get (5.4). The proof is completed.



By (5.4), we can decompose a=

 1≤i
a(ik) ,

 T ∂bik ∂bik a(ik) = 0, · · · , 0, , 0, · · · , − , 0, · · · , 0 , ∂xk ∂xi

i < k.

(5.9) Every vector field a(ik) in (5.9) is a 2-dimensional Hamiltonian on the xi -xk plane and zero in other dimensions. We call such decompositions essentially Hamiltonian decompositions. We note that the tensor potential b = (bik )1≤i,k≤n is far from uniquely determined for a given source-free field a = (a1 , · · · , an )T from Equation (5.4). For uniqueness, one may impose normalizing conditions in many different ways. One way is to impose, as done by H. Weyl in[Wey40] in 3-dimensional case: N0 : bik = 0,

|i − k| ≥ 2,

(5.10)

this condition is ineffective for n = 2. The non zero components are b12 = −b21 , b23 = −b32 , · · · , bn−1,n = −bn,n−1 . Nk : bk,k+1 |xk+1 =0 = 0, 1 < k ≤ n − 2

(5.11) (5.12)

454

10. Volume-Preserving Methods for Source-Free Systems

this condition is ineffective for n = 2, Nn−1 : bn−1,n |xn−1 =xn =0 = 0.

(5.13)

Then, simple calculations show that all bk,k+1 are uniquely determined by quadrature - x2 a1 d x2 , (5.14) b12 = -0 xk+1   ∂b (5.15) bk,k+1 = ak + k−1,k d xk+1 , 2 ≤ k ≤ n − 2, ∂ xk−1 & -0 xn % - xn−1 ∂b bn−1,n = an |xn =0 d xn−1 . (5.16) an−1 + n−2,n−1 d xn − ∂ xn−2

0

0

So, one gets an essentially Hamiltonian decomposition for a as a=

n−1 

a(k) ,

a(k) =

%

&T

0, · · · , 0,

k=1

or in components,

∂ bk,k+1 ∂b , − k,k+1 , 0, · · · , 0 ∂ xk+1 ∂ xk

,

⎧ ∂b ⎪ a1 = 12 , ⎪ ⎪ ⎪ ∂ x2 ⎪ ⎪ ⎪ ∂b ∂b ⎪ ⎪ a2 = − 12 + 23 , ⎪ ⎪ ∂ x ∂ x3 1 ⎪ ⎨ .. . ⎪ ⎪ ⎪ ∂b ∂b ⎪ ⎪ an−1 = − n−2,n−1 + n−1,n , ⎪ ⎪ ∂ x ∂ xn ⎪ n−2 ⎪ ⎪ ⎪ ∂ bn−1,n ⎪ ⎩ an = − .

(5.17)

(5.18)

∂ xn−1

10.6 Construction of Volume-Preserving Schemes In this section, we give a general way to construct volume-preserving difference schemes for source-free systems by means of essentially the Hamiltonian decompositions of source-free vector fields and the symplectic difference schemes for 2dimensional Hamiltonian systems. With this aim, we first prove: Lemma 6.1. Let a be a smooth vector field on Rn and have decomposition a=

m 

a(i) ,

(6.1)

i=1

with smooth fields a(i) :Rn → Rn (i = 1, · · · , m). Suppose that, for each i = 1, · · · , m, Gτi is an approximation of order p to eτa(i) , the phase flow of the system

10.6 Construction of Volume-Preserving Schemes

455

1 associated to the field a(i) , in the sense that lim p (Gτi (x) − eτa(i) (x)) = 0 for all τ →0 τ x ∈ Rn with some p ≥ 1. Then, we have: 1◦ For any permutation (i1 i2 · · · im ) of (12 · · · m), the compositions τ 1 Gi1 i2 ···im

τ 1 Gi1 i2 ···im

:= Gτim ◦ · · · ◦ Gτi2 ◦ Gτi1 ,

 :=

−τ 1 Gi1 i2 ···im

−1

(6.2)

are approximations, of order one, to eτa ; and the compositions τ 2 gi1 i2 ···im

τ

τ

2 2 := 1 G i1 i2 ···im ◦ 1 Gi1 i2 ···im ,

iτ1 i2 ···im 2g

τ

τ

− 2 := 1 Gi21 i2 ···im ◦ 1 G i1 i2 ···im (6.3)

are revertible approximations, of order 2, to eτa ; 2◦ If, for each i = 1, 2, · · · , m, Gτi is an approximation, of order 2, to eτa , then τ 2 Gi1 i2 ···im

τ

τ

τ

τ

τ

τ

:= Gi2m ◦ · · · ◦ Gi22 ◦ Gi21 ◦ Gi21 ◦ Gi22 ◦ · · · ◦ Gi2m

(6.4)

is an approximation, of order 2, to eτa ; and it is revertible if each Gτi is revertible; 3◦ If 2 Gτ is a revertible approximation, of order 2, to eτa , then the symmetric composition[QZ92] τ α1 τ ◦ 2 Gβ1 τ ◦ 2 Gα1 τ (6.5) 4G = 2G with

1

α1 = (2 − 2 3 )−1 ,

β1 = 1 − 2α1 < 0,

(6.6)

is a revertible approximation, of order 4, to eτa ; and generally, the symmetric composition, recursively is defined as follows, τ 2(l+1) G

with

= 2l Gαl τ ◦ 2l Gβl τ ◦ 2l Gαl τ , 1

αl = (2 − 2 (2l+1) )−1 ,

βl = 1 − 2αl < 0,

as a revertible approximation, of order 2(l + 1), to

(6.7) (6.8)

eτa .

Proof. It is only needed to prove for (i1 i2 · · · im ) = (12 · · · m). 1◦ It is easy to prove that the phase flow eta has the series expansion eta (x) = x +

∞ k  t k=1

k!

ak (x),

x ∈ Rn ,

a2 (x) =

∂a1 (x) a(x), ∂x

where a1 (x) = a(x), ak (x) =

∂ak−1 (x) a(x), ∂x

t  0,

k = 1, 2, · · · .

(6.9)

(6.10)

The assumption that for i = 1, 2, · · · , m, Gτi are approximations of order p ≥ 1, to eτa(i) implies that for all x ∈ Rn ,

456

10. Volume-Preserving Methods for Source-Free Systems

Gτi (x) = x + τ a(i) (x) + O(τ 2 ),

τ  0,

i = 1, 2, · · · , m.

(6.11)

So, from Taylor expansion, we have that for x ∈ Rn ,   (Gτ2 ◦ Gτ1 )(x) = Gτ2 Gτ1 (x) = x + τ (a(1) (x) + a(2) (x)) + O(τ 2 ),

τ  0. (6.12)

By induction for m, we get τ 1 G(12···m) (x)

= (Gτm ◦ · · · ◦ Gτ2 ◦ Gτ1 )(x) = x + τ (a(1) (x) + a(2) (x) + · · · + a(m) (x)) + O(τ 2 ) (6.13) = x + τ a(x) + O(τ 2 ), τ  0.

This implies that 1 Gτ(12···m) is an approximation, of order one, to eτa , which provides the proof needed. In[QZ92] that 2 giτ1 i2 ···im and 2 giτ1 i2 ···im , defined by Equation (6.2), are revertible approximations, of order 2, to eτa , the conclusion 1◦ of the Lemma 6.1 is proved. 2◦ By assumption, we have that for x ∈ Rn and τ ∼ 0, 1  2 Gτi (x) = x + τ a(i) (x) + τ 2 a(i) (x) + O(τ 3 ), 2

i = 1, 2, · · · , m.

(6.14)

Taylor expansion of the right hand side of Equation (6.4) with (i1 i2 · · · im ) = (12 · · · m) yields τ 2 G(12···m) (x)

=x+τ

⎛

m 

m 

⎞

1 a(i) (x) + τ 2 ⎝ a(i) a(j) ⎠ (x) + O(τ 3 ), 2 i=1 i,j=1

τ  0. (6.15)

Here, we have used the convention (ab)(x) = (a∗ b)(x) = a∗ (x)b(x),

a∗ (x) =

∂a(x) , ∂x

(6.16)

for a, b : Rn → Rn . However, we have *m + m m m      2 (i) a = a∗ a = a a(j) = (a(i) )∗ a(j) = a(i) a(j) . i=1

∗

j=1

i,j=1

(6.17)

i,j=1

So 1 eτa (x) = x + τ a(x) + τ 2 a2 (x) + O(τ 3 ) = 2 Gτ(12···m) (x) + O(τ 3 ), 2

τ ∼ 0.

This shows that 2 Gτ(12···m) is an approximation, of order 2, to eτa . By direct verification, this is revertible if each component Gτi is revertible.  The conclusion 3◦ directly follows from Qin-Zhu’s paper[QZ93] .

10.6 Construction of Volume-Preserving Schemes

457

Lemma 6.2. Given system x˙ = a(k) (x),

 T ∂b ∂b a(k) (x) = 0, · · · , 0, k,k+1 (x), − k,k+1 (x), 0, · · · , 0 , ∂xk+1

∂ xk

(6.18) with x = (x1 , · · · , xk , xk+1 , · · · , xn )T and smooth function bk,k+1 : Rn → Rn . Then any symplectic difference scheme, of order p ≥ 1, of the Hamiltonian system on the xk -xk+1 plane x˙ k =

∂ bk,k+1 , ∂ xk+1

x˙ k+1 = −

∂ bk,k+1 , ∂ xk

(6.19)

with xj , j = k, k + 1 as parameters naturally gives a volume-preserving difference scheme, of order p, of the source-free system (6.18) on the n-dimensional (x1 , · · · , xn )T -space by simply freezing the coordinates xj , j = k, k + 1 and transforming xk and xk+1 according to the symplectic difference scheme for (6.19) in which xj , j = k, k + 1 are considered as frozen parameters. Proof. It is obvious that the so-constructed difference scheme is of order p. As to the volume-preserving property, we easily prove that it is true by direct calculation of the determinant of the Jacobian of the step-transition map of the scheme, with the notice of the fact that the determinant of the Jacobian of a symplectic map is equal to one. Now, we construct volume-preserving difference schemes for source-free systems. Let a = (a1 , · · · , an )T be a source-free field. As was proved in Section 10.5, we have essentially Hamiltonian decomposition (5.17) for a with the functions bk,k+1 given from a by (5.14) – (5.16). We denote by Skτ the step transition map of a volume-preserving difference scheme with step size τ , as constructed in Lemma 6.2, % &T ∂ bk,k+1 ∂ bk,k+1 (k) = 0, · · · , 0, ,− , 0, · · · , 0 for associated to the vector field a ∂ xk+1

k = 1, 2, · · ·.

∂ xk



Then by Lemma 6.1, we have: Theorem 6.3. τ , say Sn−1

[FW94]

1◦

A simple composition of the n − 1 components S1τ , S2τ , · · · , τ 1G

τ := Sn−1 ◦ · · · ◦ S2τ ◦ S1τ

is a volume-preserving algorithmic approximation, of order one, to eτa ; and 2g

τ

 2 ◦ 1G 2 , := 1 G τ

τ

τ 2g

2 = 1G 2 ◦ 1G τ

τ

are revertible volume-preserving algorithmic approximations, of order 2. 2◦ If each Skτ is an approximation, of order 2, to eτa(k) , then the symmetric composition τ τ τ τ τ τ τ 2 2 2 2 2 2 (6.20) 2 G = Sn−1 ◦ · · · ◦ S2 ◦ S1 ◦ S1 ◦ S2 ◦ · · · ◦ Sn−1 is a volume-preserving approximation, of order 2, to eτa . 3◦ If each Skτ is revertible, then the so-constructed 2 Gτ is revertible too. 4◦ From the above constructed revertible algorithmic approximation 2 g τ or τ 2 G , we can further recursively construct revertible approximations, of all even orders, to eτa according to the process of Lemma 6.1.

458

10. Volume-Preserving Methods for Source-Free Systems

Remark 6.4. If a has essentially Hamiltonian decompositions other than (5.17) and (5.14) – (5.16), then one can construct volume-preserving difference schemes corresponding to these decompositions in a similar way to the above.

10.7 Some Special Discussions for Separable Source-Free Systems For a source-free field a = (a1 , · · · , an )T with essentially Hamiltonian decomposition  = ( x1 , · · · , x n )T as determined from (5.17), we take Skτ : x = (x1 , · · · , xn )T → x the following: ⎧ x j = xj , j = k, k + 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂b k , xk+1 , · · · , xn ), x k = xk + τ k,k+1 (x1 , · · · , xk−1 , x (7.1) ∂ xk+1 ⎪ ⎪ ⎪ ⎪ ∂b ⎪ ⎩ x k+1 = xk+1 − τ k,k+1 (x1 , · · · , xk−1 , x k , xk+1 , · · · , xn ). ∂ xk

τ ◦ · · · ◦ S2τ ◦ S1τ from Then, simple calculations show that 1 Gτ = Sn−1 ⎧ x 1 = x1 + τ a1 ( x1 , x2 , · · · , xn ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x j = xj + τ aj ( x1 , · · · , x j , xj+1 , · · · , xn ) ⎪ ⎪ ⎨ - x j  j−1 ∂ al ⎪ ⎪ ( x1 , · · · , x j−1 , t, xj+1 , · · · , xn )dt, j = 2, · · · , n − 1, +τ ⎪ ⎪ ∂ xl ⎪ x j ⎪ l=1 ⎪ ⎪ ⎪ ⎩ x1 , · · · , x n−1 , xn ), x n = xn + τ an ( (7.2)  −1  τ = 1 G−τ and 1 G is given from

⎧ x n = xn + τ an (x1 , · · · , xn−1 , x n ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j+1 , · · · , x n ) x j = xj + τ aj (x1 , · · · , xj , x ⎪ ⎪ ⎨ - x j  j−1 ∂al ⎪ ⎪ (x1 , · · · , xj−1 , t, x j+1 , · · · , x n )d t, −τ ⎪ ⎪ ⎪ xj l=1 ∂xl ⎪ ⎪ ⎪ ⎪ ⎩ x 1 = x1 + τ a1 (x1 , x 2 , · · · , x n ).

j = 2, · · · , n − 1,

(7.3) (7.2) and (7.3) are both volume-preserving difference scheme, of order 1, of the source-free system associated to the field a, with the step-transition maps 1 Gτ and τ 1 G . They can be composed into revertible volume-preserving schemes of order 2, τ  τ2 : x = (x1 , · · · , xn )T → say, 2-stage scheme with step transition map 2 gτ = 1 G 2 ◦1 G T x  = ( x1 , · · · , x n ) as follows,

10.7 Some Special Discussions for Separable Source-Free Systems

⎧ 1 1 τ  ⎪ n2 , x n2 = xn + an x1 , · · · , xn−1 , x ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ τ  ⎪ 2 2 2 ⎪ = x + a , · · · , x , x  , · · · , x  x  x n i i 1 i ⎪ i i+1 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ - x i12  ⎪ i−1 ⎪ 1 1 ⎪ τ ∂ al  ⎪ 2 ⎪ x1 , · · · , xi−1 , t, x i+1 ,···,x n2 d t, − ⎪ ⎪ 2 xi ∂ xl ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ τ  ⎪ ⎨ x 12 = x1 + a1 x1 , x 22 , · · · , x n2 ,

459

i = 2, · · · , n − 1,

2

1 1 1 ⎪ τ  ⎪ ⎪ x 1 = x 1 , x 12 + a1 x 22 , · · · , x n2 , ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ τ  ⎪ 2 ⎪ x j = x 1 , · · · , x j2 + aj x j , x j+1 ,···,x n2 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ j−1 ⎪ x j  ⎪ 1 1 ⎪ τ ∂ al  ⎪ 2 ⎪ + x 1 , · · · , x j−1 , t, x j+1 ,···,x n2 d t, 1 ⎪ ⎪ 2 x 2 ∂ xl ⎪ ⎪ j l=1 ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ τ  ⎩ x n = x 1 , · · · , x n2 + an x n−1 , x n2 .

j = 2, · · · , n − 1,

2

(7.4) Either (7.2) or (7.3) contains n − 1 implicit equations generally. But for fields a with some specific properties, it will turn into explicit. For example, ∂ ai = 0, ∂ xi

i = 1, · · · , n

(7.5)

(i.e., ai does not depend on xi ), then (7.2) turns into explicit[QZ93] ⎧ x 1 = x1 + τ a1 (x2 , · · · , xn ), ⎪ ⎪ ⎨ x1 , · · · , x j−1 , xj+1 , · · · , xn ), j = 2, · · · , n − 1, x j = xj + τ aj ( ⎪ ⎪ ⎩ x n = xn + τ an ( x1 , · · · , x n−1 , xn ).

(7.6)

For details, see the Section 10.2. We note that, for a = (a1 , · · · , an )T , a=

n 

a{k} ,

a{k} = (0, · · · , 0, ak , 0, · · · , 0)T ,

k = 1, 2, · · · , n.

(7.7)

k=1

It is easy to verify that if a = (a1 , · · · , an )T satisfies the condition (7.5), then the scheme (7.6) is just the result of composing the Euler explicit schemes of the systems associated to the fields a{k} (k = 1, · · · , n), i.e., we have τ 1G

where

= Eaτ{n} ◦ · · · ◦ Eaτ{2} ◦ Eaτ{1} ,

Eaτ{k} = I + τ a{k} ,

k = 1, 2, · · · , n,

I = identity.

(7.8) (7.9)

460

10. Volume-Preserving Methods for Source-Free Systems {k}

In fact, Eaτ{k} are the phase flows eτa{k} , since a∗ a{k} = 0 for k = 1, 2, · · · , n, which is implied by the condition (7.5). According to Theorem 6.3, we then get a 2nd order explicit revertible volume-preserving scheme, with step transition map 2G

τ

τ

τ

τ

τ

τ

τ

 2{1} ◦ E  2{2} ◦ · · · ◦ E  2{n} = Ea2{n} ◦ · · · ◦ Ea2{2} ◦ Ea2{1} ◦ E a a a =

τ

1G 2

 τ2 = 2 gτ . ◦ 1G

(7.10)

10.8 Construction of Volume-Preserving Scheme via Generating Function Not only symplectic scheme can be constructed via generating function, but volumePreserving scheme is also constructed via generating function. A. Thyagaraja and F.A. Haas[TH85,Sco91] give an important type generating function for volume-preserving mapping in 3-dimensions. It is however, not complete both in generality and in systematization. The complete results are given by Z.J. Shang[Sha94a,Sha94b] .

10.8.1 Fundamental Theorem 5

5 6 6 α α B Aα Bα A Theorem 8.1. Let α = ∈ GL(2n), α−1 = . Assume Cα Dα C α Dα that g : Rn → Rn , z = g(z) is a differentiable mapping given, in some point z0 ∈ Rn , satisfying transversality condition    ∂g(z)  C α  + D (8.1) α  = 0.  ∂z Then, in Rn neighborhood W of point w0 = Cα g(z0 ) + Dα z0 exists a unique differentiable mapping f (w) = fα,g = (f1 (w), f2 (w), · · · , fn (w)): Wn → Rn such that z = g(z) satisfying condition      ∂f (w)    ∂f (w)     (8.2)  ∂w Cα − Aα  = Bα − ∂w Dα  = 0, such that, mapping z = g(z) can reconstruct in a neighborhood V of the point z = z0 from w  = f (w) by the relation Aα z + Bα z = f (Cα z + Dα z).

(8.3)

Conversely, for any differential mapping f (w) = (f1 (w), · · · , fn (w)) : Rn → R , satisfying condition (8.2) at the point w0 ∈ Rn , we give a unique differential mapping in some neighborhood V of the point z0 = C α f (w)+Dα w0 (8.3). Moreover, the transversality condition (8.1) is satisfied for the mapping g at the point z0 = C α f (w) + Dα w0 . n

10.8 Construction of Volume-Preserving Scheme via Generating Function

461

Remark 8.2. Generally speaking, a volume-preserving mapping f is uniquely determined by the matrix α ∈ GL(2n) and mapping g as above by relation (8.3) determined by mapping f = fα,g . We called f = fα,g as generating mapping dependent on α, g. Remark 8.3. We only consider some typical types generating mapping 5 6 Ess In − Ess α(s,s) = , 1 ≤ s ≤ n, Ess In − Ess

(8.4)

where Ess denotes the n × n matrix, of which only the entry at the s-th row and the s-th column is equal to 1, and all other entries are 0. In this case, (8.2) and (8.3) have much more simple forms. For example, for α = α(1,1) , (8.2) turns into    ∂(f2 , · · · , fn )  ∂f1  = 0, =  (8.5) ∂w1 ∂(w2 , · · · , wn )  and (8.3) turns into

z1 = f1 (z1 , z2 , · · · , zn ), z2 = f2 ( z1 , z2 , · · · , zn ), (8.6)

··· z1 , z2 , · · · , zn ). zn = fn ( The same situation also applies for α(s,s) .

Remark 8.4. For such a matrix α(1,1) , generating mapping f (w) of type α(1,1) , there are n component f (w) = (f1 (w), f2 (w), · · · , fn (w)), in which n − 1 component f2 (w), · · · , fn (w) is linear independent, satisfying condition    ∂(f2 , · · · , fn )     ∂(w2 , · · · , wn )  = 0, then we can express the first component f1 of f by other n − 1 component  - w1   ∂(f2 , · · · , fn )    (ξ, w2 , · · · , wn )dξ, f1 (w1 , w2 , · · · , wn ) = C(w2 , · · · , wn ) +   w1,0 ∂(w2 , · · · , wn ) (8.7) where C is scalar function dependent n − 1 variable.  A  Aα B α  Bα  α ∈ GL(2n) and α−1 = . SupCα Dα C α Dα pose |Cα + Dα | = 0 for some M0 ∈ GL(2n), then there exists a dependent at t and α of the diffeomorphism gαt (M0 z) generating mapping, f (w, t) = fα,a (w, t), its phase flow gαt satisfying Theorem 8.5. Let α =

z˙ = a(z),

a(z) = (a1 (z), · · · , an (z))T ,

z = (z1 , · · · , zn )T ,

(8.8)

462

10. Volume-Preserving Methods for Source-Free Systems

such that ∂f = ∂t

%

& ∂f Aα − Cα a(Aα f + B α ω), ∂ω

(8.9)

f (w, 0) = (Aα + Bα )(Cα + Dα )−1 ω.

(8.10)

We call (8.9) a Hamilton–Jacobi equation. The proofs of Theorems 8.1 and 8.2 can found in [Sha94b] . Remark 8.6. If α = α(1,1) , then relations (8.9) and (8.10) turn into ∂ f1 ∂f = −a1 (w1 , f2 , · · · , fn ) 1 , ∂t ∂ w1 ∂ fk ∂f = ak (w1 , f2 , · · · , fn ) − a1 (w1 , f2 , · · · , fn ) k , ∂t ∂ w1

fk (w1 , · · · , wn , 0) = wk ,

k = 1, 2, · · · , n.

(8.11) k = 2, · · · , n, (8.12) (8.13)

When a is source-free system, i.e., div a(z) =

n  ∂ak k=1

∂zk

(z) = 0,

z ∈ Rn ,

then gαt is volume-preserving, we get    ∂(f2 , · · · , fn )  ∂f1  (w, t).  (w, t) =  ∂w1 ∂(w2 , · · · , wn ) 

(8.14)

(8.15)

From (8.11), (8.13), and (8.15), we get    ∂(f2 , · · · , fn )   (w, τ )dτ.  a1 (w1 , f2 (w, τ ), · · · , fn (w, τ1 ))  f1 (w, t) = w1 − ∂(w2 , · · · , wn )  0 (8.16) f2 , · · · , fn is independently determined by (8.12) and (8.13) (for k = 2, · · · , n) we call these as generating function type α(1,1) for source-free system (8.8). -

t

Theorem 8.7. Suppose vector field a is analytical function of z, then f (w, t) = fα,a (w, t), is solution of Cauchy problem (8.9) and (8.10), it is expressible as a convergent power series in t for sufficiently small |t|, with the recursively determined coefficients f (w, t) =

∞ 

f (k) (w)tk ,

(8.17)

k=0

(8.18) f 0 (w) = N0 w, N0 = (Aα + Bα )(Cα + Dα )−1 , 1 −1 f (w) = L0 a(E0 w), E0 = (Cα + Dα ) , L0 = Aα − N0 Cα , (8.19) for k ≥ 1, we have

10.8 Construction of Volume-Preserving Scheme via Generating Function

f

(k+1)

m k 1 ∂ f (k) (w) 1  (w) =− Cα a(E0 w) − k+1 ∂w k+1

ip ≥1



m=1 j=1 i1 +···+ij

·

∂f

+

(k−m)

(w)

∂w k 1  k+1

463

1 j! =m

j Cα Dα,E (Aα f (i1 ) (ω), · · · , Aα f (ij ) (w)) 0w



m=1 i1 +···+im =k ip 1

1 m Aα Dα,E (Aα f (i1 ) (ω), · · · , Aα f (im ) (w)), 0w m!

(8.20) (k)

(k)

where for ξ (k) = (ξ1 , · · · , ξn )T ∈ Rn (k = 1, 2, · · · , m), we get ⎡ ⎢ ⎢ ⎢ ⎢ m Dα,w (ξ (1) , · · · , ξ (m) ) = ⎢ ⎢ ⎢ ⎣

n 

∂ m a1 (w) ξα1 1 · · · ξαmm ∂z · · · ∂z α α 1 m α1 ,···,αm =1 .. . n  ∂ m an (w) ξ 1 · · · ξαmm ∂zα1 · · · ∂zαm α1 α ,···,α =1 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(8.21)

m

Proof. Under the above proposition, if generating function f (w, t) = fα,a (w, t) is dependent analytically on w and t in some neighborhood Rn for sufficient small t, then it can be expressed as a power series f (w, t) =

∞ 

f (k) (w)tk .

k=0

Differentiating it with respect to w and t, we get ∞  ∂f ∂ f (k) (w) k (w, t) = t , ∂w ∂w

(8.22)

∂f (w, t) = ∂t

(8.23)

k=0 ∞ 

(k + 1)f (k+1) (w)tk .

k=0

By (8.10), f (0) (w) = f (w, 0) = N0 w. This is (8.19). Denote E0 = Aα N0 + B α = (Cα + Dα )−1 , then Aα f (w, t) + B α w = E0 w +

∞  k=1

Expanding a(z) at z = E0 w, we get

Aα f (k) (w)tk .

464

10. Volume-Preserving Methods for Source-Free Systems * α

α

α(A f (w, t) + B w) = a E0 w +

∞ 

+ α

A f

(k)

k

(w)t

k=1

= a(E0 w) +

∞  k=1

k 

tk



m=1 i1 +···+im =k ip 1

1 m Da,E0 w (Aα f (i1 ) (w), · · · , Aα f (im ) (w)). m! (8.24)

Here, Da,E0 is multilinear operator defined by (8.21). Substituting (8.22) and (8.24) in the right hand side of Equation (8.9), substituting (8.23) in the left hand side of (8.9), and then comparing the coefficients of tk on both sides, we get the recursions (8.18) – (8.20). The proof is completed.  Remark 8.8. Let α = α(1,1) , then (8.18) – (8.20) turn into f (0) (w) = w, f (1) (w) = " a(w),

(8.25) T

" a(w) = (−a1 (w), a2 (w), · · · , an (w)) ,

(8.26)

for k ≥ 1, (k+1)

fi

(w)

=

(k) ∂f (w) 1 $1 (w) i∂w k+1 a 1

·

∂

(k−m) fi (w)

∂ w1 k 1  + k + 1 m=1 i

+

1 k+1

k−1 

m 



n 

m=1 j=1 i1 +···+ij =m α1 ,···,αj =2

1 j!

ip 1

∂ j a"1 (w) (i ) f (i1 ) (w) · · · fαjj (w) ∂ wα1 · · · ∂ wαj α1 n 



1 +···+im =k α1 ,···,αm =2 ip 1

∂ m a"i (w) 1 (i ) (im ) f 1 (w) · · · fαm (w), m! ∂wα1 · · · ∂wαm α1

i = 1, 2, · · · , n.

(8.27)

10.8.2 Construction of Volume-Preserving Schemes In this subsection, we consider the construction of volume-preserving schemes[Sha94a] for the source-free system (8.8). By Remark 8.3 of Theorem 8.1, for given time" t) : dependent scalar functions φ2 (w, t), · · · , φn (w, t) : Rn × R → R and C(w, Rn−1 × R → R, we can get a time-dependent volume-preserving mapping g"(z, t). If φ2 (w, t), · · · , φn (w, t) approximates the generating functions f2 (w, t), · · · , fn (w, t) " t), g"(w, t) of the type α(1,1) of the source-free system (8.8), then suitable choice C(w, approximates the phase flow gαt (z) = g(z, t). Fixing t as a time step, we can get a difference scheme (volume-preserving schemes) whose transition from one time step to the next is volume-preserving. By Remark 8.8 of Theorem 8.7, generating functions f2 (w, t), · · · , fn (w, t) can be expressed as power series. So, a natural way to approximate f2 (w, t), · · · , fn (w, t) is take the truncation of the series. However, we have to choose a suitable C(w, " t) in (8.7) to guarantee the accuracy of the scheme. Assume that m  (k) φm (w, t) = fi (w)tk , i = 2, · · · , n (8.28) i k=0

10.8 Construction of Volume-Preserving Scheme via Generating Function

and ψ1m (w, t) =

m 

(k)

f1 (w)tk .

465

(8.29)

k=0

Let for some fixed value w1,0 , (m)

C m (w2 , · · · , wn , t) = ψ1 and (m)

φ1 (w, t) = C (m) (w2 , · · · , wn , t) +

-

w1 w1,0

(w1,0 , w2 , · · · , wn , t)

(8.30)

  (m)  ∂ (φ(m) , · · · , φn )  2   ∂ (w2 , · · · , wn )  (ξ, w2 , · · · , wn , t)d ξ, (8.31)

then we have, Theorem 8.9. Using Theorem 8.5 and Theorem 8.7 for sufficiently small τ ≥ 0 (m) (m) as the time step, defining mapping φ(m) (w, τ ) = (φ1 (w, τ ), φ2 (w, τ ), · · · , (m) (m) φn (w, τ ))T with the components φi (w, τ )(i = 1, 2, · · · , n) given as above for m = 1, 2, · · · , then the mapping w −→ w  = φ(m) (w, τ ),

(8.32)

defines a volume-preserving scheme z = z k → z k+1 = z ⎧ k+1 k ⎨ z1k = φ(m) , z2 , · · · , znk , τ ), 1 (z1 ⎩

(m)

zik+1 = φi

(z1k+1 , z2k , · · · , znk , τ ),

i = 2, · · · , n,

(8.33)

of m-th order of accuracy of the source-free system (8.8). (m)

Proof. Since φi

(w, 0) = fi0 (w, 0) = wi (i = 2, · · · , n),    ∂(φ(m) , · · · , φ(m) )  n   2  (w, 0) = 1.   ∂(w2 , · · · , wn ) 

Therefore, for sufficiently small τ and in some neigbourhood of Rn    ∂(φ(m) , · · · , φ(m) )  n   2  (w, τ ) = 0.   ∂(w2 , · · · , wn )  By Theorem 8.1, Remark 8.3, Remark 8.4, and Equation (8.31), the relation (8.33) defines a time-dependent volume-preserving z = z k → z k+1 = z = g"(z, τ ). That is, (8.33) is a volume-preserving scheme. Noting that (m)

φi

(w, τ ) = fi (w, τ ) + O(τ m+1 ),

(m)

ψ1

(w, τ ) = f1 (w1 , τ ) + O(τ m+1 ),

i = 2, · · · , n,

466

10. Volume-Preserving Methods for Source-Free Systems

for sufficiently small τ and -

w1

f1 (w, τ ) = f1 (w1,0 , w2 , · · · , wn , τ ) +

w1,0

   ∂ (f2 , · · · , fn )     ∂ (w2 , · · · , wn )  (ξ, w2 , · · · , wn )d ξ,

we have from (8.31) (m)

φ1 (w, τ ) = f1 (w, τ ) + O(τ m+1 ). (m)

(m)

So, φ(m) (w, τ ) = (φ1 (w, τ ), · · · , φn (w, τ )) is an m-th order approximant to f (w, τ ) = (f1 (w, τ ), · · · , fn (w, τ )), the generating function of the type α1,1 of gαt and hence the volume-preserving scheme (8.33) is of m-th order of accuracy. The proof is completed.  Remark 8.10. We note that the volume-preserving scheme z = z k → z k+1 given by (8.33) is implicit for only one new variable z1k+1 and explicit for all other new variables zik+1 (i = 2, · · · , n) in terms of the old variables zik (i = 2, · · · , n). Remark 8.11. We can get volume-preserving scheme similar to the above one if we consider the types α = α(s,s) (2 ≤ s ≤ n), instead of α = α(1,1) . Example 8.12. First order scheme: ⎧ (1) k+1 k k k ⎪ ⎨ z1 = φ1 (z1 , z2 , · · · , zn , τ ), (1) k+1 k k k ⎪ ⎩ zi = φi (z1 , z2 , · · · , zn , τ ),

i = 2, · · · , n,

where (1)

φ1 (w, τ )

(1)

φi (w, τ )

−τ a1 (0, w2 , · · · , wn )  ∂ a2 ∂ a2   1+τ τ  ∂ w2 ∂ w3   ∂ a3 ∂ a3  1+τ - w1  τ  ∂ w ∂ w3 2  +  0  .. ..  . .    ∂ an ∂ an  τ  τ ∂ w2 ∂ w3

=

=

···

τ

∂ a2 ∂ wn

···

τ

∂ a3 ∂ wn .. .

···

1+τ

∂ an ∂ wn

          (ξ, w2 , · · · , wn )d ξ,        

wi + τ ai (w).

Second order scheme: ⎧ (2) k+1 k k k ⎪ ⎨ z1 = φ1 (z1 , z2 , · · · , zn , τ ), (2) k+1 ⎪ = φi (z1k+1 , z2k , · · · , znk , τ ), ⎩ zi

i = 2, · · · , n,

10.9 Some Volume-Preserving Algorithms

467

where (2) φ1 (w, τ ) (2) φi (w, τ )

=

(2) ψ1 (0, w2 , · · · , wn , τ )

=

(2) ψi (w, τ ),

-

w1

+ 0

   ∂ (ψ2(2) , · · · , ψn(2) )     ∂ (w2 , · · · , wn )  (ξ, w2 , · · · , wn )d ξ,

i = 2, · · · , n,

and  (2) T 1 ∂" a(w) (2) ψ (2) (w, τ ) = ψ1 (w, τ ), · · · , ψn (w, τ ) = w + τ" a(w) + τ 2 " a(w), 2

T



" a(w) = − a1 (w), a2 (w), · · · , an (w)

∂ w1

.

10.9 Some Volume-Preserving Algorithms In this section, we analyze and study under conditions a source-free system that has volume-preserving R–K schemes.

10.9.1 Volume-Preserving R–K Methods Consider the system dz = a(z), dt

where

5 z=

a(z) =

x y

6 ,

x ∈ Rp ,

y ∈ Rq , (9.1)

 g(y)  f (x)

.

Obviously, this is a source-free system. Its phase flow in Rp+q preserves the phase volume of (p + q) form d x1 ∧ d x2 ∧ · · · ∧ d xp ∧ d y1 ∧ d y2 ∧ · · · ∧ d yq . Only R–K and P–R–K are to be discussed. We wish, some of the phase volume is preserved. The formula of a general m-th stage P–R–K method with time step h applied to system (9.1) is read as

468

10. Volume-Preserving Methods for Source-Free Systems

ξi = x n + h

m 

dij g(ηj ), ηi = yn + h

j=1

xn+1 = xn + h

m 

m 

cij f (ξj ),

1 ≤ i ≤ m,

j=1

δj g(ηj ),

(9.2)

j=1

yn+1 = yn + h

m 

γj f (ξj ),

j=1

here ξi ∈ Rp , ηi ∈ Rq (1 ≤ i ≤ m) are auxiliary vectors used to compute updates (xn+1 , yn+1 ). [DV84] Suppose (9.2) is irreducible, that is, if i = j, then ξi = ξj or ηi = ηj . We have following Lemma of Y.B. Suris[Sur96] . Lemma 9.1. Let δ = [δ1 , δ2 , · · · , δm ]T , D = (dij ), C = (cij ), e = [1, 1, · · · , 1]T be a m-dimensional vector D− = eδT − D, C − = eδ T − C. The P–R–K method preserves phase volume for system (9.1) in arbitrary dimensions, iff dk1 l1 cl,k2 · · · dkr−1 lr−1 clr−1 kr dkr lr clr k1 − − − − − = d− k1 l1 cl1 k2 · · · dkr−1 lr−1 clr−1 kr dkr lr clr k1 .

For arbitrary 1 ≤ r ≤ m and two arbitrary ordered sets (k1 , · · · , kr ) and (l1 , · · · , lr ) of different natural numbers from (1, m), dij and cij are elements (i, j) with respect to matrix D − and C − . Next, for system (9.1), we construct some volume-preserving method by P–R–K method, using the above criteria. First we consider volume-preserving by R–K method for linear system. Linear system of ODE is read as y˙ = M y,

(9.3)

where M is n × n matrix with trace M = 0. If det M = 0, the system (9.3) can degrade to a lower stage, so we assume det M = 0. Now, we assume that M is a constant matrix. As in R–K method, (A.b.c) applied to system (9.3) takes the form Yi = yn + h

s 

aij M Yj ,

j=1

yn+1 = yn + h

s 

(9.4) bj M Yj ,

j=1

where A = (aij )s×s , b = [b1 , b2 , · · · , bs ]T . Here, we just talk about R–K method and according to Lemma 4.2, we cannot find a general volume-preserving R–K method. So, our hope is to distinguish M into

10.9 Some Volume-Preserving Algorithms

469

different classes and find out whether there are volume-preserving R–K method in any class. Now, we need the following notations: A = A ⊗ En ,

M = diag (M, M, · · · , M ) = Es ⊗ M,

b = bT ⊗ En ,

Y = [Y1 , Y2 , · · · , Ys ]T

y n = [yn , yn , · · · , yn ]T ,

(9.5)

e = e ⊗ es ,

where En is an n-stage identical matrix and e = [1, 1, · · · , 1]T is a n-dimensional vector. For R–K method to be volume-preserving, we have equivalent condition: det

∂ yn+1 ∂ yn+1 ≡ 1. So, we need to calculate the matrix ≡ 1. In matrix notations, ∂ yn ∂ yn

R–K method (9.4) reads yn+1 = yn + hM bY, (9.6)

Y = (1 − hM A)−1 y n . So, yn+1 = (En + (hM b(I − hM A)−1 e)yn ∂yn+1 =⇒ = En + hM b(I − hM A)−1 e. ∂yn

(9.7)

Lemma 9.2. Let A, D be non-degenerate m × m and n × n matrices respectively and B an m × n and C an n × m matrix, then det A det (D + CA−1 B) = det D det (A + BD−1 C).

(9.8)

The proof can be found in any textbook of linear algebra. By Lemma 9.2, it is easy to get from (9.7) % & ∂ yn+1 det (I − hM A − eM b) . det = det (I − hM A)

∂ yn

Additionally, we define the notations A− = (a− ij ),

a− ij = aij − bj ,

N = A ⊗ M,

N − = A− ⊗ M.

In these notations, (9.7) reads as & % ∂ yn+1 det (I − hN − ) = det . ∂ yn

det (I − hN )

(9.9)

(9.10)

Now, if (9.10) is identical to 1, we arrive at the criterion for R–K method (9.4) to be volume-preserving scheme as det (λI − N − ) = det (λI − N ),

∀ λ ∈ R.

(9.11)

470

10. Volume-Preserving Methods for Source-Free Systems

Theorem 9.3. If dimension of M is odd, then all the R–K methods based on high order quadrature formula such as Gauss–Legendre, Radau, and Lobatto are not volumepreserving. Proof. Note that N = A ⊗ M and N − = A− ⊗ M . If the method is volumepreserving, then det N = det(N − ) ⇐⇒ det(A ⊗ M ) = det(A− ⊗ M ) ⇐⇒ (det A)n (det(M ))s = (det(A− ))n (det M )s ⇐⇒ (det A)n = (det(A− ))n ⇐⇒ det A = det(A− ).

(9.12)

Now, we need the W -transformation proposed by Hairer and Wanner[HW81] . They introduced a generalized square matrix W defined by W = (p0 (c), p1 (c), · · · , pn−1 (c)),

(9.13)

where the normalized shifted Legendre polynomials are defined by pk (x) =

k  k  k + i   √ xi , 2k + 1 (−1)k+i i i

k = 0, 1, · · · , s − 1. (9.14)

i=0

For Gauss–Legendre, let X = W −1 AW , then ⎡ 1 −ξ1 ⎢ 2 ⎢ ξ 0 −ξ2 ⎢ 1 ⎢ ⎢ ξ2 X=⎢ ⎢ .. .. ⎢ . . ⎣

⎤

ξs−1

−ξs−1 0

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

(9.15)

1 (k = 0, 1, · · · , s − 1). 2 4k2 − 1

where ξk = √

However, X − = W −1 A− W , then ⎡ 1 −ξ1 − ⎢ 2 ⎢ ξ 0 ⎢ 1 ⎢ − ξ 2 X =⎢ ⎢ ⎢ ⎣

⎤ −ξ2 ..

.

..

.

ξs−1

−ξs−1 0

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

It is easy to verify that det X = det(X − ) ⇒ det A = det(A− ). So, Gauss–Legendre method is not volume-preserving. Using the Table 2.1 of Chapter 7, the remaining part of the proof is similar, where σ ∈ R and uσ = 0. 

10.9 Some Volume-Preserving Algorithms

471

Theorem 9.4. [QL00] If the dimension of M is even, then the R–K methods based on high order quadrature formulas such as Gauss–Legendre, Lobatto III A, Lobatto III B, Lobatto III S, Radau IB, and Radau IIB are volume-preserving, iff λ(M ) = (λ1 , λ2 , · · · , λ n2 , −λ1 , −λ2 , · · · , −λ n2 ). Proof. Assume A, B are n × n and m × m matrices respectively, and their eigenvalue are respectively{λ1 , λ2 , · · · , λn } and {μ1 , μ2 , · · · , μm }. Then according to the property of Kronecker product, we have λ(A ⊗ B) = {λi μj , i = 1, · · · , n; j = 1, · · · , m}. For R–K methods to be volume-preserving schemes, according to (9.11), N and N − must have same eigenvalue, that is to say, A ⊗ M and A− ⊗ M must have the same eigenvalues. For example, for Gauss–Legendre method, λ(A) = λ(X) and λ(A− ) = λ(X − ), however, it is obvious that λ(X) = −λ(X − ), so according to the properties of Kronecker product, we can easily verify that A ⊗ M and A− ⊗ M have the same eigenvalues.  Remark 9.5. If (9.3) is a Hamiltonian system, that is to say, M = J −1 S, where  0 In  and S  = S is an n × n invertible matrix, then J= −In 0   λ(M ) = λ1 , λ2 , · · · , λ n2 , −λ1 , −λ2 , · · · , λ n2 . So, the R–K method based on high order quadrature formula (Gauss–Legendre, Lobatto IIIA, Lobatto IIIB, Lobatto IIIS, Radau IB, and Radau IIB) are volumepreserving. The Theorem 9.4 says that for the methods to preserve volume, the system, in some sense, must be similar to a Hamiltonian system. If the matrix M similar to an infinitesimally symplectic matrix, i.e., there is an invertible matrix P , subjected to P −1 M P = JS, S T = S, then we can transform the system to a Hamiltonian system by a coordinate transformation. In this situation, the volume-preserving R–K methods and the symplectic R–K methods almost have no difference, that is, if P is a symplectic matrix, then volume-preserving R–K methods are equivalent to symplectic R–K methods; and in this case, they can be transformed to one another by a linear transformation.

10.9.2 Volume-Preserving 2-Stage P–R–K Methods In the case r = 1, if the necessary and sufficient condition of Lemma 9.5 are satisfied, then a 2-stage P–R–K method is volume-preserving. This condition is the same condition of symplecity on the class of separable Hamiltonian system. Thus for system (9.3), all 2-stage P–R–K methods proposed in[Sun95] are volume-preserving algorithms[QL00] . Example 9.6. 3th order Radau IA-IA method

472

10. Volume-Preserving Methods for Source-Free Systems

0

1 4

1 4

0

0

0

2 3

1 4

5 12

2 3

1 3

1 3

1 4

3 4

1 4

3 4

−

Example 9.7. 3th order Radau IIA-IIA method 1 12

1 3

1 3

0

3 4

1 4

1

1

0

3 4

1 4

3 4

1 4

1 3

5 12

1

−

Example 9.8. 2nd order Lobatto IIIC-IIIC method 0

1 2

−

1 2

1

1 2

1 2

1 2

1 2

0

0

0

1

1

0

1 2

1 2

Example 9.9. 4th order Gauss IA-IA method √ 1 3 − 2 6 √ 1 3 + 2 6

√ 1 3 − 2 6 √ 3 1 + 2 6

1 + 2σ 4

√ 1 − 2σ 3 − 4 6

√ 1 − 2σ 3 + 4 6

1 + 2σ 4

1 2

1 2

1 − 2σ 4

√ 3 1 + 2σ − 4 6

√ 1 + 2σ 3 + 4 6

1 − 2σ 4

1 2

1 2

10.9 Some Volume-Preserving Algorithms

473

10.9.3 Some Generalizations Method in[Sur96] can be applied to the extension of following system: x˙ = g(y),

y˙ = h(z),

z˙ = f (x),

x, y, z ∈ RP .

(9.16)

For this system, we consider multi-stage P–R–K method ξi = xn + h

m 

dij g(ηj ), ηi = yn + h

j=1

m 

cij h(wj ), wj = zn + h

j=1

m 

eij f (ξj ),

j=1

(9.17)

xn+1 = xn +h

m 

αj g(ηj ), yn+1 = yn +h

j=1

m 

βj h(wj ), zn+1 = zn +h

j=1

m 

γj (ξj ).

j=1

Theorem 9.10. A multi-stage P–R–K method is volume-preserving for a system type (9.17), iff dk1 l1 cl1 m1 em1 k2 dk2 l2 cl2 m2 em2 k3 · · · dkr lr clr mr emr kr − − − − − − − − = d− k1 l1 cl1 m1 em1 k2 dk2 l2 cl2 m2 em2 k3 · · · dkr lr clr mr emr kr

(9.18)

for arbitrary 1 ≤ r ≤ m and three arbitrary ordered sets (k1 , · · · , kr ), (l1 , · · · , lr ), and (m1 , · · · , mr ) of different natural number form [1, · · · , m], here dij , cij , eij , d− ij , − − cij , eij are defined by Lemma 9.5. Example 9.11. A multi-stage P–R–K method 0

1 2

−

1 2

1

1 2

1 2

1 2

1 2

0

0

0

a

a

0

1

1

0

b+c

b

c

1 2

1 2

b

2c

Suitably choose a, b, c, as method can get global truncation error with order O(h2 ). Remark 9.12. Dimension of x, y, z may be different. Remark 9.13. Theorem 9.10 can be extended with no difficulty to the following system: x˙ 1 = f2 (x2 )

x˙ 2 = f3 (x4 ),

···,

x˙ n = f1 (x1 ).

(9.19)

474

10. Volume-Preserving Methods for Source-Free Systems

10.9.4 Some Explanations We usually state that symplectic method is volume-preserving scheme. But this parlance is somewhat irrelevancy because symplectic scheme (satisfying symplectic condition), true only in this case, that it applied to Hamiltonian system. For P–R–K method (dij , δi , cij , γj ), if it satisfies γi dij + δi cji − γi δj = 0, we can say this integrator is symplectic. If system is not Hamiltonian, we cannot say that this P–R–K method is symplectic. The main problem is that we say a scheme is symplectic because it preserves symplectic structure for a given system. Therefore, only Hamiltonian system possesses symplectic structure. Consequently, we cannot say “volume-preserving P–R–K methods form a subset of symplectic ones”. Until now, we gave some criteria for volume-preserving by R–K and P–R–K methods. In fact, it is almost impossible based on these criteria to construct volumepreserving algorithm with high order accuracy. Indeed, we even cannot predict that there exists schemes which satisfied those criteria. We are too far to resolve these problems. It should be noted that in the above discussion, we always suppose system is not reducible. In other words, det M = 0. But in practice, some systems are reducible, for example x˙ = cy − bz, y˙ = az − cx,

a, b, c ∈ R.

z˙ = bx − ay, In fact, for this system, centered Euler scheme is volume-preserving. Furthermore, LobattoIIIA, LobattoIIIB, LobattoIIIS, RadauIB, RadauIIB etc. are also volumepreserving. With detailed analysis of the process in Subsection 10.9.2, it is easy to get the following[QL00] . Theorem 9.14. If the dimension of M is odd, then the R–K methods based on high order quadrature formulae, such as LobattoIIIA, LobattoIIIB, LobattoIIIS, RadauI, RadauIIB etc., are volume-preserving, iff   λ(M ) = λ1 , λ2 , · · · , λ n2 , 0, −λ1 , −λ2 , − · · · , −λ n2 . We also find that in Theorem 9.4, det M = 0 is not necessary. As for nonlinear systems, we cannot give some satisfactory results. A nonlinear system y˙ = f (y), t ∈ R, y ∈ Rn , n  ∂fi (y)

= 0. Such system preserves the phase ∂yi volume on the phase Rn . For these systems, we only point out the centered Euler

is said to be source-free if divf =

i=1

10.9 Some Volume-Preserving Algorithms

schemes is volume-preserving iff the Jacobian

475

∂ fi = M is, in some sense, similar to ∂ yi

an infinitesimally symplectic matrix. That is, the eigenvalues of M can be specified as   λ(M ) = λ1 , λ2 , · · · , λ n2 , −λ1 , −λ2 , − · · · , −λ n2 , or

  λ(M ) = λ1 , λ2 , · · · , λ n2 , 0, −λ1 , −λ2 , − · · · , −λ n2 .

Bibliography

[DV84] K. Dekker and J.G. Verwer: Stability of Runge–Kutta Methods for Stiff Initial Value Problems. Elesevier Science Pub. B. V., North-Holland, Amsterdam, (1984). [FS95] K. Feng and Z. J. Shang: Volume-preserving algorithms for source-free dynamical systems. Numer. Math., 71:451–463, (1995). [FW94] K. Feng and D.L. Wang: Dynamical systems and geometric construction of algorithms. In Z. C. Shi and C. C. Yang, editors, Computational Mathematics in China, Contemporary Mathematics of AMS, Vol. 163, pages 1–32. AMS, (1994). [HW81] E. Hairer and G. Wanner: Algebraically stable and implementable Runge–Kutta methods of high order. SIAM J. Numer. Anal., 18:1098–1108, (1981). [MQ04] R.I. McLachlan and G.R.W. Quispel: Explicit geometric integration of polynomial vector fields. BIT, 44:513–538, (2004). [QD97] G. R. W. Quispel and C. P. Dyt: Solving ODE’s numerically while preserving symmetries, Hamiltonian structure, phase space volume, or first integrals. In A. Sydow, editor, Proceedings of the 15th IMACS World Congress, pages 601–607. Wissenschaft & Technik, Berlin, (1997). [QD98] G. R. W. Quispel and C. P. Dyt: Volume-preserving integrators have linear error growth. Physics Letters A, 202:25–30, (1998). [QL00] M. Z. Qin and H. W. Li: Volume preserving R–K methods for linear systems. Acta Applicandae Mathematicae, 16:430–434, (2000). [QM03] G. R. W. Quispel and D. I. McLaren: Explicit volume-preserving and symplectic integrators for trigonometric polynomial flows. J. of Comp. Phys., 186(1):308–316, (2003). [Qui95] G. R. W. Quispel: Volume-preserving integrators. Physics Letters A, 206:26–30, (1995). [QZ92] M. Z. Qin and W. J. Zhu: Construction of higher order symplectic schemes by composition. Computing, 47:309–321, (1992). [QZ93] M. Z. Qin and W. J. Zhu: Volume-preserving schemes and numerical experiments. Computers Math. Applic., 26:33–42, (1993). [Sco91] C. Scovel: Symplectic numerical integration of Hamiltonian systems. In T. Ratiu, editor, The Geometry of Hamiltonian Systems, pages 463–496. Springer, New York, (1991). [Sha94a] Z Shang: Construction of volume-preserving difference schemes for source-free systems via generating functions. J. Comput. Math., 12:265–272, (1994). [Sha94b] Z. Shang: Generating functions for volume-preserving mappings and Hamilton– Jacobi equations for source-free dynamical systems. Science in China (series A), 37:1172– 1188, (1994). [Sun95] G. Sun: Construction of high order symplectic Partitioned–Runge–Kutta methods. J. Comput. Math., 13(1):40–50, (1995). [Sur96] Y. B. Suris: Partitioned–Runge–Kutta methods a phase volume preserving integrators. Physics Letters A, 220:63–69, (1996). [TH85] A. Thyagaraja and F.A. Haas: Representation of volume-preserving maps induced by solenoidal vector fields. Phys. Fluids, 28:1005, (1985). [Wey40] H. Weyl: The method of orthogonal projection in potential theory. Duke Math. J., 7:411–444, (1940).

Chapter 11. Contact Algorithms for Contact Dynamical Systems

An odd-dimensional manifold cannot admit a symplectic structure. The analogue of symplectic structure for odd-dimensional manifolds is a little less symmetric, but is also a very interesting structure – the contact structure. In this chapter, we apply the ideas of preserving Lie group and Lie algebra structure of dynamical systems in constructing symplectic algorithms for Hamiltonian systems to the study of numerical algorithms for contact dynamical systems and present so-called contact algorithms, i.e., algorithms preserving contact structure, for solving numerically contact systems.

11.1 Contact Structure The source of contact structures are manifolds of contact element of configuration spaces. It is also of basic importance in physical and engineering sciences. Contact geometry has – as does symplectic geometry – broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, and applied mathematics such as control theory.

11.1.1 Basic Concepts of Contact Geometry Contact geometry[Arn89,Arn88] is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a “maximum non-degeneracy” condition called “complete nonintegrability”. The integration of first-order partial differential equations is reduced to the integration of a system of ordinary differential equations, the so-called characteristic equations. The basic of this reduction is a simple geometric analysis of the formulation of curves. Let M be a smooth manifold and let V be a direction field on M . Definition 1.1. N ⊂ M is called an integral surface of V if the tangent plane of N contains the direction of V at every point (Fig. 1.1). Let Γ be a k-dimensional submanifold in an n-dimensional manifold M (Fig. 1.2), Γ is called a hypersurface if k = n − 1. The Cauchy problem for the direction field v with initial manifold Γ is the problem of finding a (k + 1)-dimensional integral submanifold of v containing the initial submanifold Γ . K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

478

11. Contact Algorithms for Contact Dynamical Systems z

z

6

6 .... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

........ ... ............................... ................ ... ................ .. ................ ... ...... . . .. . ... . . .. . . . ... .. ... .. .. . . . .. .. .. ... .... . ... ... . .. ...... .... ... ..... .. . .... . . .. . . . . . . . ..... .. .... ... ..... ... .... ... .... .. .... .. . . . . . . ........................ ... .......... ... .. ........ .. .. ........ .. .. ....... . . . . ...... ... ...... ... ...... ... ...... .. ...... .. ..... . . ...... .. ........

N

O M x / Fig. 1.1.

V

-y

Meaning of definition z

.. .. .. .. ... ..... ... .. .. .. .. .. .. .. .. .. ... ... ... ... .... ..

V

N

- y

O M x /

.. ... ... ... ... . ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ... .............. . . . . . . . . . . . . . . . . . ... ... .......... . . . . . . ... . . . . . . . . ... ... ............ .. ... .............. .. ... ....................... ...... ....... . . . . . . . . . . . . . ................. . ... . . . . . . ... . . . . . . . . . . . . . .. .......... . . ... . . . . . . . . . . . . .. . ... ... .. ............. .. k+1 ... ............. . . . . . . . .. . . .. ............. .. ....... .. .. .. k .. . ... .. ..

V

6

V

Γ

- y M x / Fig. 1.2.

Integral surface with initial manifold of Γ

Every point in n-dimensional space existence an (n − 1)-dimensional hyperplane, codim = 1 field of hyperplane, this means field of tangent hyperplane can be locally n  αi dxi , and described by 1-form, and α = i=1

n 

αi2 (x) = 0,

∀ x ∈ Rn .

i=1

Hyperplane in Fig. 1.3 is null space of 1-form α. Relation between hyperplane and its 1-form is not 1- to -1 correspondence. They may be different up to multiplication by a non zero constant. This multiplicator is dependent of point. We consider what a filed of hyperplane looks like in general in a neighborhood of a point in an n-dimensional manifold. For example, let n = 2. Then the manifold is a surface and field of hyperplane is a field of straight line. Such a field in a neighborhood of a point is always constructed very simply, namely, as a field of tangent to a family of parallel lines in a plane. More precisely, one of the basic results of the local theory of ODEs is that it is possible to change any smooth field of tangent lines on a manifold into a field of tangents to family of straight lines in Euclidean space by using a diffeomorphism in a sufficiently small neighborhood of any point of the manifold.

11.1 Contact Structure

z

479

...... ... ............................... ................ ... ................ ................ ... ......... .. .. . ... . . ..... ... . .. . . . .. .. .. . . ... .... .... ... .... ... .... .. . . . ... .... .... ... .... .. .... .. . . . . . ....................... ... .......... ........ ... ........ ... ....... .. ....... . ...... ... ...... ...... ... ...... ... ..... ...... .... .......

6

6

- y

x / Fig. 1.3.

Hyperplane

If n > 2, then a hyperplane is not a line. For example, if n = 3, most field of 2-dimensional tangent planes in ordinary 3-dimensional space cannot be diffeomorphically mapped onto a field of parallel planes. The reason is that there exists fields of tangent planes for which it is impossible to find integral surfaces, i.e., surface which have the prescribed tangent plane at each point. A 1-form in 3-dimensional can be written in following standard form α = xd y + d z. Every tangent hyperplane in point x, which is denoted by Πx , have: ⎤ ⎡ 0 ⎥ ⎢ [ηx , ηy , ηz ] ⎣ x ⎦ = 0, 1 and [0, x, 1] not all equal to zero, it is defined as a 2-dimensional field of hyperplane. When x = 0, ⎤T ⎡ ⎡ ⎤ 0 ηx ⎥ ⎢ ⎢ ⎥ ⎣ 0 ⎦ ⎣ ηy ⎦ = 0. 1 0 Each point with a hyperplane intersecting wall defines a direction field, see Fig. 1.4 and 1.5. Next, we prove that in R3 space, there does not exist an integral surface which can be given by the 1-form α = xd y + d z, where x, y is horizontal coordinate, z is vertical coordinate, see Fig. 1.6. Consider a pair of vectors emanating from the origin (0,0,0) and lying in the horizontal plane of our coordinate systems; another integral curve from (0,0,0) to (0,1,0), and then from (0,1,0) to (1,1,0), and another integral curve from (0,0,0) to (1,0,0), and then from (1,0,0) to (1,1, −1). As a result, these two curves cannot close up. The difference in the heights of these points is 1, this difference can be considered as a measure of the nonintegrability of the field. We have four direction fields from the origin point 0 to walls of east, south, west, and north, respectively, describing by Fig. 1.5.

480

11. Contact Algorithms for Contact Dynamical Systems

Fig. 1.4.

Defines the field of 2n-dimensional plane α = 0 in R2n+1

-

North Fig. 1.5.

:

:

:

:

:

:

West

East

~ ~ ~

~

South

Direction fields in each wall

11.1.2 Contact Structure A contact element to an n-dimensional smooth manifold at some point is an (n − 1)dimensional plane tangent to the manifold at that point, i.e., an (n − 1)-dimensional subspace of the n-dimensional tangent space at that point. At the n-dimensional space for each point there is a n − 1 dimensional hyperplane, dimensions of this hyperplane field is n − 1. We note first that a field of hyperplanes can be given locally by a differential 1-form: a plane in the tangent space gives a 1-form up to multiplication by a non zero constant. We choose this constant so that the value of the form on vertical basic vector is equal to 1. The Hyperplanes of the field are null space of the 1-form[Arn89,Arn88] . Definition 1.2. A field of hyperplanes is said to be nondegenerate at a point if the rank of the 2-form dα|ω=0 in the plane of the field passing through this point is equal to the dimension of the plane. Definition 1.3. A differential 1-form α which is nowhere equal to the zero form on a manifold M is called a contact form if the exterior derivative dα of α defines a nondegenerate exterior 2-form in every plane α = 0.

11.1 Contact Structure

Fig. 1.6.

481

Integral curves constructed for a non-integrable field of planes

Example 1.4. Consider the space R2n+1 with the contact structure by the 1-form α = d u + p d q. Where q = (q1 , · · · , qn ), u, p = (p1 , · · · , pn ), α is not equal to zero form at any point in R2n+1 , and consequently defines the field of 2n-dimensional planes α = 0 in R2n+1 . Example 1.5. The form constructed in Example 1.4 is a contact form, the exterior derivatives of the form α is equal to d α|α=0 = d q1 ∧ d p1 + · · · + d qn ∧ d pn . In the plane α = 0, (q1 , · · · , qn ; p1 , · · · , pn ) may serve as coordinate.  O −I  , where I is the The matrix of the form ω = dα|α=0 has the form I O identity matrix of order n. The determinant of this matrix is equal to 1. Consequently, the 2-form ω is nondegenerate. In other words, the rank of this form is 2n, so our field is nondegenerate at the origin and thus also in a neighborhood of the origin (in fact, this field of planes is nondegenerate at all points of the space). Definition 1.6. A contact structure on the manifold M is a field of tangent plane which are given locally as the set of zeros of a contact 1-form. The hyperplanes of the field are called contact hyperplanes. We can denote by Πx the contact hyperplane at the point x. Putting briefly, a contact structure on a manifold is a nondegenerate field of tangent hyperplane. . Definition 1.7. A field of planes is called nondegenerated on a manifold if it is nondegenerate at every point of the manifold. It should be noted that on the even-dimensional manifold there cannot be a nondegenerate field of hyperplanes, on such a manifold a hyperplane is odd-dimensional, and the rank of every skew-symmetric bilinear form on an odd-dimensional space is less than the dimension of the space. Nondegenerate field of hyperplane do exist on odd-dimensional manifold.

482

11. Contact Algorithms for Contact Dynamical Systems

Definition 1.8. A hyperplane (dimension n − 1) tangent to a manifold at some point is called a contact element, and this point is called the point of contact. The set of all contact element of an n-dimensional manifold has the structure of a smooth manifold of dimension 2n − 1. The manifold of all contact elements of an n-dimensional manifold is a fiber bundle whose base is our manifold and whose fiber is (n − 1)-dimensional projective space. Theorem 1.9. The bundle of contact element is the projectivization of the cotangent bundle: it can be obtained from the cotangent bundle by changing every cotangent n-dimensional vector space into on (n − 1)-dimensional projective space (a point of which is a line passing through the origin in the cotangent space). Proof. A contact element is given by a 1-form on the tangent space, for which this element is not zero, and it is determined up to multiplication by a non zero number. But a form on the tangent space is a vector of the cotangent space. Therefore, a non zero vector of the cotangent space, determined up to a multiplication by a non zero number, is a non zero vector of the cotangent space, determined up to a multiplication by a non zero number, i.e., a point of the projectivized cotangent space.  In this chapter, we simply consider the Euclidean space R2n+1 of 2n+1 dimensions as our basic manifold with the contact structure given by the normal form ⎡ ⎤ dx n  ⎢ ⎥ ⎥ xi d yi + d z = xd y + d z = (0, xT , 1) ⎢ (1.1) α= ⎣ dy ⎦, i=1 dz here we have used 3-symbol notation to denote the coordinates and vectors on R2n+1 x = (x1 , · · · , xn )T ,

y = (y1 , · · · , yn )T ,

z = (z).

(1.2)

A contact dynamical system on R2n+1 is governed by a contact vector field f = (a , bT , c) : R2n+1 → R2n+1 through equations T

x˙ = a(x, y, z),

y˙ = b(x, y, z),

z˙ = c(x, y, z),

· =:

d , dt

(1.3)

where the contactivity condition of the vector field f is Lf α = λf α,

(1.4)

with some function λf : R2n+1 → R, called the multiplier of f . In (1.4), Lf α denotes the Lie derivation of α with respect to f and is usually calculated by the formula (see Chapter 1 of book)[Arn88] Lf α = if d α + d if α.

(1.5)

It is easy to show from (1.4) and (1.5) that to any contact vector field f on R2n+1 , there corresponds a function K(x, y, z), called contact Hamiltonian, such that

11.1 Contact Structure

a = −Ky + Kz x,

c = K − xT Kx =: Ke .

b = Kx ,

483

(1.6)

In fact, (1.6) represents the general form of a contact vector field. Its multiplier, denoted as λf from now, is equal to Kz . Definition 1.10. A contact transformation g is a diffeomorphism on R2n+1 ⎛ ⎞ ⎛ ⎞ x x (x, y, z) ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ g: ⎜ ⎝ y ⎠ −→ ⎝ y(x, y, z) ⎠ z z(x, y, z) conformally preserving the contact structure, i.e., g ∗ α = μg α, that means + * n n   x i d yi + d z = μg xi d yi + d z , i=1

(1.7)

i=1

for some everywhere non-vanishing function μg : R2n+1 → R, called the multiplier of g. The explicit expression of (1.7) is ⎡ y x x x ⎢ (0, x T , 1) ⎢ ⎣ yx yy zx

x z

⎤

⎥ T yz ⎥ ⎦ = μg (0, x , 1).

zy

zz

t of a contact dynamical system associated A fundamental fact is that the phase flow gK 2n+1 → R is a one parameter (local) group of with a contact Hamiltonian K : R t contact transformations on R2n+1 , i.e., gK satisfies 0 = identity map on R2n+1 ; gK t+s t s = gK ◦ gK , ∀ t, s ∈ R; gK t ∗ t α, (gK ) α = μgK

(1.8) (1.9) (1.10)

2n+1 t : R for some everywhere non-vanishing function μgK → R. Moreover, we have the following relation between μG∗k and the Hamiltonian K:

t = exp μgK

0

t s (Kz ◦ gK )d s.

(1.11)

For general contact systems, condition (1.10) is stringent for algorithmic approximations to phase flows because only the phase flows themselves satisfy it. We will construct algorithms for contact systems such that the corresponding algorithmic approximations to the phase flows satisfy the condition (1.10), of course, probably, with t . We call such algodifferent, but everywhere non-vanishing, multipliers from μgK rithms as contact ones.

484

11. Contact Algorithms for Contact Dynamical Systems

11.2 Contactization and Symplectization There is a well known correspondence between contact geometry on R2n+1 and conic (or homogeneous) symplectic geometry on R2n+2 . To establish this correspondence, we introduce two spaces R2n+2 and R+ × R2n+1 . + a. We use the 4-symbol notation for the coordinates on R2n+2 ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ p0 p11 q11 ⎢ p ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ 1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ , p0 = (p0 ), q0 = (q0 ), p1 = ⎢ .. ⎥ , q1 = ⎢ ⎣ .. ⎦ . (2.1) ⎣ q0 ⎦ ⎦ ⎣ p1n q1n q1 Consider

( ) = (p0 , p1 , q0 , q1 ) ∈ R2n+2 | p0 > 0 R2n+2 +

(2.2)

as a conic symplectic space with the standard symplectic form ω = dp0 ∧ dq0 + dp1 ∧ dq1 .

(2.3)

Definition 2.1. Function φ : R2n+2 → R is called a conic function if it satisfies +  p  1 (2.4) φ(p0 , p1 , q0 , q1 ) = p0 φ 1, , q0 , q1 , ∀ p0 > 0. p0 So, a conic function on R2n+2 depends essentially only 2n + 1 variables. → R2n+2 is called a conic map if Definition 2.2. F : R2n+2 + + F ◦ Tλ = Tλ ◦ F,

∀ λ > 0,

where Tλ is the linear transformation on R2n+2 5 6 6 5 6 5 p λp p0 Tλ , = , p= q q p1

(2.5) 5

q=

q0

6 .

(2.6)

q1

The conic condition (2.5) for the mapping F : (p0 , p1 , q0 , q1 ) → (P0 , P1 , Q0 , Q1 ) can be expressed as follows:   p P0 (p0 , p1 , q0 , q1 ) = p0 P0 1, 1 , q0 , q1 > 0, p0

 P1 (p0 , p1 , q0 , q1 ) = p0 P1 1,



p1 , q0 , q1 , p0

  p Q0 (p0 , p1 , q0 , q1 ) = Q0 1, 1 , q0 , q1 ,  Q1 (p0 , p1 , q0 , q1 ) = Q1 1,

p0



p1 , q0 , q1 , p0

∀ p0 > 0.

(2.7)

11.2 Contactization and Symplectization

485

So, a conic map is essentially depending only on 2n + 2 functions in 2n + 1 variables. It should be noted that, in some cases, we also consider conic functions and conic maps defined on the whole Eucildean space. The following lemma gives a criterion of a conic symplectic map. Lemma 2.3. F : (p0 , p1 , q0 , q1 ) → (P0 , P1 , Q0 , Q1 ) is a conic symplectic map if and only if (0, 0, P0T , P1T )F∗ − (0, 0, pT0 , pT 1 ) = 0, where F∗ is the Jacobi matrix of F at the point (p0 , p1 , q0 , q1 ). Proof. For F : (p0 , p1 , q0 , q1 ) → (P0 , P1 , Q0 , Q1 ), the condition T (0, 0, P0T , P1T )F∗ − (0, 0, pT 0 , p1 ) = 0,

(2.8)

is equivalent to the condition P0 d Q0 + P1 d Q1 = p0 d q0 + p1 d q1 ,

or

P d Q = pd q,

(2.9)

where P = (P0 , P1 ), Q = (Q0 , Q1 ), p = (p0 , p1 ), q = (q0 , q1 ). Hence in matrix form, it can be written as QT p · P = 0,

QT q · P = p.

(2.10)

Notice that a function f (x1 , x2 , · · · , xn ) is homogeneous of degree k, i.e., f (λx1 , λx2 , · · · , λxn ) = λk f (x1 , x2 , · · · , xn ), if and only if 

xi fxi (x1 , x2 , · · · , xn ) = kf (x1 , x2 , · · · , xn ).

Therefore, the condition (2.7) is equivalent to Pp (p, q) · p = P (p, q),

Qp (p, q) = 0.

(2.11)

If F is conic symplectic, then T QT p Pp − Pp Qp = O,

T QT q Pq − Pq Qq = O,

T QT q Pp − Pq Qp = I.

(2.12)

Combining with (2.11), we get T T T T T p = QT q Pp p − Pq Qp p = Qq P, O = Qp Pp p − Pp Qp p = Qp P.

(2.13)

This proves the “only if” part. Conversely, if F satisfies the condition (2.8), then it satisfies (2.9), which means that it is symplectic. We know that if a matrix is symplectic, then its transpose is also symplectic. Therefore, Pq PpT − Pp PqT = O,

Qq QTp − Qp QT q = O,

T Pp QT q − Pq Qp = I.

(2.14)

486

11. Contact Algorithms for Contact Dynamical Systems

Combining with (2.10), we get T P = Pp Q T q P − Pq Qp P = Pp p, T 0 = Qq QT p P − Qp Qq P = Qq p.

(2.15) 

This means that F is conic. This finishes the proof.

b. Consider R+ × R2n+1 as the product of the positive real space R+ and the contact space R2n+1 . We use (w, x, y, z) to denote the coordinates of R+ × R2n+1 with w > 0 and with x, y, z as before. Definition 2.4. A map G: R+ × R2n+1 → R+ × R2n+1 is called a positive product map if it is composed by a map g : R2n+1 → R2n+1 and a positive function γ : R2n+1 → R+ in the form ⎡ ⎢ ⎢ ⎢ ⎣

w x y z

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ −→ ⎢ ⎦ ⎣

W X Y Z

⎤ ⎥ ⎥ ⎥, ⎦

W = w γ(x, y, z),

(X, Y, Z) = g(x, y, z).

(2.16)

We denote γ ⊗ g the positive product map composed of map g and function γ. c. Define mapping S : R+ × R2n+1 → R2n+2 + ⎡ ⎢ ⎢ ⎢ ⎣

w x y z

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ −→ ⎢ ⎦ ⎣

p0 = w p1 = wx q0 = z q1 = y

⎤ ⎥ ⎥ ⎥. ⎦

(2.17)

Then the inverse S −1 : R2n+2 → R+ × R2n+1 is given by + ⎡ ⎢ ⎢ ⎢ ⎣

p0 p1 q0 q1

⎤

⎡

w = p0

p1 ⎢ ⎥ ⎢ x= ⎥ p0 ⎢ ⎥ −→ ⎢ ⎦ ⎣ y = q1 z = q0

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

p0 = 0.

(2.18)

Lemma 2.5. [Fen93b,Fen95] Given a transformation F : (p0 , p1 , q0 , q1 ) → (P0 , P1 , Q0 , and let G = S −1 ◦ F ◦ S. Then we have: Q1 ) on R2n+2 + ◦ if and only if G is a positive product map on 1 F is a conic map on R2n+2 + R+ × R2n+1 ; in this case, if we write G = γ ⊗ g, then γ(x, y, z) = P0 (1, x, z, y), and g : (x, y, z) → (X, Y, Z) is given by

(2.19)

11.2 Contactization and Symplectization

X=

P1 (1, x, z, y) , P0 (1, x, z, y)

Y = Q1 (1, x, z, y),

Z = Q0 (1, x, z, y).

487

(2.20)

2◦ F is a conic symplectic map if and only if G is a positive product map, say γ ⊗ g, on R+ × R2n+1 with g also a contact map on R2n+1 . Moreover, in this case, the multiplier of the contact map g is just equal to γ −1 = P0−1 (1, x, z, y). Proof. The conclusion 1◦ is easily proved by some simple calculations. Below we devote to the proof of 2◦ . Let F send (p0 , p1 , q0 , q1 ) → (P0 , P1 , Q0 , Q1 ), G send (w, x, y, z) → (W, X, Y, Z). Then by using the conclusion 1◦ , we have P0 ◦ S = wP0 (1, x, z, y) = wγ, P1 ◦ S = wP1 (1, x, z, y) = wγX(x, y, z), ⎡ ⎤ ⎡ 1 0 0 0 γ wγx wγy ⎢ ⎥ ⎢ x wIn 0 0 ⎥ ⎢ ∂ (W, X, Y, Z) ⎢ ⎥ ⎢ 0 S∗ = ⎢ =⎢ ⎥ , G∗ = ⎢ 0 ∂ (w, x, y, z) ⎣ 0 0 0 1 ⎥ g∗ ⎣ ⎦ 0 0 0 In 0 ⎤ ⎡ 1 0 0 0 ⎢ X W In 0 0 ⎥ ⎥ ⎢ S∗ ◦ G = ⎢ ⎥, ⎣ 0 0 0 1 ⎦ 0

0

In

wγz

⎤ ⎥ ⎥ ⎥, ⎦

0

and compute  T ((0, 0, P0T , P1T )F∗ − (0, 0, pT 0 , p1 )) ◦ S S∗     T = (0, 0, P0T , P1T ) ◦ S (F∗ ◦ S)S∗ − (0, 0, pT 0 , p 1 ) ◦ S S∗ 

= (0, 0, wγ, wγX T )(F∗ ◦ S)S∗ − (0, 0, w, wxT )S∗ = (0, 0, wγ, wγX T )(S∗ ◦ G)G∗ − (0, 0, w, wxT )S∗     = wγ 0, (0, X T , 1)g∗ − wγ 0, γ −1 (0, xT , 1) . Noting that S is a diffeomorphism, S∗ is non-singular, w > 0, γ > 0, we obtain T T −1 (0, 0, P0T , P1T )F∗ − (0, 0, pT (0, xT , 1) ≡ 0, 0 , p1 ) ≡ 0 ⇐⇒ (0, X , 1)g∗ − γ

which proves the conclusion 2◦ .



Lemma 2.5 establishes correspondences between conic symplectic space and contact space and between conic symplectic maps and contact maps. We call the transform from F to G = S −1 ◦ F ◦ S = γ ⊗ g contactization of conic symplectic maps, the transform from G = γ ⊗ g to F = S ◦ GS −1 symplectization of contact maps and call the transform S : R+ × R2n+1 → R2n+1 symplectization of contact space, and + 2n+1 → R × R contactization of conic symplectic the transform C = S −1 : R2n+2 + + space.

488

11. Contact Algorithms for Contact Dynamical Systems

11.3 Contact Generating Functions for Contact Maps With the preliminaries of the last section, it is natural to derive contact generating function theory for contact maps from the well known symplectic analog[Fen93b,Fen95,Shu93] . The following two lemmas can be proved easily[Fen95] . Lemma 3.1. Hamiltonian φ : R2n+2 → R is a conic function if only if the associated Hamiltonian vector field aφ = J∇φ is conic, i.e., a(Tλ z) = Tλ a(z), λ = 0, z ∈ R

2n+2

, where J =

O

−In+1

In+1

O

5

p

!

.

6

5

6

p

→ C is a conic transformation on R2n+2 , q q i.e., C ◦ Tλ6 = Tλ ◦ C, if and only if the matrix C has the diagonal form C = 5 C0 O with (n + 1) × (n + 1) matrix C0 and C1 . O C1 Lemma 3.2. Linear map

Noting that the matrix in gl(2n + 2) C=

1 (I + JB), 2

B = B T ∈ Sm(2n + 2),

(3.1)

establishes a 1-1 correspondence between near-zero Hamiltonian vector fields z → a(z) ≡ J∇φ(z) and near-identity symplectic maps z → g(z) via generating relation g(z) − z = J∇φ(Cg(z) + (I − C)z),

(3.2)

and combining Lemma 3.1 and Lemma 3.2, we find that matrix 5 C=

C0

O

O

I − C0T

6 ,

C0 ∈ gl(n + 1),

(3.3)

establishes a 1-1 correspondence between near-zero conic Hamiltonian vector fields z → a(z) = J∇φ(z) and near-identity conic symplectic maps z → g(z) via generating relation (3.3). ! α βT Write C0 = with α ∈ R, β, γ ∈ Rn and δ ∈ gl(n). Then the γ δ generating relation (3.2) with generating matrix C given by (3.3) can be expressed as ⎧ p0 − p0 = −φq0 (p, q), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p1 − p1 = −φq1 (p, q), (3.4) q0 − q0 = φq0 (p, q), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ q1 − q1 = φq1 (p, q),

11.3 Contact Generating Functions for Contact Maps

5 where p =

p0 p1

6

5 and q =

q0

489

6 are given by

q1

⎧ p0 = α p0 + (1 − α)p0 + β T ( p1 − p1 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p1 = δ p1 + (I − δ)p1 + γ( p0 − p0 ),

(3.5)

⎪ ⎪ q 0 = (1 − α) q0 + αq0 − γ T ( q1 − q1 ), ⎪ ⎪ ⎪ ⎪ ⎩ q 1 = (I − δ T ) q1 + δ T q1 − β( q0 − q0 ).

Every conic function φ can be contactized as an arbitrary function ψ(x, y, z) as follows ψ(x, y, z) = φ(1, x, z, y),

(3.6)

i.e., φ(p0 , p1 , q0 , q1 ) = p0 φ(1, p1 /p0 , q0 , q1 ) = p0 ψ(p1 /p0 , q1 , q0 ),

p0 = 0,

and we have the partial derivative relation: ⎧ φq0 (p0 , p1 , q0 , q1 ) = p0 ψz (x, y, z), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ φq1 (p0 , p1 , q0 , q1 ) = p0 ψy (x, y, z),

(3.7)

⎪ ⎪ φp0 (p0 , p1 , q0 , q1 ) = ψ(x, y, z) − xT ψx (x, y, z) = ψe (x, y, z), ⎪ ⎪ ⎪ ⎪ ⎩ φp1 (p0 , p1 , q0 , q1 ) = ψx (x, y, z), where x = transforms ⎡ w ⎢ ⎢ x ⎢ S: ⎢ ⎢ ⎢ y ⎣ z

p1 , y = q1 , z = q0 on the right hand side. So, under contactizing p0

⎤

⎡

p0

⎤

⎡

w

⎤

⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ p ⎥ ⎢ wx ⎥ ⎥ ⎥ ⎢ 1 ⎥ ⎢ ⎥=⎢ ⎥, ⎥ −→ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ q0 ⎥ ⎢ z ⎥ ⎦ ⎣ ⎦ ⎦ ⎣ y q1 ⎤ ⎡ ⎡ w ⎥ ⎢ ⎢ ⎢ ⎢ x ⎥ ⎥ ⎢ ⎢ ⎥ −→ ⎢ ⎢ ⎥ ⎢ ⎢ ⎢ ⎢ y ⎥ ⎦ ⎣ ⎣ z

the generating relation (3.4) turns into

⎡

p0 p1 q0 q1

w 

⎤

⎡

p0

⎤

⎡

w 

⎤

⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ x ⎥ ⎢ p ⎥ ⎢ w ⎢  ⎥ ⎢ 1 ⎥ ⎢ x ⎥ ⎥=⎢ ⎥, ⎢ ⎥ −→ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ y ⎥ ⎢ q0 ⎥ ⎢ z ⎥ ⎦ ⎣ ⎣ ⎦ ⎦ ⎣ y q1 z ⎤ ⎡ ⎤ w ⎥ ⎢ ⎥ ⎥ ⎢ wx ⎥ ⎥ ⎢ ⎥ ⎥=⎢ ⎥, (3.8) ⎥ ⎢ ⎥ ⎥ ⎢ z ⎥ ⎦ ⎣ ⎦ y

490

11. Contact Algorithms for Contact Dynamical Systems

⎧ w  − w = −wψz (x, y, z), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ w x − wx = −wψy (x, y, z),

(3.9)

⎪ ⎪ z − z = ψe (x, y, z), ⎪ ⎪ ⎪ ⎪ ⎩ y − y = ψx (x, y, z), and Equation (3.5) turns into ⎧ w = αw  + (1 − α)w + β T (w x − wx), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ w x = δ w x + (I − δ)wx + γ(w  − w),

(3.10)

⎪ ⎪ z = (1 − α) z + αz − γ T ( y − y), ⎪ ⎪ ⎪ ⎪ ⎩ y = (I − δ T ) y + δ T y − β( z − z).

Since the p0 -axis is distinguished for the contactization in which we should always w p w p take p0 = 0, it is natural to require β = 0 in (3.5). Let μ  = = 0 and μ = = 0 , w 

we obtain from Equations (3.9) and (3.10) μ =

1 + αψz (x, y, z) , 1 − (1 − α)ψz (x, y, z)

p0

μ = 1 + αψz (x, y, z),

w

p0

(3.11)

and the induced contact transformation on the contact (x, y, z) space R2n+1 is   ⎧ x + αx , x  − x = −ψy (x, y, z) + ψz (x, y, z) (1 − α) ⎪ ⎪ ⎨ y − y = ψx (x, y, z), (3.12) ⎪ ⎪ ⎩ z − z = ψe (x, y, z), with the bar variables on the right hand side given by x, y, z ⎧ x = d1 x  + d2 x + d0 , ⎪ ⎪ ⎨ y = (I − δ T ) y + δ T y, ⎪ ⎪ ⎩ z = (1 − α) z + αz − γ T ( y − y), where

  ⎧ d1 = I − (1 − α)ψz (x, y, z) δ, ⎪ ⎪ ⎪ ⎨   d2 = I + αψz (x, y, z) (I − δ), ⎪ ⎪ ⎪ ⎩ d0 = −ψz (x, y, z)γ.

(3.13)

(3.14)

Summarizing the above discussions, we have: Theorem 3.3. Relations (3.12) – (3.14) give a contact map (x, y, z) → ( x, y, z) via  α O  contact generating function ψ(x, y, z) under the type C0 = . Vice versa. γ δ

11.3 Contact Generating Functions for Contact Maps

491

However, the difficulty in the algorithmic implementation lies in the fact that, unlike y¯ and z, which are linear combinations of y, y and z, z with constant matrix coefficients, since x = d1 x  + d2 x + d0 and d1 , d2 are matrices with coefficients depending  on ψ¯z = ψz (x, y, z) which in turn depends on (x, y, z) the combination of x from x and x is not explicitly given, the entire equations for solving x , y, z in terms of x, y, z are highly implicit. The exceptional cases are the following: (E1) α = 0, δ = On , γ = O, μ  = 1 − ψz (x, y, z), μ = 1, ⎧ , z) + x ψz (x, y, z), x  − x = −ψ y (x, y ⎪ ⎪ ⎨ y − y = ψx (x, y, z), ⎪ ⎪ ⎩ z − z = ψe (x, y, z) = ψ(x, y, z) − xT ψx (x, y, z). (E2)

(3.16)

α = 1, δ = In , γ = O, μ  = μ = 1 + ψz ( x, y, z), ⎧ x, y, z) + xψz ( x, y, z), x  − x = −ψy ( ⎪ ⎪ ⎨ y − y = ψx ( x, y, z), ⎪ ⎪ ⎩ z − z = ψe ( x, y, z) = ψ( x, y, z) − xT ψx ( x, y, z).

(E3)

(3.15)

1 2

(3.17)

(3.18)

1 2

α = , δ = In , γ = O, 1 ψz (x, y, z) 2 μ = , 1 1 − ψz (x, y, z) 2 1+

1 2

μ = 1 + ψz (x, y, z),

(3.19)

⎧ x +x ⎪ , x  − x = −ψy (x, y, z) + ψz (x, y, z) ⎪ ⎪ 2 ⎨ y − y = ψx (x, y, z), ⎪ ⎪ ⎪ ⎩ z − z = ψe (x, y, z) = ψ(x, y, z) − xT ψx (x, y, z),

(3.20)

with x=

x +x 1 − ψz (x, y, z)( x − x), 2 4

y=

y + y , 2

z=

z + z . 2

(3.21)

For ψz = λ = constant, the case (E3) reduces to 1 λ 2 , μ = 1 1− λ 2 1+

1 2

μ = 1 + λ,

(3.22)

⎧ x +x ⎪ x  − x = −ψy (x, y, z) + ψz (x, y, z) , ⎪ ⎪ 2 ⎨ y − y = ψx (x, y, z), ⎪ ⎪ ⎪ ⎩ z − z = ψe (x, y, z) = ψ(x, y, z) − xT ψx (x, y, z),

(3.23)

492

11. Contact Algorithms for Contact Dynamical Systems

with x=

x +x 1 − λ ( x − x), 2 4

y=

y + y , 2

z=

z + z . 2

(3.24)

Note that the symplectic map induced by generating function φ from the relation (3.2) can be represented as the composition of the maps, non-symplectic generally, z → z and z → z z = z + CJ∇φ(z), z = z + (I − C)J∇φ(z). Theorem 3.4. Contact map (x, y, z) → ( x, y, z) induced by contact generating function ψ from the relations (3.12)–(3.14) can be represented as the composition of the x, y, z) which are not contact generally maps (x, y, z) → (x, y, z) and (x, y, z) → ( and given, respectively, as follows ⎧ x − x = −δψy (x, y, z) + αψz (x, y, z)x − γψz (x, y, z), ⎪ ⎪ ⎨ y − y = (I − δ T )ψx (x, y, z), ⎪ ⎪ ⎩ z − z = (1 − α)ψe (x, y, z) − γ T ψx (x, y, z)

(3.25)

⎧ x  − x = −(I − δ)ψy (x, y, z) + (1 − α)ψz (x, y, z) x + γψz (x, y, z), ⎪ ⎪ ⎨ y − y = δ T ψx (x, y, z), ⎪ ⎪ ⎩ z − z = αψe (x, y, z) + γ T ψx (x, y, z).

(3.26)

and

(3.25) and (3.26) are the 2-stage form of the generating relation (3.12) of the contact   α O map induced by generating function ψ under the type C0 = . Correspondγ δ ing to the exceptional cases (E1), (E2) and (E3), the above 2-stage representation has simpler forms, we no longer use them here.

11.4 Contact Algorithms for Contact Systems Consider contact system (1.3) with the vector field a defined by contact Hamiltonian K according to Equation (1.6). Take ψ(x, y, z) = sK(x, y, z) in (3.12) – (3.14) as the generating function, we then obtain contact difference schemes with 1st order of ac  α O curacy of the contact system (1.3) associated with all possible types C0 = . γ δ The simplest and important cases are (write K x = Kx (x, y, z), etc.) as follows[Fen95] .

11.4 Contact Algorithms for Contact Systems

493

11.4.1 Q Contact Algorithm Q. Contact analog of symplectic method (p, Q)1 (α = 0, δ = 0n , γ = 0).   x  = x + s − Ky (x, y, z) + x Kz (x, y, z) , y = y + sKx (x, y, z),

1-stage form :

z = z + sKe (x, y, z); x = x, 2-stage form :

y = y + s Kx,

 K x ), x  = x + s(−K y + x

(4.1) z = z + s Ke, y = y,

z = z.

11.4.2 P Contact Algorithm P . Contact analog of symplectic method (P, q)(α = 1, δ = In , γ = O).   x, y, z) + xKz ( x, y, z) , x  = x + s − Ky ( y = y + sKx ( x, y, z),

1-stage form:

z = z + sKe ( x, y, z); x = x + s(−K y + x K z ), 2-stage form:

x  = x,

y = y + s K x ,

(4.2) y = y,

z = z,

z = z + s K e .

11.4.3 C Contact Algorithm C. Contact version of Poincar´egenerating function method similarly to symplectic  1 1 case α = , δ = In , γ = O . 2

2

2-stage form: s s s y = y + Kz , z = z + K e , 2 2 2  −1  s s s x  = x + (−K y + x K x ) = x − K y 1 − K z , 2 2 2 s s y = y + K x = 2y − y, z = z + K e = 2z − z. 2 2

x = x + (−K y + xK z ),

(4.3)

One might suggest, for example, the following scheme for (1.3): 1

For Hamiltonian system p˙ = −Hq (p, q), q˙ = Hp (p, q), the difference scheme p = p − sHq (p, q), q = q + sHp (p, q) is symplectic and we call it (p, Q) method because the pair (p, q), composed of the old variables of p and the new variables of q, emerges in the Hamiltonian. The following (P, q) method has the similar meaning.

494

11. Contact Algorithms for Contact Dynamical Systems

x  = x + sa( x, y, z),

y = y + sb( x, y, z),

z = z + sc( x, y, z).

It differs from (4.2) only in one term for x , i.e., x K( x, y, z) instead of xK( x, y, z). This minute, but delicate, difference makes (4.2) contact and the other non-contact! It should be noted that the Q and P methods are of order one of accuracy and the C method is of order two. The proof is similar to that for symplectic case. In principle, one can construct the contact difference schemes of arbitrarily high order of accuracy for contact systems, as was done for Hamiltonian systems, by suitably composing the Q, P or C method and the respective reversible counterpart[QZ92] . Another general method for the construction of contact difference schemes is based on the generating functions for phase flows of contact systems which will be developed in the next section.

11.5 Hamilton–Jacobi Equations for Contact Systems We recall that a near identity contact map g : (x, y, z) → ( x, y, z) can be generated from the! so-called generating function ψ(x, y, z), associated with a matrix C0 =

α γ

O δ

, by the relations (3.12) – (3.14). Accordingly, to the phase flow etK

of a contact system with contact Hamiltonian K, there corresponds a time-dependent x, y, z) is gengenerating function ψ t (x, y, z) such that the map etK : (x, y, z) → ( erated from ψ t by the relations (3.12) – (3.14), in which ψ is replaced by ψ t and C0 is given in advance as above. The function ψ t should be determined by K and C0 . Below we derive the relevant relations  them.  p1 between Let H(p0 , p1 , q0 , q1 ) = p0 K , q1 , q0 , p0 = 0. With this conic Hamiltonian p0

and with normal Darboux matrices C =

C0 O

O I − C0

!

, where C0 =

α γ

O δ

!

,

we get the Hamilton–Jacobi equation   ∂ t φ (u) = H u + (I − C)J∇φt (u) , ∂t

with

u = (p0 , p1 , q0 , q1 )T ,

(5.1)

t satisfied by the generating function φt (u) of the phase flow gH of the Hamiltonian t is generated from system associated with the Hamiltonian H while the phase flow gH φt by the relation

 t  t (u) − u = Jφt CgH (u) + (I − C)u . gH

(5.2)

On the other hand, according to the discussion in Section 11.3, we have φt (p0 , p1 , q0 , q1 ) = p0 ψ t (x, y, z), So, by simple calculations, we have

with x =

p1 , p0

y = q1 ,

z = q0 .

11.5 Hamilton–Jacobi Equations for Contact Systems ⎡

⎤

p0 − (1 − α)φq0

⎢ ⎢ p1 + γφq − (I − δ)φq 0 1 ⎢ u + (I − C)J∇φt (u) = ⎢ ⎢ T ⎢ q0 + αφp0 + γ φp1 ⎣ q1 + δ T φp1

⎡

p0 (1 − (1 − α)ψz )

495 ⎤

⎥ ⎢ ⎥ ⎢ p0 (x + γψz − (I − δ)ψy ) ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ z + αψe + γ T ψx ⎦ ⎣ y + δ T ψx

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

and H(u + (I − C)J∇φt (u)) & %   x − (I − δ)ψy + γψz = p0 1 − (1 − α)ψz K , y + δ T ψx , z + αψe + γ T ψx . 1 − (1 − α)ψz

Therefore, from Equation (5.1), ψ t (x, y, z) satisfies & %   ∂ t x − (I − δ)ψy + γψz T T ψ = 1 − (1 − α)ψz K , y + δ ψx , z + αψe + γ ψx . 1 − (1 − α)ψz

∂t

(5.3) t Now we claim1 . From (5.2), it follows that u  = gH (¯ u). The claim is then proved, t u)) = H(¯ u), for all u. The following equality is valid since H(gH (¯     H u + (I − C)J∇φt (u) = H u − CJ∇φt (u) . (5.4)

So, replacing C by C − I in above discussions or, equivalently, replacing α and δ by α − 1 and δ − 1 with γ unchanging in (5.3), we can derive equation satisfied by the ψ t ∂ t ψ = (1+αψz )K ∂t

%

& x + δψy + γψz , y + (δ T − I)ψx , z + (α − 1)ψe + γ T ψx . 1 + αψz (5.5)

(5.3) and (5.5) define the same function ψ t . When t = 0, etK = I, so we should impose the initial condition (5.6) ψ 0 (x, y, z) = 0, for solving the first order partial differential equation (5.3) or (5.5). We call both equations the Hamilton–Jacobi equations of the contact system associated with the contact ! Hamiltonian K and the matrix C0 =

α γ

O δ

.

Specifically, we have Hamilton–Jacobi equations for particular cases: (E1) α = 0, δ = O, γ = O. % & x − ψyt ∂ t t , y, z = K(x, y − ψxt , z − ψet ). ψ = (1 − ψz )K ∂t 1 − ψzt

(5.7)

(E2) α = 1, δ = In , γ = O. 1

Proof of the claim: let u  = u + (I − C)J∇φt (u) and u ¯ = u − CJ∇φt (u), then we have u = Cu  + (I − C)¯ u.

496

11. Contact Algorithms for Contact Dynamical Systems

∂ t ψ = K(x, y + ψxt , z + ψet ) = (1 + ψzt )K ∂t 1 2

%

x + ψyt , y, z 1 + ψzt

& .

(5.8)

1 2

(E3) α = , δ = In , γ = O. ∂ t ψ ∂t

⎛

1 t ψy 1 t 2 ,y + = 1 − ψz K ⎝ 1 2 1 − ψzt 2 ⎛ 1   x + ψyt 1 t 2 ,y − ⎝ = 1 + ψz K 1 2 1 + ψzt 2





x−

⎞ 1 t 1 ψ , z + ψet ⎠ 2 x 2

⎞ 1 t 1 ψ , z − ψet ⎠ . 2 x 2

(5.9)

Remark 5.1. On the construction of high order contact difference schemes. If K is analytic, then one can solve ψ t (x, y, z) from the above Hamilton–Jacobi equations in the forms of power series in time t. Its coefficients are recursively determined by the K and the related matrix C0 . The power series are simply given from the corresponding conic Hamiltonian generating functions φt (p0 , p1 , q0 , q1 ) by ψt (x, y, z) = ψ t (1, x, z, y), since the power series expressions of φt with respect to  p1 , q1 , q0 have been well t from the conic Hamiltonian H(p0 , p1 , q0 , q1 ) = p0 K p0

given in[FW94] . Taking a finite truncation of the power series up to order m, an arbitrary integer, with respect to the time t and replacing by the truncation the generating function ψ in (3.12)–(3.14), then one obtains a contact difference scheme of order m for the contact system defined by the contact Hamiltonian K. The proofs of these assertions are similar to those in the Hamiltonian system case, hence are omitted here.

Bibliography

[Arn78] V. I. Arnold: Ordinary Differential Equations. The MIT Press, New York, (1978). [Arn88] V. I. Arnold: Geometrical Methods In The Theory of Ordinary Differential Equations. Springer-Verlag, Berlin, (1988). [Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Berlin Heidelberg, Second edition, (1989). [Etn03] J. Etnyre: Introductory lectures on contact geometry. In Proc. Sympos. Pure Math, volume 71, page 81C107. SG/0111118, (2003). [Fen93b] K. Feng: Symplectic, contact and volume preserving algorithms. In Z.C. Shi and T. Ushijima, editors, Proc.1st China-Japan conf. on computation of differential equationsand dynamical systems, pages 1–28. World Scientific, Singapore, (1993). [Fen95] K. Feng: Collected works of Feng Kang. volume I,II. National Defence Industry Press, Beijing, (1995). [FW94] K. Feng and D.L. Wang: Dynamical systems and geometric construction of algorithms. In Z. C. Shi and C. C. Yang, editors, Computational Mathematics in China, Contemporary Mathematics of AMS, Vol 163, pages 1–32. AMS, (1994). [Gei03] H. Geiges: Contact geometry. Math.SG/0307242, (2003). [MNSS91] R. Mrugała, J.D. Nulton, J.C. Schon, and P. Salamon: Contact structure in thermodynamic theory. Reports on Mathematical Physics, 29:109C121, (1991). [QZ92] M. Z. Qin and W. J. Zhu: Construction of higher order symplectic schemes by composition. Computing, 47:309–321, (1992). [Shu93] H.B. Shu: A new approach to generating functions for contact systems. Computers Math. Applic., 25:101–106, (1993).

Chapter 12. Poisson Bracket and Lie–Poisson Schemes

In this chapter, a clear Lie–Poisson Hamilton–Jacobi theory is presented. It is also shown how to construct a Lie–Poisson scheme integrator by generating function, which is different from the Ge–Marsden[GM88] method.

12.1 Poisson Bracket and Lie–Poisson Systems Before introducing the Lie–Poisson system, let us first review more general about the Poisson system.

12.1.1 Poisson Bracket Take a system with finite dimensions as an example. Give a manifold M and two smooth functions F, G on M , i.e., F, G ∈ C ∞ (M ). If an operation {·, ·} defined on C ∞ (M ) satisfies the following 4 properties, then {·, ·} is called Poisson bracket, and (M, {·, ·}) is called Poisson manifold[Olv93] . 1.

Bilinearity {aF1 + bF2 , H} = a{F1 , H} + b{F2 , H}, {F, aH1 + bH2 } = a{F, H1 } + b{F, H2 }.

2.

Skew-Symmetry {F, H} = −{H, F }.

3.

Jacobi Identity {{F, H}, G} + {{H, G}, F } + {{G, F }, H} = 0.

4.

Leibniz Rule {F1 · F2 , H} = F1 {F2 , H} + F2 {F1 , H}. Given a Hamiltonian function H ∈ C ∞ (M ), the induced equation F˙ = {F, H},

∀ F ∈ C ∞ (M )

K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

500

12. Poisson Bracket and Lie–Poisson Schemes

is called the generalized Hamiltonian equation. The most general case of Hamiltonian system is the one with symplectic structure, whose equations have the form: 5 6 5 6 O I p z˙ = JHz , J = , z= . I O q According to Darboux theorem, a general Poisson system with finite dimensions can be transformed into a local coordinate form, whose equations may be written as z˙ = K(z)Hz ,

(1.1)

the corresponding Poisson bracket is {F, H} = (∇z F (z))T K(z)∇z H(z),

∀ F, H ∈ C ∞ (M ).

K(z) satisfies 4 properties the above , if and only if K(z) = (kij (z)) satisfies kij (z)

∂klm (z) ∂k ∂k + kil (z) mj + kim (z) jl = 0, ∂zi ∂zi ∂zi

j, l, m = 1, 2, · · · , n.

(1.2)

We remark that any antisymmetry constant matrix satisfies (1.2) and hence is a Hamiltonian operator, and the bracket defined by it is a Poisson bracket. We will discuss its algorithm in more detail in the next section. Definition 1.1. A diffeomorphism z → z = g(z) : M → M is called a Poisson mapping, if it preserves the Poisson bracket, i.e., {F ◦ g, H ◦ g} = {F, H} ◦ g,

∀ F, H ∈ C ∞ (M ).

(1.3)

Theorem 1.2. For a Poisson manifold with structure matrix K(z), Equation (1.3) is equivalent to z ), gz K(z)gzT = K( where gz is the Jacobian matrix of g with respect to z. Proof. {F ◦ g, H ◦ g} =



∇(F ◦ g)

T

  K(z)∇ H ◦ g(z)

= (F ◦ g)z K(z)(H ◦ g)T z  T  T ∂g ∂g = Fz (g(z)) K(z) Hz (g(z) ∂z

= (∇F ◦ g)T and

∂z



∂g ∂g K(z) ∂z ∂z

T

(∇H ◦ g),

{F, H} ◦ g = ∇F T K∇H(g(z)) = (∇F ◦ g)T K(g(z))(∇H ◦ g).

By comparison, we get gz (z)K(z)(gz (z))T = K(g(z)) = K( z ). The theorem is proved.



12.1 Poisson Bracket and Lie–Poisson Systems

501

A Hamiltonian system on a Poisson manifold usually refers to the following ODEs dz = K(z)∇H(z), dt

(1.4)

where H(z) is a Hamiltonian function. The phase flow of the Equation (1.4), which is expressed as g t (z) = g(t, z) = gH (t, z), is a one parameter diffeomorphism group (at least locally), i.e., g 0 = identity,

g t1 +t2 = g t1 ◦ g t2 .

Theorem 1.3. The phase flow gH (z, t) of the Hamiltonian system (1.4) is a one parameter group of Poisson maps, i.e., {F ◦ g(z, t), G ◦ g(z, t)} = {F, G} ◦ g(z, t). Proof. See[Olv93] .

(1.5) 

By Theorem 1.2, we get gz (z, t)K(z)(gz (z, t))T = K(g(z)).

(1.6)

Definition 1.4. A smooth function C(z) is called a Casimir function, if {C(z), F (z)} = 0,

∀ F ∈ C ∞ (M ).

Definition 1.5. F (z) ∈ C ∞ (M ) is a first integral of Hamiltonian system, iff {F, H} = 0. Obviously, every Casimir function is a first integral.

12.1.2 Lie–Poisson Systems The Lie–Poisson system is a type[MW83,MR99] of common Poisson systems. Its structure space is the dual space of any Lie algebra, and its bracket is called Lie–Poisson bracket. There are two types of definition for the Lie–Poisson bracket: one relies on the coordinate definition, and the other does not rely on the coordinate definition. k Lie–Poisson bracket. Let g be a r-dimensional Lie algebra, Cij (i, j, k = 1, 2, · · · , r) be the configuration constants of g w.r.t. basis v1 , v2 , · · · , vr . Let V be another rdimensional linear space, with coordinate x = (x1 , x2 , · · · , xr ). Then Lie–Poisson bracket is defined by: {F, H} =

r  i,j,k=1

k k Cij x

∂F ∂H , ∂ xi ∂ xj

∀ F, H ∈ C ∞ (R).

(1.7)

According to the notation of the Poisson system k ij (x) =

r 

l l Cij x.

l=1

It is easy to verify that {F, H} satisfies the 4 properties of a Poisson bracket. For the infinite dimensional evolution equations, there exists a corresponding coordinate definition; see the literatures[Arn89,MR99] .

502

12. Poisson Bracket and Lie–Poisson Schemes

Lie group action and momentum mapping. The Lie–Poisson system is closely related to the Hamiltonian system with symmetry. Definition 1.6. The invariant property of a Hamiltonian system under one parameter differomorphism group is called symmetry of the Hamiltonian system. Under certain circumstance, this invariant property is called momentum. The corresponding mapping is called momentum mapping. The Lie group, action on manifold M , ∀ g ∈ G, and corresponds to a self-homeomorphism φg on M . Below, we consider only the translation action of G on itself and the induced action on T G and T ∗ G. Definition 1.7. Infinitesimal generator vector field: let g be a Lie algebra of G, ξ ∈g, then exp tξ ∈ G, d  ξM =  φexp tξ (x), x ∈ M d t t=0 is called infinitesimal generator vector field of the flow Ft = φexp tξ . Definition 1.8 (Lifted action). Action φg : M → M may induce action φ"g : T ∗ M → T ∗ M , which is defined as follows: φ"g (α) = T ∗ φg−1 = (T ∗ φg )−1 (α),

∗ α ∈ Tφ(g) (x).

Thus, we can prove that the lifted mapping of a diffeomorphism is symplectic. Definition 1.9 (Momentum mapping). Let (P, ω) be a connected symplectic manifold. Let G be a Lie group, φg : P → P a symplectic action. We call J : P → g∗ (g∗ is the dual space of g) a momentum mapping, if J satisfies ∀ ξ ∈ g,

d J(ξ) = iξp ω,

where J(ξ) is defined by J(ξ)(x) = J (x), ξ,  · , ·  denotes a scalar product, and ξp is the infinitesimal generator of the action to ξ. Theorem 1.10 (Generalized Noether theorem). Let φ be a symplectic action of G on (P, ω) with a momentum mapping J . Suppose H : P → R is G-invariant, i.e., H(x) = H(φg (x)),

∀ x ∈ P,

g ∈ G,

(1.8)

then J is a first integral of XH , i.e., if Ft is the phase flow of XH , then J (Ft (x)) = J (x). Proof. See[MW83] .



Definition 1.11. A momentum mapping J is called Ad∗ -equivariant, if J (φg (x)) = Ad∗g−1 J (x), that is, the following diagram commutes

∀ g ∈ G,

12.1 Poisson Bracket and Lie–Poisson Systems

503

φg

P −−−−→ P ⏐ ⏐ ⏐ ⏐ JG JG Ad∗ g −1

g∗ −−−−→ g∗ and we call such a group action as a Poisson action[AN90] . Theorem 1.12.

[MR99]

J is Ad∗ -equivariant momentum mapping, iff {J(ξ), J(η)} = J([ξ, η]),

i.e., J is a Lie homomorphism. Corollary 1.13. Let φ be a Poisson action of G on the manifold M , and φ" be the lifted action on T ∗ (M ) = P . Then this action φ" is symplectic and has an Ad∗ -equivariant momentum mapping given by J : P −→ g∗ ,

J(ξ)(α(q)) = α(q) · ξM (q),

q ∈ M,

α(q) ∈ T ∗ M.

ξM is the infinitesimal generator of φ on M . Below, we will discuss the translation action of a Lie group on itself using the above theorem and deduction. Let G be a Lie group, φ : G × G → G be a left translation action (g, h) → gh. Then its infinitesimal generator is ξG (g) = Te Rg ξ = Rg∗ ξ. Because lifted action is symplectic, by Corollary 1.13, we can obtain the momentum mapping: J (αq )(ξ) = αq Te Rg ξ = αq Rg∗ ξ =⇒ J (αq ) = Te Rg∗ αq = Rg∗ αq , or can rewrite it as

JL (αq ) = Rg∗ αq .

Likewise, we can obtain the similar result for the right translation JR (αq ) = L∗g αq . Lie–Poisson bracket and motion equation. In the previous sections, we have introduced the Lie–Poisson bracket and equations which are expressed by the local coordinates. Below, we will introduce an intrinsic definition of Lie–Poisson bracket and its induced equation of motion. Let  · , ·  be the pairing between g∗ and g, ∀ F : g∗ → R, defined by

= δF> DF (μ)γ = γ, , δμ

γ ∈ g∗ .

δF ∈ g, μ ∈ g∗ , is δμ

504

12. Poisson Bracket and Lie–Poisson Schemes

If we regard g∗∗  g, then DF (μ) ∈ g∗∗ becomes an element of g, =  δF δG > {F, G}(μ) = − μ, , , δμ δμ where [ · , · ] is the Lie bracket on g. The above equation is usually denoted as {F, G}. It is easy to verify that { · , · } satisfies the 4 properties of Poisson bracket, and are often called as (−) Lie–Poisson bracket. They are first proposed by Lie[Lie88] and are redefined by Berezin and others thereafter. We can prove that { · , · } can be derived from the left translation reduction of a typical Poisson bracket on T ∗ G. If the right translation reduction is used, we have the Lie–Poisson bracket (+): =  δF δG > , = {F, G}+ . {F, G}(μ) = μ, δμ δμ Given a Lie–Poisson bracket, we can define the Lie–Poisson equation. Take { · , · } as an example. Proposition 1.14. If H− ∈ C ∞ (g∗ ) is a Hamiltonian function, then the evolutionary equation on g∗ is: F˙ = {F, H}− , i.e.,

μ˙ = XH− (μ) = ad∗δH μ. δμ

Proof. Because

(1.9)

= δF > F˙ (μ) = DF (μ) · μ˙ = μ, ˙ , δμ

and =  δF δH > = δF > δF > = ∗ , = μ, ad δH = ad δH μ, . {F, H− }− (μ) = − μ, δμ δμ δμ δμ δμ δμ Since F is arbitrary, we obtain

μ˙ = ad∗δH μ. δμ

Likewise, for the right invariant system, the equation is μ˙ = −ad∗δH μ. δμ

Henceforth, we will denote the system of left translation reduction as g∗+ , and the right translation reduction as g∗− . Generally speaking, the rigid body and Heavy top system belongs to the left invariant system g∗− , and the continuous systems, such as plasma  and the incompressible flow, are right invariant system g∗+ . Lemma 1.15. JL , JR are Poisson mapping. Proof. See[MW83] .



12.1 Poisson Bracket and Lie–Poisson Systems

505

From this lemma, we can obtain the following reduction theorem (it will be used in the generating function theory later). Theorem 1.16. 1◦ commutes:

∗ For the left invariant system g− , we have the following diagram GtH◦J

R T ∗G T ∗ G −−−−−→ ⏐ ⏐ ⏐ ⏐ JR G JR G

∗ g−

Gt

−−−H−→

∗ g−

where H : g∗ → R is a Hamiltonian function on g∗− , GtH is a phase flow of Hamiltonian function H on g∗− , and GtH◦JR is phase flow of Hamiltonian function H ◦ JR on T ∗ G. 2◦ Similarly for right invariant system g∗+ , we have GtH◦J

L T ∗G T ∗ G −−−−→ ⏐ ⏐ ⏐ ⏐ JL G JL G

g∗+

Gt

−−−H−→

g∗+

Theorem 1.17. The solutions of a Lie–Poisson system are a bundle of coadjoint orbits. Each coadjoint orbit is a symplectic manifold and is called symplectic leave of the Lie–Poisson system. This theorem is from literature[AM78] . For Lie–Poisson system such as Heavy Top and the compressible flows, similar set of theories can be established. The readers can refer to literature[MRW90] for more details.

12.1.3 Introduction of the Generalized Rigid Body Motion Let G be a Lie group (finite dimensional), g(t) be a movement on G. We define: Velocity: V (t) = g(t) ˙ ∈ Tg(t) G; ˙ ∈ g; Angular velocity in body description: WB (t) = T Lg(t)−1 (g(t)) Angular velocity in space description: WS (t) = T Rg(t)−1 (g(t)) ˙ ∈ g; Momentum : M (t) = Ag g, ˙ where Ag : Tg G → Tg∗ G is called a moment of inertia operator, it relates to the kinetic energy by T =

1 1 1 1 (g, ˙ g) ˙ g = (WB , WB ) = AWB , WB  = Ag g, ˙ g, ˙ 2 2 2 2

where A: g→ g∗ is a value of Ag at g = e; Angular momentum in body description: MB (t) = T ∗ Rg(t) (M (t)) ∈ g∗ ; Angular momentum in space description: MS (t) = T ∗ Lg(t) (M (t)) ∈ g∗ .

506

12. Poisson Bracket and Lie–Poisson Schemes

From the above definition, we can obtain the following conclusions: WS (t) = Adg(t) WB (t),

MS (t) = Ad∗g(t)−1 MB (t),

MB (t) = AWB (t).

By Theorem 1.10, we get: Theorem 1.18. Conservation of spatial angular momentum theorem d MS (t) = 0. dt

(1.10)

Because the system that takes kinetic energy T as the Hamiltonian function is left invariant, MS (t) is the momentum mapping exactly. Corollary 1.19. Euler equation d MB (t) = {WB (t), MB (t)} = {A−1 MB (t), MB (t)}, dt

(1.11)

where { · , · } is defined by: {ξ, a} = ad∗ξ a,

∀ ξ ∈ g,

a ∈ g∗ .

Given below are two different proofs of the Euler equation. Proof. 1◦

From the Lie–Poisson equation of the motion μ˙ = ad∗∂H μ, we can obtain ∂μ

directly 1 2

H = (WB (t), MB (t)) =

 1  −1 A MB (t), MB (t) , 2

δH = A−1 MB (t) = WB (t). δ MB

2◦

By the definition of spatial angular momentum, we have MB (t) = Ad∗g(t) Ad∗g(0)−1 MB (0) = Ad∗g(t) η.

Since

(1.12)

MS (t) = MS (0) =⇒ Ad∗g(t)−1 MB (t) = Ad∗g(0)−1 MB (0) = η.

This also indicates that the trajectory of Lie–Poisson equation lies in some coadjoint orbit. From MB (t), ξ = Ad∗g(t) η, ξ = η, Adg(t) ξ,

∀ ξ ∈ g,

taking time derivatives on two sides, we get = > H I d MB (t) , ξ = η, [T Rg(t)−1 (g(t)), ˙ Adg(t) ξ] , dt

since

d Adg(t) ξ = [T Rg(t)−1 g(t), ˙ Adg(t) ξ] dt

12.2 Constructing Difference Schemes for Linear Poisson Systems

507

(see[AM78] ), then =

d MB (t) ,ξ dt

> =

η, [T Rg(t)−1 g(t), ˙ Adg(t) ξ]

=

η, Adg(t) [T Lg(t)−1 g(t), ˙ ξ]

=

Ad∗g(t) η, adT Lg(t)−1 g(t) ξ ˙

=

ξ MB (t), adT Lg(t)−1 g(t) ˙

=

ad∗T L

=⇒

˙ g(t)−1 g(t)

MB (t), ξ

d MB (t) = ad∗T L −1 g(t) MB (t) = {WB (t), MB (t)}. ˙ g(t) dt



The proof can be obtained.

Generally speaking, an equation of motion on T ∗ G, if it has Hamiltonian function H = T, it can be expressed by g(t) ˙ = T Lg(t)

∂H ∂H = Lg(t)∗ , ∂μ ∂μ

μ(t) ˙ = ad∗∂H μ(t). ∂μ

(1.13) (1.14)

Its solution is μ(t) = Ad∗g(t) Ad∗g(0)−1 μ(0). The Equation (1.14) is called as the Lie– Poisson equation.

12.2 Constructing Difference Schemes for Linear Poisson Systems Since the phase flow of Hamiltonian system is Poisson phase flow, which preserves the Poisson structure, it is important to construct difference schemes for system (1.4) that preserve the same property. Difference scheme that preserves the Poisson bracket is called as the Poisson difference scheme. One special case of the Poisson phase flow is the symplectic phase flow. How to construct the symplectic difference schemes has already been described in the previous chapters. The reader can also refer to literatures[Fen85,FWQW89,FQ87,CS90] for more details. However, the numerical algorithm for a general Poisson phase flow is still in its infancy. So far the results are limited to cases where structure matrix K is constant[Wan91,ZQ94,AKW93,Kar04] and K(z) is linear (Lie–Poisson) only. We will discuss the results for the Lie–Poisson case in the next section. In this section, we will discuss the results when K is a constant matrix.

508

12. Poisson Bracket and Lie–Poisson Schemes

12.2.1 Constructing Difference Schemes for Linear Poisson Systems Without loss of generality, we assume that K is an odd-dimensional matrix. Because odd dimensional antisymmetric matrix is definitely degenerated, there exists a coordi! nate transformation P ∈ GL(n) such that P KP T =

J2r O

O Os

.

τ (z) is called a Poisson scheme, if and Definition 2.1. A difference scheme z = gH T only if gz Kgz = K.

Next, we have: Definition 2.2. SK (n) = {A ∈ GL(n) | AKAT = K}, then the set SK (n) has the following properties: 1◦ When the rank of K is an even number and non-singular, then K has all the properties of a symplectic matrix. 2◦ When the rank of K is an odd number, it must be degenerated. It is easy to verify that SK (n) is a group and we call it as K-symplectic group. Its Lie algebra is sK (n) = {a ∈ gl(n) | aK + KaT = 0}. According to Feng et al.[FWQ90] , we can establish the relationship between SK (n) and sK (n) via Cayley transformation. If A ∈ SK (n), namely if AKAT = K, then B = (I − A)(I + A)−1 = (I + A)−1 (I − A) is an element of sK (n). However, if B ∈ sK (n), then A = (I − B)(I + B)−1 = (I + B)−1 (I − B) is an element of SK (n) . For a generalized Cayley transformation, we have the following result similarly: Theorem 2.3. Given φ(Λ) = p(0) ˙ = 0, if B ∈ sK (n), then

p(Λ) , p(Λ) is a polynomial that satisfies p(0) = 1, p(−Λ)

A = φ(B) ∈ SK (n). Therefore, we may use Pad´e approximation and pseudo-spectral method (the Chebyshev spectral method) to construct the Poisson schemes for the linear Poisson system. The Pad´e approximation has been described in the literatures[Qin89,ZQ94,FWQ90] in detail. Below, we will briefly describe the Chebyshev spectral method to construct the Poisson scheme. The Chebyshev spectral method is a highly effective method to approximate eA . The detailed explanation of this is described in literature[TF85] . Here, we give only the result. The Chebyshev spectral method is an approach based on series expansion by Chebyshev polynomial, i.e.,

12.2 Constructing Difference Schemes for Linear Poisson Systems

ex =

∞ 

 

x , R

Ck Jk (R)Qk

k=0

|x| < R,

509

(2.1)

where x is a real number and Qk is the Chebyshev complex orthogonal multinomial. Qk satisfies the following recurrence relation: Q0 (x) = 1, Q1 (x) = x, Qk+1 (x) = Qk−1 (x) + 2xQk (x), where C0 = 1, and Ck = 2 for k > 0. Jk denotes the k-order Bessel function. Qk denotes the Chebyshev polynomial. R is chosen arbitrarily. During computing, we calculate Jk (R) first, and then calculate Qk using the above recursive procedure. Using the generalized Cayley transformation, and A

eA =

e2 e

−A 2

,

and applying the Chebyshev spectral method to the numerator and denominator respectively, we can obtain the Poisson algorithm. It was pointed out in literature[TF85] that when k > R, the series converges exponentially. Therefore, the summation in (2.1) is always finite. Where to truncate the series is determined by the size of Jk (R). Since Jk (R) converges exponentially too, only a few steps of iteration is enough. Numerical tests show that this method has high accuracy and efficiency, especially when A is a dense matrix. The above method can be applied only to the linear dynamic system, where H is a quadratic form of z, z˙ = KBz.

12.2.2 Construction of Difference Schemes for General Poisson Manifold For a general H, there are other methods to construct Poisson integrator such as method of generating function. The reader can refer to literatures[Fen85,FWQW89,FQ87] for more details. For a low-order scheme, we can construct directly using the implicit Euler scheme and verify it by criterion (1.3). Let ∂H z˙ = K , ∂z

construct z k+1 = z k + τ K∇H





1 1 (I + B)z k+1 + (I − B)z k . 2 2

Take derivative of (2.2) w.r.t. z k , zz = I + τ KHzz





1 1 (I + B) zz + (I − B) , 2 2

(2.2)

510

12. Poisson Bracket and Lie–Poisson Schemes

i.e.,



 1 1 I − τ KHzz (I + B) zz = I + τ KHzz (I − B), 2

where z

k+1

2

∂x = z, z = z, xy = , therefore, ∂y k

 −1   1 1 zz = I − τ KHzz (I + B) I + τ KHzz (I − B) . 2

2

To become a Poisson scheme, it should satisfy zz K zzT = K, i.e.,  −1   1 1 I − τ KHzz (I + B) I + τ KHzz (I − B) K 2

2

 T  −1 1 1 I − τ KHz z(I + B) · I + τ KHz z(I − B) = K. 2

2

After manipulation, we obtain KHzz (KB T + BK)Hzz K = O. Therefore, if KB T + BK = O, i.e., B T ∈ sK (n), this scheme is a Poisson scheme. When B = O, then the scheme becomes Euler midpoint scheme. Denote z k+1 = GτH,B z k , then for B = O, the scheme is of second-order, for B = O, it is only of first order. Using z k+1 = GτH,±B z k , we can construct a composite scheme, τ

τ

2 2 ◦ GH,−B, GτH,±B = GH,B τ

τ

2 2 GτH,∓B = GH,−B ◦ GH,B.

Proposition 2.4. The above scheme has the second-order accuracy and the following proposition can be easily derived. 1 If φA (z) = z T Az, where AT = A, is a conservative quantity of Hamiltonian dz

2

∂H

system = K , and if A satisfies B T A + AB T = 0, then φA (z) is also a dt ∂z conservative quantity of difference scheme GτH,−B . Proof. Because φA (z) is a conservative quantity of Hamiltonian system, 1 T 1 z A z = z T Az, 2 2

12.2 Constructing Difference Schemes for Linear Poisson Systems

then

511

1 ( z + z)T A( z + z) = 0. 2

From B T A + AB T = 0, we obtain 1 1 (B( z − z))T A( z − z) = ( z − z)B T A( z − z) 2 2 1 4

= ( z − z)T (B T A + AB)( z − z) = 0, 

T

1 1 ( z + z) + B( z − z) 2 2

A( z − z) = 0 =⇒ τ w T AKHz (w) = 0,

Let

1 2

∀ w ∈ Rn .

1 2

w = (z k+1 + z k ) + B(z k+1 − z k ), we obtain

1 k+1 T k+1 1 (z ) Az = (z k )T Az k . 2 2



The proof can be obtained.

12.2.3 Answers of Some Questions 1. Euler explicit scheme[LQ95a] For a separable Hamiltonian H in a standard Hamiltonian system, there exists an Euler explicit symplectic scheme. Similar question is raised for the Poisson system: does there exist an Euler explicit Poisson scheme for a separable H? The answer is “may not be”. We take n = 3 as an example to explain this point. Let ⎡

−c

0

⎢ K=⎢ ⎣ c

0

−b H=

z˙ = K

z k+1 = z k + τ K∇H

⎤

⎥ −a ⎥ ⎦, 0

1 2 (z + z22 + z32 ), 2 1

then To make scheme

a

b



∂H = Kz. ∂z

1 1 (I + B)z k+1 + (I − B)z k 2 2



a Poisson scheme, we should have KHzz (KB T + BK)Hzz K = K 2 B T K + KBK 2 = O,

512

12. Poisson Bracket and Lie–Poisson Schemes

i.e., K 2 B T K ∈ Sm(n) (symmetrical matrix), here KB T ∈ Sm(n), i.e., BK ∈ Sm(n). Expand scheme z k+1 = z k + τ K∇H(w),

1 2

1 2

w = (I + B)z k+1 + (I − B)z k

⎧ k+1 ⎪ z1 = z1k − cτ w2 + bτ w3 , ⎪ ⎪ ⎨ z2k+1 = z2k + cτ w1 − aτ w3 , ⎪ ⎪ ⎪ ⎩ k+1 z3 = z3k − bτ w1 + aτ w2 .

into

To make sure the scheme is explicit, w2 , w3 have to be a function of z k only. From w2 =

1 k+1 1 1 1 (z + z2k ) + b21 (z1k+1 − z1k ) + b23 (z3k+1 − z3k ) + b22 (z2k+1 − z2k ), 2 2 2 2 2

we obtain b21 = 0 = b23 , b22 = −1. Likewise, b31 = b32 = 0, b33 = −1. Then B has the form ⎤ ⎡ b1 b2 b3 ⎥ ⎢ ⎢ 0 −1 0 ⎥ , ⎦ ⎣ 0 0 −1 substituting it into BK(∈ Sm(n)), we know only when a = 0, the scheme becomes an explicit scheme. Note that when a = 0, K is degenerated to the symplectic case. Therefore, in many situations, the separable system does not have an explicit scheme. Here the explicit scheme refers to the low-order finite-difference scheme, not the explicit analytic solution. 2. Midpoint scheme and Euler scheme Below, we will answer the questions whether the midpoint scheme is a Lie–Poisson scheme of Euler equation, and whether there exists a Lie–Poisson scheme in a generalized Euler scheme[LQ95a,LQ95b] . We already know that the answer for the first question is “no”. Now, we turn to the second question. The Euler equation has the form z˙ = J(z)Hz = f (z). For the case n = 3, ⎡

0

⎢ J(z) = ⎢ ⎣ z3

H=

1 2

−z2 % 2

−z3 0 z1 z22

z2

⎥ −z1 ⎥ ⎦, 0 &

z32

z1 + + I1 I2 I3

We construct a generalized Euler scheme:

⎤

.

12.2 Constructing Difference Schemes for Linear Poisson Systems

513

z = z + τ J(w)Hz (w) = z + τ f (w), where

1 1 1 1 ( z + z) + B( z − z) = (I + B) z + (I − B)z. 2 2 2 2 The Jacobian matrix of map z → z is & % ∂w ∂ z = I + τ D∗ f (w) , A= ∂z ∂z w=

where

⎡ 0

⎢ ⎢ ⎢ I3 − I1 D∗ f (z) = D∗ J(z)Hz = ⎢ ⎢ I1 I3 z3 ⎢ ⎣ I1 − I2 z2 I1 I2

I2 − I3 z3 I2 I3

⎤

I2 − I3 z2 I2 I3 ⎥

0

I3 − I1 z1 I1 I3

I1 − I2 z1 I1 I2

0

⎥ ⎥ ⎥, ⎥ ⎥ ⎦

∂w 1 1 = (I + B)A + (I − B), ∂z 2 2

therefore A = (I − τ D∗ f (w)(I + B))−1 (I + τ D∗ f (w)(I − B)). For the Euler scheme to be a Poisson scheme, it has to be AJ(z)AT = J( z ), therefore: AJ(z)AT = (I − τ D∗ f (w)(I + B))−1 (I + τ D∗ f (w)(I − B))J(z) · (I + τ (I − B T )(D∗ f (w))T )(I − τ (I + B)(D∗ f (w))T )−1 = J( z ), i.e.,

(I + τ D∗ f (w)(I − B))J(z)(I + τ (I − B T )(D∗ f (w))T ) = (I − τ D∗ f (w)(I + B))J( z )(I − τ (I + B T )(D∗ f (w))T ),

after manipulation, J( z ) − J(z) + τ 2 D∗ f (w)[(I + B)J( z )(I + B T ) − (I − B)J(z)(I − B T )](D∗ f (w))T = τ [J(z)(I − B T ) + J( z )(I + B T )](D∗ f (w))T + τ D∗ f (w)[(I − B)J(z) + (I + B)J( z )] = J( z − z) + τ 2 D∗ f (w)[J( z − z) + BJ( z + z) + J( z + z)B T + BJ( z − z)B T ](D∗ f (w))T .

Because τ is arbitrary, and z − z = O(τ ), we can have ⎧ z + z), J( z − z) = τ J( z + z)(D∗ f (w))T + τ D∗ f (w)J( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ τ 2 D∗ f (w)[BJ( z + z) + J( z + z)B T ](D∗ f (w))T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

= τ J( z − z)B T (D∗ f (w))T + τ D∗ f (w)BJ( z − z), z − z) + BJ( z − z)B T ](D∗ f (w))T = O. D∗ f (w)[J(

514

12. Poisson Bracket and Lie–Poisson Schemes 1

When B = O, the above equation is the midpoint scheme, w = ( z + z). It is 2 easy to verify that the last equality in the above equations is dissatisfied. Hence the midpoint scheme is not a Poisson scheme. When B = O, after complex computation, we can obtain similarly that there does not exist any B ∈ gl(n) to satisfy the above 3 formulas. Therefore, there does not exist a Poisson scheme in a generalized Euler form.

12.3 Generating Function and Lie–Poisson Scheme The generating function method plays a crucial role in constructing the symplectic scheme (see the literatures[FWQW89,CS90,CG93] for details). Therefore, how to use the generating function method to construct the Lie–Poisson scheme becomes a research hot spot. The literatures in this aspect include[GM88,Ge91,CS91] . We have also investigated the generating function for Lie–Poisson system in details, and discovered that the Ge–Marsden method needs further improvement. Below is our understanding and derivation on the generating function and the Hamilton–Jacobi theory[LQ95b] .

12.3.1 Lie–Poisson–Hamilton–Jacobi (LPHJ) Equation and Generating Function According to the diagram in Section 12.1 (for the left invariant system), GtH◦J =S

T ∗ G −−−−−R−−→ T ∗ G ⏐ ⏐ ⏐ ⏐ JR G JR G g∗

Gt =P

−−−H−−−→

g∗

the phase flow determined by H on g∗ can induce a phase flow on T ∗ G determined by H ◦ JR . Let ut (q, q0 ) be a first kind generating function of the symplectic map S. Then we have the following properties. Property 3.1. If u : G × G → R is invariant under the left action of G, i.e., ut (gq, gq0 ) = ut (q, q0 ),

(3.1)

then the symplectic mapping generated by u, S : (q0 , p0 ) → (p, q), where: p0 = −

∂ut (q, q0 ) , ∂q0

p=

preserves momentum mapping JL . That is to say, JL ◦ S = JL . For the right-invariant translation on G, JR ◦ S = J R .

∂ ut (q, q0 ) , ∂q

(3.2)

12.3 Generating Function and Lie–Poisson Scheme

515

Definition 3.2. If G acts on the configuration space without fixed point, then we say G acts on G freely. Property 3.3. If G acts on G freely, and its induced symplectic mapping S preserves the momentum mapping JL , then the first-kind generating function of S is left invariant. Proof. See[GM88] .



For a left-invariant system, such as a generalized rigid body, the Hamiltonian function is left invariant, the phase flow is also left invariant, the momentum mapping JL is a first integral for this dynamics, i.e., JL is invariant under the phase flow of GtH◦JR . Therefore, if the action is free (generally speaking, the action is locally free), the firstkind generating function is left invariant. Let ut (q, q0 ) be the first-kind generating function of S, then by the left invariance ut (q, q0 ) = ut (e, q −1 q0 ) = u "t (g),

g = q−1 q0 .

By Equation (3.2), we have p0 = −

 ∂u "t (Lq−1 q0 ) ∂ ut (q, q0 ) ∂ u"t (q −1 q0 ) ∂u " =− =− = −L∗q−1 ,  ∂ q0 ∂ q0 ∂ q0 ∂ g g=q−1 q0 

p=

∂ ut (q, q0 ) ∂u "(q −1 q0 ) ∂u "(Rq0 V (q)) ∂u " = = = V ∗ Rq∗0 ,  ∂q ∂q ∂ q" ∂ g g=q−1 q0

V (q) = q −1 , V ∗ = −L∗q−1 Rq∗−1 , then p = −L∗q−1 Rq∗−1 Rq∗0 therefore, μ0 = L∗q0 p0 = −L∗q0 L∗q−1 = −L∗q−1 q0



∂u "  ,  ∂ g g=q−1 q0 

∂u "  ∂ g g=q−1 q0

and μ = L∗q p = −L∗q L∗q−1 Rq∗−1 Rq∗0 = −Rq∗−1 q0



∂u " ∂u " = −L∗g ,   ∂ g g=q−1 q0 ∂ g g=q −1 q0





∂" u  ∂g g=q−1 q0



∂" u ∂" u = −Rg∗  −1 .  ∂g g=q−1 q0 ∂g g=q q0

Through the above derivation, it is easy to prove (3.1)

516

12. Poisson Bracket and Lie–Poisson Schemes

M0 = Rq∗0 p0 = −Rq∗0 L∗q−1



∂u " ,  ∂ g g=q −1 q0

M = Rq∗ p = −Rq∗ L∗q−1 Rq∗−1 Rq∗0 



∂" u  ∂g g=q−1 q0

= −L∗q−1 Rq∗0

∂u "  ∂ g g=q −1 q0

= −Rq∗0 L∗q−1

∂u " = M0 ,  ∂ g g=q −1 q0



i.e., JL ◦ S = JL . Take g = q −1 q0 , then ⎧ "(g) ∗∂u ⎪ ⎪ ⎨ μ0 = −Lq ∂ g , ⎪ ⎪ ⎩ μ = −Rg∗ ∂ u"(g) = Ad∗g−1 μ0 ,

(3.3)

∂g

therefore ut (q, q0 ) = u "t (q −1 q0 ) = u "t (g) defines a Poisson mapping: μ0 → μ = Ad∗g−1 μ0 . We now derive the conditions that ut (q, q0 ) must meet. ut (q, q0 ) generates a symplectic map S = GtH◦J = GtH , where H = H ◦ J , and S : (p0 , q0 ) −→ (p, q), p0 = −

∂u , ∂ q0

p=

∂u . ∂q

Because pd q − p0 d q0 = du =

we have

Note that

∂u ∂u dq + d q0 , ∂q ∂ q0

∂u ∂u ∂u ∂u dq + d q0 + d t = pd q − p0 d q0 + d t, ∂q ∂ q0 ∂t ∂t

& % ∂u d t = 0. d pd q − p0 d q0 + ∂t

(3.4)

12.3 Generating Function and Lie–Poisson Scheme

517

d (pd q − p0 d q0 ) = d p ∧ d q − d p0 ∧ d q0 % & % & ∂p ∂p ∂q ∂q ∂p ∂q = d p0 + d q0 + dp0 + dq0 + dt ∧ d t − d p0 ∧ d q 0 ∂ p0 ∂q0 ∂t ∂p0 ∂ q0 ∂t % & ∂p ∂q ∂p ∂q = − d p0 ∧ d q0 − d p0 ∧ d q0 ∂ p0 ∂ q0 ∂ q0 ∂ p0 +

∂q ∂p ∂q ∂ p0 d p0 ∧ d t + d q0 ∧ d t ∂ t ∂p0 ∂t ∂ q0

−

∂p ∂q ∂p ∂p dp0 ∧ dt − d q0 ∧ d t ∂ t ∂p0 ∂ t ∂ q0

= f1 + f2 + f3 .

Since (p0 , q0 ) → (p, q) is symplectic, we have gz JgzT = J =⇒ f1 = 0. Because

∂t

therefore,

⎧ ∂H ⎪ ⎪ ⎨ f2 = ∂ p d p ∧ d t =⇒ , ⎪ ⎪ ⎩ f3 = ∂ H d q ∧ d t

⎧ ∂q ∂H ⎪ ⎪ ⎨ ∂t = ∂p ⎪ ⎪ ⎩ ∂ p = −∂ H ∂q

∂q

d p ∧ d q − d p0 ∧ d q0 =

∂H ∂H dp ∧ dt + dq ∧ dt = dH ∧ dt ∂p ∂q

%

=⇒ d H ∧ d t + d %

We have d Therefore,

H+

∂H ∂t

∂H ∂t

&

∧ d t = 0.

& ∧ d t = 0.

∂u + H(p, q, t) = c. ∂t

Taking a proper initial value, we can obtain: ∂u + H(p, q, t) = 0, ∂t

i.e.,

∂ ut (p, q) + H ◦ JR (p, q, t) = 0. ∂t

Therefore we obtain the LPHJ equations   ∂ u(g) ∂ u(g) = 0, + H − Rg∗ ∂t

∂g

g = q−1 q0 .

(3.5)

518

12. Poisson Bracket and Lie–Poisson Schemes

Remark 3.4. If we can construct a generating function u(g), we then have u(q0 , q). This function can generate a symplectic mapping on T ∗ G. By the commutative diagram, a Poisson mapping on g∗ can also be induced. This is a key point of constructing a Lie–Poisson integrator by generating function. Remark 3.5. In order that the induced phase flow is a Poisson phase flow, the phase flow on T ∗ G should be symplectic. Therefore, the condition of g = q −1 q0 cannot be discarded. Namely, when t → 0, g = q−1 q0 (unit element). Remark 3.6. Only when g = q−1 q0 is satisfied, the momentum mapping is invariant. This is because the momentum mapping is JL (p, q) = Rq∗ p = Ad∗q−1 JR (p, q). To make sure JL (p0 , q0 ) = JL (p, q) =⇒ Ad∗q−1 JR (p0 , q0 ) = Ad∗q−1 JR (p, q) 0

=⇒ JR (p, q) = Ad∗q Ad∗q−1 JR (p0 , q0 ) 0

=

Ad∗(q−1 q0 )−1 JR (p0 , q0 )

= Ad∗g −1 JR (p0 , q0 ).

If g = q −1 q0 , deriving back, we obtain the momentum mapping is invariant. Remark 3.7. The above generating function theory can be transformed into the generating function theory on g (for details see literature[CS90] ). That is to say, the above generating function theory on T ∗ G can be reformulated by the exponential mapping in terms of algebra variables, which has been done by Channell and Scovel[CS90] . Below, we list only some of their results. For g ∈ G, choose ξ ∈ g, so that g = exp (ξ). Then the LPHJ equation can be transformed into ⎧ ∂s ⎪ + H(−ds · ψ(adξ )) = 0, ⎪ ⎪ ⎨ ∂t (3.6) M0 = −ds · χ(adξ ), ⎪ ⎪ ⎪ ⎩ M = −ds · ψ(adξ ), where

⎧ ⎨ χ(adξ ) = id + 1 adξ + 1 ad2ξ + · · · , ⎩

2

ψ(adξ ) = χ(adξ ) · e

and the condition g = q

−1

12

−adξ

(3.7)

 χ(adξ ) − adξ ,

q0 is transformed into s(ξ, 0) = s0 (ξ) = s0 (I),

i.e., ξ|t=0 = id.

(3.8)

12.3 Generating Function and Lie–Poisson Scheme

519

12.3.2 Construction of Lie–Poisson Schemes via Generating Function The generating function theory to construct the symplectic scheme has been described in detail in the literatures[LQ95a,Fen86,FWQW89] . The next step is to use the generating function theory to construct the Lie–Poisson schemes. As we know, the generating function must generate identity transformation at time zero. From the previous section, the generating function should satisfy the condition (3.8), i.e., the group element becomes a unit element at t = 0. We are not able to find a generating function universally applicable to a general Lie–Poisson system after a long time pursuit. Scovel[MS96] once suggested a possible solution using the Morse bundle theory. However, for the Hamilton function of quadratic form, we can find the low-order generating function. Below, we will give a brief description: 1 The Hamiltonian for so(3)∗ is H(M ) = M I −1 M . From (3.6) and (3.7), using 2 u as the generating function, we have:   1 1 M = −d u · ψ(adξ ) = −d u 1 − adξ + ad2ξ + O(ad3ξ ) 2 12 (3.9) 1 = −d u + d u · adξ + O(ad2ξ ). 2

After substituting H into Equation (3.6) and using expansion of ψ, we have   ∂u 1 + H −d u + d u · adξ − O(ad2xi ) ∂t



2







∂u 1 1 1 −d u + du · adξ + O(ad2ξ ) I −1 −d u + d u · adξ + O(ad2ξ ) = + ∂t 2 2 2

=

∂u 1 1 + I −1 d ud u − I −1 d ud u(adξ ) + O(ad2ξ ). ∂t 2 2

(3.10) Because ξ and time τ have the same order of magnitude, the Equation (3.10) can be simplified as % & ∂u ∂u 1 ∂u ∂u 1 ∂u ∂u + H(M ) = + I −1 − I −1 adξ + O(τ 2 ) ∂t

∂t

=

2

∂ξ ∂ξ

2

∂ξ

∂ξ

∂u 1 ∂u ∂u + I −1 + O(τ 2 ) ∂t 2 ∂ξ ∂ξ

= 0. From this, we can obtain a first-order generating function. Taking u=

Iξ · ξ . 2τ

(3.11)

It can be easily verified that the Equation (3.11) satisfies (3.10) to the approximate order. Therefore, we can use u to construct the Lie–Poisson scheme.

520

12. Poisson Bracket and Lie–Poisson Schemes

We first calculate ξ by solving M0 = −Iξ · χ(ξ),

(3.12)

and then substitute it into Equation (3.6). Next, we calculate M = exp (ξ)M0 . On repeating this procedure, we can obtain a Lie–Poisson algorithm. Below, we will apply this algorithm to free rigid body. For motion of the rigid body, χ(ξ) has a closed expression (see Subsection 12.5.2). Solving nonlinear (3.12) for ξ becomes a key point. It is necessary to linearize (3.12). The iterative formula for ξ is      −1 p ) δ ξ = ξ 1 + τ c1 ξ − c3 ξ(ξ + c4 ) (I −1 M0 × ξ) + c2 (I 0 k+1 − ξk , where c1 =

2 − |ξ| sin |ξ| − 2 cos |ξ| , |ξ|4

c2 =

cos |ξ| − 1 , |ξ|2

c3 =

−2|ξ| − |ξ| cos |ξ| + 3 sin |ξ| , |ξ|5

c4 =

2|ξ| − sin |ξ| . |ξ|3

In fact, the above algorithm is applicable even when H is a polynomial. Ge–Marsden[GM88] have proposed an algorithm, which neglects the generating function condition (3.8). Therefore, it has certain flaw. Below, we will explain it from the theoretical point of view. First, we should point out that the Ge–Marsden algorithm can only give the firstorder scheme for simple system such as the free rigid body. Its second-order scheme, however, is not a second-order approximation to the original system, as we will prove it below. Generally speaking, a generating function can be given as the following equation u = u0 +

∞  (δt)n n=1

n!

un ,

(3.13)

(ξ, ξ)

where u0 = generates the identical transformation at time t = 0. Substituting 2 (3.13) into the LPHJ equation, we have u1 = −H(V ),

u2 =

∂H · du1 · ψ(adξ ), · · · . ∂V

(3.14)

Below, we will take so(3)∗ as an example to explain the flaw of this algorithm. For so(3)∗ , u0 =

ξ2 , and hence V = ξ. The first-order scheme is 2

S1 = u0 + τ u1 =

ξ2 ξ2 τ − τ H(ξ) = − ξI −1 ξ. 2 2 2

The generating function for the second-order scheme is

12.3 Generating Function and Lie–Poisson Scheme

521

τ2 ξ2 τ 2 ∂H u2 = − τ H(ξ) + · du1 · ψ(adξ ) 2 2 2 ∂V   ξ2 τ τ2 = − ξI −1 ξ − I −1 ξ I −1 ξ · ψ(ξ) . 2 2 2

S2 = S 1 +

Using the system of Equation (3.6) (for SO(3) M, M0 denote angular momentum), we get: (3.15) M − M0 = −du · adξ . Next, we will prove that S1 indeed generates a first-order Lie–Poisson scheme to the Euler equation. However, S2 actually is not a second-order approximation to the Euler equation. Furthermore, we will find that with this algorithm, it is impossible to construct difference scheme that preserves the momentum mapping. Because % 2 & ξ τ d S1 = d − ξ · I −1 ξ = ξ − τ I −1 ξ 2

2

and M0 = −dS1 · χ(adξ ) = (−ξ + τ I −1 ξ) · χ(ξ), we have ξ = −M0 + O(τ ). By Equation (3.15) and applying ξ · adξ = 0, we obtain M − M0 = (ξ − τ I −1 ξ) · adξ = −τ I −1 ξ · adξ = τ [ξ, I −1 ξ] = τ [−M0 + O(τ ), I −1 (−M0 + O(τ ))] = τ [M0 , I −1 M0 ] + O(τ 2 ). This is a first-order approximation to the Euler equation M˙ = [M, I −1 M ].

(3.16)

For the second-order generating function S2 , we first calculate χ(ξ). Let χ(ξ) = 1+a1 ξ +a2 ξ 2 , where a1 , a2 have closed analytical expression (see Subsection 12.5.2) as follows 1 − cos |ξ| a1 = , 2 2 sin |ξ| + (1 − cos |ξ|)

sin |ξ| − |ξ| (cos |ξ| − 1)2 + (sin |ξ| − |ξ|)|ξ| + |ξ|2 |ξ| , a2 = 2 sin |ξ| + (1 − cos |ξ|)2

therefore,

u2 = −I −1 ξ(I −1 ξ · ψ(ξ)) = −I −1 ξ, I −1 ξ − a2 I −1 ξ(I −1 ξ · ξ 2 ),

then d S2 = ξ − τ I −1 ξ − τ 2 (I −1 )2 ξ − by

 τ2  d a2 I −1 ξ · (I −1 ξ · ξ 2 ) , 2

522

12. Poisson Bracket and Lie–Poisson Schemes

M0 = −d S2 · χ(ξ) = −ξ + τ ξ · χ(ξ) + O(τ 2 ), we have

ξ = −M0 + τ ξ · χ(ξ) + O(τ 2 ).

From Equation (3.15), we get M − M0 = −dS2 · adξ τ2

−1 ξ)2 − = −(ξ − τ I −1 ξ − τ 2 (I d(a2 I −1 ξ · (I −1 ξ · ξ2 ))) · ξ 2  = τ [M0 , I −1 M0 ] + a1 τ 2 [[M0 , I −1 M0 ], I −1 M0 ]  +[M0 , I −1 [M0 , I −1 M0 ]]   42 ) + I −1 (I −1 M0 · M 42 )M0 +a2 I −1 M0 (I −1 M0 · M 0 0

−

 τ2  d a2 · I −1 ξ(I −1 ξ · ξ2 ) · ξ + O(τ 3 ). 2

(3.17) According to the Euler equation (3.16), its second-order approximation should be M − M0 =

τ [M0 , I −1 M0 ] +

τ2 ([[M0 , I −1 M0 ], I −1 M0 ] 2

(3.18)

+[M0 , I −1 [I −1 M0 , M0 ]]) + O(τ 3 ). As t → 0, ξ → M0 , by comparison, we found that the Equation (3.17) is not an approximation to the Equation (3.18). Therefore, the generating function S2 cannot generate the second-order approximation to the Euler equation. We have shown that S1 generates a first-order Lie–Poisson scheme. However, a momentum mapping preserving scheme should satisfy JL (q, M ) = JL (q0 , M0 ). For T ∗ SO(3), this becomes qM = q0 M0 , and hence M = q−1 q0 M0 . Therefore, it is necessary to estimate q ∈ G. If we had formula M = gM0 , a very natural idea is to make g = q −1 q0 , which leads to q = q0 g −1 . An algorithm well constructed on so(3)∗ should lead to a good approximation of q ∈ SO(3) to equation of motion. The scheme that satisfies Equations (3.6) and the condition (3.8) and is generated by our generating function theory that belongs to this type. However, the scheme constructed via algorithm[GM88] does not belong to this type. Because the condition (3.8) is neglected, it is impossible to construct the algorithm on G using algorithm[GM88] . This can be illustrated as follows. Using another representation of (3.6) M0 = −du · χ(adξ ),

M = exp (adξ )M0 ,

(3.19)

and ξ = (−M0 + τ I −1 ξ) · χ(ξ), if we let q = q0 g −1 = q0 exp (−ξ) = q0 exp ((M0 − τ I −1 ξ) · χ(ξ)), then q is not a first-order approximation to the equation of motion q˙ = 4. In fact, the algorithm[GM88] cannot produce the form of q alone to construct qI −1 M momentum mapping preserving scheme.

12.4 Construction of Structure Preserving Schemes for Rigid Body

523

12.4 Construction of Structure Preserving Schemes for Rigid Body We have already introduced the equation of motion for generalized rigid body in previous section. In this section, we will take SO(3) as an example to explain how to construct structure-preserving schemes.

12.4.1 Rigid Body in Euclidean Space Let Λ(t) ∈ SO(3), such that Λ(t)Λ(t)T = I, |Λ(t)| = 1. Then the equation of motion for the free rigid body can be formulated as ˙ 4 (t), Λ(t) = Λ(t)W

(4.1)

4 (t) ∈ so(3), so(3) is the Lie algebra of SO(3). The isomorphism relation, where W so(3)  R3 , can be realized through the following equations: 4 (t)  W (t) ∈ R3 , W ⎡

0

⎢ ⎢ w3 ⎣ −w2

−w3 0 w1

⎤

w2

⎡

w1

⎤

⎥ ⎢ ⎥ ⎢ w ⎥ ⎢ 2 ⎥, −w1 ⎥  ⎥ ⎦ ⎢ ⎦ ⎣ w 3 0

4 (t) · a = W × a, W

a ∈ R3 .

4 (t) in Equation (4.1) is called angular velocity in the body description. W 4 (t) = The W −1 ˙ Λ(t) Λ(t) is consistent with the definition of generalized rigid body. The corresponding Euler equation is M˙ = [M, W ],

M = JW,

(4.2)

where J is called inertia operator, M the body angular momentum. The body variables and the spatial variables have the following relations: ⎧ ω = AW, ⎪ ⎪ ⎨ m = ΛM, ⎪ ⎪ ⎩ a = ΛA,

4 ΛT =⇒ ω = ΛW, ω  = ΛW

here A is an acceleration. Operator “  ” has the following equalities:

524

12. Poisson Bracket and Lie–Poisson Schemes

u × v = [ u, v], u  · v = u × v, [ u, v] · w = (u × v) × w, u·v =

1 tr ( u v). 2

The equation of motion of the rigid body may be expressed on space SU (2) or SH1 (unit quaternion). Applying their equivalence (their Lie algebra is isomorphism), we may obtain different forms of the Equation (4.1) under SU (2) and SH1 . SU (2): U ∈ SU (2), satisfies U U ∗ = I,

|U | = 1.

The equation of motion becomes U˙ = U Ωu , where Ωu = U ∗ U ∈ su(2), satisfies Ωu + Ω∗u = 0, tr Ωu = 0. In su(2), we choose {(−iσ1 ), (−iσ2 ), (−iσ3 )} as a basis, where 5 σ1 =

0 1

5

6 σ2 =

, 1 0 5

σ3 =

1

0

0

−1

5

6 ,

σ0 =

0

−i

i

0

1 0

6 ,

6

0 1

are 4 Pauli matrices. It is easy to see that 3  i=1

Hence

5 ωi σ i =

−iω3

−ω2 − iω1

ω2 − iω1

iω3

6 ∈ SU (2).

Ωu = (ω1 , ω2 , ω3 ) ∈ su(2)  R3  so(3),

using the matrix notation, rewrite the equation: 5 6 5 65 σ β −iω3 σ˙ β˙ = γ˙ δ˙ ω2 − iω1 γ δ

−ω2 − iω1

6 .

iω3

∀ Q ∈ SH1 , Q = 1, Q = (q0 , q1 , q2 , q3 ) ∈ H (set of all quaternion). The equation of motion becomes Q˙ = QΩh , where Ωh = QQ˙ = Q−1 Q˙ ∈ sh1 (quaternion with zero real part). Let

12.4 Construction of Structure Preserving Schemes for Rigid Body

Ωh = ω1 i + ω2 j + ω3 k = (0, ω1 , ω2 , ω3 ),

525

ωh = (ω1 , ω2 , ω3 ).

Rewrite the equation of motion into the quaternion form (q˙0 , q˙1 , q˙2 , q˙3 )

(q0 , q1 , q2 , q3 ) · (0, ω1 , ω2 , ω3 )  q˙0 = −qω = −(q, ω), q = (q1 , q2 , q3 ). =⇒ q˙ = q0 ω T + q ω, =

The Euler equation of motion becomes so∗3 :

dM = [M, W ]; dt

su∗2 :

d Mu 1 = [Mu , Wu ] = [Mu , W ], dt 2

Mu = 2M,

ωu = W ;

sh∗1 :

d Mh 1 = [Mh , Wh ] = [Mh , W ], dt 2

Mh = 2M,

ωh = W.

1 2

1 2

If the unified Euler equation of motion is used, we have dM = [M, W ]. dt If ω is assigned using the values of SO(3), then the corresponding equation of motion becomes: 4 (t), W (t) = (ω1 , ω2 , ω3 ), Λ˙ = ΛW   ω ω ω Ωh = 0, 1 , 2 , 3 , Q˙ = QΩh , 2 2 2   ω1 ω2 ω3 . U˙ = U Ωu , Ωu = , , 2

2

2

After the above transformation, the equation of motion becomes more simpler. The number of unknowns become fewer from the original 9 (SO(3)) to 4 complex variables (SU (2)), and then reduced to 4 real variables (SH1 ). The computation storage and operation may be sharply reduced for large-scale scientific computations. More details about the relations among SO(3), SU (2) and SH1 will be given in Section 12.5.

12.4.2 Energy-Preserving and Angular Momentum-Preserving Schemes for Rigid Body With the equation of motion of rigid body, we may construct the corresponding difference scheme[LQ95a] . One type of important schemes is the structure-preserving scheme. Structure-preserving may have some different meaning for different systems. For example, it could mean to preserve the original system’s physical structure, the symmetry, or invariant physical quantities.

526

12. Poisson Bracket and Lie–Poisson Schemes

The total energy and the angular momentum, especially the angular momentum, are important invariants for the rigid motion. Many experiments indicated that the energy and the angular momentum can be well maintained, which is essential for computer simulation to have a good approximation to the real motion. The equation of motion for the rigid body is ⎧ ˙ 4 (t), = Λ(t)W ⎨ Λ(t) / ˙ M (t) = M (t) × W (t) ˙ (t) = IW (t) × W (t), ⎩ =⇒ I · W M (t) = I · W (t) where I is the inertia operator. Note that the energy function H =

1 14 4 (t) is a Hamil(M (t), W (t)) = W (t)J W 2 2

4(t) = tonian function and the spatial angular momentum M (t) = ΛM (t) ⇔ M 4(t)Λ(t)T becomes momentum mapping. To maintain the energy and the anguΛM lar momentum invariant is just to maintain the Hamilton function and the momentum mapping of the Lie–Poisson system invariant. The energy invariance is mainly manifested in solving Euler equations, and the angular momentum invariance concerns more with equations of motion on SO(3). Using relation Λn+1 Mn+1 = Λn Mn , we can derive the formula for which Λn+1 should satisfy. For Euler equation M˙ (t) = M (t) × W (t) = M (t) × I −1 M (t), the midpoint scheme preserves the Hamiltonian function, i.e., it is energy-preserving (in fact midpoint scheme preserves all functions of quadratic form). The midpoint scheme for Euler equation is Mn+1 − Mn M + Mn M + Mn = n+1 × I n+1 , δt 2 2

multiply I −1

(4.3)

Mn+1 + Mn via inner product on both sides, 2

(Mn+1 − Mn ) · (I −1 (Mn+1 + Mn )) = 0 =⇒ I −1 Mn+1 · Mn+1 = I −1 Mn · Mn , i.e., Hn+1 = Hn . Since I −1 is a symmetric operator, we have Mn · I −1 Mn+1 = Mn+1 · I −1 Mn . Rewrite scheme (4.3) into Mn+1

δt (Mn+1 + Mn ) × I −1 (Mn+1 + Mn ) 4 % & % & δt δt   =⇒ I + I −1 (Mn+1 + Mn ) Mn+1 = I − I −1 (Mn+1 + Mn ) Mn 4 4 % &−1 % & δt −1 δ t −1   I − I (Mn+1 + Mn ) Mn =⇒ Mn+1 = I + I (Mn+1 + Mn ) 4 4 =

Mn +

=

Λ−1 n+1 Λn Mn .

12.4 Construction of Structure Preserving Schemes for Rigid Body

527

By conservation of angular momentum: Λn+1 Mn+1 = Λn Mn . By comparison, we obtain  −1   δt δt   Λn+1 = Λn I − I −1 (Mn+1 I + I −1 (Mn+1 + Mn ) + Mn ) . 4 4 Since

δt δt −1  I (Mn+1 + Mn ) = Wn + O(δt2 ), 4 2

from Cayley transformation, we know this is a second-order approximation to equa4. tion Λ˙ = ΛW In brief, if we construct an energy-preserving scheme on so(3)∗ , we may obtain a scheme approximate to the equation of motion by using the conservation of an angular momentum. We remark that this highly depends on the schemes constructed on so(3)∗ . Not every scheme on so(3)∗ corresponds to a good approximation scheme to the equation of motion on SO(3). Ge–Marsden algorithm for Lie–Poisson system is a typical example.

12.4.3 Orbit-Preserving and Angular-Momentum-Preserving Explicit Scheme The orbit-preserving[LQ95a] here means the motion trajectory remains at coadjoint orbit. For rigid body this means in every time step Mn+1 = Λn Mn ,

∃ Λn ∈ SO(3).

The midpoint scheme constructed in (4.2) is a kind of implicit orbit-preserving scheme. Below, we will derive explicit orbit-preserving schemes. The equation is 4 · M, M˙ = M × W = −W × M = −W

4 ∈ SO(3), W

4 = I −1 M. W

Assume the difference scheme to be constructed has the form Mn+1 = exp (b(δt))Mn .

(4.4)

It is easy to see when b(δt) = −δtWn = −δt(I −1 Mn ), (4.4) is a first-order scheme. Expanding the scheme (4.4), we obtain Mn+1 = Mn + b(δt)Mn + Using Taylor expansion, we have

2 3 1 1 b(δt) Mn + b(δt) Mn + · · · . 2 3!

(4.5)

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12. Poisson Bracket and Lie–Poisson Schemes

 ¨  (3)  4n Mn + M δt2 + M Mn+1 = Mn − δtW δt3 + · · · 2

4n Mn + = Mn − δtW +

3!

2

δt (Mn × Wn × Wn ) 2

(4.6)

 δt2  Mn × I −1 (Mn × Wn ) + · · · . 2

Let b(δt) = δtB1 + δt2 B2 + δt3 B3 + · · · , substitute it into (4.5), and retain only the first two terms 1 2

Mn+1 = Mn + δtB1 Mn + δt2 B2 Mn + (δtB1 + δt2 B2 )2 Mn + o(δt3 ) 1 2

= Mn + δtB1 Mn + δt2 B2 Mn + δt2 B12 Mn + o(δt3 ). (4.7) Comparing the coefficients of Equation (4.6) with those of (4.7), we have 4n , B1 = −W (B12 + 2B2 )Mn = (Mn × Wn × Wn ) + (Mn × I −1 (Mn × Wn ))   4 2 Mn − I −1 (M = W n × W n ) Mn , n then 4n , B1 = −W B2 = −

 1  −1 I (M n × Wn ) . 2

Likewise, we can construct third or fourth order schemes. Here we give only the result B3 =

1  4  −1   4 + I −1 (M × W I (M × W ) + 2I −1 (M × W )W W × W) 6

   1 1  + I −1 M × I −1 (M × W ) − B1 B2 − B2 B1 . 2

2

Another way to construct the orbit-preserving scheme is the modified R–K method, which can be described as follows. If the initial value M0 is known, let:

12.4 Construction of Structure Preserving Schemes for Rigid Body

529

μ0 = M0 ,  −1 μ ) 0

μ1 = eτ c10 (−I

M0 ,

 −1 μ ) τ c (−I −1 μ ) 1 20 0

μ2 = eτ c21 (−I

e

M0 ,

··· μr = eτ cr,r−1 (−I

 −1 μr−1 )

eτ cr,r−2 (−I

 −1 μr−2 )

 −1 μ ) 0

· · · eτ cr,0 (−I

M0 ,

the (r + 1)-th order approximation of the equation is   −1 μ ) τ c −1 μr−1 ) r r−1 (−I

M = eτ cr (−I

e

 −1 μ ) 0

· · · eτ c0 (−I

M0 .

Comparing the coefficients between the above equation and the Taylor expansion (4.6), we obtain cij and cs . Take r = 1 as an example.  −1

 −1

 −1

μ1 = eτ c10 (−I μ0 ) M0 = eτ c10 (−I M0 ) M0 = e−τ c10 (I M0 ) M0   2 −1 M + τ c2 (I −1 M )2 + O(τ 3 ) M , = 1 − τ c10 I 0 0 0 10 2

−1 μ ) τ c (−I −1 μ ) τ c1 (−I 1 0 0

M = e =

e



  −1 μ −1 μ 1 −τ c0 I 0

M0 = e−τ c1 I

2

−1 μ + τ c2 (I −1 μ )2 + O(τ 3 ) 1 − τ c1 I 1 1 1

e 

2

+ 

M0

−1 M 1 − τ c0 I 0



τ2 2  c (I −1 M0 )2 + O(τ 3 ) M0 2 0

2 2 −1 M − τ c I −1 μ + τ c2 (I −1 M )2 + τ c2 (I −1 μ )2 1 − τ c0 I 0 1 1 0 1 0 2 2 1  −1 μ · I −1 M + O(τ 3 ) M +τ 2 c0 c1 I 1 0 0  −1 M − τ c I −1 M + τ 2 c c I −1 (M −1 M ) = 1 − τ c0 I 0 1 0 1 10 0×I 0

=



+

τ2 2  τ2 −1 M )2 + τ 2 c c (I −1 M )2 + O(τ 3 ) M c0 (I −1 M0 )2 + c21 (I 0 0 1 0 0 2 2

−1 M ) · M = M0 − τ (c0 + c1 )I −1 M0 · M0 + τ 2 c1 c10 I −1 (M 0×I 0 0

+

τ2 2 −1 M )2 M + O(τ 3 ). (c + c21 + 2c0 c1 )(I 0 0 2 0

By the Taylor expansion, we have ⎧ c0 + c1 = 1, ⎪ ⎧ ⎪ ⎪ ⎨ ⎨ c0 + c1 = 1, 1 =⇒ c1 c10 = , 2 ⎪ ⎩ c c = 1. ⎪ 1 10 ⎪ ⎩ 2 2 c0 + c21 + 2c0 c1 = (c0 + c1 )2 = 1,

530

12. Poisson Bracket and Lie–Poisson Schemes 1

1

Set c0 = c1 = , c10 = 1 or c0 = 0, c1 = 1, c10 = , we obtain a second-order 2 2 modified R–K method. Literature[CG93] gives the modified R–K methods for general dynamic system. The scheme on so(3)∗ constructed via the above methods can be written as Mn+1 = ΛMn . −1 . It is easy to verify that the Λn+1 = Take Λ−1 n+1 Λn = Λ, we obtain Λn+1 = Λn Λ −1 ˙ Λn Λ approximates Λ = ΛW in the same order of accuracy as scheme Mn+1 = ΛMn .

12.4.4 Lie–Poisson Schemes for Free Rigid Body We have mentioned how to construct a scheme to preserve the angular momentum and the Lie–Poisson structure. The free rigid motion is a simple Lie–Poisson system. Among existing methods, Ge–Marsden algorithm is a first-order method to preserve the Lie–Poisson structure (we thus prove that this method is unable to maintain angular momentum). In Section 12.3, we introduced a generating-function method which is slow. We will introduce a fast method in this section. It is a split Lie–Poisson method[LQ95a] . It is also an angular momentum preserving method. Because the free rigid motion’s Hamiltonian function is separable, we can use the composite method to construct Lie–Poisson scheme according to separable system’s procedure. MacLachlan introduced an explicit method[McL93] which requires analytic solution for each split subsystem at every step. The midpoint method proposed below is also an explicit Lie–Poisson method but with few computations. The rigid motion’s Lie–Poisson equation is ⎡

x˙ 1

⎢ ⎢ ⎢ x˙ 2 ⎣

x˙ 3

⎤

⎡

0

−x3

x2

x3

0

−x1

−x2

x1

0

⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎦ ⎣

⎤ ⎥ ⎥ ⎥ ⎦

⎡ ∂H ⎢ ∂x1 ⎢ ⎢ ⎢ ∂H ⎢ ⎢ ∂ x2 ⎢ ⎣ ∂H ∂ x3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(4.8)

1

where x = (x1 , x2 , x3 )T ∈ R3 is an angular momentum, H = I −1 x, x is a 2 Hamiltonian function and energy function of system. For a separable system, I is usually a diagonal matrix. Without loss of generality, 1 let H = (a1 x21 + a2 x22 + a3 x23 ). According to the decomposition rule, the fewer 2 the split steps the better. We can use Casimir function of the Lie–Poisson equation to rewrite the system’s Hamilton function, and obtain an equivalent system. Note that n  |x|2 = x2i is a first integral of the system. Let i=1

1 2

1 2

1 2

H = H − a1 |x|2 = (a2 − a1 )x22 + (a3 − a1 )x23 = H1 + H2 , 1 2

1 2

where H1 = (a2 − a1 )x22 , H2 = (a3 − a1 )x23 .

12.4 Construction of Structure Preserving Schemes for Rigid Body

531

Substituting H1 into the Lie–Poisson equation (1.1), we have ⎡ x˙ = J(x)

⎢ ∂ H1 =⎢ ⎣ ∂x

−(a2 − a1 )x2 x3 0

⎤ ⎥ ⎥, ⎦

(4.9)

(a2 − a1 )x1 x2 where

⎡

0

⎢ J(x) = ⎢ ⎣ x3 −x2

−x3 0 x1

x2

⎤

⎥ −x1 ⎥ ⎦. 0

This equation can be simplified as a standard symplectic system 

x˙ 1 = −(a2 − a1 )x2 x3 , x˙ 3 = (a2 − a1 )x1 x2 ,

(4.10)

where x2 is a constant. Among symplectic difference schemes for the standard symplectic system (4.10), only a few of them can preserve the Lie-Poisson structure of the original system (4.9). Theorem 4.1. For the system (4.9), the midpoint scheme is a Lie–Poisson scheme[LQ95a] . In order to prove the Theorem 4.1, we need the following lemma first. Lemma 4.2. For the system (4.9), a symplectic algorithm for the standard symplectic system (4.10) preserves Poisson structure, if and only if the following three conditions are satisfied ⎧ −x11 x3 + x13 x1 = − x3 , ⎪ ⎪ ⎨ x1 , x31 x3 − x33 x1 = − (4.11) ⎪ ⎪ ⎩ x12 x 1 + x32 x 3 = 0, where xi = xni , xi = xn+1 , xij = i

∂ xi . ∂ xj

Proof. By the Theorem 1.2, a scheme is of Poisson if and only if %

∂x  ∂x

&

% J(x)

Expanding the above equation, we get

∂x  ∂x

&T = J( x).

532

12. Poisson Bracket and Lie–Poisson Schemes

⎡

x11

⎢ ⎢ 0 ⎣

x31 ⎡

0 ⎢ 3 = ⎢ ⎣ x − x2 i.e., ⎡

0

⎢ ⎢ x11 x3 − x13 x1 ⎣ −a13

x12 1 x32

⎤⎡

x13

−x3

0

⎥⎢ ⎢ 0 ⎥ ⎦ ⎣ x3 −x2

x33

− x3 0

0

x2

⎤⎡

x11

⎥⎢ ⎢ −x1 ⎥ ⎦ ⎣ x12

x1

0

x13

0

x31

⎤

1

⎥ x32 ⎥ ⎦

0

x33

⎤

x 2

⎥ − x1 ⎥ ⎦,

x 1

0

−x11 x3 + x13 x1 0

⎤

a13

⎡

0

⎥ ⎢ ⎢ 3 x31 x3 − x33 x1 ⎥ ⎦=⎣ x

x33 x1 − x31 x3

− x2

0

− x3 0 x 1

x 2

⎤

⎥ − x1 ⎥ ⎦, 0

where a13 = (x12 x3 − x13 x2 )x31 + (x13 x1 − x11 x3 )x32 + (x11 x2 − x12 x1 )x33 . Since the scheme is symplectic for (4.10), we have −x13 x31 + x11 x33 = 1. So a13 can be simplified as: a13 = (x3 x31 − x1 x33 )x12 + (x13 x1 − x11 x3 )x32 + x2 . Comparing the corresponding elements of the matrix on both sides and using the condition x 2 = x2 , we have ⎧ 3 , x11 x3 − x13 x1 = x ⎪ ⎪ ⎨ x31 x3 − x33 x1 = − x1 , ⎪ ⎪ ⎩ 1 + x32 x 3 = 0. x12 x 

Thus before the lemma is proved. Now we will prove the Theorem 4.1.

Proof. The midpoint scheme for system (4.9) is (here, I = (I1 , I2 , I3 ) = (a1 , a2 , a3 )) ⎧ x  + x3 ⎪ x2 , x 1 = x1 + τ (I1 − I2 ) 3 ⎪ ⎪ 2 ⎨ x 2 = x2 , ⎪ ⎪ ⎪ x  + x1 ⎩ x 3 = x3 + τ (I2 − I1 ) 1 x2 . ⎡

x11 ⎢ Its Jacobian matrix is ⎣ 0 x31

x12 1 x32

x13

⎤

⎥ 0 ⎦, where x33

2

12.4 Construction of Structure Preserving Schemes for Rigid Body

533

⎧ τ x11 = 1 + (I1 − I2 )x2 x31 , ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ τ τ ⎪ ⎪ x3 + x3 ) + (I1 − I2 )x2 x32 , x12 = (I1 − I2 )( ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ τ τ ⎪ ⎪ ⎨ x13 = (I1 − I2 )x33 x2 + (I1 − I2 )x2 , 2

2

τ τ ⎪ ⎪ x31 = (I2 − I1 )x11 x2 + (I2 − I1 )x2 , ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ τ τ ⎪ ⎪ x32 = (I2 − I1 )( x1 + x1 ) + (I2 − I1 )x2 , ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎩ x = 1 + τ (I − I )x x . 33 2 1 2 13 2

Solving the above equations, we get ⎧ 1 − a2 ⎪ ⎪ x11 = x33 = , ⎪ ⎪ 1 + a2 ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ (I1 − I2 ) x3 ⎪ ⎪ ⎪ , ⎨ x12 = 2 2 1+a

(4.12)

⎪ 2a ⎪ ⎪ x13 = −x31 = − , ⎪ ⎪ 1 + a2 ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ x1 (I2 − I1 ) ⎪ ⎪ ⎩ x32 = 2 , 2 1+a

where a=

τ (I2 − I1 )x2 . 2

(4.13)

Substituting the system of Equations (4.12) into condition (4.11), we find that all conditions are satisfied. Therefore, by Lemma 4.2, the scheme is of Poisson.  Lemma 4.3. [FQ91] Consider dynamic system x˙ = a(x). If a can be split into a = a1 + a2 + · · · + ak , and g s  esa is phase flow of a dynamic system, then gis  esai ,

s

2nd-order,

s

s

s

∀ i =⇒ g12 ◦ · · · gk2 ◦ gk2 ◦ · · · g12  esa ,

2nd-order.

Proof. For the standard symplectic system (4.10), the generalized Euler scheme x  = x + τ J∇H(B x + (I − B)x) is symplectic, iff 1 (I + C), JC + C T J = O. (4.14) 2 It is natural to ask what kind of symplectic difference scheme for the system (4.10) is also a Poisson scheme for the system (4.9). Below we restrict our discussion to the generalized Euler scheme ! (4.14). c1 c2 , then the symplectic condition (4.14) turns into c4 = −c1 . Let C = c3 c4 Therefore, B=

534

12. Poisson Bracket and Lie–Poisson Schemes

5 1 B= 2

then

5 1 B x + (I − B)x = 2

1 + c1

c2

c3

1 − c1

6 ,

(1 + c1 ) x1 + (1 − c1 )x1 + c2 ( x3 − x 3 )

x1 − x1 ) + (1 − c1 ) x3 + (1 − c1 )x3 c3 ( 5 6 z1 1 , = 2 z3

then Euler scheme becomes



x 1 = x1 − az3 , x 3 = x3 − az1 ,

6

(4.15)

(4.16)

where a is defined by Equation (4.13), z1 , z3 are defined by Equation (4.15). After complex computations, the elements of Jacobian matrix of the solution are x11 =

(1 + ac3 )(1 − ac2 ) − a2 (1 − c1 )2 , (1 + ac3 )(1 − ac2 ) + a2 (1 − c21 )

x13 =

−2a(1 − ac2 ) , (1 + ac3 )(1 − ac2 ) + a2 (1 − c21 )

x 3 =

((1 + ac3 )(1 − ac2 ) − a2 (1 − c1 )2 )x3 + 2a(1 + ac3 )x1 , (1 + ac3 )(1 − ac2 ) + a2 (1 − c21 )

3 x11 x3 − x13 x1 = x

(see (4.11)).

Since x1 , x3 are arbitrary real number, we can get c1 = 0,

c2 = −c3 .

(4.17)

Substituting Equation (4.17) into (4.16), and recalculating the Jacobian matrix, we have 2a(1 − ac2 ) (1 − ac2 )2 − a2 x31 = 2 , x = , 33 2 2 2 a + (1 − ac2 )

x 1 =

a + (1 − ac2 )

((1 − ac2 )2 − a2 )x1 − 2a(1 − ac2 )x3 . a2 + (1 − ac2 )2

It is easy to see that one of the conditions (4.11) x1 x31 x3 − x33 x1 = − is satisfied. Likewise, we can prove that another condition of (4.11) is also satisfied. From (4.17), we have C = cJ, where c is an arbitrary constant and

12.4 Construction of Structure Preserving Schemes for Rigid Body

5

0

J=

1

−1 0

535

6 . 

Therefore, the lemma is completed.

12.4.5 Lie–Poisson Scheme on Heavy Top The Lie–Poisson algorithm as we discussed in the previous sections are based on the dual space of semi-simple Lie algebra. In practice, we often have some Lie-Poisson system whose configuration space is not based on semi-simple Lie algebra, but on the dual space of the semi-product of Lie algebra and linear space. Such systems include but not limited to heavy top and compressible hydrodynamics flows. The reader can refer to literature[MRW90] for a more detailed discussion. In such configuration space, there exists no momentum mapping as we discussed in previous sections. The angular momentum is preserved only along a specific direction. Therefore, the generating function theory is no longer valid. However, using Lie–Poisson equations under local coordinates, we can construct the Lie–Poisson algorithm and the angular momentum preserving scheme. We will illustrate this by heavy top as an example. Heavy top is a gravity body under the action of gravity with a fixed point. The free rigid body is a heavy top with the fixed point in center of mass. Its configuration space is 3 dimensional Euclid space E(3). Its Lie algebra is no longer semisimple. Its phase space e∗ (3) has 6 coordinates {x1 , x2 , x3 , p1 , p2 , p3 }. The Poisson bracket operation on e∗ (3) is {xi , xj } = εijk xk , where

 εijk =

{xi , pi } = εijk pk ,

{pi , pj } = 0,

(4.18)

(i, j, k), i, j, k is not the same, 0,

i, j, k is the same.

There are two independent Casimir functions for bracket (4.18) f1 =

3  i=1

p2i ,

f2 =

3 

pi xi .

i=1

Let H(x, p) be the Hamiltonian function of system. Introducing notation ui = ∂H ∂H , Ωi = . Then the Lie–Poisson equation has the form of Kirchhoff equation ∂ pi ∂ xi

p˙ = [p, Ω],

x˙ = [x, ω] + [p, u],

(4.19)

where square bracket denotes cross product. H is the system’s energy, x and p are angular momentum, and momentum under momentum coordinate. For a general case, energy H is of quadratic form about x, p, and positive definite, which can be given as follows

536

12. Poisson Bracket and Lie–Poisson Schemes

2H =

3 

ai x2i

+

i=1

3 

bij (pi xj + xi pj ) +

i,j=1

3 

cij pi pj .

(4.20)

i,j=1

For heavy top, the energy is often expressed as the sum of kinetic energy and potential energy, i.e., H(x, p) =

x21 x2 x2 + 2 + 3 + γ1 p1 + γ2 p2 + γ3 p3 , 2I1 2I2 2I3

(4.21)

where Ii is the main movement inertia of the rigid body, γi (i = 1, 2, 3) are three coordinates of the center of mass. It is easy to see that this is a separable system. The structure matrix of the Lie–Poisson system is 5 6 J(x) J(p) , J(p) O where J(x) is defined in Subsection 12.4.4. It is difficult to construct Lie–Poisson algorithm on heavy top than on the free rigid body because the generating function methods are no longer suitable. However, it may become easier by the composition method and the Lemma 4.3. We will first split the Hamiltonian function H of heavy top system into six part 6  Hi , where H= i=1

Hi =

x2i , 2Ii

Hi+3 = γi pi ,

i = 1, 2, 3.

We will take H1 , H4 as examples to construct Lie–Poisson scheme. First, we will take H1 as the Hamilton function, then 5 6 5 6 ⎡ ∂ H1 ⎤ x˙ J(x) J(p) ⎣ ∂x ⎦, = p˙ J(p) 0 0 after manipulating, we get

⎧ x1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x x ⎪ ⎪ x2 = 3 1 , ⎪ ⎪ I1 ⎪ ⎪ ⎪ ⎪ x2 x1 ⎪ ⎪ ⎨ x3 = − I , 1

⎪ ⎪ p1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ x p ⎪ ⎪ p2 = 1 3 , ⎪ ⎪ I1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ p3 = − x1 p2 . I1

(4.22)

12.4 Construction of Structure Preserving Schemes for Rigid Body

537

Theorem 4.4. The Midpoint scheme of (4.22) is Poisson scheme for heavy top. Proof. By Theorem 1.2, the midpoint scheme is the Poisson scheme iff mapping (x, p) −→ ( x, p) satisfies ⎡

⎤

∂x  ⎢ ∂x

⎢ ⎣ ∂ p

⎥

∂ p ⎦ ∂p

∂x

 Denote

∂ y ∂z

⎡ 6 ∂ x J(p) ⎢ ∂ x ⎣ ∂x  O

∂x  5 J(x) ∂p ⎥



J(p)

∂p

∂ p ⎤ ∂x ⎥ ∂ p ∂p

⎦=

5

J( x) J( p)

6 .

J( p)

(4.23)

O

= yz , then the expand Equation (4.23), ⎧ xT x), x x J(x) x = J( ⎪ ⎪ ⎨ x x J(x) pT x J(p) pT p), x +x p = J( ⎪ ⎪ ⎩ pT p J(p) pT x J(p) pT px J(x) x +p x +p p = 0.

(4.24)

From the results of Subsection12.4.4, the first equation of system (4.24) is obviously hold. Note also ⎤ ⎡ ⎡ ⎤ 0 p21 p31 1 0 0 ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ 0 ⎥ px = ⎢ ⎣ 0 0 ⎦ , pp = ⎣ 0 p22 p23 ⎦ , 0 where p21 = we have

0

0

0

p32

p33

∂ p2 ∂ p3 ∂ pi , p31 = , pij = (i, j = 2, 3). Through the computation, ∂ x1 ∂ x1 ∂ pj

p22 = x22 , p21

p23 = x23 ,

τ p3 = I1 2 , 1+a

τ p2 I1 = , 1 + a2 −

p31

p32 = x32 ,

p33 = x33 , (4.25)

where a is defined by the Euler scheme (4.16). Substituting (4.25) into Equation (4.24), the 2nd and 3rd equations of (4.24) are also hold.  If H4 is taken as Hamiltonian function of a system, the equation degenerates into a constant equation. Then constructing Lie–Poisson scheme is trivial. For a Hamiltonian function of general form, we need to perform a transformation, so that the equation is easier for constructing the Lie–Poisson scheme. Take the free rigid body as the example. For a quadratic form, we have

538

12. Poisson Bracket and Lie–Poisson Schemes

H = Hi + Hij =

1 1 ai x2i + aij (xi + xj )2 , 2 2

i.e., we can eliminate the mixed items and transform it into a sum of squares. Next, we can construct Lie–Poisson scheme for system with Hij as Hamiltonian function. Take H12 as an example x˙ = J(x) It is easy to see that x1 + x2 (4.26) yields ⎡ x˙ 1 ⎢ ⎢ x˙ 2 ⎣

∂ H12 . ∂x

(4.26)

is a Casimir function of system. Expanding Equation ⎤

⎡

−a12 x3 (x1 + x2 )

⎤

⎥ ⎥ ⎢ ⎥ = ⎢ a12 x3 (x1 + x2 ) ⎥ . ⎦ ⎦ ⎣

(4.27)

a12 (x21 − x22 )

x˙ 3

Since x1 + x2 is a constant, denote c = x1 + x2 , the Equation (4.27) becomes ⎧ x˙ 1 = −ca12 x3 , ⎪ ⎪ ⎨ x˙ 2 = ca12 x3 , ⎪ ⎪ ⎩ x˙ 3 = ca12 (c − 2x2 ).

(4.28)

The midpoint scheme to the above equations is no longer Lie–Poisson scheme. However, we can solve system of the Equations (4.28) analytically without difficulty.

12.4.6 Other Lie–Poisson Algorithm Apart from the Lie–Poisson algorithm described above, we have other Lie–Poisson algorithms, which include but not limited to Scovel and MacLaclan[MS95] constrained Hamiltonian algorithm, and Veselov[Ves91b,Ves88] discrete Lagrangian system approach, as well as the reduction method mentioned before. Below, we will give a brief introduction to these method. 1.

Constrained Hamiltonian algorithm The detailed description about the constrained Hamiltonian algorithm can be found in literature[MS95] and its references. Here we apply it to rigid motion only. The structure space for a rigid motion is SO(n) = N . Take a larger linear space M = gl(n). Then the constraint function of N on M is φ(q) = q T q − 1,

∀ q ∈ M.

Note that φ(q) = 0 on N , and d φ(q) = Tq (M ) → R is a differential mapping. Assume on T ∗ M , there exists a non-constraint system of Hamiltonian equations 

p˙ = −∂q H, q˙ = ∂p H.

12.4 Construction of Structure Preserving Schemes for Rigid Body

539

Then on T ∗ N , if the local coordinates (p, q) on T ∗ M is still used, we should have φ(q) = 0 =⇒ dφ · q˙ = dφ · ∂p H = {H, φ}. Therefore, on T ∗ M , there exists an embedded submanifold CM = {(p, q) ∈ T ∗ M : φ(q) = 0, {H, φ} = 0}, which can induce a mapping ψ:

CM −→ T ∗ M,

(p, q) → (p− , q),

where p− = ψ(p, q). It is easy to verify that this is an isomorphic mapping and preserving the symplectic structure. There exist constrained equations of dynamic system on CM ,  q˙ = ∂p H, (4.29) p˙ = −∂q H + dφ · μ. If it is easy to construct structure-preserving scheme for the Equation (4.29) (e.g. when (4.29) is a separable system), then we can use map ψ to induce the algorithm on T N . Take SO(n) as an example. 1 tr (qJ ˙ q˙T ). Using the Legen2 1 dre transformation, we can obtain Hamiltonian function H(p, q) = tr (pJ −1 pT ) on 2 ∗

On T M we have a Lagrangian function L(q, q) ˙ =

T M . Therefore, using (4.29), we can obtain the constrained Hamiltonian equation of the dynamic system: ⎧ ⎨ q˙ = 1 pJ −1 , 2 (4.30) ⎩ p˙ = dψ · μ = 2qμ, which is a separable Hamilton system obviously. It is easy to construct the explicit symplectic difference scheme. But on T N , the Hamiltonian function becomes 1 H(p, q) = tr (I −1 (q T p)(q T p)T ), and its Hamiltonian equations are 4

⎧ ∂H −1 T ⎪ ⎪ ⎨ q˙ = qI (q p) = ∂ p , ⎪ ⎪ ⎩ p˙ = pI −1 (qT p) = − ∂ H ,

(4.31)

∂q

where q ∈ SO(n), q T p ∈ so(n). This is not a separable Hamilton system. Therefore, constructing its symplectic difference method will be difficult and computationally complicated. However using (4.30) and maps ψ, we can construct the algorithm for 1 SO(n) easily. Note that ψ(p) = (p − qpT q) in this case. 2

Scovel and McLachlan[MS95] proved this algorithm preserves the momentum mapping. We remark that the constraint Hamiltonian system has advantage only when the

540

12. Poisson Bracket and Lie–Poisson Schemes

expansion system is separable. Otherwise this algorithm is impractical. Take the rigid body as an example. On T ∗ SO(3), if Euler equation is to be solved, there are only 6 unknowns. If we expand it to T ∗ GL(n), the number of unknown becomes 18. If system is not separable, then the computation cost will definitely increase. 2.

Veselov–Moser algorithm Veselov–Moser algorithm[MV91] is to discretize Lagrange function first and then apply Legendre transformation to the discrete Lagrange function. The constructed algorithm preserves discreted symplectic structure, thus also preserves system’s Lie– Poisson structure. The concrete procedure is as follows: 1◦ First discretize the Lagrange function. 2◦ Add constraint and find the solution for δS = 0. 3◦ Obtain the discrete equation. 4◦ Solve this equation. n  T For SO(n), S = tr (Xk JXk+1 ). The constrained Lagrange function is k=1

L=S+

n 

(Xk XkT − 1),

k=1

then δL = 0 =⇒ Xk+1 J + Xk−1 J = Λk Xk , from this, we can have a system of equations  Mk+1 = wk Mk wk−1 , Mk = wkT J − Jwk ,

∀ k ∈ Z,

wk ∈ O(n),

(4.32)

−1 Xk . It is easy to prove that this discrete system of equations conwhere wk = Xk+1 verges to continuous system of Euler-Arnold equations:  ˙ M = [M, Ω], Ω ∈ o(n). (4.33) M = JΩ + ΩJ,

To solve Equation (4.32), the key lies in solving for wk . In order to make iteration (Xk , Yk ) → (Xk+1 , Yk+1 ) symplectic, Yk = Xk+1 , we need Yk+1 J + Xk J = Λk Xk+1 ,

Λk ∈ Sm(n).

This is because T J + Xk J Yk+1 J + Xk J = Xk+1 wk+1 T T = Xk+1 (wk+1 J + Xk+1 Xk J) T = Xk+1 (wk+1 J + wk J).

12.4 Construction of Structure Preserving Schemes for Rigid Body

See also

541

T J − Jwk+1 , JwkT − wk J = Mk+1 = wk+1

then

T JwkT + Jwk+1 = wk+1 J + wk J,

T J + wk J is symmetric. Thus ∃ Λk , Λk = ΛT i.e., wk+1 k , so that T Xk+1 (wk+1 J + wk J) = Λk Xk+1 .

Therefore, Yk+1 J + Xk J = Λk Xk+1 satisfies symplectic condition. The next question is how to solve wkT J − Jwk = Mk = tmk for wk ? The numerical experiments show that not all solutions wk that satisfy Equations (4.32) are the solutions we want. To solve ωk quickly, we propose to use the Quaternion method. w ∈ SO(3) corresponds to an element q = (q0 , q1 , q2 , q3 ) in SH1 . Their relations will be given in Section 12.5. Then the second equation in Equation (4.32) becomes ⎧ 2(α2 − α1 )q2 q1 + 2(α1 + α2 )q3 q0 = −δtm3 , ⎪ ⎪ ⎪ ⎨ 2(α3 − α1 )q3 q1 − 2(α3 + α1 )q2 q0 = δtm2 , ⎪ ⎪ ⎪ ⎩ 2(α − α )q q + 2(α + α )q q = −δtm , 3 2 3 2 3 2 1 0 1 in addition,

q02 + q12 + q22 + q32 = 1.

Solving the above nonlinear equations for (q0 , q1 , q2 , q3 ) is not an easy task. We found that when iteration step size is small, q0 , q1 , q2 , q3 behaves reasonable. However, when the step size is large, the solution behaves erratically. Numerical experiments show that solving these nonlinear equations is quite time-consuming, and hence this method is not recommended in practice. 3.

Reduction method Reduction method bases on the momentum mapping discussed in previous sections. We have mentioned that the solution of a Lie–Poisson system lies in a coadjoint orbit in Section 12.2, and this orbit has non-degenerated symplectic structure. If we can construct the symplectic algorithm on this reduced orbit, then this algorithm is naturally Lie–Poisson. Moreover it preserves the Casimir function and also preserves the orbit. Below, we will take SO(3) as an example to illustrate this method. The coadjoint orbit of SO(3) is a two dimensional spherical surface S2r . On S2r , we have a symplectic structure ωμ (ξg∗ (μ), ηg∗ (μ)) = −μ[ξ, η], and Hamiltonian function 1 2

Hμ (Ad∗g −1 μ) = I −1 (Ad∗g−1 μ)2 ,

542

12. Poisson Bracket and Lie–Poisson Schemes

where Ad∗g−1 μ denotes an element on S2r . How to choose the chart and local coordinate on S2r , so that the symplectic structure becomes simple, is very important. We once selected the spherical coordinate to be the local coordinate and the corresponding symplectic structure and Hamiltonian function become very complicated. However, if the Euler angle coordinate is used, the equations become very simple. Let S2r = {(x, y, z) | x2 + y 2 + z 2 = r 2 }, where x, y, z are three angular momentums in the body description. Using Euler angle coordinate θ, ψ to do the following coordinate transformation: ⎧ x = r sin θ cos ϕ, ⎪ ⎪ ⎨ y = r sin θ sin ϕ, ⎪ ⎪ ⎩ z = r cos θ. Lie–Poisson (Euler) equation may become the following Hamiltonian equations: ⎧ 1 ∂H ⎪ ⎨ θ˙ = − r sin θ ∂ ϕ , ⎪ ⎩ ϕ˙ = 1 ∂ H ,

(4.34)

r sin θ ∂ θ

where H=

1 2

%

r2 sin2 θ cos2 ϕ r2 sin2 θ sin2 ϕ r2 cos2 θ + + I1 I2 I3

& .

We can construct a non-standard symplectic algorithm for Equations (4.34) or we can simplify the problem further by transformation (θ, ϕ) → (cos θ, ϕ) = (x1 , x2 ), then

⎧ dx 1 ∂H 1 ⎪ ⎨ dt = r ∂ x , 2 ⎪ ⎩ d x2 = − 1 ∂ H . dt

r ∂ x1

This is a Hamiltonian system with standard symplectic structure, and its symplectic algorithm is easy to construct. To sum up, constructing Lie–Poisson scheme for a Lie–Poisson system has three methods. The first method is to lift it to T ∗ G and construct the symplectic algorithm (includes constraint Hamiltonian method) on it. The second is the direct construction based on g∗ (generating function method and composition method). The third is to construct symplectic algorithm on the reduced coadjoint orbit.

12.5 Relation Among Some Special Group and Its Lie Algebra

543

12.5 Relation Among Some Special Group and Its Lie Algebra In this section, we present relation among special group and its Lie algebra.

12.5.1 Relation Among SO(3), so(3) and SH1 , SU (2) Let

Λ ∈ SO(3),

|Λ| = 1,

ΛΛ = 1,

ξ ∈ so(3) =⇒ ξ + ξT = 0, q ∈ SH1 is a normal Quaternion q = (q0 , q) = (q0 , q1 , q2 , q3 ), q = (q1 , q2 , q3 ), q02 + q2 = 1 = q02 + q12 + q22 + q32 . We assume ⎡ ∀ ξ ∈ R3 ,

−ξ3

0

⎢ ξ = (ξ1 , ξ2 , ξ3 ) =⇒ ξ = ⎢ ⎣ ξ3

0

−ξ2

ξ1

ξ2

⎤

⎥ −ξ1 ⎥ ⎦ ∈ so(3), 0

 When A  ∈ so(3), A expresses its axial quantity. ξ is called the axial quantity of ξ. 1.

Transformation between SO(3) and SH1 ∀ q ∈ SH1 , x ∈ q0  R3 , qxq −1 represents a rotation of x. Using isomorphic mapping: ! q0 + q1 i q2 + q3 i , H  C2 , A(q) = −q2 + q3 i q0 − q1 i

we can obtain ∀ q ∈ H, ∃ Λ ∈ SO(3), ∀ x ∈ R3 , we have A(qxq −1 ) = A(0, Λx). From this we can get Λ. Given q = (q0 , q1 , q2 , q3 ), then ⎡

q02 + q12 −

1 2

⎢ ⎢ Λ = 2⎢ ⎢ q1 q2 + q0 q3 ⎣ q1 q3 − q0 q2 or simplify as

q1 q2 − q0 q3

q0 q2 + q1 q3

1 − 2

q2 q3 − q0 q1

q02

+

q22

q0 q1 + q2 q3

q02 + q32 −

Λ = (2q02 − 1)1 + 2q0 q + 2q ⊗ q.

It is easy to see that, if Λ = (Λij ) is known, then

1 2

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

544

12. Poisson Bracket and Lie–Poisson Schemes 1√ 1 + tr Λ, 2 (Q − Q23 ) q1 = 32 4q0 (Q13 − Q31 ) q2 = 4q0

q0 =

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ (Q21 − Q12 ) ⎪ ⎭ q3 =

=⇒ q =

1 (Λ − ΛT ). 4q0

4q0

2.

Relation between so(3) and SO(3) The relation between so(3) and SO(3) is the relation between Lie algebra and Lie group. Let ξ be an antisymmetry matrix for the axial quantity ξ, then exp ( ε) denotes a rotation in SO(3). We have the expansion  = exp (ξ)

∞ n  1  ξ ∈ SO(3). n=0

n!

According to the properties of SO(3), this expansion has a closed form, i.e., the Rodrigue formula % & 1 ξ sin2 2  = 1 + sin ξ ξ + 1 % 2 & ξ . Λ = exp (ξ) 2 ξ 2 1 ξ 2

We have two proofs of the above formula: one is from the geometry point of view and the other is from algebra point of view. Below, we will give details on the algebraic proof. ∀ ξ ∈ so(3), the following results hold after simple calculations: 3

 ξ = −ξ,

4

2

ξ = −ξ ,

|ξ| = 1.

Substitute them into the above series expansion,  = exp (ξ · n exp (ξ) )

 n=

ξ ξ



= 1 + sin ξ n + (1 − cos ξ) n2 = 1+

2 1 2 sin ξ  1 sin 2 ξ  ξ+ % &2 ξ . ξ 2 1 ξ 2

 We can prove that ξ is the angle of rotation exp (ξ).

12.5 Relation Among Some Special Group and Its Lie Algebra

545

3.

Transformation between SO(3) and SH1 The relation between SO(3) and SH1 is manifested by the relation between so(3) and SH1 . A rotation in SO(3), (θ, n) ↔ (ξ, ξ), ∀ (ξ, ξ) ∈ SO(3) =⇒ ξ ∈ so(3) =⇒ q0 = cos ⎛

q

1 ξ, 2

⎞

1 1 ⎝ sin 2 ξ ⎠ ξ. 1 2 ξ 2

=

When ξ  1, we use sin x x2 x4 x6 =1− + − x 6 120 5040

to deal with the singularity situation.  If q02 + q2 = 1, normalization is needed, which is just divided by q02 + q2 .  Since ξ has the same direction as q, Given (q0 , q) ∈ SH1 , we need to solve for ξ. we have q ξ = ξ , q

−1

where ξ can be given by ξ = 2 sin

(q ).

12.5.2 Representations of Some Functions in SO(3) By definition, we have iex(ξ) =

∞  n=0

ξn , (n + 1) !

χ(ξ)iex(−ξ) = Idξ . For ξ ∈ so(3), from  = iex(−ξ)

%

ξ ξ

&3 =−

∞  n (−ξ) n=0

= 1+

(n + 1)!

=

ξ , we have ξ ∞  k=0

∞  (−1)k+1 ξ2k k=1

(2k + 1)! 2

= 1+

∞   2k  2k+1 (ξ) (−ξ) + (2k + 1)! (2k + 2)! k=0

%

ξ ξ

&2 +

|ξ| − sin |ξ|  cos |ξ| − 1  ξ + ξ |ξ|3 |ξ|2 2

= 1 + c1 ξ + c2 ξ ,

∞  (−1)k+1 ξ2k+1 k=0

(2k + 2)!

·

ξ ξ

546

12. Poisson Bracket and Lie–Poisson Schemes

where c1 =

cos |ξ| − 1 |ξ| − sin |ξ| , c2 = . |ξ|2 |ξ|3

  = Idξ . χ(ξ)iex(− ξ)

 can be obtained by formula χ(ξ)

2

 = 1 + a1 ξ + a2 ξ , then Let χ(ξ) 2

2

  = (1 + a1 ξ + a2 ξ )(1 + c1 ξ + c2 ξ ) χ(ξ)iex(− ξ) 2

3

= 1 + (a1 + c1 )ξ + (a1 c1 + c2 + a2 )ξ + (a1 c2 + a2 c1 )ξ + a2 c2 ξ

4

= 1 + (a1 + c1 − (a1 c2 + a2 c1 )|ξ|2 )ξ + (c2 + a2 + a1 c1 − a2 c2 |ξ|2 )ξ = Id,

therefore



a1 + c1 − (a1 c2 + a2 c1 )|ξ|2 = 0, a1 c1 + c2 + a2 − a2 c2 |ξ|2 = 0.

Solving the above equations, we have a1 =

−c1 1 − cos |ξ| = , (1 − c2 |ξ|2 )2 + c21 |ξ|2 (sin |ξ|)2 + (1 − cos |ξ|)2 2

a2 =

−c2 + c2 |ξ| + c21 (1 − c2 |ξ|2 )2 + c21 |ξ|2

sin |ξ| − |ξ| (cos |ξ| − 1)2 + + (sin |ξ| − |ξ|)|ξ| |ξ|2 |ξ| = . (sin |ξ|)2 + (1 − cos |ξ|)2

2

Bibliography

[AKW93] M. Austin, P. S. Krishnaprasad, and L.-S. Wang: Almost Poisson integration of rigid body systems. J. of Comp. Phys., 107:105–117, (1993). [AM78] R. Abraham and J. E. Marsden: Foundations of Mechanics. Addison-Wesley, Reading, MA, Second edition, (1978). [AN90] A. I. Arnold and S.P. Novikov: Dynomical System IV. Springer Verlag, Berlin Heidelberg, (1990). [Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Berlin Heidelberg, Second edition, (1989). [CFSZ08] E. Celledoni, F. Fass`o, N. S¨afstr¨om, and A. Zanna: The exact computation of the free rigid body motion and its use in splitting methods. SIAM J. Sci. Comput., 30(4):2084– 2112, (2008). [CG93] P. E. Crouch and R. Grossman: Numerical integration of ordinary differential equations on manifolds. J. Nonlinear. Sci., 3:1–33, (1993). [CS90] P. J. Channell and C. Scovel: Symplectic integration of Hamiltonian systems. Nonlinearity, 3:231–259, (1990). [CS91] P. J. Channel and J. S. Scovel: Integrators for Lie–Poisson dynamical systems. Physica D, 50:80–88, (1991). [Fen85] K. Feng: On difference schemes and symplectic geometry. In K. Feng, editor, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, pages 42–58. Science Press, Beijing, (1985). [Fen86] K. Feng: Symplectic geometry and numerical methods in fluid dynamics. In F. G. Zhuang and Y. L. Zhu, editors, Tenth International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, pages 1–7. Springer, Berlin, (1986). [FQ87] K. Feng and M.Z. Qin: The symplectic methods for the computation of Hamiltonian equations. In Y. L. Zhu and B. Y. Guo, editors, Numerical Methods for Partial Differential Equations, Lecture Notes in Mathematics 1297, pages 1–37. Springer, Berlin, (1987). [FQ91] K. Feng and M.Z. Qin: Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study. Comput. Phys. Comm., 65:173–187, (1991). [FWQ90] K. Feng, H.M. Wu, and M.Z. Qin: Symplectic difference schemes for linear Hamiltonian canonical systems. J. Comput. Math., 8(4):371–380, (1990). [FWQW89] K. Feng, H. M. Wu, M.Z. Qin, and D.L. Wang: Construction of canonical difference schemes for Hamiltonian formalism via generating functions. J. Comput. Math., 7:71–96, (1989). [Ge91] Z. Ge: Equivariant symplectic difference schemes and generating functions. Physica D, 49:376–386, (1991). [GM88] Z. Ge and J. E. Marsden: Lie–Poisson–Hamilton–Jacobi theory and Lie–Poisson integrators. Physics Letters A, pages 134–139, (1988). [HV06] E. Hairer and G. Vilmart: Preprocessed discrete Moser–Veselov algorithm for the full dynamics of the rigid body. J. Phys. A, 39:13225–13235, (2006). [Kar04] B. Karas¨ozen: Poisson integrator. Math. Comput. Modelling, 40:1225–1244, (2004). [Lie88] S. Lie: Zur theorie der transformationsgruppen. Christiania, Gesammelte Abh., Christ. Forh. Aar., 13, (1988).

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[LQ95a] S. T. Li and M. Qin: Lie–Poisson integration for rigid body dynamics. Computers Math. Applic., 30:105–118, (1995). [LQ95b] S. T. Li and M. Qin: A note for Lie–Poisson– Hamilton–Jacobi equation and Lie– Poisson integrator. Computers Math. Applic., 30:67–74, (1995). [McL93] R.I. McLachlan: Explicit Lie–Poisson integration and the Euler equations. Physical Review Letters, 71:3043–3046, (1993). [MR99] J. E. Marsden and T. S. Ratiu: Introduction to Mechanics and Symmetry. Number 17 in Texts in Applied Mathematics. Springer-Verlag, Berlin, Second edition, (1999). [MRW90] J.E. Marsden, T. Radiu, and A. Weistein: Reduction and hamiltonian structure on dual of semidirect product Lie algebra. Contemporary Mathematics, 28:55–100, (1990). [MS95] R. I. McLachlan and C. Scovel: Equivariant constrained symplectic integration. J. Nonlinear. Sci., 5:233–256, (1995). [MS96] R. I. McLachlan and C. Scovel: A Survey of Open Problems in Symplectic Integration. In J. E. Mardsen, G. W. Patrick, and W. F. Shadwick, editors, Integration Algorithms and Classical Mechanics, pages 151–180. American Mathematical Society, New York, (1996). [MV91] J. Moser and A. P. Veselov: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Communications in Mathematical Physics, 139:217– 243, (1991). [MW83] J.E. Marsden and A. Weinstein: Coadjoint orbits, vortices and Clebsch variables for incompressible fluids. Phys D, 7: (1983). [MZ05] R.I. McLachlan and A. Zanna: The discrete Moser–Veselov algorithm for the free rigid body. Foundations of Computational Mathematics, 5(1):87–123, (2005). [Olv93] P. J. Olver: Applications of Lie Groups to Differential Equations. GTM 107. SpringerVerlag, Berlin, Second edition, (1993). [Qin89] M. Z. Qin: Cononical difference scheme for the Hamiltonian equation. Mathematical Methodsand in the Applied Sciences, 11:543–557, (1989). [TF85] H. Tal-Fzer: Spectral method in time for hyperbolic equations. SIAM J. Numer. Anal., 23(1):11–26, (1985). [Ves88] A.P. Veselov: Integrable discrete-time systems and difference operators. Funkts. Anal. Prilozhen, 22:1–33, (1988). [Ves91] A.P. Veselov: Integrable maps. Russian Math. Surveys,, 46:1–51, (1991). [Wan91] D. L. Wang: Symplectic difference schemes for Hamiltonian systems on Poisson manifolds. J. Comput. Math., 9(2):115–124, (1991). [ZQ94] W. Zhu and M. Qin: Poisson schemes for Hamiltonian systems on Poisson manifolds. Computers Math. Applic., 27:7–16, (1994). [ZS07a] R.van Zon and J. Schofield: Numerical implementation of the exact dynamics of free rigid bodies. J. of Comp. Phys., 221(1):145–164, (2007). [ZS07b] R.van Zon and J. Schofield: Symplectic algorithms for simulations of rigid body systems using the exact solution of free motion. Physical Review E, 50:5607, (2007).

Chapter 13. KAM Theorem of Symplectic Algorithms

Numerical results have shown the overwhelming superiority of symplectic algorithms over the conventional non-symplectic systems, especially in simulating the global and structural dynamic behavior of the Hamiltonian systems. In the class of Hamiltonian systems, the most important and better-understood systems are completely integrable ones. Completely integrable systems exhibit regular dynamic behavior which corresponds to periodic and quasi-periodic motions in the phase spaces. In this chapter, we study problems as to whether and to what extent symplectic algorithms can simulate qualitatively and approximate quantitatively the periodic and quasi-periodic phase curves of integrable Hamiltonian systems.

13.1 Brief Introduction to Stability of Geometric Numerical Algorithms Among the various kinds of equations of mathematical physics, only a few can be integrated exactly by quadrature and the rest are unsolvable. However, even an approximate solution is also valuable in many scientific and engineering problems. In a wide range of applications, the most powerful and perhaps the only practically feasible approximation is the numerical method — this is the case, especially in the computer era. A question arises accordingly: Whether a numerical method can reflect the real information of exact solutions of original problems properly or simulate accurately? To a problem described by time evolutionary equations, the solutions can often be represented by a flow (or semi-flow), which is locally defined on a phase space. Curves on the phase space which are invariant under the action of the flow (or semi-flow) are called invariant curves (or positively invariant curves) of the flow (or semi-flow). There is a natural correspondence between the solutions of the equations and the invariant curves (or positively invariant curves) of the flow (or semi-flow). The invariant curves, or positively invariant curves, are called solution curves of the equations. The qualitative analysis concerns with problems about understanding topological structures of the solution curves and their limit sets, which are often sub-manifolds of the phase space. The aim of the numerical method, in principle, not only pursues an optimal quantitative approximation to the real solution of the considered problem locally but also preserves as well as possible the topological and even geometrical properties of K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

550

13. KAM Theorem of Symplectic Algorithms

the solution curves and their limit sets globally. The latter constitutes the main content of qualitative analysis of the numerical method. Qualitative analysis becomes important in the study of numerical methods because instability phenomena take place very often even in the numerical simulations of very stable systems. Numerical treatments of stiff problems show that explicit methods have a severe time-step restriction, which lead to Dahlquist’s pioneering work about A-stability[Dah63] . Various notions of stability for numerical methods have been established since then, classifying different types of stable methods for different problems. The celebrated linear stability theory (A-stability, A(α)-stability, and L-stability)[Wid76,Ehl69] is based on the scalar linear equation1 y˙ = λy

(1.1)

and turns out to be powerful for the numerical study of all linear-dissipation-dominated problems. G-stability, which was also developed by Dahlquist[Dah75] , is characterized by retaining the contractivity property of any two solutions of nonlinear “contractive” systems y˙ = f (y). (1.2) Here, the “contractivity” of the system (1.2) is defined by the condition f (u) − f (v), u − v ≤ 0, d

(1.3)

which implies that u(t) − v(t) ≤ 0 for any two solutions u(t) and v(t) of (1.2). dt It is remarkable that for linear multistep one-leg methods, G-stability is equivalent to A-stability[Dah78] . Butcher extended Dahlquist’s idea and developed B-stability theory for Runge–Kutta methods[But75] . An elegant algebraic criterion of B-stability for Runge–Kutta methods was given by Burrage and Butcher, who further suggested the notion of algebraic stability of Runge–Kutta methods by only using the algebraic conditions of the criterion[BB79] . The algebraically stable Runge–Kutta methods can inherit very important dynamic properties of dissipative systems[HS94] . In many cases, algebraic stability is equivalent to B-stability[HS81] . Many notions and results about stability were also generalized to the general linear methods[HLW02,HNW93] . In almost the whole latter half of the last century, one of the central tasks of numerical analysis was the construction and analysis of numerical methods, satisfying these various stability conditions. Stable methods have no stringent step-size restriction in their own applicable ranges. They can preserve the dynamic stability properties (fixed points, periodic curves, and attractors) of most of dissipative systems[SH96] . Even explicit methods can also properly reflect the key dynamics of dissipative systems if sufficiently small step sizes are used[Bey87,SH96] . It was also proved that Runge–Kutta methods (including Euler methods), with small step sizes, can preserve the topological structures of dynamic trajectories of many structure-stable systems[Li99,Gar96] . The application of a numerical method to a generic system definitely changes the structure of the system. On the other hand, conventional methods do not change the 1

Dahlquist test equation.

13.2 Mapping Version of the KAM Theorem

551

topological structures of most dynamic trajectories of typical stable systems (e.g., dissipative systems having motion stability and Morse–Smale systems and Axiom A systems having structure-stability)[SH96,Li99] . However, this remarkable advantage of the conventional methods does not carry over to conservative systems. Most of the stable methods introduce artificial dissipation into conservative systems. They produce illusive attractors and therefore destroy the qualitative character of the conservative systems even if sufficiently small step sizes of the numerical method are used. The approximate solutions of conservative systems ask for new numerical methods which require more stringent stability. Geometric numerical integration theory for conservative systems has been developed rapidly in recent twenty years. The monographs[SSC94,HLW02,FQ03,LR05] summarize the main developments and important results of this theory. Qualitative behavior of geometric integrators has been investigated by many authors[Sha99,Sha00b,HL97,Sto98a,HLW02] . For symplectic integrators applied to Hamiltonian systems, some stability results, either in the spirits of the KAM theory or based on the backward analysis, have been well established[Sha99,Sha00b,HL97,CFM06,DF07] . The typical stable dynamics of Hamiltonian systems, e.g., quasi-periodic motions and their limit sets — minimal invariant tori, can be topologically preserved and quantitatively approximated by symplectic discretizations. In this chapter, we give a review about these results. For more details, readers refer to the relevant references[Sha99,Sha00b,HLW02,CFM06,DF07] .

13.2 Mapping Version of the KAM Theorem In this section, we introduce the mapping version of the celebrated KAM theorem. The main results of the theorem stem from answering a question about the stability of motions of planets in the solar system. This question attracted many great scientists in history and the culminated breakthrough was given by Kolmogorov (1954), Arnold (1963) and Moser (1962). The monograph[HLW02] gives a nice introduction to the KAM theorem based on the Hamiltonian perturbation theory. We present some results about differentiable Cantorian foliation structures of invariant tori in phase space of an integrable symplectic mapping under perturbations. We give relevant estimates explicitly in terms of the diophantine constant and nondegeneracy parameters of the frequency map of the integrable system. As a direct application of these estimates, we state a generalization of Moser’s small twist theorem to higher dimensions, which can be applied to prove a numerical version of the KAM theorem for symplectic algorithms.

13.2.1 Formulation of the Theorem Consider an exact symplectic mapping S : (p, q) → ( p, q) to be defined in the phase space I × Tn p = p − ∂2 H( p, q), q = q + ∂1 H( p, q), (2.1)

552

13. KAM Theorem of Symplectic Algorithms

where H : I × Tn → R is the generating function, I is an open and usually bounded set of Rn and Tn is the standard n-torus. In (2.1), ∂1 and ∂2 denote the gradient operators with respect to the first n and the last n variables respectively. When H(p, q) = H0 (p) does not depend on q, then (2.1) represents an integrable mapping p, q) = (p, q + ω(p)) with the frequency map S = S0 : (p, q) → ( ω(p) = ∂H0 (p),

p ∈ I,

(2.2)

where ∂ denotes the gradient operator with respect to p. Under the mapping S0 , the phase space I × Tn is completely foliated into invariant n-tori {p} × Tn , p ∈ I. On each torus, the iterations of S0 are linear with frequencies ω = ω(p). This is a typical integrable case. When a perturbation h(p, q) is added to H0 , i.e., H(p, q) = H0 (p) + h(p, q), (2.1) does not define an integrable mapping generally. However, KAM theorem shows that the perturbed mapping S still exhibits to a large extent the integrable behavior in the phase space if the frequency map ω is nondegenerate in some sense (see[Arn63,AA89,Arn89,Kol54b,Mos62] for Kolmogorov’s nondegeneracy and[CS94,R¨us90] for weak nondegeneracy) and the perturbation h is sufficiently small in some function space. In this chapter, we consider the following nondegeneracy condition for ω : I → Ω: θ |p1 − p2 | ≤ |ω(p1 ) − ω(p2 )| ≤ Θ |p1 − p2 |

(2.3)

for some 0 < θ ≤ Θ. Here I and Ω are the domains of action variables and the corresponding frequency values respectively. We always assume that I and Ω are open in Rn and ω is analytic and can be analytically extended to some complex domain, say I + r, of the real domain I, where r is the extension radius. We assume (2.3) is satisfied for p1 , p2 ∈ I + r with |p1 − p2 | ≤ r. Note that this nondegeneracy condition implies that the frequency map ω is invertible in any ball of radius r and centered in I, which is stronger than the standard Kolmogorov’s nondegeneracy assumption of the following (this was already noticed by P¨oschel in[P¨os82] , θ|d p| ≤ |d ω(p)| ≤ Θ|d p|

for p ∈ I + r.

(2.4)

An invariant torus of the integrable system is naturally specified by its frequency vector. Those tori are rationally dependent and even Liouville frequency vectors are generally destroyed by perturbations (Poincar`e and Mather[Mat88] ). The invariant tori of KAM type are specified by the so-called diophantine frequency vectors ω = (ω1 , · · · , ωn ),   ik,ω γ e − 1 ≥ τ for 0 = k = (k1 , · · · , kn ) ∈ Zn (2.5) |k|

with some constants γ > 0, τ > 0, where k, ω =

n  j=1

kj ωj and |k| =

n  j=1

|kj | for

integers k ∈ Zn . We introduce some notations. For an open or closed set I ⊂ Rn and for a ≥ 0, denote by C a (I ×Tn ), the class of isotropic differentiable functions of order a defined

13.2 Mapping Version of the KAM Theorem

553

on I ×Tn in the sense of Whitney. The norm of a function u ∈ C a (I ×Tn ) is denoted by ua,I×Tn . Since we also get the anisotropic differentiability of the foliations of invariant tori, we need the class C ν1 ,ν2 (I ×Tn ), of anisotropic differentiable functions of order (ν1 , ν2 ), with the norm denoted by uν1 ,ν2 ;I×Tn for a function u in the class. These two classes, endowed with the corresponding norms, are both Banach spaces. We also use another norm  · ν1 ,ν2 ;I×Tn ,ρ for ρ > 0 defined by uν1 ,ν2 ;I×Tn ,ρ = u ◦ σρ ν1 ,ν2 ;σρ−1 (I×Tn )

(2.6)

for u ∈ C ν1 ,ν2 (I × Tn ), where σρ denotes the partial stretching (x, y) → (ρx, y) for (x, y) ∈ I × Tn . Note that the following relation between these two norms is valid for 0 < ρ ≤ 1: uν1 ,ν2 ;ρ ≤ uν1 ,ν2 ≤ ρ−ν1 uν1 ,ν2 ;ρ , (2.7) where we dropped the domains to simplify the notations. Take Ω = ω(I) and denote by Ωγ the set of those frequencies, in Ω, which satisfy the diophantine condition (2.5) for given γ > 0 and whose distance to the boundary # of Ω is at least equal to 2γ. The set Ωγ is a Cantor set2 and the difference Ω \ Ωγ γ>0

is a zero set if τ > n + 1. Therefore Ωγ is large for small γ. The main results of this section are stated as follows: Theorem 2.1. Given positive integer n and real number τ > n+1, consider mapping p, q) = H0 ( p)+h( p, q), where H0 is S defined in phase space I ×Tn by (2.1) with H( analytic in I + r with r > 0 and h( p, q) belongs to the Whitney’s class C αλ+λ+τ (I × Tn ) for some λ > τ + 1 and α > 1,   i + j : i, j ≥ 0 integer . α∈ /Λ= λ

Suppose the frequency map ω = ∂H0 : I → Ω satisfies the nondegeneracy condition (2.3) for p1 , p2 ∈ I + r with |p1 − p2 | ≤ r where the constants θ and Θ satisfy 0 < θ ≤ Θ, then there exists a positive constant δ0 , depending only on n, τ , λ and α,   1 such that for any 0 < γ ≤ min 1, rΘ , if 2

hαλ+λ+τ,I×Tn ;γΘ−1 ≤ δ0 γ 2 θΘ−2 ,

(2.8)

then there exists a Cantor set Iγ ⊂ I, a surjective map ωγ : Iγ → Ωγ of C α+1 class and a symplectic injection Φ : Iγ × Tn → Rn × Tn of C α,αλ class, in the Whitney’s sense, such that 1◦ Φ is a conjugation from S to R. That is, the following equation holds: S ◦ Φ = Φ ◦ R,

(2.9)

where R is the integrable rotation on Iγ × Tn with frequency map ωγ , i.e., R(P, Q) = (P, Q+ωγ (P )) for (P, Q) ∈ Iγ ×Tn . Moreover, Equation (2.9) may be differentiated as often as Φ allows. 2

A subset of Rn is called a Cantor set if it is nowhere dense and complete in Rn .

554

13. KAM Theorem of Symplectic Algorithms

2◦ If Ω is a bounded open set of type D in the Arnold’s sense3 , then we have the following measure estimate  −n   mEγ ≥ 1 − c4 θΘ−1 γ m E,

(2.10)

where Eγ = Φ(Iγ × Tn ) is the union of invariant tori Φ({P } × Tn ), P ∈ Iγ , of S and m denotes the invariant Liouville measure on the phase space E = I × Tn ; c4 is a positive constant depending on n, τ , a and the geometric property of the domain Ω. 3◦ If h is of C βλ+λ+τ class with α ≤ β not in Λ, then we have further that ωγ ∈ C β+1 (Iγ ) and Φ ∈ C β,βλ (Iγ × Tn ). Moreover, ? ? ? −1 ? , γ −1 ωγ − ωβ+1;γΘ−1 ≤ c5 γ −2 Θ hβλ+λ+τ ;γΘ−1 ?σγΘ−1 ◦ (Φ − I)? β,βλ;γΘ −1

(2.11) with constant c5 depending on n, τ , λ and β, here we have dropped the domains in the notation of norms. 4◦ For each ω ∗ ∈ Ωγ , there exists p∗ ∈ I and P ∗ ∈ Iγ such that ω(p∗ ) = ωγ (P ∗ ) = ω ∗ and  −1 |P ∗ − p∗ | ≤ c6 γθΘ−1 hαλ+λ+τ,I×Tn ;γΘ−1 ,

(2.12)

where c6 is a positive constant depending on n, τ , λ and α. Theorem 2.2. If the frequency map ω satisfies the nondegeneracy condition (2.4), then the conclusions of Theorem 2.1 are still true with the same estimates (2.10) – (2.12) under the following smallness condition for h, hαλ+λ+τ,I×Tn ;γΘ −1 ≤ δ0 γ 2 θ 2 Θ −3 ,

(2.13)

where δ0 > 0 is depending only on n, τ , λ and α and is sufficiently small. Remark 2.3. The above two theorems are stated for the case when h is finitely many times differentiable. If h is infinitely many times differentiable or analytic, we have the following conclusions, which are easily derived by similar remarks to those of[P¨os82] . 1◦ If h ∈ C ∞ (I × Tn ), then ωγ ∈ C ∞ (Iγ ) and Φ ∈ C ∞ (Iγ × Tn ) and the estimates (2.8) hold for any β ≥ α. 2◦ If h ∈ C ω (I × Tn ), then we have further Φ ∈ C ∞,ω (Iγ , Tn ) under an additional smallness condition for δ0 which also depends on the radius of analyticity of h with respect to angle variables. Here C ω denotes the class of real analytic functions.

13.2.2 Outline of the Proof of the Theorems In this section, we give an outline of the proof of Theorem 2.1. For detailed arguments refer to [Sha00a] . 3

For example, a domain with piece-wise smooth boundary is of type D in the Arnold’s sense.

13.2 Mapping Version of the KAM Theorem

555

a. We transform the mapping S, by a partial coordinates stretching σρ : (x, y) → x, y). The mapping T is determined (p, q) = (ρx, y), to T = σρ−1 ◦ S ◦ σρ : (x, y) → ( by x  = x − ∂2 F ( x, y), y = y + ∂1 F ( x, y), (2.14) where F (x, y) = F0 (x) + f (x, y)

(2.15)

is well defined on Iρ × T with n

F0 (x) = ρ−1 H0 (ρx), and

f (x, y) = ρ−1 h(ρx, y)

(2.16)

Iρ = ρ−1 I = {x ∈ Rn |ρx ∈ I}.

(2.17)

For the time being, ρ is regarded as a free parameter. F0 (x) is real analytic in Iρ + rρ and f belongs to the class C a (Iρ × Tn ) where rρ = ρ−1 r, a = αλ + λ + τ . So the new mapping T satisfies the assumptions of Theorem 2.1 in which only I, r, H, H0 , and h are replaced by Iρ , rρ , F , F0 , and f respectively. Accordingly, the frequency map of the integrable mapping associated to the generating function F0 turns into ω " (x) = ∂F0 (x), x ∈ Iρ , and the nondegeneracy condition for the mapping turns out to be ρθ |x1 − x2 | ≤ |" ω (x1 ) − ω " (x2 )| ≤ ρΘ |x1 − x2 |

(2.18)

for x1 , x2 ∈ Iρ + rρ with |x1 − x2 | ≤ rρ . In addition, from (2.16), we have f a,Iρ ×Tn = ρ−1 ha,I×Tn ;ρ . 1 2

From now on, we fix ρ = γΘ−1 . Then the assumption 0 < γ ≤ rΘ in Theorem 1

2.1 implies that 0 < ρ ≤ r. Hence, rρ ≥ 2. Let Iρ∗ be the set of points in Iρ with the 2 distance to its boundary at least one and let Iρ;γ = ω " −1 (Ωγ ) ∩ Iρ .

(2.19)

Then, from (2.18) and the definition of Ωγ it follows that (Iρ;γ + 1) ∩ Rn ⊂ Iρ∗ ⊂ (Iρ∗ + 1) ∩ Rn ⊂ Iρ ,

(2.20)

and γμ |x1 − x2 | ≤ |" ω (x1 ) − ω " (x2 )| ≤ γ |x1 − x2 | ,

μ = θΘ −1

(2.21)

for x1 , x2 ∈ Iρ + 2 with |x1 − x2 | ≤ 2. b. We approximate f by real analytic functions. Let sj = s0 4−j ,

rj = sλj ,

j = 0, 1, 2, · · ·

(2.22)

556

13. KAM Theorem of Symplectic Algorithms

with fixed λ > τ + 1 and s0 > 0. Let Uj = Iρ × Tn + (4sj , 4sj ) be the complex extended domain of Iρ × Tn with extended widths 4sj of Iρ and Tn respectively[P¨os82] . By an approximation lemma [P¨os82] , there exists real analytic functions fj defined on U0 with f0 = 0 such that, in case f ∈ C b (I × Tn ) with b ≥ a, |fj − fj−1 |Uj ≤ sbj cb f b;Iρ ×Tn f − fj b ,I ∗ ×Tn −→ 0 ρ

j = 1, 2, · · · ,

(j −→ ∞)

for 0 < b < b,

(2.23)

where cb is a positive constant only depending on b, n and s0 but not depending on the domain Iρ and hence not depending on the parameter ρ. Moreover, we may require fj to be 2π-periodic in the last n variables. In (2.23), | · |Uj denotes the maximum norm of analytic functions on the complex domain Uj . c. We give the KAM iteration process which essentially follows P¨oschel [P¨os82] in x, y) the Hamiltonian system case. For each fj , we define a mapping Tj : (x, y) → ( by x  = x − ∂2 Fj ( x, y), y = y + ∂1 Fj ( x, y) (2.24) x, y) = F0 ( x)+fj ( x, y). For each j, the function Fj ( x, y) is well-defined and with Fj ( real analytic on Uj if 4sj ≤ rρ = ρ−1 r — this inequality is satisfied for j = 0, 1, · · ·  1

if we choose 0 < s0 ≤ 4−1 noting that 0 < γ < rΘ . We can show that each Tj 2 for j ≥ 0 is a well-defined analytic mapping on a domain of complex extension of the phase space Iρ × Tn , which is appropriate for the KAM iterations if h is bounded by (2.8) with a sufficiently small δ0 > 0. It follows from (2.23) that Tj converges to T in C b−1−κ -norm for any κ > 0 on some subdomain Iρ∗ × Tn of the phase space Iρ × Tn , where T is well-defined. The central problem is to find transformations Φj and integrable rotations Rj , defined on a sequence of nested complex domains that intersect a nonempty Cantor set, say I"ρ;γ × Tn , such that the following holds as j → ∞ on I"ρ;γ × Tn in some Whitney’s classes, " " Cj = Rj−1 ◦ Φ−1 j ◦ Tj ◦ Φj −→ identity, Φj −→ Φ, Rj −→ R,

(2.25)

" and R " are well-defined on I"ρ;γ × Tn . In this case, we have where Φ "=Φ " ◦R " T ◦Φ

on I"ρ;γ × Tn .

(2.26)

Transforming the mapping T back to S by the partial coordinates stretching σρ , " and R " to Φ and R respectively, we have and meanwhile transforming Φ S◦Φ=Φ◦R where

on Iγ × Tn ,

Iγ = ρI"ρ;γ = {p ∈ Rn | ρ−1 p ∈ I"ρ;γ }

13.2 Mapping Version of the KAM Theorem

557

is a Cantor set of I. In fact, due to the nondegeneracy of the frequency map ω in the sense of (2.3), we may keep the frequencies prescribed by (2.5) fixed in the above approximation process. As a result, we have ωγ (Iγ ) = Ωγ , where ωγ is the frequency map of the integrable rotation R on Iγ ×Tn . This is just the conclusion (1) of Theorem 2.1. The construction of Φj and Rj uses the KAM iteration, which is described as follows. Assume (2.27) |fj − fj−1 | ∼ εj , j = 1, 2, · · · , where εj is a decreasing sequence of positive numbers . Suppose we have already found a transformation Φj and a rotation Rj with frequency map ω (j) such that Cj = Rj−1 ◦ Φ−1 j ◦ Tj ◦ Φ j

(2.28)

|Cj − I| ∼ εj+1 .

(2.29)

satisfies Then, we construct a transformation Ψj and a new rotation Rj+1 with frequency map ω(j+1) such that (2.30) Φj+1 = Φj ◦ Ψj and (2.29) is also true for the next index j + 1 with Cj+1 defined by (2.28) in which j is replaced by j + 1. As was remarked by P¨oschel in[P¨os82] , “for this procedure to be successful it is essential to have precise control over the various domains of definition”. We define transformation Ψj : (ξ, η) → (x, y) implicitly with the help of a generating function ψj by x = ξ + ∂2 ψj (ξ, y),

y = η − ∂1 ψj (ξ, y).

(2.31)

To define ψj , we consider mapping Bj = Rj−1 ◦ Φ−1 j ◦ Tj+1 ◦ Φj .

(2.32)

Bj is near identity and is assumed to be given implicitly from its generating function bj by x  = x − ∂2 bj ( x, y), y = y + ∂1 bj ( x, y). (2.33) The function ψj is then determined from bj by the following homological equation ψj (x, y + ω (j) (x)) − ψj (x, y) + "bj (x, y) = 0,

(2.34)

where "bj (x, y) = bj (x, y) − [bj ](x) with [bj ] being the mean value of bj with respect to the angle variables over Tn . Define ω (j+1) (x) = ω (j) (x) + ∂[bj ](x). Then Rj+1 : (x, y) → ( x, y) is just given by

(2.35)

558

13. KAM Theorem of Symplectic Algorithms

x  = x,

y = y + ω (j+1) (x).

(2.36)

With the so defined Ψj and Rj+1 , we easily show that, formally, Ψ−1 j ◦ Rj ◦ Bj ◦ Ψj = Rj+1 ◦ Cj+1 . Formal calculations similar to those in[P¨os82] show that (2.29) is valid if we replace j by j + 1. We do not solve the Equation (2.34) exactly. Instead, we will solve an approximate equation by truncating the fourier expansion of "bj with respect to angle variables to some finite order so that “only finitely many resonances remain, and we obtain a real analytic solution ψj defined on an open set”[P¨os82] . This idea was first successfully used by Arnold[Arn63,AA89] . For an earnest proof, we need more precise arguments by carefully controlling the domains of definition of functions and mappings in the iterative process. Readers can refer to[Sha00a] for details.

13.2.3 Application to Small Twist Mappings In this section, we state a theorem of KAM type for small twist mappings. Such a theorem first appeared in Moser’s celebrated paper[Mos62] for 2-dimensional areapreserving mappings. Its generalization to higher dimensions was given in[Sha00a] , as a direct application of the theorems of the last section. The result may be formulated as follows: Theorem 2.4. Consider one parameter family of mappings St : (p, q) → ( p, q) with S0 = I and S1 = S, defined in phase space I × Tn by  p = p − t∂2 H( p, q) = p − t∂2 h( p, q), (2.37) p, q) = q + tω( p) + t∂1 h( p, q), q = q + t∂1 H( where H( p, q) = H0 ( p) + h( p, q) and ω( p) = ∂H0 ( p). Under the assumptions of Theorem 2.1 (Theorem 2.2) for H, if h satisfies the smallness condition of Theorems 2.1 (Theorem 2.2), then the corresponding conclusions of the theorem are still valid for St (0 < t ≤ 1), only with the following remarks: 1◦ Ωγ is replaced by    tγ Ωt,γ = ω ∈ Ω∗ : eik,tω − 1 ≥ |k|τ

 for k ∈ Zn \ {0} ,

(2.38)

where Ω∗ denotes the set of points in Ω with distance to its boundary at least equal to 2γ. Accordingly, Iγ , ωγ , Φ, and R are replaced by It,γ , ωt,γ , Φt and Rt ,respectively. 2◦ If Ω is a bounded open set of type D in the Arnold’s sense[Arn63] , then we have the following Lebesgue measure estimate m(Ω \ Ωt,γ ) ≤ DγmΩ

(2.39)

13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems

559

for t ∈ (0, 1], with constant D only depending on n, τ and the geometry of Ω. So in this case, Ωt,γ is still a large Cantor set in Ω when γ is small enough. 3◦ If h ∈ C ∞ (B × Tn ), then ωγ,t ∈ C ∞ (Bγ,t ) and Φt ∈ C ∞ (Bγ,t × Tn ) which satisfy the estimates (2.11) for any β ≥ α. 4◦ If h is analytic with the domain of analyticity containing ( ) S(r, ρ) = (p, q) ∈ C2n : |p − p | < r, |Im q| < ρ with p ∈ B and Re q ∈ Tn for some r > 0 and ρ > 0 (Re q and Im q denote the real and imaginary parts of q respectively) and if h satisfies hr,ρ =

sup (p,q)∈S(r,ρ)

|h(p, q)| ≤ δ0 γ 2 θ 2 Θ−3

(2.40)

for some sufficiently small δ0 > 0 depending on n, τ , r and ρ, then all the conclusions of Theorem 2.1 (Theorem 2.2) are still true with ωγ,t ∈ C ∞ (Bγ,t ), Φt ∈ C ∞,ω (Bγ,t × Tn ) and the estimate (2.11) holds for any β ≥ 0. We have presented the results about the existence of differentiable foliation structures in the sense of Whitney of invariant tori for nearly integrable symplectic mappings and for mappings with small twists. Such a result was proved first by Lazutkin in 1974[Laz74] for planar twist maps and was generalized to higher dimensions by Svanidze in 1980[Sva81] . For the case of Hamiltonian flows of arbitrary dimensions, the generalizations were given by J. P¨oschel in 1982[P¨os82] , Chierchia and Gallavotti in 1982[CG82] . The perturbation and measure estimates in terms of γ were studied by R¨ussmann in 1981[R¨us81] , Svadnidze in 1981[Sva81] , Neishtadt in 1982[Nei82] and P¨oschel in 1982[P¨os82] . The estimates in terms of θ and Θ were given by Shang in 2000[Sha00a] , which are also crucial in the small twist mapping case.

13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems In this section, we study stability of symplectic algorithms when applied to typical nonlinear Hamiltonian systems. We introduce a numerical version of the KAM theorem. Such a theorem was already suggested by Channel and Scovel in 1990[CS90] , Kang Feng 1991[Fen91] , and Sanz-Serna and Calvo in 1994[SSC94] . Its rigorous formulation and proof were given by Shang in 1999 and 2000[Sha99,Sha00b] based on the thesis[Sha91] . The main results consist of the existence of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable Hamiltonian system with arbitrarily many degrees of freedom if the time-step size of the algorithm is sufficiently small and falls in a Cantor set of large measure. This existence result also implies that the algorithm, when it is applied to a generic integrable system of n degrees of freedom, possesses n independent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in the sense of Whitney. The invariant tori are just the level sets of these functions. Quantitative analysis shows that the numerical invariant tori of a symplectic algorithm can approximate the corresponding exact invariant tori of the systems.

560

13. KAM Theorem of Symplectic Algorithms

13.3.1 Symplectic Algorithms as Small Twist Mappings We consider a Hamiltonian system with n degrees of freedom in canonical form x˙ = −

∂K (x, y), ∂y

y˙ =

∂K (x, y), ∂x

(x, y) ∈ D,

(3.1)

where D is a connected bounded, open subset of R2n ; x and y are both n-dimensional Euclidean coordinates with x˙ and y˙ the derivatives of x and y with respect to the time “t” respectively; K : D → R1 is the Hamiltonian. A symplectic algorithm that is compatible with the system (3.1) is a discretization scheme such that, when applied to the system (3.1), it uniquely determines one parameter family of symplectic step-transition maps GtK that approximates the phase flow t in the sense that gK  1 t lim s GtK (z) − gK (z) = 0 for any z = (x, y) ∈ D (3.2) t→0 t for some s ≥ 1, here t > 0 is the time-step size of the algorithm and s, the largest integer such that (3.2) holds, is the order of accuracy of the algorithm approximating " t, the continuous systems. Note that the domain in which GtK is well-defined, say D depends on t generally and converges to D as t → 0 — this means that any z ∈ D " t when t is sufficiently close to zero. may be contained in D From (3.2), we may assume t t GtK (z) = gK (z) + ts RK (z),

where t (z) = RK

(3.3)

 1 t t (z) GK (z) − gK s t

" t ⊂ D and has the limit zero as t → 0 for z ∈ D. Below, we is well-defined for z ∈ D prove the main results of this chapter by simply regarding the approximation GtK to t of the above form as a symplectic discretization scheme of order s. the phase flow gK We assume that the system (3.1) is integrable. That is, there exists a system of action-angle coordinates (p, q) in which the domain D can be expressed as the form B × Tn and the Hamiltonian depends only on the action variables, where B is a connected bounded, open subset of Rn and Tn the standard n-dimensional torus. Let us denote by Ψ : B × Tn → D the coordinate transformation from (p, q) to (x, y), then Ψ is a symplectic diffeomorphism from B × Tn onto D. The new Hamiltonian K ◦ Ψ(p, q) = H(p),

(p, q) ∈ B × Tn

(3.4)

only depends on p. Therefore, in the action-angle coordinates (p, q), (3.1) takes the simple form ∂H (p) (3.5) p˙ = 0, q˙ = ω(p) = ∂p t is just the one parameter group of rotations (p, q) → (p, q + and the phase flow gH tω(p)) which leaves every torus {p} × Tn invariant.

13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems

561

Assume K is analytic and, without loss of generality, assume the domain of analyticity of K contains the following open subset of C2n ( ) Dα0 = z = (x, y) ∈ C2n : d(z, D) < α0 , (3.6) with some α0 > 0, where

|z − z  | d(z, D) = inf  z ∈D

denotes the distance from the point z ∈ C2n to the set D ⊂ C2n in which |z| = max |zj | for z = (z1 , · · · , z2n ). Also, we assume that Ψ extends analytically to the

1≤j≤2n

following complex domain ( ) S(r0 , ρ0 ) = (p, q) ∈ C2n : d(p, B) < r0 , Re q ∈ Tn , |Im q| < ρ0

(3.7)

with r0 > 0, ρ0 > 0 and has period 2π in each component of q. In (3.7), B is " 0 , ρ0 ) = considered as a subset of C2n . Without loss of generality, we suppose D(r  Ψ S(r0 , ρ0 ) ⊂ Dα0 and further that Ψ is a diffeomorphism from S(r0 , ρ0 ) onto " 0 , ρ0 ). So the Equation (3.4) is valid for (p, q) ∈ S(r0 , ρ0 ) and D(r t t Ψ−1 ◦ gK ◦ Ψ = gH

(3.8)

on this complex domain of coordinates (p, q). Checking the existing available symplectic algorithms, we find that GtK is always analytic if the Hamiltonian K is analytic. Note that the domain in which GtK is wellt as t approaches zero. We may defined converges to the domain of definition of gK t assume, without loss of generality, GK is well-defined and analytic in the complex domain Dα0 for t sufficiently close to zero. Moreover, in the analytic case, we have     t t (z) ≤ ts+1 M (z, t) GK (z) − gK with an everywhere positive continuous function M : Dα0 × [0, δ1 ] → R for some sufficiently small δ1 > 0. " t = Ψ−1 ◦ Gt ◦ Ψ is Lemma 3.1. There exists δ2 > 0 such that for t ∈ [0, δ2 ], G K K r ρ  well-defined and real analytic on the closed complex domain S 0 , 0 and 2

 t  " (p, q) − g t (p, q) ≤ M ts+1 , G K H

(p, q) ∈ S





r0 ρ0 , , 2 2

2

t ∈ [0, δ2 ],

(3.9)

where M is a positive constant depending on r0 , ρ0 , α0 , δ1 , Ψ and K, not on t.      r ρ r ρ Proof. Let U1 = S 0 , 0 and V1 = Ψ S 0 , 0 . Since U1 is a closed subset 2 2 2 2 of S(r0 , ρ0 ) and Ψ is a diffeomorphism from S(r0 , ρ0 ) onto Dα0 , V1 is closed in Dα0 . Let ξ be the distance from V1 to the boundary of Dα0 , then ξ > 0. The compactness 

t of V1 implies that there exists 0 < δ1 < δ1 such that gK maps V1 into V1 +

ξ for 2

562

13. KAM Theorem of Symplectic Algorithms



t ∈ [0, δ1 ], where V1 + ξ 2

ξ denotes the union of all complex open balls centered in V1 2 

with radius . Since M (z, t) is continuous and positive for (z, t) ∈ V1 × [0, δ1 ], there 

exists a constant M 0 > 0 which is an upper bound of M (z, t) on V1 × [0, δ1 ]. Let   D  ξ δ2 = min 1, δ1 , . Then for t ∈ [0, δ2 ], GtK maps V1 into Dα0 and hence 4M0

"t = Ψ−1 ◦ Gt ◦ Ψ is well-defined on U1 . The real analyticity of the map follows G K K from the real analyticity of Ψ and K. To verify Equation (3.9) , we first note that the analyticity of Ψ−1 on V1 +

3ξ ⊂ Dα0 implies that 4

 −1   ∂Ψ  (z) ≤ M1  ∂z

3ξ

for all z ∈ V1 + with some constant M1 > 0,and then Taylor formula gives 4 Ψ(p, q) ∈ V1 and     t ξ t RK (Ψ(p, q)) = GtK (Ψ(p, q)) − gK (Ψ(p, q)) ≤ M0 ts+1 ≤

4

for (p, q) ∈ U1 and t ∈ [0, δ2 ]. Therefore, " t (p, q) − g t (p, q)| = |Ψ−1 (g t (Ψ(p, q)) + Rt (Ψ(p, q))) − Ψ−1 (g t (Ψ(p, q)))| |G K H K K K ≤ 2nM1 M0 ts+1 . 

Let M = 2nM1 M0 , then (3.9) is verified.

"t is an approximant to the one parameter group The above lemma shows that G K t of integrable rotations gH up to order ts+1 as t approaches zero. To apply Theorem " t so that it can be expressed by 2.4, we need to verify the exact symplecticity of G K globally defined generating function. Because Ψ is not necessarily exact symplectic, "t = Ψ−1 ◦ Gt ◦ Ψ is not trivially observed. the exact symplecticity of G K K Lemma 3.2. Let G be an exact symplectic mapping of class C 1 from D into R2n where D is an open subset of R2n and let Ψ be a symplectic diffeomorphism from B × Tn onto D. Then Ψ−1 ◦ G ◦ Ψ is an exact symplectic mapping in the domain in which it is well-defined. Proof. Let ( p, q) = Ψ−1 ◦ G ◦ Ψ(p, q) and let γ be any given closed curve in the " =: Ψ−1 ◦ G ◦ Ψ, which is an open subset of B × Tn . The domain of definition of G " exact symplecticity of G will be implied by[Arn89] I(γ) = p d q − p d q = 0. (3.10) γ

γ

Now we verify (3.10). Let (x, y) = Ψ(p, q) and ( x, y) = Ψ( x, y) =  p, q). Then ( G(x, y). Since G is an exact symplectic, we have γ x  d y − γ x d y = 0, where x,

13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems

563

y, x , y are considered as functions of (p, q), which vary over γ. Therefore, with these conventions and with γ  = Ψ−1 ◦ G ◦ Ψ(γ), I(γ) = p d q − x  d y + x d y − p d q γ γ γ γ pdq − xdy + xdy − pdq = γ Ψ(γ  ) Ψ(γ) γ pdq − x d y. (3.11) = γ  −γ

Ψ(γ  )−Ψ(γ)

Note that G is exact and hence it is homotopic to the identity. This implies that Ψ−1 ◦ G ◦ Ψ is homotopic to the identity too. So γ  and γ belong to the same homological class in the fundamental group of the manifold B × Tn . Therefore, one may find a 2-dimensional surface, say σ, in the phase space B × Tn , which is bounded by γ  and γ. Ψ(σ) is then a 2-dimensional surface in D bounded by Ψ(γ  ) and Ψ(γ). By stokes formula and from (3.11), we get dp ∧ dq − d x ∧ d y, I(γ) = σ

Ψ(σ)

which is equal to zero because Ψ preserves the two form d p ∧ d q.



Checking the existing available symplectic algorithms, we observe that they are generally constructed by discretizing Hamiltonian systems, therefore, they generate exact symplectic step transition maps. In our case, this means that GtK is a one"t . As a result, parameter family of exact symplectic mappings. By Lemma 3.2, so is G K t " GK can be re-expressed by generating function. On the other hand, by Lemma  3.1, we  r ρ t t s+1 " on S 0 , 0 see that G is near the identity and approximates g up to order t K

H

2

2

for t ∈ [0, δ2 ]. A simple argument of implicit function theorem, with the notice of the " t , will show the following: exact symplecticity of G K Lemma 3.3. There exists a function ht which depends on  the time step t such that it is  r0 ρ0 for t ∈ [0, δ3 ] with δ3 being well-defined and real analytic on the domain S , 4 4 t " : (p, q) → ( a sufficiently small positive number so that G p, q) can be expressed by K ht as follows: p = p − ts+1

∂ht ( p, q), ∂q

q = q + tω( p) + ts+1

∂ht ( p, q). ∂ p

(3.12)

Proof. It follows immediately from Lemmas 3.1 and 3.3 that ? t? ? t? ? ∂h ? ? ∂h ? ? ? ? ? ≤ M, ≤ M. ? ? ? ? ∂ p

r0 4

,

ρ0 4

∂q

r0 4

,

ρ0 4

p0 , q0 ) = 0. For any ( p, q) ∈ S Fix ( p0 , q0 ) ∈ D and let ht ( the exact differential one form





r0 ρ0 , , integrating 4 4

∂ht ∂ht d p + d q along one of the shortest curves from ∂ p ∂q

564

13. KAM Theorem of Symplectic Algorithms

  r ρ ( p0 , q0 ) to ( p, q) in S 0 , 0 and then taking the maximal norm of the integration 4 4  r0 ρ 0 for ( p, q) over S , we obtain the estimate , 4

4

ht  r40 , ρ40 ≤ 2nM L,

for t ∈ [0, δ3 ],

(3.13)

where M is the constant in Lemma 3.1 and L is  bound of the length of  an upper r ρ the shortest curves from ( p0 , q0 ) to points of S 0 , 0 , which is clearly a finite 4 4 positive number. Note that B is a connected bounded, open subset of Rn and therefore,   r ρ  S 0 , 0 is bounded too. 4

4

13.3.2 Numerical Version of KAM Theorem We formulate the main result of this chapter as follows. Theorem 3.4. Let Hamiltonian system (3.1) be integrable in a connected bounded, open domain D of R2n , and let K be real analytic and nondegenerate in the sense of Kolmogorov after expressed as action-angle variables. For an analytic symplectic algorithm4 compatible with the system, as long as the time-step t of the algorithm is small enough, most nonresonant invariant tori of the integrable system do not vanish, but are only slightly deformed, so that in the phase space D, the symplectic algorithm also has invariant tori densely filled with phase orbits winding around them quasiperiodically, with a number of independent frequencies equal to the number of degrees of freedom. These invariant tori are all analytic manifolds and form a Cantor set, say Dt . The Lebesgue measure mDt of the Cantor set Dt tends to mD as t tends to zero. Moreover, on Dt , the algorithm is conjugate to a one parameter family of rotations of the form (p, q) → (p, q + tωt (p)) by a C ∞ -symplectic conjugation Ψt : Bt × Tn → Dt , where (p, q) are action-angle coordinates and ωt is the frequency map defined on a Cantor set Bt ⊂ Rn of actions. More quantitative results hold. For any given and sufficiently small γ > 0, if the time step t is sufficiently small, then there exists closed subsets Bγ,t of Bt and Dγ,t of Dt such that Dγ,t = Ψt (Bγ,t × Tn ) and the following hold: 1◦ mDγ,t ≥ (1 − c1 γ)mD, where c1 is a positive constant not depending on t and γ; 2◦ Ψt − Ψβ,βλ;Bγ,t ×Tn , ωt − ωβ+1;Bγ,t ≤ c2 γ −(2+β) · ts for any β ≥ 0, where s is the accuracy order of the algorithm, λ > n + 2, c2 is a positive constant not depending on γ and t. The norms here are understood in the sense of Whitney; 3◦ Every numerical invariant torus in Dγ,t is ts -close to the invariant torus of the same frequencies of the original integrable system (3.1) in the sense of Hausdorff 5 . 4

5

An analytic algorithm is an algorithm generating an analytic step-transition map whenever the Hamiltonian is analytic. Note that all the existing available symplectic algorithms are analytic in this sense. [Bey87] The Hausdorff distance of two sets d(A, B) =  A and B is defined as max

sup dist(x, B), sup dist(y, A) , where dist(x, B) = inf |x − y|.

x∈A

y∈B

y∈B

13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems

565

"t . The Proof. Now the analytic version of Theorem 2.4 can be applied to St = G K conditions required by Theorem 2.4 are satisfied clearly according to the assumptions of Theorem 2.1. For example, the nondegeneracy of the integrable system in the sense of Kolmogorov means that the frequency map ω : B → Rn is nondegenerate and therefore, there exists positive constants θ ≤ Θ such that ω satisfies (2.4) with some positive numbers r ≤ r0 . We assume r = r0 here without loss of generality. In Theorem 2.4, the function h is replaced by ts ht which satisfies the estimate (2.40) r ρ with r = 0 and ρ = 0 if we choose 4

4

d

γ = γt =: Γ t ,

s with 0 < d ≤ and Γ = 2

D

2nM L −1 32 θ Θ δ0

(3.14)

and if t is sufficiently small, where δ0 is the bound given by (2.40) of Theorem 2.4.  1  It is clear that the so chosen γ satisfies the condition γ ≤ min 1, rΘ required by 2 Theorem 2.4 for t sufficiently close to zero. By Theorem 2.4, we then have the Cantor sets Bt = Bγ,t ⊂ B and Ωt = Ωγ,t ⊂ ω(B), a surjective map ωt = ωγ,t : Bt → Ωt of class C ∞ and a symplectic mapping Φt : Bt × Tn → Rn × Tn of class C ∞,ω , in the sense of Whitney, such that the conclusions (1) – (4) of Theorem 2.4 hold with " t fill out a set Et = Eγ,t = Φt (Bt × Tn ) in γ = Γ td . From (2.10), invariant tori of G K n phase space E = B × T with measure estimate   −n d  t mE. mEt ≥ 1 − c4 Γ θΘ−1

(3.15)

From (2.11), with the notice of (2.7) and the fact that ht βλ+λ+τ ≤ c7 ht  r40 , ρ40 by Cauchy’s estimate for derivatives of an analytic function, we have Φt − Iβ,βλ;Bt ×Tn ≤



γΘ−1

−β

−1 σγΘ −1 ◦ (Φt − I)β,βλ;γΘ −1

≤ c5 c7 γ −(2+β) Θ 1+β ts ht  r40 , ρ40 ≤ c8 θ2+β Θ −(2+β/2) · ts−(2+β)d

(3.16)

for t sufficiently close to zero, where β

1+ β 2

c8 = c5 c7 (2nM L)− 2 δ0

.

In the last inequality of (3.16), we have used the estimate (3.13) for ht . From (2.11), we also get ωt − ωβ+1;Bt ≤ (γΘ−1 )−(β+1) ωt − ωβ+1;γΘ−1 ≤ c8 θ2+β Θ −(1+β/2) · ts−(2+β)d .

(3.17)

Let Ψt = Ψ ◦ Φt and Dt = Ψ(Et ), then GtK ◦ Ψt = Ψt ◦ Rt , which means that Ψt realizes the conjugation from GtK to Rt : (p, q) → (p, q + tωt (p)) and for any

566

13. KAM Theorem of Symplectic Algorithms

fixed P ∈ Bt , Ψt (P, Tn ) is an invariant torus of GtK , which is an analytic Lagrangian manifold since Ψt is a symplectic diffeomorphism and analytic with respect to the angle variables. On the torus, the iterations of GtK starting from any fixed point are quasi-periodic with frequencies tωt (p) which are rationally independent and satisfy the diophantine condition (4.3) with ω = ωt (p) and γ = Γ td . These invariant tori distribute C ∞ -smoothly in the phase space due to the C ∞ -smoothness of the conjugation Ψt . Moreover, we have the same estimates for the measure of Dt and for the closeness of Ψt to Ψ as (3.15) and (3.16), with larger constants c4 and c8 , in which Et , E, Φt and I are replaced by Dt , D, Ψt and Ψ respectively. For β ≥ 0, if we choose d satisfying s , (3.18) 0
(3.19)

with constant " c4 > 0 not depending on γ and t, where Dγ,t = Ψ(Eγ,t ) with Eγ,t = Φt (Bγ,t × Tn ) and with Bγ,t being the subset of B as indicated above. Note that Bγ,t is a closed subset of Bt and Dγ,t a closed subset of Dt if t is sufficiently small. Moreover, the estimate c8 γ −(2+β) Θ 1+β · ts Ψt − Ψβ,βλ;Bγ,t ×Tn ≤ "

(3.20)

c8 γ −(2+β) Θ 2+β · ts ωt − ωβ+1;Bγ,t ≤ "

(3.21)

and hold for any β ≥ 0 with " c8 > 0 not depending on γ and t. The conclusions (1)  −n and (2) of the last part of Theorem 3.4 are proved if we set c1 = " c4 θΘ −1 and c8 ·max(Θ 1+β , Θ2+β ). From (3.19), it follows that for a sufficiently small γ > 0, c2 = " Dγ,t has a positive Lebesque measure. From (2.12), it follows that for any ω ∗ ∈ Ωγ,t , there exists p∗ ∈ B and P ∗ ∈ Bγ,t such that ω(p∗ ) = ωt (P ∗ ) = ω∗ and  −1 s |P ∗ − p∗ | ≤ 2nM Lc6 c7 γθΘ−1 ·t , which implies that   $1 c6 c7 γθΘ −1 −1 · ts , |Ψ(P ∗ , q) − Ψ(p∗ , q)| ≤ 4n2 M LM

13.3 KAM Theorem of Symplectic Algorithms for Hamiltonian Systems

567

$1 is an upper bound of the norm of ∂Ψ (p, q) for uniformly for q ∈ Tn , where M ∂z r ρ  0 0 (p, q) ∈ S , . This estimate, together with (3.20), proves the third conclusion 2 2 of the second part of Theorem 3.4. Theorem 3.4 is completely proved.  A natural corollary of the above theorem is: Corollary 3.5. Under the assumptions of the above theorem, there exists n functions F1t , · · · , Fnt which are defined on the Cantor set Dt and of class C ∞ in the sense of Whitney such that: 1◦ F1t , · · · , Fnt are functionally independent and in involution (i.e., the Poisson bracket of any two functions vanishes on Dt ); 2◦ Every Fjt (j = 1, · · · , n), is invariant under the difference scheme and the invariant tori are just the intersection of the level sets of these functions; 3◦ Fjt (j = 1, · · · , n) approximate n independent integrals in involution of the integrable system, with a suitable order of accuracy with respect to the time-step t which will be explained in the proof. Proof. By Theorem 3.4, we have GtK ◦ Ψt (p, q) = Ψt ◦ Rt (p, q),

for (p, q) ∈ Bt × Tn ,

(3.22)

where Rt is the integrable rotation (p, q) → (p, q + tωt (p)) and admits n invariant functions, say, p1 , · · · , pn , analytically defined on Bt × Tn . Let Fit = pi ◦ Ψ−1 t ,

i = 1, · · · , n,

then they are well-defined on the Cantor set Dt and of class C ∞ in the sense of on Dt . Moreover, we easily verify by Whitney due to the C ∞ -smoothness of Ψ−1 t (3.22) that Fit ◦ GtK = Fit , i = 1, · · · , n, and this means that Fit (i = 1, · · · , n) are n invariant functions of GtK . These n invariant functions are functionally independent because pi (i = 1, · · · , n) are functionally independent and Ψt is a diffeomorphism. The claim that Fit and Fjt are in involution for 1 ≤ i, j ≤ n simply follows from the fact that pi and pj are in involution and Ψt is symplectic. Note that the Poisson bracket is invariant under symplectic coordinate transformations. Finally, it is observed from the proof of Theorem 3.4 that for each of j = 1, · · · , n, Fjt approximates Fj = pj ◦ Ψ−1  as t → 0, with the order of accuracy equal to ts−(2+β)d 0 < d <



s is given on 2+β

the set Dt (note that this set depends also on d by definition) and equal to ts on Dγ,t , a subset of Dt , in the norm of the class C β for any given β ≥ 0. It is clear that the functions Fj (j = 1, · · · , n) are integrals of the integrable system and that any two of them are in involution by the symplecticity of Ψ−1 . Corollary 3.5 is then proved. 

568

13. KAM Theorem of Symplectic Algorithms

13.4 Resonant and Diophantine Step Sizes It is observed from the proof of Theorem 3.4 that the preserved invariant tori have frequencies of the form ωt = tω, where t is the step size of the algorithm and ω belongs to the frequency domain of the system, which the algorithm applies to. The frequency tω is required to satisfy the diophantine condition    exp (ik, tω) − 1 ≥ tγ , τ |k|

0 = k ∈ Zn

(4.1)

with some γ > 0 and τ > 0, where u, v denotes the inner product of vectors u and v in Rn . Note that t > 0 may be arbitrarily small. For any fixed ω ∈ Rn , even if it is a diophantine vector, there exists some t in any small neighborhood of the origin such that (4.1) does not hold for any γ > 0 and any τ > 0. In fact, one can choose t to satisfy the resonance relation exp (ik, tω) = 1

(4.2)

for some 0 = k ∈ Zn . In the next section, we will show that such t forms a dense set in R. We note that a one-step algorithm, when applied to system of differential equations, can be regarded as a perturbation of the phase flow of the system. On the other hand, according to Poincar´e, arbitrarily small perturbations in the generic case may destroy those resonant invariant tori of an integrable system. Therefore, to simulate the invariant torus with a given frequency of some Hamiltonian system by symplectic algorithms, one is forced to be very careful to select step sizes, say, to keep them away from some dense set. Some questions arise: is it possible to simulate an invariant torus of an integrable system by symplectic algorithms? If possible, how does one select the step sizes and what structure does the set of those admitted step sizes have? In this paper, we try to answer these questions.

13.4.1 Step Size Resonance For any frequency vector, step size resonance may take place very often. Lemma 4.1. For any ω ∈ Rn , there exists a dense subset, say D(ω), of R such that for any t ∈ D(ω), the resonance relation (4.2) holds for some 0 = k ∈ Zn . Proof. If k, ω = 0 for some 0 = k ∈ Zn , then D(ω) = R. If k, ω = 0 for any 0 = k ∈ Zn , then   2πl D(ω) = t = : 0 = k ∈ Zn , l ∈ Z , k, ω

(4.3)

which is clearly dense in R and the resonance relation (4.2) holds for any t ∈ D(ω). The proof of the lemma is completed. 

13.4 Resonant and Diophantine Step Sizes

569

Definition 4.2. D(ω) is called the resonant set of step sizes with respect to the frequency ω ∈ Rn . Any t ∈ D(ω) is called a resonant step size with respect to ω. From Lemma 3.1, If ω ∈ Rn is a resonant frequency, i.e., k, ω = 0 for some 0 = k ∈ Zn , then D(ω) = R. In other words, each step size is resonant with respect to a resonant frequency. If ω ∈ Rn is a nonresonant frequency, i.e., k, ω = 0 for any 0 = k ∈ Zn , then D(ω) is a countable and dense set of R. Because a resonant torus may be destroyed by arbitrarily small Hamiltonian perturbations (Poincar´e), any invariant torus with frequency ω of a generic integrable system may not be preserved by symplectic algorithms with step sizes in D(ω). To simulate an invariant torus of the frequency ω, it is natural to consider those step sizes which are far away from the resonant set D(ω). Note that if ω is of at least 2 dimensions, the resonant set D(ω) is “denser” than the rational numbers in R because the set D(ω) consists of all real numbers in the case when ω is resonant and consists of all numbers of the form αk r in the case when ω is nonresonant, where r takes any rational number and, for 2π k ∈ Zn \ {0}, αk = which may be arbitrarily small and large, and moreover, k, ω there are arbitrarily many pairs of rationally independent numbers in αk . Anyway, for nonresonant ω, D(ω) is countable.

13.4.2 Diophantine Step Sizes Even though the step size may encounter resonance densely, we can still have a big possibility to select step sizes to keep away from resonance. We discuss this as follows. Definition 4.3. A number t ∈ R is said to be of diophantine type with respect to the nonresonant frequency ω ∈ Rn , if  2πl  λ  (4.4) t −  ≥ μ τ , 0 = k ∈ Zn , 0 < l ∈ Z k, ω l |k| for some constants λ > 0, μ and τ > 0. We denote by Iλ,μ,τ (ω), the set of numbers τ satisfying (4.4) for given constants λ > 0, μ and τ > 0. Then, Iλ,μ,τ (ω) is a subset of R which is far away from resonance with respect to ω. For this set, we have: Lemma 4.4. For any nonresonant frequency ω ∈ Rn , and for any λ > 0, any μ and any τ > 0, the set Iλ,μ,τ (ω) is nowhere dense and closed in R. Moreover, if μ > 1 and τ > n, then we have   (4.5) meas R \ Iλ,μ,τ (ω) ≤ cλ, where c is a positive number depending only on n, μ and τ . Proof. The nowhere denseness and the closedness of Iλ,μ,τ (ω) follow from the fact that the complement of the set is both open and dense in R for any λ > 0, μ and τ > 0. It remains to prove (4.5). Since

570

13. KAM Theorem of Symplectic Algorithms

'

  2πl   t ∈ R : t − <

,

R \ Iλ,μ,τ (ω) =

k, ω

0
λ lμ |k|τ

/ ,

0=k∈Zn

we have 

  meas R \ Iλ,μ,τ (ω) ≤

0
0=k∈Zn

Define cμ =

∞



l=1

0=k∈Zn

 1 2λ ≤ 2λ · μ τ l |k| lμ

1 . |k|τ

∞  1 . lμ l=1

Then cμ < ∞ when μ > 1 and  0=k∈Zn

−τ

|k|

∞ ∞   1 1 n+m−1 n n = · #{k ∈ Z : |k| = m} ≤ 2 C τ τ m m m m=1 m=1 2n−1

≤2

∞ 

1 = 22n−1 cτ −n+1 < ∞ τ −n+1 m m=1

when τ > n, here #S denotes the number of the elements of the set S and Csk are binomial coefficients. (4.5) is verified with c = 4n cμ cτ −n+1 . Therefore, the lemma is completed.

(4.6) 

Remark 4.5. We may define Iλ,μ,τ (ω) to be empty for any resonant frequency ω and any λ > 0, any μ and any τ > 0 because no number t satisfies (4.4) in this case. It is possible that the set Iλ,μ,τ (ω) may still be empty even for nonresonant frequencies ω if here the numbers μ and τ are not properly chosen. Anyway, the above lemma shows that if μ > 1 and τ > n, then the set Iλ,μ,τ (ω) has positive Lebesgue measure and hence is nonempty for any λ > 0. Remark 4.6. If λ1 > λ2 > 0, then Iλ1 ,μ,τ (ω) ⊂ Iλ2 ,μ,τ (ω). Therefore, if ω is a nonresonant frequency and μ > 1 and τ > n, then the set of all real numbers t satisfying (4.4) for some λ > 0 has full Lebesgue measure in any measurable set of R. It should be an interesting number theoretic problem to study the cases when μ ≤ 1 or τ ≤ n. In numerical analysis, the step sizes are usually considered only in a bounded interval. We take the interval [−1, 1] as illustration without loss of generality. Lemma 4.7. For a nonresonant frequency ω = (ω1 , ω2 , · · · , ωn ), assume 0 < λ < 2π with |ω| = max |ωj |. If −1 ≤ μ ≤ 1 and μ + τ > n + 1, then we have |ω| 1≤j≤n

  meas [−1, 1] \ Iλ,μ,τ (ω) ≤ " cλ,

where " c is a positive number depending not only on n, μ and τ but also on |ω|.

(4.7)

13.4 Resonant and Diophantine Step Sizes

571

Proof. The set [−1, 1] \ Iλ,μ,τ (ω) is contained in the union of all subintervals % & λ λ 2πl 2πl − μ τ, + μ τ k, ω l |k| k, ω l |k| for those l ∈ Z, l > 0 and k ∈ Zn \ {0} such that 2πl λ <1+ μ τ. |k, ω| l |k|

(4.8)

Since −1 ≤ μ ≤ 1, we have that τ > n + 1 − μ ≥ n and (3.30) implies that 2π − λ l. Therefore, |k| > |ω|

∞     meas [−1, 1] \ Iλ,μ,τ (ω) ≤ l=1 |k|>Nl,λ

 where Nl,λ =

∞



l=1

m>Nl,λ

 1 2λ ≤ 4n λ μ τ l |k| lμ

1 , mτ −n+1



2π − λ l which is positive for positive l. We will use the following |ω|

estimate which is easy to prove: ⎧ ⎨ cτ −n+1 ,  1 1 ≤ , ⎩ mτ −n+1 m>N (τ − n)(N − 1)τ −n

0 < N ≤ 1, (4.9)

N > 1.

Assume lλ is the integer such that Nlλ ,λ ≤ 1 and Nlλ +1,λ > 1. Then (4.7) is verified with ⎛ ⎞ lλ ∞ ⎜   1 1 ⎜ + " c = 4n ⎜cτ −n+1 lμ τ −n ⎝ l=1

l=lλ +1 lμ

%%

⎟ 1 ⎟ & &τ −n ⎟ ⎠ 2π −λ l−1

(4.10)

|ω|

which is finite because the conditions μ + τ > n + 1 and 0 < λ <

2π guarantee the |ω|

lλ  1 =0 lμ l=1 and hence the first term in the bracket of Equation (4.10) disappears in this case. Note that here the number " c depends also on λ, but this dependence is not fatal essentially 2π because the only harmful case is when λ is close to . However, this case is not of |ω| π in interest and may always be avoided. For example, we simply assume 0 < λ ≤ |ω| the lemma. The proof of Lemma 4.7 is completed. 

convergence of the infinite summation in (4.1). If lλ = 0, then we take

Therefore, to guarantee the positiveness of the Lebesgue measure of the set Iλ,μ,τ (ω), it is not necessary to assume μ > 1. One may require μ to only satisfy μ ≥ −1. In the case −1 ≤ μ ≤ 1, however, one has to additionally require

572

13. KAM Theorem of Symplectic Algorithms

μ + τ > n + 1, which automatically implies that τ > n. One may also consider how big the set Iλ,μ,τ (ω) is in other unit intervals with integer endpoints, but we do not go further in this direction. Remark 4.8. It remains to study the set Iλ,μ,τ (ω) in other cases: μ < −1 or τ ≤ n or μ + τ ≤ n + 1. I believe the Lebesgue measure of the set is zero in each of these cases. It is also an interesting problem to calculate the Hausdorff dimensions of the set Iλ,μ,τ (ω) in all of these cases. The cases when −1 ≤ μ ≤ 1 and ν = n − μ + 1 and when μ > 1 and τ = n should be particularly interesting. In all other cases, I intend to believe the set is empty. Note that a special case when n = 1, μ = 0 and τ = n − μ + 1 = 2 with ω = 2π just corresponds to the classical diophantine problem on approximating an irrational number by rational ones. To any nonresonant frequency ω in Rn , we have associated a 3-parameter family of sets Iλ,μ,τ (ω) on the real line. The set Iλ,μ,τ (ω) has positive Lebesgue measure and hence is nonempty if μ ≥ −1, τ > n, μ + τ > n + 1 and λ > 0 suitably small (in the case when μ > 1 and τ > n, Iλ,μ,τ (ω) has positive Lebesgue measure for any λ > 0). But to guarantee an invariant torus of the frequency tω for symplectic algorithms with the step size t, it seems that the only way is to require tω satisfy a diophantine condition of the type (1.1) (J. Mather showed in [Mat88] ) that for any exact area-preserving twist mapping, an invariant circle with any Liouville frequency can be destroyed by arbitrarily small perturbations in C ∞ -topology). This is the case when one requires both ω be a diophantine frequency and t be a diophantine step size with respect to the ω, as the following lemma shows. Lemma 4.9. Let γ > 0 and 0 < λ ≤ 1. Then for any ω ∈ Ωγ (τ1 )6 and any t ∈ [−1, 1] ∩ Iλ,μ,τ2 (ω), we have  ik,tω  e − 1 ≥

|t|" γ , |k|μ+τ1 +τ2

where γ "=



2λγ

0 = k ∈ Zn ,

π 1+

μ .

1 |ω| 2π

(4.11)

(4.12)

Proof. It is easy to prove that for k ∈ Zn , k = 0, there exists l ∈ Z such that   ik,tω  2 e − 1 ≥ k, tω + 2πl. π

We have two cases: 1◦ l = 0. Since ω ∈ Ωγ (τ1 ),  ik,tω   2|t|γ 2 e − 1 ≥ k, tω ≥ ; π π|k|τ1 6

We denote by Ωγ (τ ), the set of all vectors ω ∈ Rn satisfying the diophantine condition of the form γ |k, ω| ≥ , 0=  k ∈ Zn . |k|τ

13.4 Resonant and Diophantine Step Sizes

2◦

573

l = 0. Since t ∈ Iλ,μ,τ2 (ω) and ω ∈ Ωγ (τ1 ), J     K   ik,tω  2  2πl 2 2πl   e − 1 ≥  k, t + ω  = t +  k, ω k, ω π k, ω π ≥

λγ 2 · . π lμ |k|τ1 +τ2

But |2πl| ≤ |k, tω + 2πl| + |k, tω| ≤

 π  ik,tω e − 1 + |t||k, ω| 2

≤ π + |t| |ω||k|, therefore,

  ik,tω e − 1 ≥ %

2λγ &μ . 1 |t| π + |ω| |k|μ+τ1 +τ2 2 2π

Combining the two cases, (4.11) is verified and hence Lemma 2.4 is proved.    # From the above lemmas and the fact that meas Rn \ Ωγ (τ ) = 0 for τ > γ>0

n − 1, we conclude that for almost all ω ∈ Rn and almost all t ∈ [−1, 1], tω satisfies a diophantine condition of the mapping type (2.5). As the step size of a difference scheme, however, t may fall into an arbitrarily small neighbourhood of the origin. The next lemma shows that for a nonresonant frequency ω ∈ Rn and for μ ≥ −1, 2π τ > n + 1, μ + τ > n + 1 and 0 < λ < , the set Iλ,μ,τ (ω) has large measure near |ω|

the origin of the real line.

Lemma 4.10. Let ω be a nonresonant frequency of Rn . Let λ > 0, μ ≥ −1, τ > n and μ + τ > n + 1. For any δ > 0, let δ (ω) = (−δ, δ) \ Iλ,μ,τ (ω). Jλ,μ,τ

If λ + δ <

2π , then |ω|

where

δ (ω) ≤ dδ τ −n , meas Jλ,μ,τ ∞

d=

4n λ  τ −n l=1

%% lμ

1 & &τ −n < ∞. 2π −λ l−δ

(4.13)

(4.14)

|ω|

Consequently, if in addition τ > n + 1, then   meas Iλ,μ,τ (ω) ∩ (−δ, δ) = 1. lim meas(−δ, δ) δ→0+

(4.15)

574

13. KAM Theorem of Symplectic Algorithms

δ Proof. Let t ∈ Jλ,μ,τ . By definition, we have

−δ −

λ

λ 2πl ≤δ+ μ τ k, ω l |k|

≤

lμ |k|τ

(4.16)

for some k ∈ Zn and 0 < l ∈ Z. Fix l ∈ Z, l = 0, denoted by Klδ the set of k ∈ Zn satisfying (4.16). If k ∈ Klδ , then 2πl λ δ+ μ l |k|

2πl

≤

δ+

which implies that

* 2π |k| >

≤ |k, ω| ≤ |k| |ω|,

λ lμ |k|τ

|ω|

−λ δ

+ . l = Nlδ

) ( since μ ≥ −1. This shows that Klδ ⊂ k ∈ Zn : |k| > Nlδ and therefore, δ (ω) meas Jλ,μ,τ

≤

∞   l=1

Because 0 < δ + λ <

k∈Zn

|k|>Nlδ

∞  2λ 1  1 n ≤ 4 λ . τ −n+1 lμ |k|τ lμ m δ l=1

m>Nl

2π , we have Nlδ > 1. (4.13) follows from (4.9) with the |ω|

constant d defined by (4.14), which is finite because τ > n and μ + τ − n > 1. (4.15) is true if, in addition, τ > n + 1. 

13.4.3 Invariant Tori and Further Remarks Now, we summarize the main result of this section as follows. Theorem 4.11. Given an analytic, nondegenerate and integrable Hamiltonian system of n degrees of freedom, and given a frequency ω, in the domain of the frequencies of the system, which satisfies the diophantine condition of the form γ , 0 = k = (k1 , · · · , kn ) ∈ Zn (4.17) |k, ω| ≥ |k|τ for some γ > 0 and τ > 0, there exists a Cantor set I(ω) of R, for any symplectic algorithm applied to the system, there exists a positive number δ0 such that if the step size t of the algorithm falls into the set (−δ0 , δ0 ) ∩ I(ω), then the algorithm, if applied to the integrable system, has an invariant torus of frequency tω. The invariant torus of the algorithm approximates the invariant torus of the system in the sense of Hausdorff with the order equals to the order of accuracy of the algorithm. The Cantor set I(ω) has density one at the origin in the sense that   m (−δ, δ) ∩ I(ω) = 1. (4.18) lim m(−δ, δ) δ→0+

13.4 Resonant and Diophantine Step Sizes

575

Proof. For the given ω, we define I(ω) = Iλ,μ,"τ (ω) for some λ > 0, μ > 1 and τ" > n + 1. By Lemma 3.7, we have for any t ∈ [−1, 1] ∩ I(ω),    ik,tω  − 1 ≥ e

|t|" γ , |k|μ+τ +"τ

0 = k ∈ Zn

with γ " given by (4.12). The analytic version of Theorem 2.4 may be applied and therefore, for a symplectic algorithm applied to the given system7 , we can find a positive number δ0 , which depends on the numbers n, γ, τ , λ, μ, τ" and |ω| and on the nondegeneracy and the analyticity of the system and, of course, also on the algorithm, such that the algorithm has an invariant torus of the frequency tω with the required approximating property to the corresponding invariant torus of the system if the step size t falls into the set [−δ0 , δ0 ] ∩ I(ω). It follows from Lemma 3.8 that the set I(ω) has density one at the origin because we have chosen μ > 1 and τ" > n + 1.  Remark 4.12. In practical computations, one would like to choose big step sizes. It is interesting to look at how the δ0 in Theorem 4.11 depends on the nonresonance property of the frequency ω and how the δ0 relates to the size of the diophantine set I(ω) of step sizes. It is known that the parameters γ and ν describe the nonresonance property of the frequency ω and the parameters λ, μ and ν" determine the size of the set I(ω). Among them, the most interesting are γ and λ because we may fix all others in advance without loss of generality. For a given ω, we define γ to be the biggest one such that (4.17) holds for a fixed τ > n − 1. It is easy to see, from Lemma 4.9 2 and Theorem 2.4, that δ0 may be chosen to be proportional to (γλ) s , where s is the order of accuracy of the algorithm considered in Theorem 4.11. Note that the more nonresonant the ω is, the bigger γ will be and therefore the bigger δ0 is admitted. On the other hand, for a given ω, the bigger step size is taken, the bigger λ has to be chosen and in this case, the set I(ω) turns out to be smaller. But anyway, the set I(ω) is of density one at the origin. Consequently, to simulate an invariant torus, one has much more possibilities to select available small step sizes than to select available big ones. Remark 4.13. It is interesting to make some comparisons between Theorem 3.4 and Theorem 4.11. Theorem 3.4 shows that a symplectic algorithm applied to an analytic nondegenerate integrable Hamiltonian system has so many invariant tori that the tori form a set of positive Lebesgue measures in the phase space if the step size of the algorithm is sufficiently small and fixed in an arbitrary way. No additional nonresonance or diophantine condition is imposed on the step size. But the set of frequencies of the invariant tori depends on the step size and, therefore, changes in general as the step size changes. It is a fact that the measure of the set of frequencies of the invariant tori becomes larger and larger as the step size gets smaller and smaller. These sets, however, may not intersect at all for step sizes taken over any interval near the origin. Therefore, the invariant tori of any frequencies may not be guaranteed for any symplectic algorithm with step size randomly taken in any neighbourhood of the origin. Theorem 7

So far, the available symplectic algorithms are exact symplectics when they are applied to global Hamiltonian systems and analytics when applied to analytic systems.

576

13. KAM Theorem of Symplectic Algorithms

4.11 shows that an invariant torus with any fixed diophantine frequency of an analytic nondegenerate integrable Hamiltonian system can always be simulated very well by symplectic algorithms for any step size in a Cantor set of positive Lebesgue measure near the origin. The following theorem shows that one can simulate simultaneously any finitely many invariant tori of given diophantine frequencies by symplectic algorithms with a sufficiently big probability to select available step sizes. The step sizes, of course, also have to be restricted to a Cantor set. Theorem 4.14. Given an analytic, nondegenerate and integrable Hamiltonian system of n degrees of freedom. Given N diophantine frequencies ω j (j = 1, 2, · · · , N ), in the domain of the frequencies of the system, there exists a Cantor set I of R, depending on the N frequencies, such that for any symplectic algorithm applied to the system, there exists a positive number δ0 such that if the step size t of the algorithm falls into the set (−δ0 , δ0 ) ∩ I, then the algorithm has N invariant tori of the frequencies τ ω j (j = 1, 2, · · · , N ) when it applies to the integrable system. These invariant tori approximate the corresponding ones of the system in the sense of Hausdorff with the order equal to the order of accuracy of the algorithm. The Cantor set I has density one at the origin. Proof. The proof of Theorem 4.14 follows from Theorem 4.11 and



Lemma 4.15. For any integer N ≥ 1, any ω j ∈ Ωγ (τ ) (j = 1, 2, · · · , N ) and any δ > 0, put AN = (ω 1 , ω2 , · · · , ω N ) and N N Iλ,μ," τ (A )

=

N L

δ N N N Jλ,μ," τ (A ) = (−δ, δ) \ Iλ,μ," τ (A )

Iλ,μ,"τ (ω j ),

j=1

with given λ > 0, μ ≥ −1, τ" > n + 1 and μ + τ" > n + 1. Then we have δ N τ" −n meas Jλ,μ," τ (A ) ≤ N dδ

if λ + δ <

2π , where |AN | = max |ω j | and d is defined by (4.14) where τ is |AN | 1≤j≤N

N N replaced by τ" and |ω| replaced by |AN |. Consequently, the set Iλ,μ," τ (A ) has density N N one at the origin. Moreover, for any t ∈ [−1, 1] ∩ Iλ,μ," τ (A ), we have

  ik,tω j  e − 1 ≥

|t|" γ |k|μ+τ +"τ

,

0 = k ∈ Zn ,

j = 1, 2, · · · , N

with γ " given by (4.12) where |ω| replaced by |AN |. Proof. Lemma 4.15 is a natural corollary of Lemmas 4.9 and 4.10.



Remark 4.16. There have been some works about exponential stability of symplectic algorithms in simulating invariant tori with given diophantine frequencies of integrable or nearly integrable systems (Benettin and Giorgilli (1994)[BG94] , Hairer and Lubich in 1997[HL97] and Stoffer in 1998[Sto98b] ). The result, for example, of Hairer and

13.4 Resonant and Diophantine Step Sizes

577

Lubich[HL97] shows that during a very long interval of iteration steps (exponentially long in 1/t ), the numerical orbits of a symplectic algorithm approximate the exact orbits of some perturbed Hamiltonian system8 with a very small error (exponentially small in −1/t ) if the starting values of the numerical orbits and the exact ones are the same and are taken on the invariant torus of the perturbed system (the invariant torus is guaranteed by the KAM theorem)[HL97] (Corollary 7) or taken in a neighbourhood of the invariant torus with the radius of order t2n+2 (this is easily derived from Hairer and Lubich (1997, Corollary 8)), here n is the degrees of freedom of the Hamiltonian system and t is the step size of the algorithm which is assumed to be sufficiently small. Theorems 4.11 and 4.14 show that one may generate quasi-periodic (therefore, perpetually stable) numerical orbits using a symplectic algorithm which approximate exact quasi-periodic orbit of an analytic nondegenerate integrable Hamiltonian system if the step sizes of the algorithm fall into a Cantor set of large density near the origin. As the step size in this Cantor set gets smaller and smaller, more and more stable numerical orbits appear. For such a stability consideration, Theorem 3.4 shows much more: the perpetually stable numerical orbits take up a large set of the phase space so that the Lebesgue measure of the set approaches the Lebesgure measure of the phase space as the step size approaches zero. Due to the well-known topological confinement of the phase plane between invariant closed curves, this implies the perpetual stability of symplectic algorithms applied to one degree of freedom systems for any initial values if the step size is small. Remark 4.17. Generally speaking, it is difficult to check the diophantine condition for a step size with respect to a nonresonant frequency vector. An obvious fact is fortunately, however, that step sizes N −1 , with N being integers, satisfy the diophantine condition (4.4) with respect to frequency vectors satisfying diophantine condition (2.8). This fact was checked by Dujardin and Faou in 2007[DF07] for the 1 + 1 dimensional linear Schr¨odinger equation with a periodic potential, where a spatially 1 periodic solution can be stably simulated using nonresonant step size t = = 0.2, but is quickly violated using resonant step size t =

8

2π ∼ 0.196. 62 − 22

5

The perturbed Hamiltonian system approximates the symplectic algorithm and is determined uniquely, in the setting of the backward analysis, by the algorithm and the Hamiltonian system which the algorithm applies to[Hai94] .

Bibliography

[AA89] V. I. Arnold and A. Avez: Ergodic Problems of Classical Mechanics. Addison-Wesley and Benjamin Cummings, New York, (1989). [Arn63] V.I. Arnold: Proof of A. N. kolmogorov’s theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian,. Russian Math. Surveys, 18(5):9–36, (1963). [Arn89] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, GTM 60, Berlin, Heidelberg, Second edition, (1989). [BB79] K. Burrage and J.C. Butcher: Stability criteria for implicit Runge–Kutta methods. SIAM J. Numer. Anal., 16:46–57, (1979). [Bey87] W. J. Beyn: On invariant closed curves for one-step methods. Numer. Math., 51:103– 122, (1987). [BG94] G. Benettin and A. Giorgilli: On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys., 74:1117–1143, (1994). [But75] J. C. Buther: A stability property of implicity Runge–Kutta methods. BIT, 15(3):358– 361, (1975). [CD09] F. Castella and G. Dujardin: Propagation of gevrey regularity over long times for the fully discrete Lie-Trotter splitting scheme applied to the linear Schr¨odinger equation. ESIAM: Mathmatical Modelling and Numericalo Analysis, 43(4):651–676, (2009). [CFM06] P. Chartier, E. Faou, and A. Murua: An algebraic approach to invariant preserving integrators; the case of quadratic and Hamiltonian invariants. Numer. Math., 103(4):575– 590, (2006). [CG82] L. Chierchia and G. Gallavotti: Smooth prime integralsfor quasi-integrable Hamiltonian systems. IL Nuovo Cimento, 67, 277– 295, (19820. [CS90] P.J. Channell and C. Scovel: Symplectic integration of Hamiltonian systems. Nonlinearity, 3:231–259, (1990). [CS94] C. G. Cheng and Y. S. Sun: Existence of KAM tori in degenerate Hamiltonian systems. J. Differential Equations, 114(1):288–335, (1994). [Dah63] G. Dahlquist: A special stability problem for linear multistep methods. BIT, 3(1):27– 43, (1963). [Dah75] G. Dahlquist: Error analysis for a class of methods for stiff nonlinear initial value problems. In G.A. Watson, editor, Lecture notes in Mathematics, Vol. 506, Numerical Analysis, Dundee, pages 60–74. Springer, Berlin, (1975). [Dah78] G. Dahlquist: G-stability is equivalent to A-stability. BIT, 18:384–401, (1978). [DF07] G. Dujardin and E. Faou: Normal form and long time analysis of splitting schemes for the linear Schr¨ odinger equation with small potential. Numer. Math., 108(2):223–262, (2007). [Ehl69] B. L. Ehle: On Pad´e approximations to the exponential function and A-stable methods for the numerical solution of initial value problems. Technical Report, Research Rep. No. CSRR 2010, University of Waterloo Dept. of Applied Analysis and Computer Science, (1969).

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Chapter 14. Lee-Variational Integrator

In the 1980s, Lee proposed an energy-preserving discrete mechanics with variable time steps by taking time (discrete) as a dynamical variable [Lee82,Lee87] . On the other hand, motivated by the symplectic property of Lagrangian mechanics, a version of discrete Lagrangian mechanics has been developed and variational integrators that preserve discrete symplectic 2-form have been obtained [MPS98,MV91,Ves88,Ves91a,WM97] , but variational integrators obtained in this way fix the time steps and consequently, they are not energy-preserving in general. Obviously, energy-preserving discrete mechanics and variational integrators are more preferable, since solutions of the Euler–Lagrange equations of conservative continuous systems are not only symplectic but also energy-preserving. To attain this goal, we should study some discrete mechanics with discrete energy conservation and symplectic variational integrators. Recently, Kane, Marsden, and Ortiz have employed appropriate time steps to conserve a defined energy and developed what they called symplectic energy-momentum-preserving variational integrators in [KMO99] . Although their approach is more or less related to Lee’s discrete mechanics, the discrete energypreserving condition is not derived by the variational principle.

14.1 Total Variation in Lagrangian Formalism The purpose of this section is to generalize or improve the above mentioned approaches as well as to explore the relations among discrete total variation, Lee’s discrete mechanics, and Kane–Marsden–Ortiz integrators. We will present a discrete total variation calculus with variable time steps and a discrete mechanics that is discretely symplectic, energy-preserving and has the correct continuous limit. In fact, this discrete variation calculus and mechanics is a generalization of Lee’s discrete mechanics in symplectic-preserving sense and can derive directly the variational symplecticenergy-momentum integrators of Kane, Marsden, and Ortiz.

14.1.1 Variational Principle in Lagrangian Mechanics Before beginning this section, we will recall very briefly the ordinary variational principle in Lagrangian mechanics for later use. Suppose Q denotes the extended configuration space with coordinates (t, q i ) and Q(1) the first prolongation of Q with K. Feng et al., Symplectic Geometric Algorithms for Hamiltonian Systems © Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Hangzhou and Springer-Verlag Berlin Heidelberg 2010

582

14. Lee-Variational Integrator

coordinates (t, q i , q˙i )[Olv93] . Here t denotes time and q i (i = 1, 2, · · · , n) denote the positions. Consider a Lagrangian L : Q(1) → R. The corresponding action functional is defined by - b S(q i (t)) = L(t, q i (t), q˙i ) d t, (1.1) a i

2

where q (t) is a C curve in Q. Hamilton’s principle seeks a curve q i (t) denoted by Cab with endpoints a and b, for which the action functional S is stationary under variations of q i (t) with fixed endpoints. Let ∂ (1.2) V = φi (t, q) i ∂q be a vertical vector field on Q, here q = (q 1 , · · · , q n ). By a vertical vector field we ∂

mean a vector field on Q which does not involve terms of form ξ(t, q) , for example, ∂t time t does not undergo variation. Let F ε be the flow of V , i.e., a one-parameter group of transformations on Q : ε F (t, q i ) = (t˜, q˜i ).

where

t˜ = t, q˜i = g i (ε, t, q),

(1.3) (1.4)

d  g i (ε, t, q) = φi (t, q) := δq i (t).  d ε ε=0

(1.5)

In other words, the deformation (1.3) – (1.4) transforms the curve q i (t) into a family of curves q˜i (ε, t˜) in Q denoted by Cεba which are determined by t˜ = t, q˜i = gi (ε, t, q(t)).

(1.6) (1.7)

Thus, we obtain a (sufficiently small) set of curves Cεba around Cab . Corresponding to this set of curves there is a set of Lagrangian and action functionals -

b

S(q (t)) −→ S(˜ q (ε, t˜)) = i

i

a

d L(˜ q i (ε, t˜), q˜i (ε, t˜)) d t˜. d t˜

Now, we can calculate the variation of S at q(t) as follows:  d q i (ε, t˜)) δS =  S(˜ dε ε=0 & ! - b %  ∂L d ∂L ∂ L i b i − φ φ d t + =  . i i i a

∂q

d t ∂ q˙

∂ q˙

a

(1.8)

(1.9)

For the fixed endpoints, φi (a, q(a)) = φi (b, q(b)) = 0, the requirement of Hamilton’s principle, δS = 0, yields the Euler–Lagrange equation for q(t)

14.1 Total Variation in Lagrangian Formalism

∂L d ∂L − = 0. ∂ qi d t ∂ q˙i

583

(1.10)

If we drop the requirement of φi (a, q(a)) = φi (b, q(b)) = 0, we can naturally obtain the Lagrangian 1-form on Q(1) from the second term in (1.9): θL = where d q i are dual to



∂ ∂ , d qi ∂ qj ∂ qj



∂L d qi, ∂ q˙I

(1.11)

= δji . Furthermore, it can be proved that the

solution of (1.10) preserves the Lagrangian 2-form ωL := dθL . On the other hand, introducing the Euler–Lagrange 1-form ' / ∂L d ∂L E(q i , q˙i ) = − d qi , ∂q i d t ∂ q˙i

(1.12)

(1.13)

the nilpotency of d leads to d E(q i , q˙i ) +

d ωL = 0, dt

(1.14)

namely, the necessary and sufficient condition for symplectic structure preserving is that the Euler–Lagrange 1-form is closed[GLW01a,GLWW01,GW03] .

14.1.2 Total Variation for Lagrangian Mechanics Consider a general vector field on Q V = ξ(t, q)

∂ ∂ + φi (t, q) i , ∂t ∂q

(1.15)

here q = (q 1 , · · · , q n ). Let F ε be the flow of V . The variations of (t, q i ) ∈ Q are described in such a way (t, qi ) −→ F ε (t, q i ) = (t˜, q˜i ),

(1.16)

q˜i = g i (ε, t, q)

(1.17)

where t˜ = f (ε, t, q), with d   f (ε, t, q) = ξ(t, q) := δt, dε ε=0

d   g i (ε, t, q) = φi (t, q) := δq i . dε ε=0

(1.18)

The deformations (1.17) transform a curve q i (t) in Q denoted by Cab into a set of ˜ curves q˜i (ε, t˜) in Q denoted by Cεba˜ , determined by

584

14. Lee-Variational Integrator

t˜ = f (ε, t, q(t)),

q˜i = g i (ε, t, q(t)).

(1.19)

Before calculating the total variation of S, we will introduce the first-order prolongation of V denoted as pr1 V pr1 V = ξ(t, q)

∂ ∂ ∂ + φi (t, q) i + αi (t, q, q) ˙ , ∂t ∂q ∂ q˙i

(1.20)

here αi (t, q, q) ˙ = Dt φi (t, q) − q˙i Dt ξ(t, q),

(1.21)

where Dt denotes the total derivative with respect to t, for example, Dt φk (t, q i ) = φkt + φkqi q˙i ,

φkt =

∂φk . ∂t

For prolongations of the vector field and the related formulae, refer to[Olv93] . Now, we let us calculate the total variation of S straightforwardly: δS = = = = =

- ˜b   d  d  d i ˜ q (ε, t)) =  L t˜, q˜i (ε, t˜), q˜i (ε, t˜) d t˜  S(˜ dε ε=0 dε ε=0 a˜ dt˜ - b      d dt˜ d L t˜, q˜i (ε, t˜), q˜i (ε, t˜) d t t˜ = f (ε, t, q(t))  ˜ dε ε=0 a dt dt - b - b    d i ˜ d  L t, q i (t), q˙i (t) Dt ξ d t q˜ (ε, t) d t +  L t˜, q˜i (ε, t˜), ˜ dt a d ε ε=0 a - b - b  ∂L ∂L ∂L LDt ξ d t ξ + i φi + i (Dt φi − q˙i Dt ξ) d t + ∂t ∂q ∂ q˙ a a ! - b    ∂L d  ∂L i d ∂L  i ∂L + q ˙ − L ξ + − dt φ ∂t dt ∂ q˙i ∂q i dt ∂ q˙i a !  ∂L ∂L  b + L − i q˙i ξ + i φi  . (1.22) ∂ q˙ ∂ q˙ a

Here we have made use of (1.18), (1.20), (1.21) and d d  ˜ d t˜ d  =   t = Dt ξ. d ε ε=0 d t d t d ε ε=0 If ξ(a, q(a)) = ξ(b, q(b)) = 0 and φi (a, q(a)) = φi (b, q(b)) = 0, the requirement of δS = 0 yields the equation from ξ, the variation along the base manifold, i.e., the time.  d  ∂L i ∂L + q ˙ − L = 0, (1.23) ∂t dt ∂ q˙i and the Euler–Lagrange equation from φi , the variation along the fiber, i.e., the configuration space, ∂L d ∂L − = 0. (1.24) ∂q i dt ∂ q˙i

14.1 Total Variation in Lagrangian Formalism

585

Here ξ and φi are regarded as independent components of total variation. However, there is another decomposition for the independent components, i.e., the vertical and horizontal variations; see Remark1.2 below. ∂L = 0, then (1.23) If L does not depend on t explicitly, i.e., L is conservative, ∂t becomes the energy conservation law d H = 0, dt

H :=

 ∂L ∂ q˙

 i q ˙ − L . i

By expanding the left-hand side of (1.25), we obtain % & % & d ∂L d ∂L i ∂L q˙ − L = − − q˙i . dt ∂ q˙i ∂q i dt ∂ q˙i

(1.25)

(1.26)

Thus, for a conservative L, energy conservation is a consequence of Euler–Lagrange equation. This agrees with Noether theorem which states that the characteristic of an infinitesimal symmetry of the action functional S is that of a conservation law for the Euler–Lagrange equation. For a conservative L,

∂ is an infinitesimal symmetry of ∂t

the action functional S, and its characteristic is −q˙i . From Noether theorem, there exits a corresponding conservation law in the characteristic form & % d ∂L ∂L − (1.27) q˙i = 0. − ∂ qi d t ∂ q˙i If we drop the requirement ξ(a, q(a)) = ξ(b, q(b)) = 0,

φi (a, q(a)) = φi (b, q(b)) = 0,

(1.28)

we can define the extended Lagrangian 1-form on Q(1) from the second term in (1.22) % & ∂L i ∂L q˙ dt + d qi . (1.29) ϑL := L − i ∂ q˙ ∂ q˙i Suppose g i (t, vqi ) is a solution of (1.24) depending on the initial condition vqi ∈ Q(1) . Restricting q˜i (ε, t˜) to the solution space of (1.24) and using the same method in[MPS98] , it can be proved that the extended symplectic 2-form is preserved: (pr1 g i )∗ ΩL = ΩL ,

ΩL := dϑL , (1.30)  d i g (s, vqi ) denotes the first-order prolongawhere pr1 g i (s, vqi ) = s, g i (s, vqi ), 

tion of gi (s, vqi )[Olv93] .

ds

Remark 1.1. If ξ in (1.15) is independent of q, the deformations in (1.17) are called fiber-preserving. In this case, the domain of definition of q˜i (ε, t˜) only depends on the deformations in (1.17). While in the general case, the domain of definition of q˜i (ε, t˜) depends on not only the deformations in (1.17) but also on q i (t).

586

14. Lee-Variational Integrator

Remark 1.2. Using the identity & % & % ∂L d ∂L ∂L d ∂L i q˙ − L = − − q˙i , + ∂t dt ∂ q˙i ∂q i dt ∂ q˙i

(1.31)

the Equation (1.22) becomes ! & - b% b d ∂L ∂L i ∂L b i i i δS = − − ξ q ˙ )d t + (φ − ξ q ˙ ) (φ  + (Lξ)a . (1.32) i i i ∂q d t ∂ q˙ ∂ q˙ a a According to (1.18), φi = δq i should be regarded as the total variation of q i , δq i = δV q i + δH q i , since the variation of t also induces the variation of q i , denoted as δH qi , the horizontal variation of q i . Substituting ξ = δt in (1.18), the horizontal variation of q i should be δH q i = ξ q˙i , and consequently φi − ξ q˙i is interpreted as vertical variation δV q i , i.e., the variation of q i (t) at the moment t (for e.g., see[CH53] ) . Therefore, the first two terms in (1.32) come from vertical variation δV q i and the last term comes from horizontal variation δt. The horizontal variation of S with respect to the horizontal variation δH q i = ξ q˙i gives rise to the identity (1.31).

14.1.3 Discrete Mechanics and Variational Integrators In this subsection, by calculus of discrete total variations, we will develop a discrete Lagrangian mechanics, which includes the boundary terms in Lee’s discrete mechanics that give rise to the discrete version of symplectic preserving. The discrete variation calculus is mainly analog to Lee’s idea that time (discrete) is regarded as a dynamical variable, i.e., the time steps are variable[Lee82,Lee87] . The vertical part of this discrete variation calculus is similar to the one in[KMO99,MV91,Ves88,Ves91a,WM97] . Using this calculus for discrete total variations we naturally derive the Kane–Marsden–Ortiz integrators. We use Q × Q to denote the discrete version of the first prolongation for the extended configuration space Q. A point (t0 , q0 ; t1 , q1 ) ∈ Q × Q 1 , corresponds to a q − q0 tangent vector 1 . A discrete Lagrangian is defined to be L : Q × Q → R and t1 − t0 the corresponding action as S=

N −1 

L(tk , qk , tk+1 , qk+1 )(tk+1 − tk ).

(1.33)

k=0

The discrete variational principle in total variation is to extremize S for variations of both qk and tk with fixed end points (t0 , q0 ) and (tN , qN ). This discrete variational principle determines a discrete flow Φ : Q × Q → Q × Q by Φ(tk−1 , qk−1 , tk , qk ) = (tk , qk , tk+1 , qk+1 ).

(1.34)

Here (tk+1 , qk+1 ) are calculated from the following discrete Euler–Lagrange equation, i.e., the variational integrator, and the discrete energy conservation law (for conservative L) 1

In this section, q is an abbreviation of (q 1 , q 2 , · · · , qn ).

14.1 Total Variation in Lagrangian Formalism

587

(tk+1 − tk )D2 L(tk , qk , tk+1 , qk+1 ) + (tk − tk−1 )D4 L(tk−1 , qk−1 , tk , qk ) = 0, (1.35) and (tk+1 − tk )D1 L(tk , qk , tk+1 , qk+1 ) + D3 L(tk−1 , qk−1 , tk , qk )(tk − tk−1 ) (1.36) −L(tk , qk , tk+1 , qk+1 ) + L(tk−1 , qk−1 , tk , qk ) = 0, for all k ∈ {1, 2, · · · , N − 1}. Here Di denotes the partial derivative of L with respect to the i-th argument. The Equation (1.35) is the discrete Euler–Lagrange equation. The Equation (1.36) is the discrete energy conservation law for a conservative L. The integrator (1.35) – (1.36) is the Kane–Marsden–Ortiz integrator. Using the discrete flow Φ, the Equations (1.35) and (1.36) become (tk+1 − tk )D2 L ◦ Φ + (tk − tk−1 )D4 L = 0, ((tk+1 − tk )D1 L − L) ◦ Φ + D3 L + L = 0,

(1.37) (1.38)

respectively. If (tk+1 − tk )D2 L and (tk+1 − tk )D1 L − L are invertible, the Equations (1.37) and (1.38) determine the discrete flow Φ under the consistency condition ((tk+1 −tk )D1 L−L)−1 ◦(D3 L+L) = ((tk+1 −tk )D2 L)−1 ◦(tk −tk−1 )D4 L. (1.39) Now, we will prove that the discrete flow Φ preserves a discrete version of the extended Lagrange 2-form ΩL . As in continuous case, we will calculate d S for variations with variable end points. d S(t0 , q0 , · · · , tN , qN ) · (δt0 , δq0 , · · · , δtN , δqN ) =

N −1 

(D2 L(tk , qk , tk+1 , qk+1 )δqk + D4 L(tk , qk , tk+1 , qk+1 )δqk+1 )(tk+1 − tk )

k=0

+

N −1 

(D1 L(tk , qk , tk+1 , qk+1 )δtk + D3 L(tk , qk , tk+1 , qk+1 )δtk+1 )(tk+1 − tk )

k=0

+

N −1 

L(tk , qk , tk+1 , qk+1 )(δtk+1 − δtk )

k=0

=

N −1 

D2 L(tk , qk , tk+1 , qk+1 )(tk+1 − tk )δqk

k=0

+

N 

D4 L(tk−1 , qk−1 , tk , qk )(tk − tk−1 )δqk

k=1

+

N −1 

D1 L(tk , qk , tk+1 , qk+1 )(tk+1 − tk )δtk

k=0

+

N  k=1

D3 L(tk−1 , qk−1 , tk , qk )(tk − tk−1 )δtk

588

14. Lee-Variational Integrator

+

N −1 

L(tk , qk , tk+1 , qk+1 )(−δtk ) +

k=0

=

N −1 

N 

L(tk−1 , qk−1 , tk , qk )δtk

k=1

(D2 L(tk , qk , tk+1 , qk+1 )(tk+1 − tk )

k=1

+D4 L(tk−1 , qk−1 , tk , qk )(tk − tk−1 ))δqk +

N −1 

(D1 L(tk , qk , tk+1 , qk+1 )(tk+1 − tk )

k=1

+D3 L(tk−1 , qk−1 , tk , qk )(tk − tk−1 )) (1.40) +L(tk−1 , qk−1 , tk , qk ) − L(tk , qk , tk+1 , qk+1 ))δtk +D2 L(t0 , q0 , t1 , q1 )(t1 − t0 )δq0 + D4 L(tN −1 , qN −1 , tN , qN )(tN − tN −1 )δqN +(D1 L(t0 , q0 , t1 , q1 )(t1 − t0 ) − L(t0 , q0 , t1 , q1 ))δt0 +(D3 L(tN −1 , qN −1 , tN , qN )(tN − tN −1 ) + L(tN −1 , qN −1 , tN , qN ))δtN . We can see that the last four terms in (1.40) come from the boundary variations. Based on the boundary variations, we can define two 1-forms on Q × Q θL− (tk , qk , tk+1 , qk+1 ) = (D1 L(tk , qk , tk+1 , qk+1 )(tk+1 − tk ) − L(tk , qk , tk+1 , qk+1 ))dtk +D2 L(tk , qk , tk+1 , qk+1 )(tk+1 − tk )dqk , (1.41) and θL+ (tk , qk , tk+1 , qk+1 )

= (D3 L(tk , qk , tk+1 , qk+1 )(tk+1 − tk ) + L(tk , qk , tk+1 , qk+1 ))d tk+1 +D4 L(tk , qk , tk+1 , qk+1 )(tk+1 − tk )d qk+1 ,

(1.42)

having employed the notations in [MPS98] . We regard the pair (θL− , θL+ ) as the discrete version of the extended Lagrange 1-form ϑL defined in (1.29). Now, we parameterize the solutions of the discrete variational principle by the initial condition (t0 , q0 , t1 , q1 ) and restrict S to that solution space. Then Equation (1.40) becomes dS(t0 , q0 , · · · , tN , qN ) · (δt0 , δq0 , · · · , δtN , δqN ) = θL− (t0 , q0 , t1 , q1 ) · (δt0 , δq0 , , δt1 , δq1 ) +θL+ (tN −1 , qN −1 , tN , qN ) · (δtN −1 , δqN −1 , δtN , δqN ) = θL− (t0 , q0 , t1 , q1 ) · (δt0 , δq0 , δt1 , δq1 ) +(ΦN −1 )∗ θL+ (t0 , q0 , t1 , q1 )(δt0 , δq0 , δt1 , δq1 ). From (1.43), we obtain

d S = θL− + (ΦN −1 )∗ θL+ .

The Equation (1.44) holds for arbitrary N > 1. By taking N = 2, we get

(1.43) (1.44)

14.1 Total Variation in Lagrangian Formalism

d S = θL− + Φ∗ θL+ .

589

(1.45)

By exterior differentiation of (1.45), we obtain Φ∗ (d θL+ ) = −d θL+ .

(1.46)

From the definition of θL− and θL+ , we know that θL− + θL+ = d (L(tk+1 − tk )).

(1.47)

By exterior differentiation of (1.47), we obtain d θL+ = −d θL− . Define ΩL ≡ d θL+ = −d θL− .

(1.48)

Finally, we have shown that the discrete flow Φ preserves the discrete extended Lagrange 2-form ΩL Φ∗ (ΩL ) = ΩL . (1.49) Now, the variational integrator (1.35), the discrete energy conservation law (1.36), and the discrete extended Lagrange 2-form ΩL converge to their continuous counterparts as tk+1 → tk , tk−1 → tk . Consider a conservative Lagrangian L(q, q). ˙ For simplicity, we choose the discrete Lagrangian as & % qk+1 − qk . (1.50) L(tk , qk , tk+1 , qk+1 ) = L qk , tk+1 − tk The variational integrator (1.35) becomes % ∂L 1 ∂L (qk , Δt qk ) − (qk , Δt qk ) − ∂ qk

tk+1 − tk

∂ Δt qk

&

∂L (qk−1 , Δt qk−1 ) ∂ Δt qk−1

q − qk q − qk−1 , Δt qk−1 = k . where Δt qk = k+1 tk+1 − tk tk − tk−1

= 0, (1.51)

It is easy to see that, as tk+1 → tk , tk−1 → tk , the Equation (1.51) converges to d ∂L ∂L − = 0. ∂ qk d t ∂ q˙k

(1.52)

The discrete energy conservation law (1.36) becomes Ek+1 − Ek = 0, tk+1 − tk where

& % ∂L qk+1 − qk , Ek+1 = Δt qk − L qk , ∂Δt qk tk+1 − tk % & ∂L qk − qk−1 Ek = Δt qk−1 − L qk−1 , . ∂Δt qk−1 tk − tk−1

The Equation (1.53) converges to

(1.53)

590

14. Lee-Variational Integrator

d dt

%

& ∂L q˙k − L = 0 ∂ q˙k

(1.54)

as tk+1 → tk , tk−1 → tk . Now, we will consider the discrete extended Lagrange 2-form ΩL defined by (1.48). By discretization of (1.50), the discrete extended Lagrange 1-form θL+ defined in (1.42) becomes % & ∂L ∂L + θL = L(qk , Δt qk ) − Δt qk d tk+1 + d qk+1 . (1.55) ∂ Δt qk ∂ Δt qk From (1.55), we can deduce that θL+ converges to the continuous Lagrangian 1-form ϑL defined by (1.29) as tk+1 → tk , tk−1 → tk . Thus, we obtain ΩL = dθL+ −→ dϑL = ΩL ,

tk+1 → tk ,

tk−1 → tk .

(1.56)

In general, the variational integrator (1.35) with fixed time steps does not exactly conserve the discrete energy, and the computed energy will not have secular variation[GM88,SSC94] . In some cases, such as in discrete mechanics proposed by Lee in [Lee82,Lee87] , the integrator (1.35) is required to conserve the discrete energy (1.36) by varying the time steps. In other words, the steps can be chosen according to (1.36) so that the integrator (1.35) conserves the discrete energy (1.36). The resulting integrator also conserves the discrete extended Lagrange 2-form dθL+ . This fact had not been discussed in Lee’s discrete mechanics. Example 1.3. Let us consider an example. For the classical Lagrangian 1 L(t, q, q) ˙ = q˙2 − V (q), 2 we choose the discrete Lagrangian L(tk , qk , tk+1 , qk+1 ) as &2 % & % 1 qk+1 − qk qk+1 − qk L(tk , qk , tk+1 , qk+1 ) = . −V 2 tk+1 − tk 2

(1.57)

(1.58)

The discrete Euler–Lagrange equation (1.35) becomes & % qk − qk−1 V  (¯ qk )(tk+1 − tk ) + V  (¯ qk−1 )(tk − tk−1 ) qk+1 − qk − + = 0, tk+1 − tk tk − tk−1 2 (1.59) which preserves the Lagrange 2-form & % 1 t − tk  + k+1 V (¯ qk ) d qk+1 ∧ d qk , (1.60) tk+1 − tk

q +q

4

q

+q

k+1 k where q¯k = k , q¯k−1 = k−1 . 2 2 If we take fixed variables tk+1 − tk = t