Variations, Geometry and Physics: In Honour of Demeter Krupka's Sixty-fifth Birthday

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VARIATIONS , G EOMETRY AND P HYSICS No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

VARIATIONS , G EOMETRY AND P HYSICS IN HONOUR OF

D EMETER K RUPKA’ S

SIXTY- FIFTH BIRTHDAY

O LGA K RUPKOV A´ AND

DAVID S AUNDERS

Nova Science Publishers, Inc. New York

c 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter cover herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal, medical or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Library of Congress Cataloging-in-Publication Data Variations, geometry & physics / Olga Krupkova and David Saunders (editor). p. cm. ISBN 978-1-61324-186-8 (eBook) 1. Calculus of variations. 2. Global analysis (Mathematics) 3. Mathematical physics. I. Krupkov, Olga, 1960- II. Saunders, D. J., 1964- III. Title: Variations, geometry, and physics. QA315.V38 2008 510–dc22 2008024454

Published by Nova Science Publishers, Inc. ✜ New York

Contents vii

Preface

PART I

VARIATIONAL P RINCIPLES ON J ET B UNDLES

1

Chapter 1

Lepage Forms in Variational Theories: From Lepage’s Idea to the Variational Sequence Jana Musilov´a and Michal Lenc

3

Chapter 2

Lepage Forms in the Calculus of Variations Olga Krupkov´a

Chapter 3

On a Generalization of the Poincar´e-Cartan Form in Higher-Order Field Theory D. R. Grigore

57

Krupka’s Fundamental Lepage Equivalent and the Excess Function of Wilkins D. J. Saunders

77

Chapter 4

27

Chapter 5

Lepage Congruences in Discrete Mechanics Antonio Fern´andez and Pedro L. Garc´ıa

85

Chapter 6

Finite Order Variational Sequences: a Short Review Raffaele Vitolo

99

Chapter 7

Concatenating Variational Principles and the Kinetic Stress-Energy-Momentum Tensor M. C. L´opez, M. J. Gotay and J. E. Marsden

117

A Geometric Hamilton-Jacobi Theory for Classical Field Theories M. de Le´on, J. C. Marrero and D. M. de Diego

129

Chapter 8

vi

Contents

PART II

N ATURAL B UNDLES AND D IFFERENTIAL I NVARIANTS

141

Chapter 9

Natural Lagrangian Structures Josef Janyˇska

143

Chapter 10

Connections on Higher Order Frame Bundles and Their Gauge Analogies Ivan Kol´arˇ

Chapter 11

Natural Lifts in Riemannian Geometry Oldˇrich Kowalski and Masami Sekizawa

Chapter 12

Invariant Variational Problems and Invariant Flows via Moving Frames Peter J. Olver

Chapter 13

Differential Invariants of the Motion Group Actions Boris Kruglikov and Valentin Lychagin

PART III

D IFFERENTIAL E QUATIONS S TRUCTURES

Chapter 14

AND

167

189

209

237

G EOMETRICAL

Remarks on the History of the Notion of Lie Differentiation Andrzej Trautman

Chapter 15

Second-Order Differential Equation Fields with Symmetry M. Crampin and T. Mestdag

Chapter 16

Dimensional Reduction of Curvature-Dependent Central Potentials on Spaces of Constant Curvature J. F. Cari˜nena, M. F. Ra˜nada and M. Santander

253

255

261

277

Chapter 17

Direct Geometrical Method in Finsler Geometry L. Tam´assy

293

Chapter 18

Linear Connections Along the Tangent Bundle Projection W. Sarlet

315

Chapter 19

On the Inverse Problem for Autoparallels G. E. Prince

341

PART IV

A PPENDIX

353

Demeter Krupka: List of Publications

355

Index

363

Preface This book is a commemorative volume to celebrate the sixty-fifth birthday of Demeter Krupka, an internationally recognized researcher in several fields of mathematics and mathematical physics. Demeter was born on 22 January 1942 in Levoˇca, Slovakia. He studied first at Masaryk University, Brno and in 1976 he received a Ph.D. degree from Charles University, Prague; during his studies he also spent a year at Warsaw University under the supervision of Professor Andrzej Trautman. In 1981 he was awarded the degree of Doctor of Sciences in geometry and topology by the Czechoslovak Academy of Sciences, Prague. Demeter has held posts at Masaryk University and at the Silesian University in Opava, and is currently Professor in the Department of Algebra and Geometry at Palack´y University, Olomouc. He also holds an honorary position as an IAS Distinguished Fellow at the Institute for Advanced Study, La Trobe University, Melbourne. His main research areas are global variational analysis, differential geometry, tensor algebra and mathematical physics. Among his recognized achievments we should mention the development of a global theory of variational functionals in fibred spaces in the early 1970s, and its higher-order generalisations in the early 1980s, based on his concept of a Lepage form and the Euler–Lagrange form. This approach provides a completely intrinsic setting for Euler–Lagrange theory and the invariance of variational functionals, including Noether theorem and its generalisations. It also facilitates the study of regular variational functionals, casting new light on the concept of regularity in the calculus of variations, Hamilton theory and Hamilton–Jacobi theory. He also was among the first to find necessary and sufficient conditions for a system of higher-order PDEs in covariant form to be variational, and he solved the local problem of the structure of higher-order null Lagrangians. In 1990 Demeter published his key paper on variational sequences, a finite order alternative to Vinogradov’s C -spectral sequence and the variational bicomplex introduced by, for example, Dedecker and Tulczyjew. Another important impact of his work concerns the geometric foundations of theory of differential invariants developed in the 1970s and 1980s, based on the concept of a higher order differential group, and the creation of a rigorous setting for natural variational principles. While solving technical problems appearing in the higher order variational calculus, he also contributed to linear algebra by studying and

viii

Preface

solving the trace decomposition problem in general tensor spaces over the real numbers. During his career, Demeter has written numerous articles for publication (a list is given in the Appendix to this volume) and taught a wide variety of topics in mathematics and mathematical physics; other highlights include being a founding editor of two major journals (Journal of Geometry and Physics and Differential Geometry and its Applications) and holding the post of Rector at the Silesian University in Opava. Most recently he has been joint editor of a major reference work (Handbook of Global Analysis). To celebrate Demeter’s sixty-fifth birthday, a Colloquium was held in Olomouc on 25th and 26th August, 2007 with twenty-one invited speakers, all with established reputations in their field. Many of the speakers have prepared versions of their talks for the present volume, and we have also been able to include some articles from authors who were not able to attend. We have divided the articles into three groups, covering three major areas of Demeter’s research interests: Variational principles on jet bundles, Natural bundles and differential invariants and Differential equations and geometric structures. We are grateful to all the authors who have provided these articles, and we are sure that they will be of great interest to the community.

Olga Krupkov´a, David Saunders (Editors) Olomouc, January 2008

Part I

VARIATIONAL P RINCIPLES ON J ET B UNDLES

1

In: Variations, Geometry and Physics Editors: O. Krupkov´a and D. Saunders, pp. 3-26

ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc.

Chapter 1

L EPAGE F ORMS IN VARIATIONAL T HEORIES : F ROM L EPAGE ’ S I DEA TO THE VARIATIONAL S EQUENCE∗ Jana Musilov´a† and Michal Lenc‡ Institute of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotl´aˇrsk´a 2, 611 37 Brno, Czech Republic

Abstract In the presented paper the development of the concept of Lepage forms is followed from the initial idea given by Lepage in 1930’s to its important role in the contemporary geometrical analysis and variational theories in physics. The attention is paid especially to the role of the concept of Lepage forms for the main formula of calculus of variations (called the first variational formula) as well as to their meaning as geometrical objects resulting by a natural way from the theory of variational sequences on finite order prolongations of fibered manifolds. As a possible application Lepage forms are studied in the context of our results concerning the representation of the variational sequence by differential forms on the one hand, and in concrete physical examples on the other. For the illustration we show that the concept of Lepage forms can be effectively used in such physical situations in variational field theories, as are e.g. variational problems in the classical string and brane theories.

1.

Introduction

The contemporary general concept of Lepage forms is closely related to fundamentals of local and global aspects of the modern calculus of variations. It represents differential forms of a special type defined on finite order jet prolongations of fibered manifolds. These manifolds are chosen as appropriate underlying geometrical structures for the majority of ∗ This paper is dedicated to 65th birthday of our colleague and friend professor Demeter Krupka, whose contribution to the development of the concept of Lepage forms as the key geometrical objects of modern calculus of variations and its application in physical theories is undoubtedly important. † E-mail address: [email protected] ‡ E-mail address: [email protected]

4

Jana Musilov´a and Michal Lenc

physical theories. Theories developed on fibered manifolds with one-dimensional bases are called “mechanics”, while for n-dimensional bases of underlying fibered manifolds we speak about “field theories”. There are specific geometrical (i.e. coordinate free) objects defined on a fibered manifold and its jet prolongations, closely connected with the fibered structure of the manifold: projectable and vertical vector fields, horizontal and contact differential forms. These objects play the key role in formulation of the fundamental variational problem which lies in searching such sections of the underlying fibered manifold, e.g. “trajectories” of studied physical systems, representing critical (stationary) points of a functional appropriately chosen with respect to a given theory. Such a functional is, for a theory formulated on a fibered manifold (Y, π, X) with the n-dimensional base X, the (m + n)-dimensional total space Y , and the projection π, defined by the integral of a horizontal n-form λ = L ω0 , called r-th order Lagrangian, where L is a function on the r-th prolongation (J r Y, πr , X) of the fibered manifold and ω0 is the volume element on the base, in local coordinates ω0 = dx1 ∧ . . . ∧ dxn . The general questions most intensively studied in last decades are especially: ⊲ the variational equivalence problem leading to the concept of Lepage equivalents of a given Lagrangian, i.e. differential forms giving the same variational integral as the corresponding Lagrangian, and in general to the concept of Lepage n-forms, ⊲ the structure of the Euler-Lagrange mapping assigning to a Lagrangian its EulerLagrange form representing the equations of motion; the study of this structure enables us to find trivial Lagrangians, leading to identically zero Euler-Lagrange forms, ⊲ the structure of Helmholtz-Sonin mapping; this enables us to solve the inverse problem of calculus of variations, i.e. stating whether given equations of motion arise from a variational problem, and leads to the concept of Lepage (n + 1)-forms and their equivalence with respect to given equations of motion, ⊲ the theory of the finite-order variational sequences and the problem of their representation by differential forms, both inspired by the above exposed equivalence problems, namely by the close relation between the Euler-Lagrange mapping and the exterior derivative operator, ⊲ a cohomological structure of the variational sequence in which both Euler-Lagrange and Helmholtz-Sonin mappings are included, as an effective tool for describing global aspects the theory, ⊲ an idea of the quite general definition of Lepage forms arising naturally from the structure of the variational sequence itself. Both primary ideas leading to the concept of Lepage forms (the importance of the exterior derivative operator and the idea of equivalence of forms in variational problems) were ´ Cartan and Th. de Donder, formulated by Lepage in [35] and [36], and presented by Elie respectively. They were later developed by Paul Dedecker in [6]. In Sec. 2 we present the brief review of considerations concerning these “historical” ideas and their consequences. The concept of a Lepage equivalent of a Lagrangian λ = L ω0 as a form leading to the same variational problem, strictly based on the geometrical structure of fibered manifolds

Lepage Forms in Variational Theories

5

and their jet prolongations, was introduced by Krupka (see e.g. his key paper [16]), and then developed up to a coherent theory. His studies of the Euler-Lagrange mapping in 1970’s and 1980’s inspired the later considerations concerning the variational sequence. (See e.g. in [17], [18], [19], [20], [21], [22], [24], a complete review with contemporary results see in [26].) The concept of a Lepage (n + 1)-form and Lepage class of a dynamical (n + 1)form E = Eσ ω σ ∧ ω0 , important from the point of view of equations of motion Eσ ◦ J s γ (variational or non-variational) for trajectories γ : X → Y , U ⊂ X, π ◦ γ = IdU of a dynamical system, was introduced by Krupkov´a for mechanics, i.e. n = 1 (see [30], [31], [32]). It was well elaborated to a coherent theory by the same author (see the key book [33]). The concept was later extended to non-holonomic constrained mechanical systems in [34]. Main ideas and results obtained in studies of Lepage forms made by Krupka and Krupka’s coworkers are summarized in Sec. 3. The above mentioned special type of a Lepage form, Lepage equivalent of a Lagrangian, is important namely for so called first variation formula. This formula represents an appropriate decomposition of the basic object of the calculus of variations – the variational functional (the integral first variation formula), or the decomposition of the integrand itself with respect to the fibered structure of the underlying manifold (the infinitesimal first variation formula). Moreover, the first variational formula gives information concerning some properties of symmetry transformations of variational problems. Thus, the application of Lepage equivalents of Lagrangians for formulation of variational physical theories is straightforward. In Sec. 4 we give some examples of such an application in modern variational field theories in physics – string and brane theories. We present there the Lepage equivalents in three physical situations: a relativistic particle (first order mechanics), the classical variational functional for a boson string (first order theory, see e.g. [5]), the Polyakov functional for D-dimensional space and its special case for Minkowski space (first order field theory). With help of results following from the first variational formula we use these Lepage equivalents for obtaining quantities important for conservation laws. The quite general definition of Lepage forms arises by a natural way from the structure of the variational sequence on finite order jet prolongations of fibered manifolds. Especially, it is closely related to the problem of representation of the variational sequence by forms. The idea of the finite order variational sequence was exposed by Krupka in 1989 (see [23]), who studied the structure of morphisms of the sequence important for calculus of variations, as well as the cohomology aspects important for global questions. Recall that the infiniteorder counterpart of Krupka’s theory is the theory of variational bicomplex, formulated in 1970’s and 1980’s by Anderson and Duchamp in [3], Dedecker and Tulczyjew in [7], [43], Takens in [42], and Vinogradov and his school in [44], [45], [46], [4], [11]. Both theories, the variational sequence as well as variational bicomplex, are based on the analysis of the exterior derivative operator d. (Recall that the r-th order variational sequence is the quotient of the well-known De Rham sequence by its exact subsequence of forms of certain kind of contactness or “variational irrelevance”, the quotient being exact as well.) However, there are some differences resulting from the mentioned different choice of basic structures as domains of definition, finite order jet prolongations J r Y of a fibered manifold Y and infinite jets J ∞ Y . The main difference lies in the fact that the finite order theory keeps the order of all sequence operation fixed (contrary to the case of bicomplex, where the order can be increased arbitrarily). This gives a deeper understand-

6

Jana Musilov´a and Michal Lenc

ing of the role of the order, of the analytic structure of fundamental variational concepts. Moreover, it is more appropriate for application in physical situations, always described as finite order problems. The instructive comparison of both theories is presented in [25] and [48]. The finite order variational sequence was intensively studied from 1990’s to the present by Krupka and coauthors or coworkers, and by other authors, mainly Grigore [9], [10] and Vitolo [47], [48], [49], [50]. (The practically complete bibliography related to both the variational bicomplex and mainly to the variational sequence is presented in [50].) A specific problem connected with the finite order variational sequence is its representation by differential forms. The idea lies in the requirement to represent every class [̺] of differential forms in the r-th order variational sequence by an appropriately chosen form with specific properties: R-linearity, coordinate invariance, exactness of the sequence of representatives, projection property of the operator assigning to a class its representative. The representation problem was also studied namely by Krupka’s school and important partial results were obtained (except for cited Krupka’s works see also e.g. [37], [12], [13], [14], [28], [38], [39], [29], [41].) A complete solution of the representation problem in r-th order field theory was inspired by works of Anderson [1], [2]. The complete solution of the representation problem for finite order variational sequence in field theories is presented in [15]. It was inspired by the Euler representation operator introduced by Anderson for the variational bicomplex: the Anderson’s concept of this operator was adapted to the finite order case and the required properties of the obtained representation were proved by taking into account specific features of the finite-order problem. It appeared that the general definition of Lepage forms arises immediately from the structure of the variational sequence representation. Note that the general concept of Lepage forms was introduced by an alternative way in [29] for the case of mechanics. Some aspects of the representation of the variational sequence in the context with the general concept of Lepage forms are discussed in Sec. 5.

2.

Lepage’s Idea – a Brief Historical Review

The famous mathematician Th´eophile Lepage gained the degree of Doctor of Science de ´ Cartan. He had 11 docl’Universit´e de Liege in 1924. He was probably the student of Elie toral students and 168 descendants in the period 1937–1973. He was the dean of the Facult´e des Sciences de l’Universit´e Libre de Bruxelles in 1953–1955. His bibliography amounts to 19 scientific papers published in 1929–1955, mostly devoted to geometry and calculus of variations. As a certain curiosity we can mention that Th´eophile Lepage had introduced a symplectic analog of Hodge theory before the Hodge theory itself was presented. His paper [36] notifying the concept of future Lepage forms was published in 1936. However, in 1933 some Lepage’s ideas was published in Comptes Rendus des S´eances de ´ l’ Acad´emie des Sciences, being presented on the seance 18-th December 1933 by Elie Cartan (see [35]). In the presentation called Sur certaines formes diff´erentielles ext´erieures et la variation des int´egrales doubles (corresponds to the field theory on a fibered manifold with the two-dimensional base) a predecessor of a contact 1-form is introduced and some factorization of 2-forms with respect to contact forms is made. Let us now present some results from the Lepage’s key paper [36] important for the concept of Lepage forms and discuss them from the point of the contemporary theory. In [36] a double variational integral

Lepage Forms in Variational Theories

7

defined by the formula I(zi ) =

Z Z

∂zn ∂z1 ∂zn ∂z1 ,..., ; ,..., f x, y; z1 , . . . , zn ; ∂x ∂x ∂y ∂y

dx dy

(2.1)

is studied. There x and y are independent variables, zi (x, y) are declared as unknown functions. It is evident that this corresponds to a first order variational problem on a fibered manifold (Y, π, X), where dim X = 2, dim Y = n + 2, dim J 1 Y = 2 + n + 2n = 3n + 2. A 2-form is defined (in the Lepage’s terminology called the symbolic quadratic form) ω = f (x, y; z1 , . . . , zn ; p1 , . . . , pn ; q1 , . . . , qn ) dx dy,

(2.2)

where function f is differentiable. (The exterior product “∧” is not explicitly written but it is evidently taken into account.) This form corresponds to a Lagrangian defined on J 1 Y . Thus, the integral (2.1) is a variational functional of the type Z

J 1 γ ∗ ω,

Ω

γ : (x, y) → (z1 γ(x, y), . . . , zn γ(x, y))

and our aim is to find its stationary paths γ. The forms ωi = dzi − pi dx − qi dy,

1 ≤ i ≤ n,

(2.3)

are introduced as Pfaffian forms, being the predecessor of today’s contact forms on J 1 Y . Moreover, congruences (i.e. equivalence classes) are introduced as follows: Denote θi = Xi dx + Yi dy + Aij ωj ,

Aij + Aji = 0,

1 ≤ i, j ≤ n,

Xi , Yi , Aij being differentiable functions of (x, y, zj , pj , qj ). The relation Ω = f dx dy + θi ωi ,

Ω ≡ ω (mod ω1 , . . . , ωn )

(2.4)

defines the equivalence relation of forms giving the same variational integral (2.1). Then there is requirement to calculate all forms Ω defining congruences dΩ ≡ 0

(mod ω1 , . . . , ωn ).

(2.5)

The result is as follows: Ω = f dx dy +

∂f ∂f dx − dy ∂qi ∂pi

ωi + Aij ωi ωj .

(2.6)

It can be easily verified that from the point of view of the contemporary theory the form Θω given by first two summands of (2.6), i.e. Θω = f dx dy +

∂f ∂f dx − dy ∂qi ∂pi

ωi ,

is exactly the principal component of the Lepage equivalent of the Lagrangian ω (compare with the relation (3.5) for n = 2, s = 1), while the form Aij ωi ω2 is 2-contact in the

8

Jana Musilov´a and Michal Lenc

present terminology. (Note that the aim of the paper is to study so called geodesic vector fields relative to the form Ω. These fields given by functions v : (x, y, z1 , . . . , zn ) −→ (pi (x, y, z1 , . . . , zn ), qi (x, y, z1 , . . . , zn )) are defined by the condition d[Ω] = 0, where [Ω] = Ω ◦ v and [ωi ] = ωi ◦ v.) Finally let us mention an important contribution of Paul Dedecker to the concept of Lepage forms. It is presented in the paper [6]. The notation used therein is close to that used at present. The variational integral in mechanics is introduced by I1 =

Z c

L(t, q i , q˙i ) dt,

1 ≤ i ≤ n,

(2.7)

where c is “an arc in the space W with local coordinates (t, q i , q˙i ) satisfying the equations ω i = 0”. In fact, this means that c can be interpreted as a first prolongation J 1 γ of a section γ of a fibered manifold with one-dimensional base. The so called Pfaffian form is defined by ∂L ω = L dt + i ω i , ω i = dq i − q˙i dt. (2.8) ∂ q˙ We can see that from the point of view of the today’s theory the form (2.8) is exactly the Lepage equivalent of the first order Lagrangian in mechanics. The semibasic forms are then introduced as θ = L dt mod ω i and it is stated and proved that the the relation (2.8) gives the unique semibasic form such that dθ ≡ 0 mod ω i . The procedure is then generalized for the field theory and a version of the first variational formula is derived.

3.

Lepage Forms on Fibered Manifolds

The concept of a Lepage form and the Lepage equivalent of a Lagrangian on fibered manifolds and their jet prolongations was introduced by Krupka in one of his basic works [16] in the connection of properties of the variational integral as well as the so called first variational formula. This formula is closely related to the fibered structure of the underlying manifold. The definition and properties of both a Lepage form and a Lepage equivalent of a Lagrangian were then studied by Krupka in [22]. In this section we follow the development and improvement of the concept of Lepage forms culminating in Krupka’s latest work on this topic [26]. First let us briefly expose basic geometrical structures for the variational integral, variational principle and the first variation formula. We use the standard definitions and standard notations given e.g. in [26]. Moreover, we often present definitions of geometrical objects by their chart expressions, which is more appropriate for their practical use.

Lepage Forms in Variational Theories

3.1.

9

Lagrange Structures, Variational Functionals

Let (Y, π, X), in shortened notation Y or π, be a fibered manifold with the n-dimensional base X, (m + n)-dimensional total space Y , m-dimensional fibers π −1 ({x}), x ∈ X and the projection π : Y → X (surjective submersion). Note that for n = 1 or n > 1 this structure represents the geometrical background for mechanics or field theories, respectively. Denote by (J r Y, πr , X), r ≥ 0, the r-th jet prolongation of (Y, π, X), where we put J 0 Y = Y , π0 = π. Canonical projections are denoted by πs,r : J s Y → J r Y for s > r ≥ 0. Let (V, ψ) be a fibered chart on Y , V being an open subset of Y , ψ = (xi , y σ ), 1 ≤ i ≤ n, 1 ≤ σ ≤ m. The pair (U, ϕ), U = π(V ), ϕ = (xi ), is the associated chart on X. A smooth mapping γ : U → Y such that π ◦ γ = Id U is called a section of π defined on U . Denote by ΓU a set of all such sections. The pair (Vr , ψr ) where Vr = πr−1 (V ), ψr = (xi , y σ , yiσ1 , . . . , yiσ1 ...ir ),

yiσ1 ...ik =

∂ k yσ γ , ∂xi1 . . . ∂xik

1 ≤ k ≤ r, 1 ≤ i1 , . . . , ik ≤ n, is the associated fibered chart on (J r Y, πr , X). A vector field field ξ on Y is called π-projectable, if there exists a vector field ξ0 on X such that T π · ξ = ξ0 ◦ π. It is called π-vertical, if ξ0 = 0. A π-projectable vector field is expressed as ∂ ∂ ξ = ξ i (xj ) i + Ξσ (xj , y ν ) σ . ∂x ∂y For π-vertical vector field ξ we have ξ i = 0, 1 ≤ i ≤ n. Definitions of πs -projectable or πs,r -projectable vector fields, and definitions of πs -vertical or πs,r -vertical vector fields on J s Y , s > r ≥ 0, are quite analogous. A vector field on J s Y is of the form ξ = ξi

s X ∂ ∂ σ ∂ + Ξ + Ξσi1 ...ik σ , i σ ∂x ∂y ∂yi1 ...ik k=1

where the components of ξ are functions on J s Y , in general. Let ξ be a π-projectable vector field in Y . Then its s-jet prolongation is the vector field on J s Y given by the chart expression J s ξ = ξ i (xj )

s X ∂ ∂ ∂ σ j ν + Ξ y ) + Ξσi1 ...ik (xj , y ν , . . . , yjν1 ...jk ) σ (x , , i σ ∂x ∂y ∂yi1 ...ik k=1

Ξσi1 ...ik = where dj f =

dΞσi1 ...ik−1 dxik

s X df ∂f = + dxj ∂xj k=0

− yiσ1 ...ik−1 j

X

1≤i1 ≤...≤ik

dξ j , dxik

∂f yiσ1 ...ik j σ ∂y i1 ...ik ≤n

is the total derivative of a function f on J s Y with respect to xj . A differential form ω on J s Y is called πr -horizontal if iξ ω = 0 for every πs -vertical vector field ξ. A form ω on J s Y is called πs,r -horizontal if iξ ω = 0 for every πs,r -vertical

10

Jana Musilov´a and Michal Lenc

vector field ξ. A form ω on J s Y is called contact if J s γ ∗ ω = 0 for every section γ of π. Differential forms ω σ = dy σ − yiσ dxi , ωiσ1 = dyiσ1 − yiσ1 j dxj ,

ωiσ1 ...is−1 = dyiσ1 ...is−1 − yiσ1 ...is−1 j dxj , dyiσ1 ...is ,

(3.1)

where 1 ≤ σ ≤ m, 1 ≤ i1 , . . . , is ≤ n, form the basis of 1-forms called the basis adapted to the contact structure. Note that these forms are contact with the exception of dyiσ1 ...is . ∗ A form ω on J s Y is called k-contact if every summand of the chart expression of πs+1,s ω contains the exterior product of exactly k 1-contact factors of the type (3.1). Let ω be a differential q-form on J s Y . Then there exists a unique decomposition ∗ πs+1,s ω = hω + pω =

q X

pq ω,

p0 ω = hω,

(3.2)

k=0

where h ω is the horizontal form called the horizontal or 0-contact component of ω and p ω is the contact form called the contact component of ω. The contact component p ω is uniquely decomposed into its k-contact components, 1 ≤ k ≤ q. Let (V, ψ) be a fibered chart on Y . We denote by Ωs0 V the ring of functions on Vs , and by Ωsq V a Ωs0 V -module of q-forms on Vs . Note that the sets Ωsq,X V of πs -horizontal q-forms and Ωsq,Y V of πs,0 -horizontal q-forms on Vs are submodules of Ωsq V . Now, let us introduce the concept of variational functional. Let W ⊂ Y be an open set. −1 A horizontal n-form Λ defined on Wr = πr,0 (W ) is called r-th order Lagrangian. It holds Λ = L(xi , y σ , yiσ1 , . . . , yiσ1 ...ir ) ω0 ,

ω0 = dx1 ∧ . . . ∧ dxn .

The pair (π, Λ) is so called Lagrange structure. Let U = π(W ). Let Ω ⊂ U be a compact n-dimensional submanifold of X with the boundary ∂Ω. Denote by ΓΩ,W a set of all sections γ ∈ ΓU such that supp γ = Ω and γ(Ω) ⊂ W . The mapping ΛΩ (γ) : ΓΩ,W ∋ γ −→ ΛΩ (γ) =

Z

J r γ∗Λ

(3.3)

Ω

is called the variational functional or variational integral over Ω. In physics it is often called the action function. For a π-projectable vector field ξ on Y the expression (∂J r ξ Λ)Ω (γ) =

Z

J r γ ∗ ∂J r ξ Λ

(3.4)

Ω

represents the variation of the variational functional, called its variational derivative or the first variation induces by the vector field ξ. A section γ ∈ ΓΩ,W is called the extremal of the variational functional (3.3) if it is its stationary point, i.e. the variational derivative of ΛΩ vanishes for γ. The aim of the calculus of variations is to find equations for extremals (in physics equations of motion of physical systems – particles and fields) and in general to study properties of variational functionals.

Lepage Forms in Variational Theories

3.2.

11

Lepage Equivalents of Lagrangians

The definition of the Lepage equivalent of a Lagrangian as one of concepts of coherent theory on fibered manifolds was introduced by Krupka in [16]. It was based on primary ideas of Lepage and Dedecker and on some generalizations of Sniatycki [40]. In the above cited Krupka’s paper horizontal and contact differential forms on jet prolongations of fibered manifolds are introduced practically in the today’s form, being called horizontal and pseudovertical forms. Especially, for a differential p-form ̺ on J r Y a form h(̺) on J r+1 Y is defined (the notation of the time inclusive) by the relation hh(̺)(jxr+1 γ), ξ1 × · · · × ξp i

= h̺(jxr γ), Tx j r γ · T πr+1 · ξ1 × · · · × Tx j r γ · T πr+1 · ξp i, for r-jets of all sections γ of π at x, and every collection of vector arguments ξ1 , . . . , ξp on J r+1 Y . By this relation the form h(̺) is defined. A form with the property that the above value is zero whenever one of vector arguments is vertical, i.e. T πr+1 · ξi = 0, is ∗ called horizontal. Inspired by the relation j r+1 γ ∗ p(̺) = 0, where p(̺) = πr+1,r ̺ − h(̺) r r ∗ a pseudovertical p-form η on J Y , is defined by the property j γ η = 0 for any section γ of π. Then, the properties of the mapping h : ̺ → h(̺) for n-forms are studied. The definition of the horizontalization mapping is extended for (n + 1)-form by the relation ∗ h(i(ξ) πr+1,r ̺) = i(ξ) ̺

for every πr+1 -vertical vector field ξ on J r+1 Y , proving the existence and uniqueness of ̺. The symbol i(ξ) η denotes the contraction of the form η by the vector field ξ. Finally, a ˜ : ̺ → h(̺) ˜ mapping h = ̺ is defined. A Lepage n-form is then defined as follows: Let ̺ be an n-form on J r Y . ̺ is called ˜ Lepagian form, if the (n+1)-form h(d̺) is horizontal with respect to πr+1,0 . The properties of Lepagian forms (in today’s terminology Lepage forms) are studied and explicit formula ˜ for h(d̺) is derived for the second order r = 2 and for a n-form ̺ on J 1 Y . The relation of ˜ to Lepage congruencies [36] is mentioned and the connections of Lepage mappings h and h forms with equations of extremals of Lagrangians is studied, giving rise to Euler-Lagrange mapping by the formula ΩnX (J 1 Y ) ∋ λ → E(λ) ∈ Ω1Y (J 2 Y ),

˜ h(d̺) = E(h(̺)) ∧ ω,

where ω = F ω0 is the volume element on X and F > 0 is a function. Note that here E ∈ Ω1Y (J 2 Y ) is a uniquely defined pseudovertical 1-form. The Lepage equivalent of the Lagrangian λ is in fact introduced by an example, but it is not explicitly defined including the terminology. (Note that ΩqX (J s Y ) or Ω1Y (J s Y ) in the previous definition of the mapping λ → E(λ) means Ωsq,X V or Ωs1,Y V in the today’s notation.) Let us now present the contemporary definition of Lepage equivalents of Lagrangians, as presented in [26]. The concept of Lepage forms is introduced by the following lemma. Lemma. Let W ⊂ Y be an open set, and let ̺ ∈ Ωsn W . The following conditions are equivalent: (1) The (n + 1)-form p1 d̺ is πs+1,0 horizontal.

12

Jana Musilov´a and Michal Lenc −1 (2) For every πs,0 -vertical vector field ξ on W s = πs,0 (W ) it holds h iξ d̺ = 0, ∗ (3) The form πs+1,s ̺ has the chart expression ∗ πs+1,s ̺ = f0 ω0 +

s X

k=0

fσi,j1 ...jk ωjσ1 ...jk ∧ ωi + η,

ωi = i

∂ ∂xi

ω0 ,

∂f0 j ,j ...j − dp fσp,j1 ...jk − fσk 1 k−1 = 0, sym(j1 . . . jk ), 1 ≤ k ≤ s, ∂yjσ1 ...jk ∂f0 − fσjs+1 ,j1 ...js = 0, sym(j1 . . . js+1 ). ∂yjσ1 ...js+1 Any form satisfying the Lemma is called the Lepage form. The structure of Lepage forms is given by the following explicit formulas. Theorem. Let W ⊂ Y be an open set. A form ̺ ∈ Ωsn W is a Lepage form if and only ∗ if for every fibered chart (V, ψ), ψ = (xi , y σ ), on Y such that V ⊂ W , πs+1,s ̺ has an expression ∗ πs+1,s ̺ = Θ + dµ + η,

Θ = f0 ω0 +

s X

k=0

s−k X

∂f0

ℓ

(−1) dp1 . . . dpℓ

ℓ=0

where

∂yjσ1 ...jk p1 ...pℓ i

!

ωjσ1 ...jk ∧ ωi ,

where f0 is a function, defined in coordinates by h̺ = f0 ω0 , µ is a contact (n − 1)-form, and η is a form of contactness ≥ 2. The Lepage equivalent of a Lagrangian Λ = L ω0 is defined as such a Lepage form ̺ for which h̺ = Λ up to a possible projection. This leads to the conclusion that the function f0 in the preceding relation is equal to L, i.e. ΘΛ = L ω0 +

s X

k=0

s−k X

∂L

ℓ

(−1) dp1 . . . dpℓ

ℓ=0

∂yjσ1 ...jk p1 ...pℓ i

!

ωjσ1 ...jk ∧ ωi .

(3.5)

For mechanics (n=1) and Λ of order r we have r−1 X

dk ΘΛ = L dτ + (−1) dτ k k=0 k

∂L σ ∂qk+1

!

ωkσ .

(3.6)

The Euler-Lagrange mapping assigning to every Lagrangian Λ its Euler-Lagrange form defined by the standard way as a dynamical form σ

EΛ = Eσ (L) ω ∧ ω0 =

r X

∂L (−1) dj1 . . . djℓ σ ∂yj1 ...jℓ ℓ=0 ℓ

!

ω σ ∧ ω0

is closely related to the concept of Lepage equivalents. For every Lepage equivalent of a Lagrangian Λ it holds p1 d̺ = EΛ , and, moreover, the following theorem can be proved:

Lepage Forms in Variational Theories

13

Theorem. Let Λ ∈ Ωrn,X W be a Lagrangian and let ̺ ∈ Ωsn W be its Lepage equivalent. Then the section γ ∈ ΓΩ,W is the extremal of the corresponding variational functional (3.3) if and only if for every π-vertical vector field ξ defined on W such that the support supp (ξ ◦ γ) ⊂ Ω it holds J s γ ∗ iJ s ξ d̺ = 0, or equivalently, the Euler-Lagrange form of Λ vanishes along J 2 γ. Another important result lies in the first variational formula resulting directly from properties of the Lepage equivalent ̺ of the Lagrangian Λ, the expression for the variational derivative (3.4) of the variational functional and the formula for Lie derivative ∂Ξ η = iΞ dη + diΞ η. Let γ be a section of π, ξ a π-projectable vector field on Y . Then it holds J r γ ∗ ∂J r ξ Λ = J s γ ∗ iJ s ξ d̺ + dJ s γ ∗ iJ s ξ ̺, (3.7) and using the Stokes theorem Z

J r γ ∗ ∂J r ξ Λ =

Ω

Z

J s γ ∗ iJ s ξ d̺ +

Ω

Z

J s γ ∗ iJ s ξ ̺,

(3.8)

∂Ω

The relations (3.7) and (3.8) give the infinitesimal first variational formula and the integral first variational formula, respectively. There are very important consequences resulting from the first variational formula for applications in variational physical theories. π-projectable vector field ξ on V is called the generator of invariance transformations of Lagrangian Λ ∈ Ωrn,X V , if ∂J r ξ Λ = 0. (3.9) For such a vector field the left hand side of the integral first variational formula (3.8) vanishes. Let γ be an extremal of the functional ΛΩ , Ω ⊂ π(V ), i.e. J s γ ∗ iJ s ξ d̺ = 0. Then the first variational formula gives Z

J s γ ∗ iJ s ξ ̺ = 0.

(3.10)

∂Ω

On the other hand, the integral (3.10) represents the flow of the quantity corresponding to the integrand through the boundary ∂Ω of Ω. It holds ∗ J s γ ∗ iJ s ξ ̺ = J s+1 γ ∗ πs+1,s iJ s ξ ̺ = h iJ s ξ ̺.

So, we can interpret the expression Ψ = h iJ s ξ ̺ ◦ J s γ

(3.11)

as a quantity obeying a conservation law along the extremal (so called elementary flow). At the end of this section let us note that a concept of Lepage forms introduced primarily for n-forms was later extended by Krupkov´a (see e.g. [33]) for (n + 1)-forms. However, in this paper we don’t discuss the properties of Lepage (n + 1)-forms separately. On the other hand in Sec. 5 we mention the quite general definition of a Lepage form as resulting from the structure of the variational sequence.

14

4.

Jana Musilov´a and Michal Lenc

Application – Lepage Equivalents of Lagrangians for String Theories

Relativistic particles and strings are physical systems typically described by variational theories. So, their equations of motion are derived from conditions for stationary sections of a variational functional. These two situations are appropriate for application of the concept of Lepage equivalents of Lagrangians, representing the case of first order mechanics and first order field theory, respectively. There are of course physical arguments for the choice of variational integral (for strings see [5]). Apart from these arguments, the obtained results enables us to formulate these variational functionals as variational integrals for a certain Lagrange structure of an appropriately chosen fibered manifold. In this section we give corresponding Lagrange structures (π, Λ) and calculate Lepage equivalents ̺. Moreover, using the infinitesimal first variational formula we can study invariance transformations of Lagrangians. Let the underlying fibered manifold be again (Y, π, X). We denote fibered charts on Y , associated charts on X and associated fibered chart on J 1 Y by the standard way (V, ψ), (U, ϕ), (V1 , ψ1 ). We will use the standard relativistic notation. Recall that considering the infinitesimal first variational formula in the form (3.7) we can see that quantities obeying conservation laws along the prolongations of extremals of the Lagrangian are (n − 1)-forms given by (3.11), where ξ is a invariance transformation of Λ. From the physical point of view it is suitable to take into account the “full” Lepage equivalent, i.e. including a trivial Lagrangian of the same order as the Lagrangian given by the physical situation itself. The last assumption is induced by the standard form of boundary conditions of the variational problem. As a good example we can mention the Lagrangian L = (1/2)mx˙ 2 +U (x) of a non-relativistic particle with one degree of freedom (the motion along x-axis). For non-constant potential energy U (x) only the time translation is the invariance transformation of the Lagrangian. However, adding an appropriately chosen trivial Lagrangian we can show that for linear potential energy U (x) = Kx + C (a homogeneous field) the translation along x is invariance transformation as well.

4.1.

Relativistic Particle

In the case of relativistic mechanics the base X of the underlying manifold is one-dimensional (n=1), the unique coordinate being a “non-physical parameter”. Every fiber is a time-space with metric g = gαβ dxα ⊗ dxβ . Thus the fiber dimension is m = 4. Thus, we have dim Y = 5. So on (Y, π, X) we have coordinates ϕ = (τ ),

ψ = (τ, xα ),

ψ1 = (τ, xα , x˙ α ),

0 ≤ α ≤ 3.

The Lagrange structure (π, Λ) with “classical” Lagrangian is Λ1 = L1 dτ,

q

L1 = −mc gαβ x˙ α x˙ β .

The Lagrange structure appropriate for both the zero and nonzero mass is given by the Lagrangian ! 1 gαβ x˙ α x˙ β 2 2 + λ(τ )m c , Λ2 = L2 dτ, L2 = − 2 λ(τ )

Lepage Forms in Variational Theories

15

where λ(τ ) is a function of the parameter. Both Lagrangians are reparametrizable. The unique Lepage equivalent of this Lagrangian given by the general expression (3.6) is of the form ∂L ∂L ∂L ̺ = L dτ + ν ω ν = L − ν x˙ ν dτ + ν dxν . ∂ x˙ ∂ x˙ ∂ x˙ By the simple calculation we finally obtain gµν x˙ ν

̺1 = −mc q

gαβ x˙ α x˙ β

1 ̺2 = 2

dxµ ,

(4.1)

!

gαβ x˙ α x˙ β gαβ x˙ β α − λ(τ )m2 c2 dτ − dx . λ(τ ) λ(τ )

(4.2)

Now, let us study conservation laws resulting from invariance transformations of “full” Lagrangians (Λ + Λ0 ) for both cases, Λ1 and Λ2 , Λ0 being a trivial Lagrangian. For our situation it holds r = s = 1, and thus J 1ξ = ξ

dξ µ dξ ξ˜µ = − x˙ µ . dτ dτ

∂ ∂ ∂ + ξ µ µ + ξ˜µ µ , ∂τ ∂x ∂ x˙

Every trivial Lagrangian of the first order has the form h df for a function f = f (τ, xµ ). Using Lepage equivalents (4.1) and (4.2) we obtain for “full” Lagrangians the resulting general expressions for quantities given by (3.11) Ψ1 = and 1 Ψ2 = 2

4.2.





x˙ ν

∂f ∂f mc gµν ξ − q − µ  ξµ, ∂τ ∂x α β gαβ x˙ x˙

gαβ x˙ α x˙ β ∂f − λ(τ )m2 c2 + λ(τ ) ∂τ

!

ξ−

gαβ x˙ β ∂f − α λ(τ ) ∂x

(4.3)

!

ξα.

(4.4)

Classical Bosonic String

Strings can be studied within the field theory. The base of the underlying manifold is in case of strings two dimensional, i.e. n = 2. The base coordinates denoted by τ 0 and τ 1 have the meaning of the time evolution parameter and the position on the string, respectively. Every fiber is again time-space, i.e. m = 4. Metric is denoted again by g = gαβ dxα ⊗ dxβ . For such a structure Lagrangian has a form Λ = L(ξ i , xµ , xµi ) dτ 0 ∧ dτ 1 . The principal component of Lepage equivalent is, following (3.5) ΘΛ = L dτ 0 ∧ dτ 1 +

ΘΛ = L −

∂L µ ω ∧ i ∂ (dτ 0 ∧ dτ 1 ), ∂xµi ∂ξi

∂L µ ∂L µ ∂L ∂L x − x dτ 0 ∧ dτ 1 + µ dxµ ∧ dτ 1 − µ dxµ ∧ dτ 0 . (4.5) ∂xµ0 0 ∂xµ1 1 ∂x0 ∂x1

16

Jana Musilov´a and Michal Lenc For an example we consider a bosonic string. Let us denote h = hij dτ i ⊗ dτ j = gαβ xαi xβj dτ i ⊗ dτ j . The Lagrangian of a string is defined as √ Λ = L dτ 0 ∧ dτ 1 = −T −det h dτ 0 ∧ dτ 1 , L = −T

q

(gαβ xα0 xβ1 )2 − (gαλ xα0 xλ0 )(gβν xβ1 xν1 )

= −T

q

(gαβ gλν − gαλ gβν ) xα0 xλ0 xβ1 xν1 .

(4.6)

Note that in the original derivation of the variational functional of the string the NambuGoto idea formulated e.g. in [8] is based on the variational integral (action) S = −T

Z

dΣ,

Σ

where dΣ is understood as the “elementary surface area”. This means that dS is, correctly speaking, the (unique) volume element of the relativistic time-space. Thus, in this approach h can be understood as the induced metric for a mapping γ : (τ 0 , τ 1 ) −→ (x0 (τ 0 , τ 1 ), x1 (τ 0 , τ 1 ), x2 (τ 0 , τ 1 ), x3 (τ 0 , τ 1 )), h = γ ∗ g = (gαβ ◦ γ)

∂xα ∂xβ i dτ ⊗ dτ j . ∂ξ i ∂ξ j

However, in the approach based on Lagrange structures formalism on fibered manifolds and their jet prolongations, the object h is defined on first jet prolongation of (Y, π, X). Using (4.5) for Lagrangian (4.6), we obtain after some simple calculations the principal component of Lepage equivalent √ ΘΛ = T −det h dτ 0 ∧ dτ 1 T +√ (gαµ gβν − gαβ gµν ) dxµ ∧ (xα0 xβ0 xν1 dτ 0 − xα1 xβ1 xν0 dτ 1 ). (4.7) −det h

Now, let us study Lepage equivalents of trivial Lagrangians for the considered physical situation. Taking into account that every trivial Lagrangian of first order has the form h dχ (see [27]), where χ is a (n − 1)-form on J r−1 Y , we can obtain the corresponding principal component of Lepage equivalents. In coordinates

Λ0 = h dχ = di χj + xµj di Xµ dτ i ∧ dτ j , where

(4.8)

χ = χi dτ i + Xµ dxµ . Thus, we have Λ0 = L0 ω0 , where ∂χ1 ∂Xν ∂χ1 ∂χ0 + xν0 − − 0 1 ∂τ ∂τ ∂xν ∂τ 1 ∂χ0 ∂Xν ∂Xν ν µ −xν1 − + (x x − xν0 xµ1 ) . ∂xν ∂τ 0 ∂xµ 1 0 L0 =

(4.9)

Lepage Forms in Variational Theories

17

The corresponding Lepage equivalent has the form ̺0 = Θ0 + η + dµ, where η is a 2-contact 2-form and µ is a contact 1-form. For conservation laws only the principal component Θ0 is important, because h dµ has the character of a special type of trivial Lagrangian and J s γ ∗ η = 0. Θ0 =

∂χ1 − ∂τ 0 ∂χ1 + ∂xµ ∂χ0 + ∂xµ

∂χ0 ∂Xµ µ ν + (x x − xµ1 xν0 ) dτ 0 ∧ dτ 1 ∂τ 1 ∂xν 0 1 ∂Xµ ∂Xµ ν ∂Xν − + x1 − dxµ ∧ dτ 1 ∂τ 1 ∂xµ ∂xν ∂Xµ ∂Xµ ν ∂Xν − + x0 − dxµ ∧ dτ 0 . ∂τ 0 ∂xµ ∂xν

(4.10)

Let

∂ ∂ ∂ + ξ 1 1 + Ξµ µ 0 ∂τ ∂τ ∂x be an invariance transformation of the Lagrangian (Λ + Λ0 ), where Λ0 is a trivial Lagrangian. We obtain the corresponding elementary flow ξ = ξ0

Ψ = ΨΛ + Ψ0 = h iJ 1 ξ ΘΛ + h iJ 1 ξ Θ0 ,

(4.11)

where √ T ΨΛ = −T −det h ξ 1 + √ (gαµ gβν − gαβ gµν ) · −det h

h

io

· xα0 xβ0 xν1 Ξµ − xµ0 xα0 xβ0 xν1 ξ 0 − xα1 xβ1 xν0 ξ 1 dτ 0 √ T + T −det h ξ 0 + √ (gαµ gβν − gαβ gµν ) · −det h h

· −xα1 xβ1 xν0 Ξµ − xµ1 xα0 xβ0 xν1 ξ 0 − xα1 xβ1 xν0 ξ 1 and

i

dτ 1

(4.12)

(4.13)

Ψ0 = A00 ξ 0 + A01 ξ 1 + B0µ Ξµ dτ 0 + A10 ξ 0 + A11 ξ 1 + B1µ Ξµ dτ 1 , where

A00 = A01 = B0µ = A10 = A11 = B1µ =

∂Xµ ∂χ0 ∂Xµ ∂Xν − µ + xµ0 xν0 − , ∂τ 0 ∂x ∂xν ∂xµ ∂χ0 ∂χ1 ∂Xµ ∂χ1 − + xµ0 − µ , ∂τ 1 ∂τ 0 ∂τ 1 ∂x ∂χ0 ∂Xµ ∂Xµ ν ∂Xν − + x0 − , ∂xµ ∂τ 0 ∂xµ ∂xν ∂χ1 ∂χ0 ∂Xµ ∂χ0 − + xµ1 − µ , 0 1 ∂τ ∂τ ∂τ 0 ∂x ∂χ1 ∂Xν µ ∂Xµ µ ν ∂Xµ x1 − µ + x1 x1 − , ∂τ 1 ∂x ∂xν ∂xµ ∂χ1 ∂Xµ ∂Xν ∂Xµ − + xν1 − . µ 1 µ ∂x ∂τ ∂x ∂xν

xµ0

18

Jana Musilov´a and Michal Lenc

The invariance transformations ξ of the Lagrangian (Λ + Λ0 ) given by (4.6) and (4.9) can be obtained from (3.9) in principle. Conservation laws can be then studied using (4.11).

4.3.

Polyakov Action for D-Dimensional Space

The Polyakov action is a variational integral arising from generalization of string Lagrangian to a D-dimensional space by the following way: Λ = L(τ i , xµ , xµi ) dτ 0 ∧ dτ 1 ,

0 ≤ i, j ≤ 1,

0 ≤ µ ≤ D − 1,

T p −det f f ij gµν xµi xνj . (4.14) 2 Here f is the metric on the base, without a relation to g. Using again relation (4.5) we can calculate the principal component of Lepage equivalent of the Lagrangian (4.14) in the following form. L=−

T p −det f f ij gαβ xαi xβj dτ 0 ∧ dτ 1 2p + T −det f gαβ xαi dxβ ∧ (f 1i dτ 0 − f 0i dτ 1 ).

ΘΛ =

(4.15)

Note that the quantities Ψ for conservation laws can be calculated for the Polyakov action by the quite analogous procedure as for a bosonic string.

5.

Lepage Forms in Variational Sequences

In previous sections a concept of a Lepage equivalent of a Lagrangian was discussed and its usefulness for application to concrete situations occurring in variational theories in physics was demonstrated. In this section we present the generalized concept of a Lepage form in the context of the variational sequence representation (for the complete solution of the representation problem in field theories see [14] and [15], for mechanics see [29]). We follow the general ideas and structures employed by Krupka in [23]. The main results are formulated in the language of the theory of sheaves. (We use sheaves of germs of C ∞ differential q-forms.)

5.1.

Variational Sequence

Let Ωrq , q ≥ 0, be the direct image of the sheaf of smooth q-forms over J r Y by the jet projection πr,0 (functions are labeled as 0-forms). Denote by Ωrq,c

=

(

ker p0 ker pq−n

for 1 ≤ q ≤ n, for n + 1 ≤ q ≤ dim J r Y,

(5.1)

where p0 and pq−n are morphisms of sheaves induced by mappings p0 and pq−n , assigning to a form ̺ its horizontal and (q − n)-contact component, respectively. The dimension of n+r r J Y is N = m n + n. We further denote Θrq = Ωrq,c + d Ωrq−1,c ,

(5.2)

Lepage Forms in Variational Theories

19

d Ωrq−1,c is the image sheaf of Ωrq−1,c by d. Let us consider the sequence {0} → Θr1 → · · · → Θrn → Θrn+1 → Θrn+2 → · · · → ΘrP → {0},

(5.3)

with P = m n+r−1 + 2n − 1 being the maximal nontrivial degree of forms of the type n (5.2). The arrows (except the first one) are given by exterior derivatives d. It can proved that the sequence (5.2) is an exact subsequence of the de Rham sequence of forms

{0} → Ωr1 → · · · → Ωrn → Ωrn+1 → Ωrn+2 → · · · → ΩrN → {0}. The resulting quotient sequence is called the variational sequence of order r. The situation is shown in the following figure. The variational sequence is, of course, also exact. We denote the quotient mappings as follows Eqr : Ωrq /Θrq ∋ [̺] → Eqr ([̺]) = [d ̺] ∈ Ωrq+1 /Θrq+1 .

(5.4)

r The mappings Enr and En+1 correspond to the Euler-Lagrange mapping and HelmholtzSonin mapping of calculus of variations, respectively. The global properties of the variational sequence are described using the abstract de Rham theorem and can be found in [23]:

(1) Each sheaf Ωrq is soft. (2) The variational sequence (concisely {0} → RY → V) is an acyclic resolution of the constant sheaf RY over Y . (3) For every q ≥ 0 it holds H q (Γ(RY , V)) = H q (Y, R), where Γ(Y, V) : {0} → Γ(Y, RY ) → Γ(Y, Ωr0 ) → · · · → Γ(Y, ΩrN ) →{0} is the cochain complex of global sections and H q (Γ(RY , V)) denotes its q-th cohomology group.

5.2.

Representation of the Variational Sequence

In this subsection we present the representation of the r-th order variational sequence by forms. All details and proofs of theorems can be found in [15]. By the representation we mean a mapping assigning to every class [̺] of the variational sequence of the r-th order an appropriately chosen representative with specific properties mentioned in Sec. 1: Rlinearity, coordinate invariance, exactness of there sequence of representatives, projection property of the operator assigning to a class its representative. Such a mapping is given by Euler operator. Its construction was inspired by the idea of Anderson in [1], [2] for the variational bicomplex. For finite order variational sequences the construction uses the concept of the Lie derivative of differential forms with respect of vector fields along maps, and is based on the generalized integration by parts procedure, both introduced in [15]. Let us give here main results only. Integration by parts is described by the following lemma. (Note that J = (j1 . . . jℓ ), 0 ≤ ℓ ≤ r, is a multiindex and |J| = ℓ is its length.)

{0}

{0}

{0}

R

Ωr0

d0

{0}

{0}

Θr1

d1

Θr2

d2

Ωr1

d1

Ωr2

d2

···

···

{0}

dq−1

Θrq

dq

dq−1

Ωrq

dq

···

···

dP −1

ΘrP

dP

dP −1

ΩrP

dP

E0

{0}

E1

Ωr2 /Θr2

{0}

E2

···

Ωr /Θrq Eq−1 q Eq

{0}

···

Ωr /ΘrP EP −1 P

{0}

ΩrP +1

dP +1

EP +1

EP

Ωr1 /Θr1

{0}

···

dN −1 EN −1

ΩrN

dN EN

{0}

Lepage Forms in Variational Theories

21

Lemma. Let (V, ψ), ψ = (xi , y σ ) be a fibered chart on Y and ̺ ∈ Ωrn+k V a form. Let pk ̺ be expressed as r X

pk ̺ =

|J|=0

ωJσ ∧ ησJ .

Then there exists the decomposition pk ̺ = I(̺) + pk d pk R(̺) where σ

σ

I(̺) = ω ∧ ζσ = ω ∧

r X

(5.5)

(−1)|J| dJ ησJ .

(5.6)

|J|=0

and R(̺) is a local k-contact (n + k − 1)-form. Moreover, we can write the operator I in the form I:

Ωrn+k W

∋ ̺ → I(̺) =

1 σ kω

∧

r X

|J|

(−1)

|J|=0

dJ

∂ y pk ̺ ∂yJσ

!

∈ Ω2r+1 n+k W,

where we used the obvious identity pk ̺ =

1 k

r X

ωJσ

|J|=0

∧

∂ y pk ̺ ∂yJσ

!

We can see that I is clearly R-linear. (Note that the operator “y” stands for the contraction, i.e. ξ y η = iξ η means the contraction of a form η by the vector field ξ.) It can be proved that the decomposition 5.5 is valid globally (the proof see in [15]) and following the terminology in [2] we call the operator I interior Euler operator. The following Theorem summarizes all properties of the interior Euler operator important for the representation of the variational sequence. Theorem. Let W ⊂ Y be an open set and let ̺ ∈ Ωrn+k W , 1 ≤ k ≤ N − n, be a form. Then (1) (π2r+1,r )∗ ̺ − I(̺) ∈ Θ2r+1 n+k W,

(2) (3) (4)

I(pk d pk R(̺)) = 0, 2

∗

I (̺) = (π4r+3,2r+1 ) I(̺), ker I =

Θrn+k W.

(5.7) (5.8) (5.9) (5.10)

Now let us define a family of mappings Rq : Rq : Ωrq W/Θrq W ∋ [̺] → Rq ([̺]) ∈ Ωsq W, as follows Rq ([̺]) =

   p0 ̺

I(̺)

  ̺

for 0 ≤ q ≤ n, s = r + 1 for n + 1 ≤ q ≤ P, s = 2r + 1 for P + 1 ≤ q ≤ N, s = r

(5.11)

22

Jana Musilov´a and Michal Lenc

Evidently, in all three cases the mapping Rq assigns to every class [̺] of q-forms on J r Y a correctly defined representative Rq ([̺]). In this sense, every class [̺] is represented by a form Rq ([̺]). Consider a sequence {0} → R0 (Ωr0 W ) → R1 (Ωr1 W/Θr1 W ) → · · · → RP (ΩrP W/ΘrP W ) →

→ RP +1 (ΩrP +1 W ) → · · · → RN (ΩrN W ), (5.12)

where arrows denote the mappings E q : Rq (Ωrq W/Θrq W ) → Rq+1 (Ωrq+1 W/Θrq+1 W )

(5.13)

induced by the commutativity of diagrams Ωrq W/Θrq W

Eqr

(5.14)

Rq+1

Rq

Rq (Ωrq W/Θrq W )

Ωrq+1 W/Θrq+1 W

Eq

Rq+1 (Ωrq+1 W/Θrq+1 W )

i.e. E q ◦Rq ([̺]) = Rq+1 ◦ Eqr ([̺]) = Rq+1 ([d ̺]). The sequence (5.12) is called the representation sequence of the r-th order variational sequence. The representation sequence is exact, as it is proved in [15]. There are practical applications od the representation of the variational sequence. Especially, the representatives Rq ([̺]) of classes of q-forms [̺] for q = n − 1, n, n + 1, n + 2 are the well-known forms on calculus of variations, i.e. r-th order currents, r-th order Lagrangians, r-th order dynamical forms and r-th order Helmholtz-Sonin forms. So, the exactness of the representation sequence (5.12) enables us e.g. to solve the problem of trivial Lagrangians and the inverse problem of calculus of variations.

5.3.

Generalized Lepage Forms

One may generalize the concept of Lepage forms to all (n + k)-forms, k ≥ 0. Such a generalization is directly induced by the structure of the representation of the variational sequence. An (n + k)-form ̺ on J r Y is called Lepage if pk+1 d ̺ = I(d ̺),

i.e.

pk+1 d ̺ = I(pk+1 d ̺).

(5.15)

The equivalent definition resulting from the decomposition (5.5) can be given in the form pk+1 d pk+1 R(d ̺) = pk+1 d R(pk+1 d ̺) = 0. A Lepage form ̺ is also called the Lepage equivalent of the class [̺]. The definition of Lepage forms as well as the general definition of Lepage equivalents was introduced also in [29] for mechanics. There the properties of Lepage forms for mechanics are studied in details and useful examples of chart expressions are presented. Note that in the terminology of [29] the Lepage equivalent of the class [̺] is called the Lepage equivalent of the source form. This is a form η ∈ [̺] such that pk η = I(η), i.e. the representative Rn+k [̺] (see (5.11)) in our notation.

Lepage Forms in Variational Theories

6.

23

Conclusion

The aim of the paper was to show the brief history of the concept of Lepage forms from the initiating Lepage’s idea up to the most general definition resulting by the quite natural way from geometrical structures built on underlying fibered manifolds and their jet prolongations. The development of this concept supports the unquestionable meaning of Lepage forms for the calculus of variations. Moreover, numerous examples following from variational physical theories, as e.g. classical Newtonian and relativistic mechanics, theory of fields, string theories, show the usefulness of Lepage forms, especially Lepage equivalents of Lagrangians, for many physical applications, as e.g. studies of conservation laws, etc. It is evident that the relation (5.15) represents the most general possibility to define the concept of a Lepage form. It ought to be mentioned that this definition inspired primarily by Lepage is undoubtedly based on Krupka’s complex theoretical considerations on Lagrange structures and Lepage equivalents of Lagrangians and on his idea and studies of the finiteorder variational sequence.

Acknowledgements The research is supported by the grant 201/06/0922 of the Czech Grant Agency and by the grant MSM 0021622409 of the Ministry of Education, Youth and Sports of the Czech Republic.

References [1] I. M. Anderson, The Variational Bicomplex (book preprint, technical report of the Utah State University, 1989; www.math.usu.edu/-fg mp). [2] I. M. Anderson, Introduction to the variational bicomplex, Contemporory Math. 132 (1992) 51–73. [3] I. M. Anderson and T. Duchamp, On the existence of global variational principles, Amer. J. Math. 102 (1980) 781–867. [4] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khorkova, I. S. Krasilschik, A. V. Samokhin, Yu. N. Thorkov, A. M. Verbovetsky and A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics ((I. S. Krasilschik and A. M. Vinogradov, Eds.) Amer. Math. Soc., 1999). [5] L. Brink, P. di Vecchia and P. Howe, A locally supersymmetric and reparametrization invariant action for the spinning string, Phys. Lett. 65B(5) (1976) 471–474. [6] P. Dedecker, A property of differential forms in calculus of variations, Pac. J. Math. 7 (1957) 1545–1549. [7] P. Dedecker and W. M. Tulczyjew, Spectral sequences and the inverse problem of calculus of variations, In: Differential Geometric Methods in Mathematical Physics (Proc. Internat. Coll., Aix-en-Provence, France, 1979, Lecture Notes in Math. 836, Springer, Berlin, 1980) 498–503.

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[8] T. Goto, Prog. Theor. Phys. 46 (1971) 1560; Derived also in unpublished lectures of Y. Nambu presented in Copenhagen, 1970. [9] D. R. Grigore, The variational sequence on finite order bundle extensions and the Lagrange formalism, Diff. Geom. Appl. 10 (1999) 43–77. [10] D. R. Grigore, Variationally trivial Lagrangians and locally variational equations of arbitrary order, Diff. Geom. Appl. 10 (1999) 79–105. [11] I. S. Krasilschik, V. V. Lychagin and A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations (Advanced Studies in Contemporary Mathematics 1, Gordon & Breach, New York, 1986). [12] M. Krbek, J. Musilov´a and J. Kaˇsparov´a, The variational sequence: Local and global properties, In: Proceedings of the Seminar on Differential Geometry (Math. Publications 2, Silesian University, Opava, 2000) 15–38. [13] M. Krbek, J. Musilov´a and J. Kaˇsparov´a, The representation of the variational sequence in field theories, In: Steps in Differential Geometry (Proc. Colloq. Differential Geometry, Debrecen, Hungary, 2000, (L. Kozma, P. T. Nagy and L. Tam´assy, Eds.) Debrecen, 2001) 147–160. [14] M. Krbek and J. Musilov´a, Representation of the variational sequence, Rep. Math. Phys. 51 (2003) 251–258. [15] M. Krbek and J. Musilov´a, Representation of the Variational Sequence by Differential Forms, Acta Appl. Math. 88 (2005) 177–199. [16] D. Krupka, Some geometric aspects of variational problems in fibred manifolds, Folia Fac. Sci. Nat. Univ. Purk. Brunensis XIV (10) (1973) pp. 65. [17] D. Krupka, On the structure of Euler-Lagrange mapping, Arch. Math. (Brno) 10 (1974) 353–358. [18] D. Krupka, A map associated to the Lepagean forms of the calculus of variations in fibered manifolds, Czechoslovak Math. J. 27 (1977) 114–118. [19] D. Krupka, Natural Lagrangian Structures, In: Differential geometry (Semester in Differential Geometry, Banach Center, Warsaw, 1979, Banach Center Publications 12, 1984) 185–210. [20] D. Krupka, On the local structure of Euler-Lagrange mapping of the calculus of variations, In: Differential Geometry and its Applications (Proc. Conf., Nov´e Mˇesto na Moravˇe, Czechoslovakia, 1980, (O. Kowalski, Ed.) Charles University, Praha, 1981) 181–188. [21] D. Krupka, Regular lagrangians and lepagean forms, In: Differential Geometry and ˇ its Applications (Proc. Conf., Brno, Czechoslovakia, 1986, (D. Krupka and A. Svec, Eds.) D. Reidel, Dordrecht, 1986) 85–101.

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[22] D. Krupka, Geometry of Lagrangean structures 2, 3, In: 14th Winter School on Abstract Analysis (Proc. Conf., Srn´ı, Czechoslovakia, 1986, Suppl. Rend. del Circ. Mat. di Palermo 14, 1987) 178–224. [23] D. Krupka, Variational sequences on finite order jet spaces, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czechoslovakia, 1989, (J. Janyˇska and D. Krupka, Eds.) World Scientific, Singapore, 1990) 236–254. [24] D. Krupka, The Geometry of Lagrange Structures, Preprint Series in Global Analysis GA 7 (1997), Mathematical Institute, Silesian University, Opava 1987, pp. 82. [25] D. Krupka, Variational sequences and bicomplexes, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czech Republic, 1998, (O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Masaryk University, Brno, 1998) 525–532. [26] D. Krupka, Global variational theory in fibred spaces, In: Handbook of Global Analysis ((D. Krupka and D. Saunders, Eds.) Elsevier, Amsterdam, 2008) 773–836. [27] D. Krupka and J. Musilov´a, Trivial Lagrangians in Field Theory, Diff. Geom. and Appl. 9 (1998) 393–505. [28] D. Krupka and J. Musilov´a, Recent results in variational sequence theory, In: Steps in Differential Geometry (Proc. Colloq. Differential Geometry, Debrecen, Hungary, 2000, (L. Kozma, P. T. Nagy and L. Tam´assy, Eds.) Debrecen, 2001) 161–186. ˇ enkov´a, Variational sequences and Lepage forms, In: Differential [29] D. Krupka and J. Sedˇ Geometry and its Applications (Proc. Conf., Prague, Czech Republic, 2004, (J. Bureˇs, O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Charles University, Prague, 2005) 617– 627. [30] O. Krupkov´a, Lepagean 2-forms in higher order Hamiltonian mechanics. I. Regularity, Arch. Math. (Brno) 22 (1986) 97–120. [31] O. Krupkov´a, Lepagean 2-forms in higher order Hamiltonian mechanics. II. Inverse problem, Arch. Math. (Brno) 23 (1987) 155–170. [32] O. Krupkov´a, Variational analysis on fibered manifolds over one-dimensional bases, PhD Thesis, Silesin University, Opava and Charles University, Prague, 1992, pp. 67. [33] O. Krupkov´a, The Geometry of Ordinary Variational Equations (Lecture Notes in Mathematics, 1678, Springer, Berlin, 1997). [34] O. Krupkov´a, Mechanical systems with non-holonomic constraints, J. Math. Phys. 38 (1997) 5098–5126. [35] Th. H. J. Lepage, Calcul des Variations. Sur certaines formes diff´erentielles ext´erieures et la variation des int´egrales doubles, Comptes rendus des s´eances de l’Acad´emie des sciences, Tome cent-quatre-vingt-dix-septieme (1933) 1718–1719. [36] Th. H. J. Lepage, Sur les champs g´eodesiques du Calcul des Variations I, II, Bull. Acad. Roy. Belg., Cl. Sci. 22 (1936) 716–729, 1036–1046.

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[37] J. Musilov´a, Variational sequence in higher order mechanics, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czech Republic, 1995, (J. Janyˇska, I. Kol´aˇr and J. Slov´ak, Eds.) Masaryk Unversity, Brno, 1996) 611–624. [38] J. Musilov´a and M. Krbek, A note to the representation of the variational sequence in mechanics, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czech Republic, 1998, (I. Kol´aˇr, D. Krupka, O. Kowalski and J. Slov´ak, Eds.) Brno, 1999) 511–623. ˇ enkov´a, On the invariant variational sequences in mechanics, In: Rend. Cont. [39] J. Sedˇ Mat. Palermo (Proc. 22-nd Winter School Geometry nad Physics, Srn´ı, Czech Republic, 2002) 185–190. [40] J. Sniatycki, On the geometric structure of classical field theory in Lagrangian formulation, Proc. Camb. Phil. Soc. 68 (1970) 475–484. ˇ anek, A representation of the variational sequence in higher order mechanics, [41] J. Stef´ In: Differential Geometry and its Applications (Proc. Conf., Brno, Czech Republic, 1995, (J. Janyˇska, I. Kol´aˇr and J. Slov´ak, Eds.) Masaryk University, Brno, 1996) 469– 478. [42] F. Takens, A global version of the inverse problem of calculus of variations, J. Diff. Geom. 14 (1979) 543–562. [43] W. M. Tulczyjew, The Euler-Lagrange resolution, In: Differential Geometric Methods in Mathematical Physics (Proc. Internat. Coll., Aix-en-Provence, France, 1979, Lecture Notes in Math. 836, Springer, Berlin, 1980) 22–48. [44] A. M. Vinogradov, On the algebro-geometric foundations of Lagrangian field theory, Soviet. Math. Dokl. 18 (1977) 1200–1204. [45] A. M. Vinogradov, A spectral sequence associated with a non-linear differential equation, and algebro-geometric foundations of Lagrangian field theory with constraints, Soviet. Math. Dokl. 18 (1978) 144–148. [46] A. M. Vinogradov, The C-spectral sequence, Lagrangian formalism and conservation laws I and II, J. Math. Anal. Appl. 100 (1) (1984) 1–129. [47] R. Vitolo, Finite order Lagrangian bicomplexes, Math. Proc. Cambridge Philos. Soc. 125 (1999) 321–333. [48] R. Vitolo, On different geometric formulations of Lagrangian formalism, Diff. Geom. Appl. 10 (3) (1999) 225–255. [49] R. Vitolo, Finite order formulation of Vinogradov’s C-spectral sequence, Acta Appl. Math. 70 (1-2) (2002) 133–154. [50] R. Vitolo, Variational sequences, In: Handbook of Global Analysis ((D. Krupka and D. Saunders, Eds.) Elsevier, Amsterdam, 2008) 1115–1163.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 27-55

Chapter 2

L EPAGE F ORMS IN THE C ALCULUS OF VARIATIONS∗ Olga Krupkov´a† Department of Algebra and Geometry, Faculty of Science, Palack´y University, Tomkova 40, 779 00 Olomouc, Czech Republic and Department of Mathematics, La Trobe University, Bundoora, Victoria 3086, Australia

Abstract Lepage forms represent a fundamental concept in the global calculus variations. Inspired by the work of Lepage, they were introduced by Demeter Krupka in 1973 in his seminal paper Some Geometric Aspects of Variational Problems in Fibered Manifolds, published in the journal of Brno University Folia Fac. Sci. Nat. Univ. Purk. Brunensis (see also arXiv:math-ph/0110005 for an electronic transcription). In this paper we recall Lepage forms, their recent generalisations, the role they play in the current global variational analysis, and their applications in the geometric theory of differential equations, the theory of variational sequences, and higher-order mechanics and field theory.

2000 Mathematics Subject Classification. 58-02, 70-02, 58E15, 70H50. Key words and phrases. Lagrangian, Lepage form, the first variation formula, Euler– Lagrange form, Noether theorem, conservation law, Hamilton equations, regular Lagrangian, the variational sequence, null-Lagrangian, the inverse problem of the calculus of variations, Helmholtz form.

1.

Introduction

This paper is a survey of the theory of Lepage forms, its current development and numerous applications in the calculus of variations on fibred manifolds. ∗ †

Paper dedicated to Demeter Krupka on the occasion of his 65th birthday E-mail address: [email protected]

28

Olga Krupkov´a

The concept of Lepage form (more precisely, Lepage n-form where n is the dimension of the base manifold) was introduced in the early 1970’s by D. Krupka, inspired by a classical paper of Th. Lepage [61]. D. Krupka was among the first who realised that fibred and prolongation structures introduced by Ehresmann [14] represent an appropriate background for the investigation of first and higher order variational functionals on manifolds (cf. [7, 8, 17, 22, 36, 77]). In his seminal paper [36] published 35 years ago he introduced the fundamental concept of the Lepage equivalent of a Lagrangian, and since that time he has systematically developed techniques and concepts of a global higher order variational theory on fibred spaces where Lepage forms and variational sequences play a central role. I am grateful to Demeter that from the beginning of my university education I had the possibility of joining his seminar, to learn his theory of Lepage forms in jet bundles, and to participate in the fascinating process of discovering geometric structures in the calculus of variations.

Figure 1. Demeter Krupka in the early 1980’s lecturing on Lepage forms. This paper, reviewing the current status of the theory of Lepage forms, starts by briefly recalling the calculus of vector fields and horizontal and contact forms on jet bundles over fibred manifolds [36, 37, 41], including the fundamental Krupka’s decomposition formula of a differential form into contact components, concepts and techniques that are essential for understanding intrinsic constructions and properties of geometric objects appearing in “fibred” variational analysis. Based on papers [36, 37, 38, 39, 41, 42, 43, 45, 49], the next section is devoted to Lepage n-forms and their role in the theory of global variational functionals. Lepage n-forms are fundamental for obtaining the intrinsic first variation formula, that gives various geometric formulations of the Euler–Lagrange equations on one hand, and, combined with the concepts of invariant variational functionals, of Noether theorems on the other hand. Moreover, it is demonstrated that this setting provides a new look at Hamiltonian theory and regularity of variational problems, that has not yet been completely explored. The reader should be aware that the family of Lepage equivalents of a Lagrangian contains the well-known Cartan form [5, 78], and the Poincar´e–Cartan form [11, 17, 22, 36] and its higher order generalisations [11, 15, 18, 31, 33, 41], that are very popular in vari-

Lepage Forms in the Calculus of Variations

29

ous geometric formulations of the calculus of variations (see e.g. [22, 12, 16, 18, 33, 34], or the quite recent multisymplectic formalism [13, 21, 24, 62] and others). It should be stressed, however, that “Cartan-like” forms, though often preferred, are not the only possibility; in some situations they are too restrictive (exterior differential systems for variational equations, Hamilton theory, regularity), or even apparently unavailable (higher order field theories, homogeneous Lagrangians, non-fibred structures), so that some other Lepage equivalents of a Lagrangian need to be considered (we refer to [4, 39, 68] for constructions and [8, 36, 41, 54, 58, 59, 65, 71] for discussions and applications). The last section of the present paper is concerned with generalisations of Lepage forms. Roughly speaking, a motivation for introducing Lepage forms of higher degrees is to “transfer” variational operators to the exterior derivative [8, 40]; as a first step, to introduce Lepage equivalents of Euler–Lagrange forms [50] in such a way that the following diagram should be commutative: Lagrangian   y

Euler–Lagrange mapping

−−−−−−−−−−−−−→ Euler–Lagrange form d

  y

Lepage n-form −−−−−−−−−−−−−−−→ Lepage (n + 1)-form

The idea comes from the solution of the inverse problem of the calculus of variations and of the problem of the structure of null Lagrangians: with help of Lepage n-forms these problems are transferred to the application of the Poincar´e Lemma [40, 41]. Lepage equivalents of Euler–Lagrange forms introduced in [50] and further studied in [25, 28, 29, 53, 54], play a fundamental role in investigating variational equations by means of exterior differential systems methods, enabling the study of the geometry of these equations and structure of their solutions, symmetries and conservation laws, and exact methods of integration (see e.g. [25, 51, 52, 55, 56, 57]). In particular, a new approach to Hamiltonian theory and regular variational problems is achieved, associating Hamilton equations directly to the Euler–Lagrange form (not to individual Lagrangians) [50, 51, 53, 54]. Finally, it is shown how the concept of Lepage form can be generalised to arbitrary k-forms (k ≥ n), and how it fits in with Krupka’s variational sequence [35, 44, 48, 57].

2.

Horizontal and Contact Forms on Fibred Manifolds

Throughout the paper, all manifolds and mappings are smooth and summation over repeated indices is understood. We refer to [66] for the theory of jet bundles and to [36, 41] or [45, 51] for details on the calculus of vector fields and differential forms on fibred manifolds. We denote by π : Y → X a fibred manifold, dim X = n, dim Y = m + n, πr : J r Y → X the r-jet prolongation of π, (V, ψ), ψ = (xi , y σ ), 1 ≤ i ≤ n, 1 ≤ σ ≤ m, a fibred chart on Y , and (Vr , ψr ), ψr = (xi , y σ , yjσ1 , . . . , yjσ1 ...jr ) the associated chart on J r Y . If dim X = 1, we write (t, q σ ), and (t, q σ , q1σ , . . . qrσ ), respectively, (t, q σ , q˙σ , q¨σ ) if r = 2. Next we put ω0 = dx1 ∧ dx2 ∧ ... ∧ dxn ,

ωi1 ...ik = i∂/∂xik ωi1 ...ik−1 , 1 ≤ k ≤ n,

for a local volume on X and its contractions, and ∂ ∂ ∂ σ ∂ di = + yiσ σ + yji + · · · + yjσ1 ...jr i σ σ i ∂x ∂y ∂yj ∂yj1 ...jr

(2.1)

(2.2)

30

Olga Krupkov´a

if dim X > 1, respectively, d ∂ ∂ ∂ ∂ σ = + q1σ σ + q2σ σ + · · · + qr+1 dt ∂t ∂q ∂q1 ∂qrσ

(2.3)

if dim X = 1, the operators of total derivative. A mapping γ : U → Y defined on an open subset U ⊂ X is called a section of the fibred manifold π, if π ◦ γ = idU . A section δ of πr is called holonomic if δ = J r γ, i.e., the r-jet prolongation of a section γ of π. A vector field ξ on J r Y , r ≥ 0, is called πr -projectable if there exists a vector field ξ0 on X such that T πr .ξ = ξ0 ◦ π, and πr -vertical if ξ projects onto a zero vector field on X, i.e., T πr .ξ = 0. Quite similarly one can define a πr,s -projectable or a πr,s -vertical vector field on J r Y , where r > s. Local flows of projectable vector fields transfer sections into sections; consequently, π-projectable vector fields on Y can be naturally prolonged to vector fields on J r Y as follows. Given a π-projectable vector field ξ on Y with projection ξ0 , denote by {φu } and {φ0u } the corresponding local one-parameter groups. For every u, the mapping φu is an isomorphism of the fibred manifold π, i.e. π ◦ φu = φ0u ◦ π. Then for every section γ, the composition γ¯ = φu ◦ γ ◦ φ−1 0u is a section of π and we can define the r-jet prolongation of φu by J r φu (Jxr γ) = Jφr0u (x) (φu γφ−1 0u ).

(2.4)

Then J r φ is a local flow corresponding to a vector field on J r π, denoted by J r ξ and called the r-jet-prolongation of ξ. The vector field J r ξ is both πr -projectable and πr,s -projectable for 0 ≤ s < r, and its πr -projection, (resp. πr,s -projection) is ξ0 (resp. ξ, resp. J s ξ, 1 ≤ s ≤ r − 1). A q-form η on J r Y is called contact if J r γ ∗ η = 0 for every section γ of π. The ideal of contact forms on J r Y is generated by local contact 1-forms ωjσ1 ...jk = dyjσ1 ...jk − yjσ1 ...jk l dxl ,

0 ≤ k ≤ r − 1,

(2.5)

and 2-forms dωjσ1 ...jr−1 . If dim X = 1 we write σ ωkσ = dqkσ − qk+1 dt,

0 ≤ k ≤ r − 1.

(2.6)

It is important to notice that contact 1-forms (2.5) give rise to a local basis of 1-forms on J r Y adapted to the contact structure, namely, (dxi , ω σ , . . . , ωjσ1 ...jr−1 , dyjσ1 ...jr ). In coordinate expressions and computations in fibred coordinates this basis is much more convenient than the canonical basis (dxi , dy σ , . . . , dyjσ1 ...jr ). A q-form η on J r Y is called πr -horizontal (or 0-contact) if the contraction iξ η vanishes for every πr -vertical vector field on J r Y . Similarly, η is called πr,s -horizontal if it vanishes whenever at least one of its arguments is a πr,s -vertical vector field. A contact q-form η on J r Y is called i-contact, 1 ≤ i ≤ q, if for every πr -vertical vector field ξ on J r Y , iξ η is (i − 1)-contact. By Krupka’s theorem, the module of πr+1,r -horizontal q-forms on J r+1 Y is the direct sum of the submodules of i-contact forms, 0 ≤ i ≤ q, meaning that every πr+1,r -horizontal q-form on J r+1 Y has a unique decomposition into a sum of i-contact forms, 0 ≤ i ≤ q.

Lepage Forms in the Calculus of Variations

31

If we denote by h and pi , 1 ≤ i ≤ q, projectors onto the submodules of the horizontal and i-contact forms, where 1 ≤ i ≤ q, we obtain the above theorem in the following form: Every q-form η on J r Y admits a unique decomposition ∗ πr+1,r η = hη + p1 η + · · · + pq η

(2.7)

into a sum of i-contact forms, 0 ≤ i ≤ q, on J r+1 Y . The operators h and pi are called the horizontalisation and i-contactisation, respectively. We also say that hη is the horizontal part of η and pi η is the i-contact part of η. Note that every q-form where q > n is contact; it is called strongly contact if pq−n η = 0.

3. 3.1.

Lepage Forms and the First Variation Variations

Let us start with Lagrangians and variations, following [36]. Consider a fibred manifold π : Y → X, where X is an orientable manifold of dimension n. A Lagrangian of order r for π is defined to be a horizontal n-form λ on J r Y . In a fibred chart (V, ψ), ψ = (xi , y σ ), on Y , λ = L ω0 , −1 where L is a real function on Vr = πr,0 (V ), i.e., L = L(xi , y σ , yjσ1 , . . . , yjσ1 j2 ...jr ). Denote by Ω a piece of X, i.e. a compact n-dimensional submanifold of X with boundary, and ΓΩ (π) the set of smooth sections of π over Ω. The real valued function

ΓΩ (π) ∋ γ →

Z

Ω

J r γ∗λ ∈ R

(3.1)

is called the variational function or action function of the Lagrangian λ over Ω. Since J r γ ∗ η = 0 for any contact n-form η on J r Y , the action function remains the same if one considers the form λ + η instead of λ. In other words, for any n form ρ such that hρ = λ, Z

J r γ∗λ =

Ω

Z

J r γ ∗ ρ.

(3.2)

Ω

Let γ : U → Y be a section defined on an open set U ⊂ X. Let ξ be a π-projectable vector field defined in a neighborhood of γ(U ), and ξ0 the projection of ξ. If αt is the local 1-parameter group of ξ and α0t is the projection of αt (i.e., α0t is the local 1-parameter −1 group of ξ0 ), then γt = αt ◦ γ ◦ α0t is a 1-parameter family of sections of π, depending smoothly on the parameter t. The family {γt } is called the variation or deformation of the section γ induced by the vector field ξ. Often the vector field ξ itself is called a variation of γ. Now, taking the action function (3.1) for a fixed section γ ∈ ΓΩ (π), and considering a variation of γ, we get the following real-valued function, defined in a neighborhood I of the point 0 in R: Z I∋t→

α0t (Ω)

J r γt∗ λ ∈ R.

32

Olga Krupkov´a

Differentiating this function at the point t = 0 we arrive at nd Z

dt

α0t (Ω)

J r γt∗ λ

o

0

=

Z

Ω

J r γ ∗ LJ r ξ λ.

(3.3)

The real number (3.3) is called variation of the action function at the point γ, induced by the vector field ξ. The function ΓΩ (π) ∋ γ →

Z

Ω

J r γ ∗ LJ r ξ λ ∈ R

(3.4)

is then called the first variation of the action function (3.1) by the vector field ξ. Let us recall the key idea that lead Krupka to the discovery of Lepage forms: The problem is to find an intrinsic form of the first variation formula, i.e. a decomposition of the integral in (3.4) to a sum of a term characterising extremals (Euler–Lagrange term) and a boundary term (connected with conservation laws). To this end, one would like (and should) utilize the well-known Cartan formula for the decomposition of the Lie derivative. Unfortunately, the result cannot be obtained by a direct application of this formula to LJ r ξ λ, since iJ r ξ dλ depends not only upon the variation ξ itself, but also upon prolongations (“derivations”) of ξ. However, taking into account property (3.2) of the action function, one can study the following question: Given a Lagrangian λ, is it possible to add to λ a contact form ν in such a way that after the decomposition of LJ r ξ (λ + ν) in (3.4) the first term would depend upon the variation vector field ξ only? Then, for such forms, Cartan’s formula would provide the desired intrinsic first variation formula, and, consequently, a geometric form of the Euler–Lagrange equations and of conservation laws.

3.2.

Lepage n-Forms and the First Variation Formula

Recall that n = dim X, and let s ≥ 0. Formalization of the above idea leads to the following definition [36]: Definition 3.1. A n-form ρ on J s Y is called a Lepage n-form of order s if for every πs+1,0 vertical vector field ξ on J s Y hiξ dρ = 0. (3.5) Note that if ρ is a Lepage n-form on J s Y then hρ is a Lagrangian on J s+1 Y . We write hρ = λ = Lω0 .

(3.6)

From the definition we obtain three Krupka theorems on the structure of Lepage n-forms [36, 41]: Theorem 3.2. Let ρ be an n-form on J s Y . The following conditions are equivalent: (1) ρ is a Lepage form of order s. (2) The (n + 1)-form p1 dρ is πs+1,0 -horizontal. (3) dρ satisfies ∗ πs+1,s dρ = E + F,

where E is 1-contact and πs+1,0 -horizontal, and F is at least 2-contact.

(3.7)

Lepage Forms in the Calculus of Variations

33

∗ (4) In every fibred chart, πs+1,s ρ is expressed as follows: ∗ πs+1,s ρ = L ω0 +

s X

k=0

fσi, j1 j2 ...jk ωjσ1 j2 ...jk ∧ ωi + η,

(3.8)

where η is an arbitrary at least 2-contact n-form, and fσ(js+1 , j1 j2 ...js ) = (j , j1 j2 ...jk−1 )

fσ k

=

∂L , ∂yjσ1 j2 ...js+1 ∂L ∂yjσ1 j2 ...jk

(3.9)

− di fσi, j1 j2 ...jk ,

1 ≤ k ≤ s,

where (. . . ) means symmetrisation in the indicated indices, i.e. (i, j1 j2 . . . jp ) = sym(i, j1 j2 . . . jp ). Theorem 3.3. A form ρ on J s Y is Lepage n-form if and only if in every fibred chart ∗ πs+1,s ρ = Θλ + dν + µ,

(3.10)

where Θλ = L ω0 +

s s−k X X

(−1)l dp1 dp2 . . . dpl

k=0 l=0

∂L ∂yjσ1 j2 ...jk p1 p2 ...pl i

ωjσ1 j2 ...jk ∧ ωi ,

(3.11)

ν is a contact (n − 1)-form, and µ is an at least 2-contact form. Note that ∗ πs+1,s ρ = Θλ + dν + µ = Θλ + p1 dν + η,

(3.12)

and that in this decomposition, the forms Θλ + p1 dν and η are global (being the at most ∗ 1-contact and the at least 2-contact part of πs+1,s ρ, respectively). We stress that, on the contrary, the decomposition Θλ + p1 dν need not be invariant under the change of fibred coordinates, meaning that in general Θλ is not a global differential form. Theorem 3.4. If ρ is a Lepage n-form then the 1-contact part p1 dρ of dρ depends only on hρ = λ = Lω0 and reads p1 dρ = Eλ =

∂L

∂y σ

−

s+1 X

(−1)l+1 dp1 dp2 . . . dpl

l=1

∂L ∂ypσ1 p2 ...pl

ω σ ∧ ω0 .

(3.13)

Definition 3.5 ([36]). Given a Lagrangian λ, we say that a form ρ is a Lepage equivalent of λ if ρ is a Lepage n-form and hρ = λ. The (n + 1)-form Eλ = p1 dρ is then called the Euler–Lagrange form of λ, and its fibred-chart components are called Euler–Lagrange expressions. Theorem 3.6 (Krupka [36, 41], Marvan [63]). Every Lagrangian has a Lepage equivalent. Taking into account the above theorems we obtain:

34

Olga Krupkov´a

Corollary 3.7. (1) Every Lagrangian λ of order r has a Lepage equivalent of order 2r − 1, that is generally nonunique. Every Lepage equivalent of λ takes the form as described by theorem 3.3 or 3.2 (4). (2) Every Lagrangian λ has a Lepage equivalent that is at most 2-contact. If λ is of order r that Lepage equivalent is of order ≤ 2r − 1, is generally nonunique, and reads Θ = Θλ + p1 dν,

(3.14)

where Θλ is given by formula (3.11) and ν is a contact (n − 1)-form. In particular, every Lagrangian of order r has (global) 2-contact Lepage equivalents that are π2r−1,r−1 horizontal; they take the form (3.14) where ν is an arbitrary 1-contact π2r−1,r−1 -horizontal (n − 1)-form. (3) To every Lagrangian λ there exists a uniquely determined Euler–Lagrange form Eλ . If λ is defined on J r Y then Eλ is generally on J 2r Y and is given by formula Eλ =

∂L

∂y σ

−

r X

(−1)l+1 dp1 dp2 . . . dpl

l=1

∂L ∂ypσ1 p2 ...pl

ω σ ∧ ω0 = p1 dρ,

(3.15)

where ρ is any Lepage equivalent of λ. It should be stressed that the splitting of Θ in (3.14) into the term Θλ that is determined by the Lagrangian, and an auxiliary term p1 dν generally is not invariant under changes of fibred coordinates. This means that the expression (3.11) may have only a local meaning, not providing a differential form on J 2r−1 Y . If Θλ happens to be a global differential form, we speak about the (higher-order) Poincar´e–Cartan form associated to the Lagrangian λ, or, about the Poincar´e–Cartan equivalent of λ. Equipped with Lepage n-forms, we are able to write down the intrinsic first variation formula. Theorem 3.8 (Krupka [36, 41]). Let λ be a Lagrangian of order r, ρ its Lepage equivalent, ξ a π-projectable vector field on Y . Then (i) the Lie derivative LJ r ξ λ is expressed by the formula LJ r ξ λ = hiJ 2r−1 ξ dρ + hdiJ 2r−1 ξ ρ,

(3.16)

J r γ ∗ LJ r ξ λ = J 2r−1 γ ∗ iJ 2r−1 ξ dρ + dJ 2r−1 γ ∗ iJ 2r−1 ξ ρ,

(3.17)

(ii) for any section γ of π,

(iii) for any piece Ω of X with boundary ∂Ω, and any section γ of π such that Ω ⊂ dom γ, Z

Ω

r ∗

J γ LJ r ξ λ =

Z

Ω

J

2r−1 ∗

γ iJ 2r−1 ξ dρ +

Z

∂Ω

J 2r−1 γ ∗ iJ 2r−1 ξ ρ,

(3.18)

and, in all the above formulas, the first term on the right-hand side depends on the vector field ξ only (not on its prolongations). Formula (3.16) or (3.17) is called the infinitesimal first variation formula, (3.18) is the integral first variation formula.

Lepage Forms in the Calculus of Variations

3.3.

35

Examples of Lepage Equivalents of Lagrangians

We have seen that to a Lagrangian one has in general many Lepage equivalents. However, a few particular cases are known, when either the Lepage equivalent is unique, or, the family of Lepage equivalents contains some interesting distinguished representatives. When discussing special cases, first of all, one has to mention classical and higher order mechanics. In the case dim X = 1 we denote local fibred coordinates on Y by (t, q σ ), and the associated higher order coordinates by (t, q σ , q1σ , . . . , qrσ ), or (t, q σ , q˙σ , q¨σ , . . . ). The σ dt, 0 ≤ j ≤ r. Taking into corresponding local contact 1-forms then read ωjσ = dqjσ − qj+1 account that now ρ is a 1-form, i.e., at most 1-contact, and ν is a contact function (i.e. = 0), we immediately obtain: Theorem 3.9. Let dim X = 1. To every Lagrangian λ on J r Y there exists a unique Lepage equivalent. In fibred coordinates where λ = Ldt, it reads r−1 ∂L d ∂L d2 ∂L r−1 d + − · · · + (−1) ωσ ∂q1σ dt ∂q2σ dt2 ∂q3σ dtr−1 ∂qrσ ∂L d ∂L σ ∂L σ − ωr−2 + σ ωr−1 . + ··· + σ σ ∂qr−1 dt ∂qr ∂qr

Θλ = L dt +

∂L

−

(3.19)

The above form Θλ is called (higher order) Cartan form. For a first order Lagrangian the formula is reduced to the well-known one, introduced to the calculus of variations and classical mechanics by Whittaker [78] and Cartan [5]: Θλ = Ldt +

∂L σ ∂L σ ∂L ω = L − q˙ dt + σ dq σ . ∂ q˙σ ∂ q˙σ ∂ q˙

(3.20)

Another important special case is dim X = n > 1 and r = 1, the so-called first order field theory. If λ is a first order Lagangian then theorem 3.3 gives all first-order Lepage equivalents of λ as follows: ρ = Θλ + p1 dν + η ∂L ik σ ij ijk σ = Lω0 + − d g k σ ω ∧ ωi − (gσ + dk gσ ) ∧ ωj ∧ ωi + η, ∂yiσ

(3.21)

where η is an arbitrary at least 2-contact n-form, and ν is an arbitrary 1-contact (n − 1)form, ν = 21 gσij ω σ ∧ ωij + 16 gσijk dω σ ∧ ωijk , where we assume the components be functions skewsymmetric in the upper indices. Obviously the part of ρ, uniquely determined by the Lagrangian, i.e. ∂L Θλ = Lω0 + σ ω σ ∧ ωi , (3.22) ∂yi is invariant with respect to changes of fibred coordinates; it is called the Poincar´e–Cartan form. We can see that the form Θλ is uniquely characterised and intrinsically defined as the Lepage equivalent of λ that is π1,0 -horizontal and at most 1-contact. It is known that behind the Poincar´e–Cartan form, the family (3.21) contains the following two global Lepage forms, completely determined by the Lagrangian:

36

Olga Krupkov´a – Carath´eodory form ∂L σ1 ∂L σn n ∧ · · ω · ∧ L dx + ω σ 1 Ln−1 ∂y1 ∂ynσn 1 ∂L ∂L σ1 ∂L ω ∧ ω σ 2 ∧ ωj 1 j 2 + · · · , = L ω0 + σ ω σ ∧ ωj + ∂yj 2L ∂yjσ11 ∂yjσ22

1

ρCλ =

L dx1 +

(3.23)

this form is invariant under changes of any (not only fibred) coordinates (cf. [65]), and – Krupka form [39] (later also found by Betounes [4]) ρK λ

= L ω0 +

n X

k=1

1 ∂kL ω σ1 ∧ · · · ∧ ω σk ∧ ωj1 ···jk . σ 1 (k!)2 ∂yj1 · · · ∂yjσkk

(3.24)

This form has an important property dρK λ = 0 if and only if the Euler-Lagrange form of λ vanishes identically. As pointed out in [36], an interesting situation arises when dim X = n > 1 and r = 2 (second order Lagrangians in field theory). First of all, in this case Θλ is no longer intrinsically characterised similarly as for r = 1, i.e. as the Lepage equivalent of λ that is π3,1 -horizontal and at most 1-contact: indeed, such form is no longer unique and is given by formula Θ = L ω0 +

∂L

∂yiσ

− dk

∂L

σ ∂yik

+ gσik

ω σ ∧ ωi +

∂L

σ ∂yij

− gσij ωjσ ∧ ωi ,

(3.25)

where gσjk are arbitrary functions skewsymmetric in the upper indices. However, one can check by a direct calculation that the part of Θ determined by the Lagrangian, i.e. Θλ = L ω0 +

∂L

∂yiσ

− dk

∂L σ ∂L ω ∧ ωi + σ ωjσ ∧ ωi , σ ∂yik ∂yij

(3.26)

is invariant under fibred coordinate transformations [41]. This means that for second order Lagrangians we still have a well-defined Poincar´e–Cartan form. Also the question on the existence of a higher order Carath´eodory form has an affirmative answer; the form is defined on J 3 Y by the following formula [68]: ρCλ =

1

∂L σ1 ∂L σ1 ω + σ σ 1 ωj 1 ∧ . . . Ln−1 ∂y1j11 ∂y1j 1 ∂L ∂L σn ∂L σn n ∧ L dx + − d j n σ n ω + σ n ωj n . ∂ynσn ∂ynjn ∂ynjn

L dx1 +

∂L

∂y1σ1

− d j1

(3.27)

Quite a different story concerns a possible Krupka form ρK λ for second order Lagrangians: yet, the question of whether such a (global) form may be found is still open, however, some arguments coming from the geometry of homogeneous Lagrangian systems support the suggestion that it does not exist [68]. For dim X = n > 1 and r > 2 (higher order field theory) the situation is even more complicated. Surprisingly, in the class Θ we have no global differential form determined by the Lagrangian, meaning that a “higher-order Poincar´e–Cartan form” Θλ does not exist;

Lepage Forms in the Calculus of Variations

37

so we have no distinguished Lepage equivalent of λ of “Cartan type”. The non-existence of a global higher-order Poincar´e–Cartan form initiated studies of geometric constructions providing a “globalization” of the expression Θλ (see e.g. ([6, 15, 18, 23, 31, 33]). From the point of view of the general theory of Lepage forms the problem of “globalization of Θλ ” is very simple: a desired form always exists and is obtained by adding to Θλ an appropriate term p1 dν where ν is 1-contact and horizontal with respect to the projection π2r−1,r−1 (cf. corollary 3.7).

3.4.

Extremals and Euler–Lagrange Equations

As a first application of Lepage forms one obtains different global expressions for the Euler– Lagrange equations [36, 42]. A section γ ∈ ΓΩ (π) is called a stable point of the action function of λ with respect to a variation ξ if Z Ω

J r γ ∗ LJ r ξ λ = 0.

(3.28)

γ is said to be an extremal of λ on Ω if (3.28) holds for all variations ξ of γ such that supp(ξ ◦ γ) ⊂ Ω; here supp(ξ ◦ γ) denotes the support of the vector field ξ ◦ γ along the section γ, defined as the closure of the set {x ∈ dom γ|(ξ ◦ γ)(x) 6= 0}. Finally, γ is called an extremal of the Lagrangian λ if it is an extremal of λ on every Ω in the domain of definition of γ. Necessary and sufficient conditions for a section be an extremal of a Lagrangian follow from the first variation formula: Theorem 3.10. Let λ be a Lagrangian on J r Y , ρ a Lepage equivalent of λ. Let γ be a section of π. The following conditions are equivalent: (1) γ is an extremal of λ. (2) For every vector field ξ on J 2r−1 Y J 2r−1 γ ∗ iξ dρ = 0 .

(3.29)

(3) For every π-projectable vector field ξ on Y J 2r−1 γ ∗ iJ 2r−1 ξ dρ = 0 .

(3.30)

(4) For every π-vertical vector field ξ on Y J 2r−1 γ ∗ iJ 2r−1 ξ dρ = 0 .

(3.31)

(5) J 2r−1 γ is an integral section of the exterior differential system generated by the system of n-forms iξ dρ, where ξ runs over all vertical vector fields on J 2r−1 Y .

(3.32)

38

Olga Krupkov´a

(6) The Euler-Lagrange form Eλ vanishes along J 2r γ, i.e., Eλ ◦ J 2r γ = 0 .

(3.33)

(7) In every fibred chart, γ satisfies the following system of differential equations r X

k=0

k

(−1) dj1 · · · djk

∂L ∂yjσ1 ···jk

!

◦ J 2r γ = 0,

1 ≤ σ ≤ m.

(3.34)

Any of the equivalent conditions (2)–(7) above may be called the Euler-Lagrange equations of the Lagrangian λ. Note that (4) means that although in place of variations one can take arbitrary vector fields, extremals are determined only by the vertical part of variations σ , . . . ). Thus among defined on Y (they do not depend upon components at ∂/∂yjσ , ∂/∂yjk equations (2)–(4), (4) represents the most simple and most frequently used form of intrinsic Euler–Lagrange equations. We stress that for dim X = 1 (3.32) is a system of 1-forms, annihilating a distribution on J 2r−1 Y , called the Euler–Lagrange distribution [46]. Notice that for any Lepage equivalent ρ of λ, dρ = dΘ + F , where F is at least 2-contact. Hence iξ F is contact and vanishes along J 2r−1 γ. This means that for the Euler– Lagrange equations (3.31) the at most 2-contact part of a Lepage equivalent is essential, so that we can simply consider them in the form J 2r−1 γ ∗ iJ 2r−1 ξ dΘ = 0,

∀ξ.

(3.35)

Finally, recall that Lagrangians λ1 , λ2 are said to be equivalent if Eλ1 = Eλ2 (possibly up to a projection). A Lagrangian λ is called null if Eλ = 0. Obviously, λ1 and λ2 are equivalent if and only if λ1 − λ2 is null.

3.5.

Hamilton Equations, Regular Lagrangians

We have seen that Euler–Lagrange equations of a Lagrangian λ have a geometric meaning as equations for holonomic integral sections of the exterior differential system (3.32), where ρ is a Lepage equivalent of λ. However, one can study all integral section of the corresponding EDS, and explore that for different Lepage equivalents of λ different exterior differential systems are obtained. Let λ be a Lagrangian on J r Y , ρ its Lepage equivalent. We call the EDS defined by (3.32) Hamiltonian system of ρ. Equations for integral sections of a Hamiltonian system, i.e., δ ∗ iξ dρ = 0, ∀π2r−1 -vertical ξ, (3.36) are called Hamilton equations, the integral sections are then called Hamilton extremals of ρ [42]. Obviously, the set of prolongations of extremals of λ is a subset of the set of Hamilton extremals. Properties of this subset, however, depend both upon properties of the Lagrangian λ and the choice of its Lepage equivalent ρ: due to Dedecker [8] and Krupka and his collaborators, this understanding is a key to the concept of a regular variational problem [49, 53, 54, 58, 59, 70, 71].

Lepage Forms in the Calculus of Variations

39

In higher-order mechanics we associate to λ a unique Lepage equivalent, Θλ , and hence a unique Hamiltonian system, generated by 1-forms iξ dΘλ , where ξ runs over vertical vector fields. A well-known result says that if a Lagrangian λ on J r Y satisfies the “regularity condition” ∂2L 6= 0, (3.37) det ∂qrσ ∂qrν then every Hamilton extremal of Θλ is the (2r − 1)-th prolongation of an extremal of λ, i.e. the sets of extremals and Hamilton extremals of λ are in 1-1 correspondence; moreover, σ σ , p , . . . pr−1 ), where pi are components the mapping (t, q σ , . . . , q2r−1 ) → (t, q σ , . . . , qr−1 σ σ σ σ of Θλ at dqi , 0 ≤ i ≤ r − 1, is a local diffeomorphism (called Legendre transformation) [9, 46]. In the case dim X = n > 1, the Hamiltonian theory is richer, since the Lepage equivalent of λ is no longer unique. Going back to De Donder and Goldschmidt and Sternberg [11, 22], the most commonly considered Hamilton equations are those based on the Poincar´e–Cartan form Θλ (cf. also Hamilton equations in the “multisymplectic formalism” [13, 21, 24, 62]); we call these Hamilton equations De Donder–Hamilton equations. The result now is completely analogous to that of mechanics: if a Lagrangian λ on J 1 Y satisfies the “regularity condition” ∂2L det (3.38) 6= 0, ∂yiσ ∂yjν then every Hamilton extremal of Θλ is the prolongation of an extremal of λ, meaning that De Donder–Hamilton equations and Euler–Lagrange equations are equivalent. Writing Θλ = L −

∂L σ ∂L yi ω0 + σ dy σ ∧ ωi = −Hω0 + piσ dy σ ∧ ωi , σ ∂yi ∂yi

(3.39)

we get the Hamiltonian and momenta of λ. The above regularity condition then guarantees that (xi , y σ , yjσ ) → (xi , y σ , pjσ ) is a local coordinate transformation on J 1 Y (Legendre transformation). In Legendre coordinates, De Donder–Hamilton equations read ∂y σ ∂H = j, j ∂x ∂pσ

∂pjσ ∂H = − σ. j ∂x ∂y

(3.40)

A higher-order version of this result was first considered by De Donder [11]. However, the generalisation is not so straightforward, since Θλ may be not globally well-defined, and if “globalised”, is non-unique. Saving the property of being determined completely by the Lagrangian, one has to resign on global Hamilton equations. Given a Lagrangian λ of order r, and its local Poincar´e–Cartan equivalent Θλ (3.11), Hamilton equations now read δ ∗ iξ dΘλ = 0

for every π2r−1 -vertical vector field ξ on J 2r−1 Y

(3.41)

and are defined on the domain W of the coordinates (xi , y σ , yjσ1 , yjσ1 j2 , . . . , yjσ1 j2 ...jr ). Put Θλ = −Hω0 + piσ dy σ ∧ ωi + pjσ1 i dyjσ1 ∧ ωi + · · · + pjσ1 ...jr−1 i dyjσ1 ...jr−1 ∧ ωi ,

(3.42)

40

Olga Krupkov´a

where pjσ1 ...jk i =

r−k−1 X

(−1)l dp1 dp2 . . . dpl

l=0

H = −L +

r X

∂L

, ∂yjσ1 ...jk p1 ...pl i

0 ≤ k ≤ r − 1,

(3.43)

pjσ1 ...jk yjσ1 ...jk ,

k=1

and denote by [q1 . . . qs ] the number of all different sequences arising by permuting the sequence q1 , . . . , qs . As proved by Shadwick [69], if the rank of all the matrices ∂2L 1 · σ [j1 . . . j2r−s (pr+1 . . . ps ] [p1 . . . pr )] ∂yj1 ...j2r−s (pr+1 ...ps ∂ypν1 ...pr )

!

(3.44)

is maximal, where r ≤ s ≤ 2r − 1, the σ, j1 ≤ · · · ≤ j2r−s label columns, ν, p1 ≤ · · · ≤ ps label rows, and the brackets (· · · ) denote symmetrization in the indicated indices, then every Hamilton extremal δ of Θλ passing in W is of the form π2r−1,r ◦ δ = J r γ where γ is an extremal of λ. Moreover, the system of functions xi , y σ , yjσ1 , . . . , yjσ1 ...jr−1 , pjσ1 ...jr , . . . , pjσ1 ,

j1 ≤ · · · ≤ jr

(3.45)

forms a part of a coordinate system on W (called Legendre coordinates). In Legendre coordinates De Donder–Hamilton equations read ∂yjσ1 ...jk ∂H = j1 ...jk i , i ∂x sym{j1 ...jk i} ∂pσ

∂pjσ1 ...jk l ∂H =− σ , l ∂x ∂yj1 ...jk

(3.46)

where 0 ≤ k ≤ r − 1, and in the second set of equations, summation over l takes place. We note that Shadwick’s regularity condition above can be put into a geometric form and can be expressed equivalently by means of certain bilinear foms or by a linear mapping [20, 34]. Considering in place of Θλ a global form Θ = Θλ +p1 dν we get to λ non-unique global De Donder–Hamilton equations. There arises the question of whether Shadwick’s regularity condition still can guarantee an analogous nice correspondence between extremals and Hamilton extremals, According to Krupka [43] and Gotay [24] the answer is affirmative: If Shadwick’s regularity condition is satisfied then every solution δ of De Donder–Hamilton equations of Θ is of the form π2r−1,r ◦ δ = J r γ where γ is an extremal of λ. However, note that De Donder–Hamilton equations of Θ in Legendre coordinates may have a more complicated form compared to (3.46). For other interesting aspects of the theory see e.g. [1, 10, 16, 24, 34, 42, 43, 67]. It is important to notice that for higher-order regular Lagrangians De Donder–Hamilton equations are no longer equivalent with the Euler–Lagrange equations: the subset of Hamilton extremals that is in bijective correspondence with extremals consists of sections that are holonomic up to the order r (note r = order of the Lagrangian). Within De Donder–Hamilton theory the concept of regularity can be reconsidered to give regularity conditions for higher order Lagrangians different from Shadwick’s regularity condition. The idea proposed in our joint paper [49] was that the “true order” of the De Donder–Hamilton equations in essential. Namely, for some Lagrangians of order r ≥ 2,

Lepage Forms in the Calculus of Variations

41

their Poincar´e–Cartan form is π2r−1,s -projectable, where s < 2r − 1; in this case, it is apparently inappropriate to apply the standard procedure leading to considering Hamilton equations of order 2r − 1. The problem should instead be studied as a problem of order s. In the above mentioned paper this idea was applied to the class of second order Lagrangians with π3,1 -projectable Θλ . This concerns the following second order Lagrangians affine in the second derivatives: λ = Lω0 where i σ ν L = L0 (xi , y σ , yjσ ) + hpq ν (x , y ) ypq .

(3.47)

In [49] we called a Lagrangian regular if all its Hamilton extremals were holonomic, and we proved the following theorem: ˇ ep´ankov´a). Let λ be a Lagrangian of the form (3.47). If the Theorem 3.11 (Krupka and Stˇ condition ! ∂ 2 L0 ∂hik ∂hki σ ν 6= 0 (3.48) det − − ∂yiσ ∂ykν ∂y ν ∂y σ is satisfied then λ is regular, its Euler–Lagrange and De Donder–Hamilton equations are equivalent, and the mapping (xi , y σ , yjσ ) → (xi , y σ , pjσ ),

pjσ =

∂hjk ∂L0 ∂hjk ∂hkj σ σ ν − − + yν ∂yjσ ∂xk ∂y ν ∂y σ k

(3.49)

is a local coordinate transformation on J 1 Y (“Legendre transformation”). It should be stressed that for second order Lagrangians (3.47) the momenta (3.49) and Hamiltonian ∂L0 ∂hjk H = −L0 + σ yjσ − σν yjσ ykν , (3.50) ∂yj ∂y are first order functions. De Donder–Hamilton equations expressed in Legendre coordinates (3.49) then take the “usual” form (3.40). The above results directly apply to the Einstein– Hilbert Lagrangian (scalar curvature) of the General Relativity Theory (see [49] for explicit computations). Thus, within this setting, gravity naturally appears as a first order regular theory (without constraints). Later the above ideas were applied to study also some other kinds of higher order Lagrangians with projectable Poincar´e–Cartan forms by Garcia and Mu˜noz [19, 20]. We have seen that regularity is a property of a Lepage form, rather than of a Lagrangian itself, and it carries a geometric content that all Hamilton extremal are holonomic (for first order Hamiltonian systems), or holonomic up to a proper order (for higher order systems). However, in field theory we have for a given Lagrangian a family of Hamiltonian systems, defined by different Lepage equivalents of λ. In this way we expect to obtain regularity conditions depending on λ and some auxiliary functions coming from parts of ρ that are not uniquely determined by the Lagrangian, and we may consider the Hamiltonian theory in a completely new setting: instead of asking whether a Lagrangian is regular (that usually means regularity of its De Donder–Hamilton system) we may ask a question if the family of Hamiltonian systems associated with a Lagrangian contains a Hamiltonian system that is regular. This possibility was first observed by Dedecker [8], and recently studied for

42

Olga Krupkov´a

Lepage equivalents of first and higher order Lagrangians in [53, 54, 58, 59, 70, 71]. In [59] a procedure of “regularisation” of conventionally singular Lagrangians was applied to important physical field Lagrangians (Dirac field and electromagetic field Lagrangian), and “corrected” Hamiltonians and momenta were found, providing Hamilton equations without constraints.

3.6.

Symmetries and Conservation Laws, Noether Theorem

Lepage forms play an important role also in the theory of invariant variational functionals developed by Krupka in the early 1970’s [36, 38]. Let λ be a Lagrangian on J r Y , Eλ its Euler-Lagrange form. An isomorphism φ of the fibred manifold π : Y → X is called an invariance transformation of λ, respectively, generalised invariance transformation of λ if J r φ∗ λ = 0,

respectively,

J 2r φ∗ Eλ = 0.

(3.51)

If ξ is a π-projectable vector field on Y and {φu } is a local one-parameter group of ξ such that φu is an invariance transformation, respectively, generalised invariance transformation, for every u, we get the following infinitesimal version of the above invariance conditions: LJ r ξ λ = 0,

respectively,

LJ 2r ξ Eλ = 0.

(3.52)

The above conditions are called the Noether equation and the Noether–Bessel-Hagen equation, respectively. Noether and Noether-Bessel-Hagen equations can be used to find the group of invariance or generalised invariance transformations of a given Lagrangian, or conversely, to find all Lagrangians (Euler–Lagrange forms) invariant with respect to a given group of transformations of Y ; of course, solving the Noether–Bessel-Hagen equation in this case we obtain invariant differential equations that need not come from a Lagrangian: variationality is an additional property to be satisfied. We have an important theorem due to Krupka [38]: Theorem 3.12. Let λ be a Lagrangian on J r Y and Eλ its Euler–Lagrange form. Given an isomorphism φ of the fibred manifold π : Y → X, respectively, a π-projectable vector field ξ on Y , it follows that J 2r φ∗ Eλ = EJ r φ∗ λ ,

LJ 2r ξ Eλ = ELJ r ξ λ .

(3.53)

Corollary 3.13. Every invariance transformation of a Lagrangian λ is its generalised invariance transformation. Corollary 3.14. If LJ 2r ξ Eλ = 0 then LJ r ξ λ is a null Lagrangian. As proved in [26, 47] (see also the next section), the above condition means that around every point in J r−1 Y there is an (n − 1)-form η such that LJ r ξ λ = hdη.

Equipped with the concept of invariant Lagrangian, we obtain a classical result of Emmy Noether [64] and its generalisation, that in the setting of Lepage forms appear as an easy consequence of the first variation formula (Krupka [38]):

Lepage Forms in the Calculus of Variations

43

Theorem 3.15 (Noether Theorem). Let λ be a Lagrangian on J r Y , ρ its Lepage equivalent. If a π-projectable vector field ξ on Y generates invariance transformations of λ, and if γ is an extremal then d(J 2r−1 γ ∗ iJ 2r−1 ξ ρ) = 0. (3.54) Theorem 3.16 (Generalised Noether Theorem). Let λ be a Lagrangian on J r Y , ρ its Lepage equivalent. If a π-projectable vector field ξ on Y generates generalised invariance transformations of λ, and if γ is an extremal then d(J 2r−1 γ ∗ (iJ 2r−1 ξ ρ − η)) = 0,

(3.55)

where η is a (local) (n − 1)-form such that LJ r ξ λ = hdη. Equation (3.54), respectively (3.55), is called a conservation law, and the (n − 1)form iJ 2r−1 ξ ρ respectively, iJ 2r−1 ξ ρ − η, that is closed along prolongations of extremals, is called a conserved current. Using the Poincar´e lemma we get around every point x ∈ X an (n − 2)-form ϕ for which dϕ = f i ωi is a conserved current on X, so that the corresponding conservation law d(f i ωi ) = 0 takes a “divergence form” div f = 0. In mechanics (dim X = 1) the situation is simpler: a “conserved current” is a function, F , and a conservation law reads F ◦ J 2r−1 γ = const. Therefore, F is called constant of the motion. Since in field theory the Lepage equivalent of λ is not unique, there arises a question on how a conserved current depends upon a choice of a Lepage equivalent ρ of λ. Using formulas (3.10) and (3.14), i.e. ρ = Θ + µ ¯ where µ ¯ is an at least 2-contact form, and taking into account that contraction of µ ¯ gives a contact form that vanishes along prolongations of sections of π, we can see that the conservation law corresponding to ρ depends merely upon the at most 2-contact part of ρ; hence, for any Lepage equivalent ρ of an invariant Lagrangian λ, the conserved current is the (n − 1)-form iJ 2r−1 ξ Θ. Of course, in higher order field theory we must take into account the non-uniqueness of Θ and the non-existence of a global Poincar´e–Cartan form Θλ , discussed in the previous sections.

4. 4.1.

Lepage Forms and Differential Equations The Inverse Problem of the Calculus of Variations

By a dynamical form of order s we understand a 1-contact and πs,0 -horizontal (n + 1)form E on J s Y [51]. This means that in every fibred chart (V, ψ), ψ = (xi , y σ ) on Y , −1 E = Eσ ω σ ∧ ω0 , where the components Eσ are functions on Vs = πs,0 (V ). A section γ of π satisfying E ◦ J sγ = 0 (4.1)

is called a path of E. Equation (4.1) in fibred coordinates takes the form Eσ ◦ J s γ = 0, 1 ≤ σ ≤ m, that is a system of m partial, respectively ordinary, differential equations of order s if dim X > 1, respectively if dim X = 1. Let E be a dynamical form on J s Y . We have the following important definitions [40]: E is called locally variational if to every point in J s Y there exists a neighbourhood U , an integer r ≤ s, and a Lagrangian λ defined on πs,r (U ) such that E|U = Eλ . E is called

44

Olga Krupkov´a

globally variational if λ is defined on J r Y , i.e., if E is locally variational and has a global Lagrangian. Paths of a locally variational form E are called extremals of E. Note that extremals of a globally variational form coincide with extremals of any of its global Lagrangians. On the other hand, if E is locally but not globally variational, the family of extremals of E contains, but is not equal to, the family of extremals of any individual Lagrangian of E. Remark 4.1. We have seen that a dynamical form on J s Y determines a system of m sth order differential equations (4.1). Conversely, given a system of m sth-order differential equations Eσ ◦ J s γ = 0, 1 ≤ σ ≤ m, for graphs of mappings from Rn to Rm (n ≥ 1) we can represent it by a dynamical form E defined on an open subset of J s (Rn × Rm ) setting E = Eσ ω σ ∧ dt if dim X = 1, respectively, E = Eσ ω σ ∧ ω0 if dim X > 1. Such equations are called variational if the associated dynamical form E is locally variational. There is a close connection between locally variational forms and closed forms: Theorem 4.2 (Krupka [40, 41]). Let E be a dynamical form on J s Y . E is locally variational if and only if to every point x ∈ J s Y there exists a neighbourhood W and an at least 2-contact form FW on W such that the form αW = E + FW is closed. Since dαW = 0, we have by the Poincar´e lemma to every point z ∈ W a neighborhood U and a form ρ on U such that dρ = αU . However, the one-contact part p1 dρ of dρ equals the dynamical form E (restricted to U ), i.e. p1 dρ is horizontal with respect to the projection onto Y . This means that ρ is a Lepage n-form. Now, λ = hρ is a Lagrangian such that Eλ = E|U . In this way we can obtain a local Lagrangian for E simply by means of the Poincar´e lemma. Moreover, the geometric structure of J s Y enables one to introduce a modified homotopy operator, A, adapted to the contact structure [41] with the following properties: (1) Adα + dAα = dα, (2) if α is k-contact then Aα is (k − 1)-contact. Then, however, λ = hAα = Ap1 α = AE,

(4.2)

in fibered coordinates L = qσ

Z

1

Eσ (t, uq ν , uq1ν , . . . , uqsν )du

(4.3)

Eσ (xi , uy ν , uyjν , . . . , uyjν1 ...js )du

(4.4)

0

if dim X = 1, and L=y

σ

Z

0

1

if dim X > 1, respectively. The formula for L was obtained in [74] and [75], and is called Tonti–Vainberg Lagrangian. Necessary and sufficient conditions for a dynamical form be locally variational were first studied by Helmholtz for the case of second order ordinary differential equations [30].

Lepage Forms in the Calculus of Variations

45

A generalisation to ordinary differential equations of an arbitrary order is due to Vanderbauwhede [76], and to higher-order partial differential equations to Anderson and Duchamp [3], and Krupka [40, 41]. The celebrated conditions read as follows: Theorem 4.3. A dynamical form E on J s Y is locally variational if and only if its components in every fibred chart satisfy the following conditions: dim X = 1 :

s X k ∂Eσ l ∂Eν − (−1) − (−1)k ∂qlν ∂qlσ l k=l+1

!

dk−l ∂Eν = 0, dtk−l ∂qkσ

(4.5)

dim X = n : s X (4.6) ∂Eσ ∂Eν k ∂Eν l − (−1) − (−1)k = 0, djl+1 djl+2 . . . djk σ ∂yjν1 j2 ...jl ∂yjσ1 j2 ...jl k=l+1 l ∂yj1 j2 ...jk

!

for all σ, ν, and 0 ≤ l ≤ s. To a locally variational form a local Lagrangian is given by formula (4.3), respectively, (4.4).

4.2.

Lepage Equivalents of Locally Variational Forms

Now we are able to extend the concept of Lepage n-form to (n + 1)-forms as proposed in [50, 53]. Theorem 4.2 shows that for a dynamical form be locally variational it is essential that it can be completed (at least locally) to a closed form. This property was a motivation for the following definitions: Definition 4.4. Let s ≥ 0. A closed (n + 1)-form α on J s Y is called a Lepage (n + 1)form if p1 α is a dynamical form. A Lepage (n + 1)-form α is called Lepage equivalent of a dynamical form E if p1 α = E. With the help of the Poincar´e lemma, and taking account of the definition of a Lepage n-form, we immediately obtain relations between Lepage (n + 1)-forms and n-forms: Theorem 4.5. (1) If α is a Lepage (n + 1)-form then locally α = dρ where ρ is a Lepage n-form. (2) If α is a Lepage (n + 1)-form then the dynamical form E = p1 α is locally variational. (3) Every Lepage equivalent α of a locally variational form E locally equals to dρ where ρ is a Lepage equivalent of a Lagrangian λ for E. Conversely, if ρ is a Lepage equivalent of a Lagrangian λ then dρ is a Lepage equivalent of the Euler–Lagrange form Eλ . Note that we can say that Lepage (n + 1)-forms are closed counterparts of variational equations. With help of Lepage (n + 1)-forms we realise that the difference between local and global variationality is the same as the difference between local and global exactness of (n+1)-forms. In the following theorem we denote by H n+1 (Y ) the (n+1)-st cohomology group of Y . We also note that due to the structure of the fibres of the bundle J s Y → Y , the cohomology groups of J s Y coincide with the corresponding cohomology groups of Y .

46

Olga Krupkov´a

Theorem 4.6. (1) A locally variational form E is globally variational if and only if it has an exact Lepage equivalent. (2) If H n+1 (Y ) = {0} then every locally variational form on J s Y is globally variational. The question of the existence of Lepage equivalents of locally variational forms for general n and s is still open. However, it is known that if n ≥ 1 and the order of E is ≥ 2, a Lepage equivalent, if exists, is nonunique [53, 54], and splits into the sum α = αE + dη,

(4.7)

where αE is determined uniquely by the components Eσ of E. On the other hand, for n = 1, s arbitrary, and n > 1, s = 1 every locally variational form has a unique minimal-order Lepage equivalent [50] (cf. also [32]), and [29]: If n = 1, s ≥ 1, α = Eσ ω σ ∧ dt +

s−1 X

j,k=0

jk σ ωj ∧ ωkν , Fσν

(4.8)

where jk Fσν

1 = 2

s−j−k−1 X l=0

j+l

(−1)

j+l l

!

dl ∂Eσ , ν dtl ∂qj+k+l+1

(4.9)

jk = 0 otherwise. Note that the Lepage equivalent of a whenever 0 ≤ j + k ≤ s − 1, and Fσν dynamical form of order s is projectable onto J s−1 Y . If n > 1, s = 1, the unique minimal order Lepage equivalent of E is defined on Y and reads

α = E σ ω σ ∧ ω0 +

n X

k=1

1 ∂ k Eσ ω σ ∧ ω ν1 ∧ · · · ∧ ω νk ∧ ωj1 ···jk . (4.10) k!(k + 1)! ∂yjν11 · · · ∂yjνkk

We refer the reader to the article [27] by Grigore (in this volume), where the structure of closed (n + 1)-forms that are counterparts of variational equations (on higher order Grassmann bundles) is clarified.

4.3.

Lepage (n + 1)-Forms in the Geometry of Variational Equations

Lepage (n + 1)-forms are useful not only in the study of the inverse variational problem. They play an even more important role for investigations of the structure and symmetries of variational equations, and are essential for the formulation of exact integration methods. In this section we just mention very briefly a few important applications in the geometric theory of differential equations. However, for more details we refer to [29, 51, 52, 53, 54, 55, 56] and references therein. Let E be a locally variational form on J s Y (s ≥ 1). Given a Lepage equivalent α of E on J r Y , we define a Hamiltonian system of α to be an exterior differential system on J r Y , locally generated by n-forms iξ α, where ξ runs over all πr -vertical vector fields on J r Y .

Lepage Forms in the Calculus of Variations

47

Due to the nonuniqueness of α we usually have many Hamiltonian systems associated with a given locally variational form (Euler–Lagrange equations), however, all have “the same” holonomic integral sections. Moreover, for any Lepage equivalent α of E, holonomic integral sections of the Hamiltonian system of α coincide with (prolongations of) extremals of E. Therefore equations for holonomic integral sections of a Hamiltonian system, i.e., J r γ ∗ iξ α = 0, ∀πr -vertical vector fields ξ, (4.11) are Euler–Lagrange equations, and equations for (all) its integral sections, i.e., δ ∗ iξ α = 0, ∀πr -vertical vector fields ξ,

(4.12)

are called Hamilton equations. We stress that every Hamiltonian system of a locally variational form provides a complete family of extremals (global solutions of the Euler–Lagrange equations) and a complete family of Hamilton extremals (global solutions of Hamilton equations (even if the locally variational form is not globally variational). Now one can study Euler–Lagrange and Hamilton equations and their solutions from a geometric point of view by investigating properties of the corresponding exterior differential systems. Let us mention at least regularity (more generally classification of equations with respect to geometric properties of their solutions), and symmetries and conservations laws, including corresponding integration methods. Note that the above geometric representation by means of exterior differential systems demonstrates a deep difference between ordinary and partial differential equations: For n = 1 the Lepage equivalent of E is unique, hence to ordinary variational equations we have unique Hamilton equations, determined by E. This means that, in particular, we get a Hamiltonian and momenta determined by the Euler–Lagrange expressions (not by an individual Lagrangian). Moreover, the Hamiltonian EDS is locally generated by 1-forms, this means that Hamilton extremals are integral sections of a distribution, called the Euler– Lagrange distribution, uniquely determined by the Euler–Lagrange form E, and defined on J s−1 Y , where s is the order of E. We obtain a natural concept of a regular variational system: a locally variational form E is called regular if the rank of its Euler–Lagrange distribution = 1. Hence, we get the regularity condition expressed by means of the Euler– Lagrange expressions. For n > 1 Hamilton extremals are no longer described by a distribution, moreover, we have a family of Hamilton equations associated with an Euler–Lagrange form. Hamilton equations (Hamiltonians, momenta) are (similarly as in the case of dim X = 1) associated directly to the Euler–Lagrange form, not to individual Lagrangians. They do, however, depend upon a choice of a Lepage equivalent of E. In this case, regularity is understood as a property of a Hamiltonian EDS, i.e., again it does not depend upon a choice of an individual Lagrangian for E, but depends however upon a choice of α for E. Thus, in field theory, we can look for a regularisation of given variational equations: the aim is to find a Hamiltonian system for E that is regular. The EDS description also helps us look for symmetries and conservation laws for Euler– Lagrange and Hamilton equations. In particular, we can easily study symmetries of Lepage n and (n + 1)-forms (i.e., vector fields along which the Lie derivative of ρ, or α vanishes).

48

Olga Krupkov´a

Moreover, every symmetry of a Lepage n-form ρ (resp. (n + 1)-form α) generates a conserved current (i.e. a closed n-form, belonging to the Hamiltonian EDS of dρ (resp. α)). For n = 1 this gives functions constant along extremals of E.

5.

Lepage Forms in the Variational Sequence

Within the theory of varational sequences developed by Krupka in [44], the concept of Lepage equivalent can be generalised to arbitrary k-forms, k ≥ n. Denote by Ωrq the sheaf q-forms on J r Y , Ωr0,c = {0}, and Ωrq,c the sheaf of contact q-forms if q ≤ n, respectively strongly contact q-forms if q > n, on J r Y . Set Θrq = Ωrq,c + dΩrq−1,c where dΩrq−1,c is the image sheaf of Ωrq−1,c by the exterior derivative d. There arises an exact sequence of soft sheaves 0 → Θr1 → Θr2 → Θr3 → · · · , where the morphisms are the exterior derivative, called contact sequence. It is a subsequence of De Rham sequence 0 → R → Ωr0 → Ωr1 → Ωr2 → Ωr3 → · · · . The quotient sequence 0 → R → Ωr0 → Ωr1 /Θr1 → Ωr2 /Θr2 → Ωr3 /Θr3 → · · · which is also exact, is called the r-th order variational sequence on π. It is important to stress that elements of the quotient sheaf Ωrq /Θrq are not forms, but classes of (local) qforms of order r. We denote by [ρ]rv an element of Ωrq /Θrq , that is the (variational) class of ρ ∈ Ωrq . The quotient mappings are denoted by Eqr : Ωrq /Θrq → Ωrq+1 /Θrq+1 . As proved by Krupka in [44], the variational sequence is an acyclic resolution of the constant sheaf R over Y . Due to the Abstract De Rham theorem, the cohomology groups of the cochain complex of global sections of the variational sequence are identified with the De Rham cohomology groups H q Y of Y . Quotient sheaves Ωrq /Θrq are determined up to natural isomorphisms of Abelian groups. In this way classes in Ωrq /Θrq admit representations by differential forms. Source forms [73] for the quotient sheaf Ωrq /Θrq arise by applying to q-forms the so-called interior Euler– Lagrange operator I [2, 35, 48]. Source forms for q = n and q = n + 1 are called Lagrangians and dynamical forms, respectively. In the source forms representation, the quotient mapping Enr : Ωrn /Θrn → Ωrn+1 /Θrn+1 coincides with the Euler-Lagrange mapping E : λ → Eλ . The next mapping is called Helmholtz mapping: if E is a dynamical form representing a class in Ωrn+1 /Θrn+1 then the image of E is an (n + 2)-form HE , called the Helmholtz form of E [44] (components of the Helmhotz form are the left-hand-sides of variationality conditions (4.6)). The variational sequence helps us to learn the structure of the Euler–Lagrange mapping and of null Lagrangians as follows: Condition HE = 0 for elements of the quotient sheaves r ([η]r ) = 0, and by exactness of the variational sequence means that there exists reads En+1 v a class [ρ]rv ∈ Ωrn /Θrn such that [η]rv = [dρ]rv . The source form λ = hρ for [ρ]rv is then a local Lagrangian for E. If H n+1 Y = {0}, ρ may be chosen globally defined on J r Y , so

Lepage Forms in the Calculus of Variations

49

we get a global Lagrangian for E. Similarly, if Eλ = 0, i.e., Enr ([η]rv ) = 0, there exists a class [ρ]rv ∈ Ωrn−1 /Θrn−1 such that [η]rv = [dρ]rv , for source forms, λ = hη = hdρ. If H n (Y ) = {0} then every null Lagrangian is globally null (there is a ρ defined on J r−1 Y ). The motivation for the generalisation of the concept of Lepage form is to obtain a representation of classes in the variational sequence such that the sequence morphisms would be the exterior derivatives [48, 57]: A q-form ρ, q ≥ n, is called a Lepage form if pq−n+1 dρ = Idρ (i.e. is a source form). If σ is a source q-form, we say that ρ is a Lepage equivalent of σ if ρ is a Lepage q-form and pq−n ρ = σ. Note that for q = n, and q = n + 1 and σ locally variational, we get definitions introduced in the previous sections. Summarising, we can express the source form representation and the Lepage form representation of the variational sequence as follows: En−1

En+1

E

En+2

· · · −−−−→ Ωrn /Θrn −−−n−→ Ωrn+1 /Θrn+1 −−−−→ Ωrn+2 /Θrn+2 −−−−→ · · · · · · −−−−→ d

· · · −−−−→

 

Iy

Ωsn

  Lepy

Ωkn

E

−−−−→ d

−−−−→

 

Iy

Ω2s n+1   Lepy

Ωkn+1

H

−−−−→ d

−−−−→

 

Iy

Ω4s n+2   Lepy

Ωkn+2

−−−−→ · · · d

−−−−→ · · ·

Generalisations of Lepage forms are studied also within the theory of non-holonomic systems [60, 72].

Acknowledgments ˇ 201/06/0922 of the Czech Science Foundation, and Research supported by grants GACR MSM 6198959214 of the Czech Ministry of Education, Youth and Sports. The author also highly appreciates support and hospitality of the Mathematics Department and the Institute for Advanced Study at La Trobe University, Melbourne, Australia.

References [1] V. Aldaya and J. de Azc´arraga, Higher order Hamiltonian formalism in field theory, J. Phys. A: Math. Gen. 13 (1982) 2545–2551. [2] I. Anderson, The Variational Bicomplex (Utah State University, Technical Report, 1989). [3] I. Anderson and T. Duchamp, On the existence of global variational principles, Am. J. Math. 102 (1980) 781–867. [4] D. E. Betounes, Extension of the classical Cartan form, Phys. Rev. D 29 (1984) 599– 606. ´ Cartan, Lec¸ons sur les invariants int´egraux (Hermann, Paris, 1922). [5] E.

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[6] M. Crampin and D. J. Saunders, The Hilbert-Carath´eodory and Poincar´e-Cartan forms for higher-order multiple-integral variational problems, Houston J. Math. 30 (2004) 657–689. [7] P. Dedecker, Calcul des Variations et Topologie Algebrique, Thesis, Univ. de Li`ege, Facult´e des Sciences, 1957. [8] P. Dedecker, On the generalization of symplectic geometry to multiple integrals in the calculus of variations, In: Lecture Notes in Math. (570, Springer, Berlin, 1977) 395–456. [9] P. Dedecker, Le th´eor`eme de Helmholtz-Cartan pour une int´egrale simple d’ordre sup´erieur, C. R. Acad. Sci. Paris, S´er. A 288 (1979) 827–830. ´ Cartan pour une int`egrale [10] P. Dedecker, Sur le formalisme de Hamilton–Jacobi–E. multiple d’ordre sup´erieur, C. R. Acad. Sci. Paris, S´er. I 299 (1984) 363–366. [11] Th. De Donder, Th´eorie Invariantive du Calcul des Variations (Gauthier–Villars, Paris, 1930). [12] M. de Le´on and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory (North-Holland, Amsterdam, 1985). [13] A. Echeverria-Enriquez, M. C. Mu˜noz-Lecanda and N. Rom´an-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys. 41 (2000) 7402–7444. [14] Ch. Ehresmann, Les prolongements d’une space fibr´e diff´erentiable, C. R. Acad. Sci. Paris 240 (1955) 1755–1757. [15] M. Ferraris, Fibered connections and global Poincar´e–Cartan forms in higher order calculus of variations, In: Geometrical Methods in Physics (Proc. Conf. on Diff. Geom. and Appl. Vol. 2, Nov´e Mˇesto na Moravˇe, Sept. 1983, (D. Krupka, Ed.) J. E. Purkynˇe Univ. Brno, Czechoslovakia, 1984) 61–91. [16] M. Ferraris and M. Francaviglia, On the global structure of the Lagrangian and Hamiltonian formalisms in higher order calculus of variations, In: Geometry and Physics (Proc. Int. Meeting, Florence, Italy 1982, (M. Modugno, Ed.) Pitagora, Bologna, 1983) 43–70. [17] P. L. Garcia, The Poincar´e-Cartan Invariant in the Calculus of Variations, Symp. Math. XIV (1974) 219–246. [18] P. L. Garcia and J. Mu˜noz, On the geometrical structure of higher order variational calculus, In: Modern Developments in Analytical Mechanics I: Geometrical Dynamics (Proc. IUTAM-ISIMM Symposium, Torino, Italy 1982, (S. Benenti, M. Francaviglia and A. Lichnerowicz, Eds.) Accad. delle Scienze di Torino, Torino, 1983) 127–147.

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[76] A. L. Vanderbauwhede, Potential operators and variational principles, Hadronic J. 2 (1979) 620–641. [77] A. M. Vinogradov, A spectral sequence associated with a non-linear differential equation, and algebro-geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978) 144–148. [78] E. T. Whittaker, A Treatise on Analytical Dynamics of Particles and Rigid Bodies (The University Press, Cambridge, 1917).

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 57-76

Chapter 3

O N A G ENERALIZATION ´ -C ARTAN F ORM OF THE P OINCAR E IN H IGHER -O RDER F IELD T HEORY D.R. Grigore∗ Dept. of Theor. Phys., National Inst. Phys. Nucl. Engeneering “Horia Hulubei”, Bucharest-M˘agurele, P. O. Box MG 6, Romˆania

Abstract We present here a possible generalization of the Poincar´e-Cartan form in classical field theory to the most general case: arbitrary dimension, arbitrary order of the theory and the absence of a fiber bundle structure. We use for the kinematical description of the system the (r, n)-Grassmann manifold associated to a given manifold X, i.e. the manifold of r-contact elements of n-dimensional submanifolds of X. The idea is to define globally a n + 1 form on this Grassmann manifold, more precisely its class with respect to a certain subspace and to write it locally as the exterior derivative of a n form which is a kind of Poincar´e-Cartan form in the higher-order and non-fibred situation.

2000 Mathematics Subject Classification. 53A55, 77S25, 58A20. Key words and phrases. Grassmann bundles, Lagrangian Formalism.

1.

Introduction

It is widely accepted that the variational principles should be given in a coordinate independent formulation. This idea was first realized for a dynamical system with a finite number of degrees of freedom (i.e. particle mechanics), using a differential 1-form instead of the Lagrangian, by Poincar´e and Cartan [43], [8]. There are a number of generalizations of this idea for classical field theory [27], [5], [6], [44], [45], [24], [11], [10], [12], [46], [42]. A related concept is that of Lepage equivalent of a Lagrangian form (see for instance [28], [29], ∗

E-mail address: [email protected], [email protected]

58

D.R. Grigore

[32]). All these generalizations use as geometric framework for classical field theory the jet bundle formalism (more explicitly the space-time and field variables are local coordinates on a fiber bundle X over a “space-time manifold” M ) and the derivative of the fields, up to order s, are variables in the s-th order jet bundle extension J s X of X. It was later [26], [47] suggested that it is more convenient to work with the exterior differential of the above Poincar´e-Cartan form. For the case of finite number of degrees of freedom this 2-form is in general presymplectic and was used by Souriau and others [47], [23] to obtain the phase space as a symplectic manifold in a deductive way. The idea is to consider that the fundamental mathematical object for a Lagrangian system must be this 2-form and not the Lagrangian function or the Poincar´e-Cartan 1-form. This point of view leads to the main features of the Lagrangian and the Hamiltonian formalism and to a natural definition of the Noetherian symmetries. For higher-order mechanics this approach has been developed in [33], [34], [35] (see also [38]). One can generalize this Lagrange-Souriau form without using the fibration hypothesis mentioned above in two particular but important cases: for classical field theory of first order [13] and for systems with a finite number of degrees of freedom and of arbitrary order [14]. Moreover, this Lagrange-Souriau form can be locally written as the exterior differential of a Poincar´e-Cartan form related to some chosen local chart. This Poincar´e-Cartan form is the same as that given by Krupka [27], Betounes [5]-[6] and Rund [44]. The Lagrangian is locally determined up to a variationally trivial Lagrangian, i.e. a Lagrangian giving trivial Euler-Lagrange equations. As a consequence, one can define in a geometrically nice way the Noetherian symmetries using the Lagrange-Souriau form. In this paper we give a generalization of the Lagrange-Souriau and of the Poincar´eCartan forms in the most general case used in higher order field theory. We consider an arbitrary manifold X without a fiber bundle structure over some space-time manifold so instead of the s-th order jet bundle extension one must use the s-th order Grassmann bundle Pns X associated to X which was recently considered in the literature [20]. In the next Section we will summarize the main features of this construction. Next, in Section 3, we will be able to define globally a n + 1 differential form, but one will be able to see that, in general, one cannot determine this form uniquely. Fortunately one can consider the equivalence class of this form to a certain globally defined subspace of differential forms. This equivalence class is the “physical” object we are looking for. It is interesting to note that this subspace of differential forms is in fact identically zero exactly in the two particular cases mentioned above (s = 2, n arbitrary and n = 1, s arbitrary). Some combinatorial tricks introduced in [16] must be used to simplify the analysis of some tensorial identities. In Section 4 we locally exhibit the n + 1 differential form as the exterior derivative of a locally defined Poincar´e-Cartan n-form and in this way the (local) Lagrangian function appears also. Some functions called hyper-Jacobians [7], [41] emerge naturally in this context and can be used to provide the most general expression for a variationally trivial Lagrangian of arbitrary order already obtained in [17] by a different method. Finally, in Section 5 we present two particular but very important cases, namely n = 1 s arbitrary and s = 2 n arbitrary. Some ideas related to the ones from this paper also appear in [31]. In the fibered situation, similar (n + 1)-forms were studied in [39], [40] and [22]. We will skip some proofs which are similar to the proofs from [16]-[19] and can be found in the web version of this paper [15].

On a Generalization of the Poincar´e-Cartan Form

2. 2.1.

59

Grassmann Manifolds The Basic Constructions of the Grassmann Manifolds

In this Section we present the basic construction of Grassmannian manifolds following [20] and [18]. We will skip all the proofs. We consider N , n ≥ 1 and r ≥ 0 integers such that n ≤ N , and let X be a smooth manifold of dimension N which is the mathematical model for the kinematical degrees of freedom of a certain classical field theory. Let U ⊂ Rn be a neighborhood of the point 0 ∈ Rn , x ∈ X and let Γ(0,x) be the set of smooth maps γ : U → X such that γ(0) = x. On Γ(0,x) one has the the equivalence relationship “γ ∼ δ” iff there exists a chart (V, ψ) ψ = (xA ), A = 1, . . . , N on X such that the functions ψ ◦ γ, ψ ◦ δ : Rn → RN have the same partial derivatives up to order r in the point 0. The equivalence class of γ will be denoted by j0r γ and it is called a r (r, n)-velocity. The set of (r, n)-velocities at x is denoted by T(0,x) (Rn , Y ) ≡ Γ(0,x) / ∼. We denote [ r Tnr X = T(0,x) (Rn , X), x∈X

and define surjective mappings τnr,s : Tnr X → Tns X, where 0 < s ≤ r, by τnr,s (j0r γ) = j0s γ and τnr,0 : Tnr X → X, where 1 ≤ r, by τnr,0 (j0r γ) = γ(0). If (V, ψ), ψ = (xA ), is a chart on X we define the couple (Vnr , ψnr ) where Vnr = r,0 A (πn )−1 (V ), ψnr = (xA , xA j , · · · , xj1 ,j2 ,...,jr ) 1 ≤ j1 ≤ j2 ≤ · · · ≤ jr ≤ n, and r xA j1 ,...,jk (j0 γ)

∂k A x ◦ γ ≡ j1 , ∂t . . . ∂tjk 0

0 ≤ k ≤ r.

(2.1)

r The expressions xA j1 ,···jk (j0 γ) are defined for all indices j1 , . . . , jr in the set {1, . . . , n} but because of the symmetry property r A r xA jP (1) ,...,jP (k) (j0 γ) = xj1 ,...,jk (j0 γ)

(k = 2, ..., n)

(2.2)

for all permutations P ∈ Pk of the numbers 1, . . . , k we consider only the independent components given by the restrictions 1 ≤ j1 ≤ j2 ≤ · · · ≤ jr ≤ n. This allows one to use A multi-index notations i.e. ψnr = (xA |J| = 0, ..., r where by definition xA J ), ∅ ≡ x . The same comment is true for the partial derivatives ∂xA ∂ . j1 ,...,jk

The couple (Vnr , ψnr ) is a chart on Tnr X called the associated chart of the chart (V, ψ) and the system of charts give a smooth structure on this set; moreover Tnr X is a fiber bundle over X with the canonical projection τnr,0 . The set Tnr Y endowed with the smooth structure defined by the associated charts defined above is called the manifold of (r, n)-velocities over X. In the chart (Vnr , ψnr ) one introduces the following differential operators: ∆jA1 ,...,jk ≡

r1 ! . . . rn ! ∂ , k! ∂xA j1 ,...,jk

j1 , . . . , jk ∈ {1, . . . , n}

where rk is the number of times the index k shows up in the sequence j1 , . . . jk .

(2.3)

60

D.R. Grigore The combinatorial factors are such that the following relation is true: ∆iA1 ,...,ik xB j1 ,...,jl

=

ik i1 BS+ δA j1 ,...,jk δj1 . . . δjk 0

(

if k = l if k 6= l.

(2.4)

Here we use the notations from [14], namely Sj±1 ,...,jk are the symmetrization (for the sign +) and respectively the antisymmetrization (for the sign −) projector operators defined by 1 X Sj±1 ,...,jk fj1 ,...,jk ≡ ǫ± (P )fjP (1) ,...,jP (k) (2.5) k! P ∈P k

where the sum runs over the permutation group Pk of the numbers 1, . . . , k and ǫ+ (P ) ≡ 1,

ǫ− (P ) ≡ (−1)|P | ,

∀P ∈ Pk ;

here |P | is the signature of the permutation P . In this way one takes care of overcounting the indices. More precisely, for any smooth function on V r , the following formula is true: df =

r X

(∆jA1 ,...,jk f )dxA j1 ,...,jk =

k=0

X

(∆IA f )dxA I

(2.6)

|I|≤r

where we have also used the convenient multi-index notation. The formal derivatives are: Dir ≡

r−1 X

j1 ,...,jk xA = i,j1 ,...,jk ∆A

k=0

X

J xA iJ ∆A .

(2.7)

|J|≤r−1

The last expression uses the multi-index notation; if I and J are two such multi-indices we mean by IJ the juxtaposition of the two sets I, J. When no danger of confusion exists we simplify the notation putting simply Di = Dir . The formal derivatives give a conveniently expression for the change of charts on the velocity manifold induced by a change of charts on X. By definition the differential group of order r is the set r Lrn ≡ {j0r α ∈ J0,0 (Rn , Rn )|α ∈ Dif f (Rn )}

(2.8)

i.e. the group of invertible r-jets with source and target at 0 ∈ Rn . The group multiplication in Lrn is defined by the jet composition Lrn × Lrn ∋ (j0r α, j0r β) 7→ j0r (α ◦ β) ∈ Lrn . The canonical (global) coordinates on Lrn are defined by aij1 ,...,jk (j0r α)

∂ k αi = j1 , ∂t . . . ∂tjk 0

j1 ≤ j2 ≤ · · · ≤ jk ,

where αi are the components of a representative α of j0r α. We denote a ≡ (aij , aij1 ,j2 , . . . , aij1 ,...,jk ) = (aiJ )1≤|J|≤r

k = 0, ..., r

(2.9)

On a Generalization of the Poincar´e-Cartan Form

61

and notice that one has det(aij ) 6= 0.

(2.10)

A r (x · a)A I ≡ xI (j0 (γ ◦ α))

(2.11)

The group Lrn is a Lie group. The manifolds of (r, n)-velocities Tnr Y admits a (natural) smooth right action of the differential group Lrn , defined by the jet composition

i where the connection between xA I and γ is given by (2.1) and the connection between aI and α is given by (2.9). The group Lrn has a natural smooth left action on the set of smooth real functions defined on Tnr X , namely for any such function f we have:

(a · f )(x) ≡ f (x · a).

(2.12)

We say that a (r, n)-velocity j0r γ ∈ Tnr X is regular, if γ (or any other representative) is an immersion. We have the central result: Theorem 2.1. The set Pnr X ≡ ImmTnr X/Lrn has a unique differential manifold structure such that the canonical projection ρrn is a submersion. The group action defines on ImmTnr X the structure of a right principal Lrn -bundle. A point of Pnr X containing a regular (r, n)-velocity j0r γ is called an (r, n)-contact element, or an r-contact element of an n-dimensional submanifold of X, and is denoted by [j0r γ]. As in the case of r-jets, the point 0 ∈ Rn (resp. γ(0) ∈ X) is called the source (resp. the target) of [j0r γ]. The manifold Pnr is called the (r, n)-Grassmannian bundle, or simply a higher order Grassmannian bundle over X. Besides the quotient projection ρrn : ImmTnr X → Pnr we have for every 1 ≤ s ≤ r, the r s canonical projection of Pnr X onto Pns X defined by ρr,s n ([j0 γ]) = [j0 γ] and the canonical r r r projection of Pn X onto X defined by ρn ([j0 γ]) = γ(0). On Pnr X there are total differential operators; as expected, in the chart ρrn (W I,r ) they have the expression: ∂ r1 ! . . . rn ! (2.13) ∂σj1 ,...jk ≡ σ k! ∂yj1 ,...,jk We note for further use the following formula: ∂σi1 ,...,ik y¯jν1 ,...,jk = Sj+1 ,...,jk Qij11 . . . Qijkk Qνσ ,

k = 1, ..., r;

(2.14)

here we have defined: Qσν ≡ ∂ν y¯σ − y¯iσ (∂ν x ¯i )

(2.15)

and Q is the inverse of the matrix: Pji ≡ dj x ¯i ,

Pji Qjl = δli .

(2.16)

Next we define the total derivative operators on the Grassmann manifold: di ≡

r−1 X X ∂ ∂ σ σ J + yi,j ∂ j1 ,...,jk = + yiJ ∂σ . 1 ,...,jk σ i i ∂x ∂x k=0 |J|≤r−1

(2.17)

62

D.R. Grigore We note that: (ρrn )∗ (zij Dj ) = di .

(2.18)

In particular, we have for any smooth function f on ρrn (W r ) the following formula: Di (f ◦ ρrn ) = xji (dj f ) ◦ ρrn .

(2.19)

The formula for the chart change on Pnr X. can be written with this operators: let us consider two overlapping charts: (ρrn (V r ), (xi , yIσ )) and respectively (ρrn (V¯r ), (¯ xi , y¯Iσ )); then we have on the overlap: σ y¯iI = Qji dj y¯Iσ . |I| ≤ r − 1

(2.20)

Pij d¯j = di .

(2.21)

We also note that:

2.2.

Contact Forms on Grassmann Manifolds

By a contact form on Pnr X we mean any form ρ ∈ Ωrq (P X) verifying [j r γ]∗ ρ = 0

(2.22)

for any immersion γ : Rn → X. We denote by Ωrq(c) (P X) the set of contact forms of degree q ≤ n. Here [j r γ] : Rn → Pnr is given by: [j r γ] (t) ≡ [jtr γ] . We mention some of properties verified by these forms. If one considers only the contact forms on an open set ρrn (V r ) ⊂ Pnr X then we emphasize this by writing Ωrq(c) (V ). The ideal of all contact forms is denoted by C(Ωr ). By elementary computations one finds out that, as in the case of a fiber bundle, for any chart (V, ψ) on X, every element of the set Ωr1(c) (V ) is a linear combination of the following expressions: σ ωjσ1 ,...,jk ≡ dyjσ1 ,...,jk − yi,j dxi , k = 0, ..., r − 1 (2.23) 1 ,...,jk or, in multi-index notations: σ ωJσ ≡ dyJσ − yiJ dxi ,

|J| ≤ r − 1.

(2.24)

σ dωJσ = −ωJi ∧ dxi ,

|J| ≤ r − 2.

(2.25)

We have the formula

Any form ρ ∈ Ωrq (P X), q = 2, ..., n is contact iff it is generated by ωJσ , |J| ≥ r−1 and dωIσ , |I| = r − 1. In the end we present the transformation formula relevant for change of charts. ¯ two overlapping charts on X and let (W r , Φr ), Proposition 2.2. Let (V, ψ) and (V¯ , ψ) r i σ i r r r ¯ ,Φ ¯ ), Φ¯ = (¯ Φ = (x , yI , xI ) and (W xi , y¯Iσ , x ¯iI ) the corresponding charts on Tnr X. ¯ r ) ⊂ P r X: Then the following formula is true on ρrn (W r ∩ W n ω ¯ Iσ =

|I| X

|J|=1

(∂νJ y¯Iσ )ωJν − QσI,ν ω ν ,

1 ≤ |I| ≤ r − 1.

(2.26)

On a Generalization of the Poincar´e-Cartan Form

63

where we have defined: σ QσI,ν ≡ ∂ν y¯Iσ − y¯jI (∂ν x ¯j ),

0 ≤ |I| ≤ r − 1

(2.27)

and ω ¯ σ = Qσν ω ν

(2.28)

where Qσν is given by the formula (2.15). As a consequence we have: Corollary 2.3. If for a q-form has the expression X

ρ=

X

X

p+s=k |J1 |,...,|Jp |≤r−1 |I1 |=...=|Is |=r−1 σ ωJσ11 · · · ∧ ωJpp ∧ dωIν11

p ,I1 ,...,Is · · · ∧ dωIνss ∧ ΦJσ11,...,J ,...,σp ,ν1 ,...,νs ,

k ≤ q (2.29)

is valid in one chart, then it is valid in any other chart. This corollary allows us to define for any q = 1, dots, dim(J r Y ) = m n+r a contact n form with order of contactness k to be any ρ ∈ Ωrq such that it has in one chart (thereafter in any other chart) the expression above. We denote these forms by Ωrq,k .

3.

A Lagrange-Souriau Form on a Grassmann Manifold

3.1.

Some Invariant Conditions

As in the preceding Section we consider a differential manifold X and the associated (s, n)Grassmann manifold Pns X. We start we the following general result: Proposition 3.1. Let α ∈ Ωsq (X) a q-differential form on Pns X verifying: iξ α = 0

(3.1)

for any ρs,s−1 -vertical vector field ξ on Pns X (i.e. (ρs,s−1 )∗ ξ = 0). Then this form has the n n local expression: α=

q X

X

k=0 |I1 |,...,|Ik |≤s−1

,...,Ik TσI11,...,σ ω σ1 ∧ · · · ∧ ωIσkk ∧ dxik+1 ∧ · · · ∧ dxiq k ,ik+1 ,...,iq I1

(3.2)

,...,Ik where TσI11,...,σ are smooth functions depending on the variables k ,ik+1 ,...,iq σ i σ σ (x , y , yj , . . . , yj1 ,...,js ) and are antisymmetric in the couples (Ip , σp ) p = 1, . . . k and in the indices ik+1 , . . . , iq .

We denote the space of these forms by Ωsq,ξ (X). By elementary computations from (2.26) and (2.28) we can obtain the transformation formulas for the coefficients ,...,Ik TσI11,...,σ in the overlap of two charts. As a consequence of these transformation k ,ik+1 ,...,iq

64

D.R. Grigore

,...,Ik formulas we can prove that some constraints on the coefficients TσI11,...,σ of the k ,ik+1 ,...,iq form α are in fact globally defined. These relations are: ,...,Ik TσI11,...,σ = 0, k ,ik+1 ,...,iq

|I1 | + · · · + |Ik | ≥ t

TσI11,i2 ,...,iq = 0,

∀t ∈ N,

∀I1 6= ∅.

Ti1 ,...,iq = 0.

(3.3) (3.4) (3.5)

and the tracelessness condition ,...,Ik TσlI11,...,σ =0 k ,l,ik+2 ,...,iq

k = 1, . . . , q.

(3.6)

Some notations will be usefull. We denote the subset of the forms verifying (3.3) by Ωs,t q,ξ (X). If the condition (3.4) is fulfilled we say that the form α verifies the Lepage condition. We denote the subset of the forms verifying the conditions (3.3) and (3.4) by Ωs,t,Lep (X) and q,ξ s,t s Ωs,t q,ξ,k (X) ≡ Ωq,k (X) ∩ Ωq,ξ (X);

these are contact forms with the order of contactness equal to k and the definition is globally true. We call the subset of the forms verifying the conditions (3.3), (3.4) (i.e Lepage) and (3.6) (i.e. tracelessness) by Ωs,t,Lep q,ξ,tr (X). We finally stress again that all the spaces of the type Ω... are globally defined. ...

3.2.

The Definition of the Lagrange-Souriau Form

We need to consider one more condition on the Lepage forms, namely closeness. First we have: s,s−1 -projectable, i.e. there Proposition 3.2. Let α ∈ Ωs,s−1 q,ξ (X) be closed. Then α is ρn exists a q-form α0 ∈ Ωs−1 q (X) such that

α = (ρs,s−1 )∗ α0 . n

(3.7)

Moreover, the form α0 is closed. Proof. We exhibit the dependence of the form α on the highest-order derivatives; according to the preceding corollary these derivatives can appear in two places: in the coefficients Tσ∅,...,∅ , k = 0, . . . , q and in the contact forms ωIσ , |I| = s − 1. It is not very 1 ,...,σk ,ik+1 ,...,iq hard to write now α as follows: α=

q X

k=0 q X

X

k=1 |I1 |=s−1

Tσ∅,...,∅ ω σ1 ∧ · · · ∧ ω σk ∧ dxik+1 ∧ · · · ∧ dxiq + 1 ,...,σk ,ik+1 ,...,iq

,∅,...,∅ kTσI11,...,σ ω σ1 ∧ ω σ2 · · · ∧ ω σk ∧ dxik+1 ∧ · · · ∧ dxiq + α′ k ,ik+1 ,...,iq I1

-projectable. where the form α′ is ρs,s−1 n The expression above is, as said before, at most linear in the highest-order derivatives. One computes explicitly the coefficient of this derivatives and finds out that they are zero. This proves the first assertion. The closeness of the form α0 follows from the surjectivity . of the map ρs,s−1 n

On a Generalization of the Poincar´e-Cartan Form

65

s,s−1,Lep It is clear that there are strong conditions on any closed form α ∈ Ωq,ξ,tr (X). We will give a structure theorem for such a form in the case q = n + 1 which is relevant for physical applications. First we note that in this case we have ∅ Tσ,i = Tσ εi1 ,...,in 1 ,...,in

(3.8)

for some smooth functions Tσ on Pns X. Here εi1 ,...,in is the completely antisymmetric tensor. Then the general structure formula is: Theorem 3.3. Let α ∈ Ωs,s,Lep n+1,ξ,tr (X) be closed. Then α admits the following decomposition: α = T0 + dT1 (3.9) where: s,s−1 - T1 ∈ Ωs,s−2 n,ξ,2 (X) and dT1 ∈ Ωn+1,ξ,2 (X).

- T0 ∈ Ωs,s−1,1 n+1,ξ (X) has the local structure given by the formula (3.2) with the tensors Tk given by formulas of the type Tk = Pk Tσ ,

k = 2, . . . , n + 1

(3.10)

with Pk some linear differential operators which can be recursively determined. The proof relies heavily on induction on k and uses some creation and annihilation operators introduced in [16] and [17]. As a corollary of this formula we can make now the connection with the Lagrangian formalism. Namely, we have: Corollary 3.4. The expressions Tσ defined according to (3.8) verify the generalized Helmholtz equations. Proof. We write explicitly the closeness condition and select only those equations contain∅ ing the expressions Tσ,i . As a result one obtains the following set of equations 1 ,...,in ∂σI 1 Tσ2

|I|

= (−1)

X

|J|

(−1)

|J|≤s−|I|

!

|J| + |I| dJ ∂σIJ2 Tσ1 |J|

(3.11)

for |I| = s, |I| = 1, . . . , s−1 and |I| = 0 respectively. But (3.11) are exactly the Helmholtz equations (see [1], [3], [4], [9], [30]). From the theorem above we also obtain: Proposition 3.5. The decomposition (3.9) proved in the preceding theorem determines in an unique way the form T0 . Proof. It is sufficient to prove that α = T0 + dT1 = 0 =⇒ T0 = 0. Indeed, from α = 0 we have in particular Tσ,i1 ,...,in = 0 and it follows from the formula (3.10) that we have T0 = 0.

66

D.R. Grigore

As a corollary, let us denote by [α] the equivalence class of the form α ∈ Ωs,s,Lep n+1,ξ,tr (X) s,s−2 s,s−1 modulo dΩn,ξ,2 (X) ∩ Ωn+1,ξ,2 (X). Then we have [α] = 0 ⇐⇒ Tσ = 0

(3.12)

which says that the class of α is uniquely determined by the so-called Euler-Lagrange components of α: Tσ , σ = 1, . . . , m ≡ N − n. We call the globally defined class [α] of a certain form α ∈ Ωs,s,Lep n+1,ξ,tr (X) a LagrangeSouriau class. We note in closing this Section that there are two particular but important cases when the class of the form α is formed only from the form α; obviously this happens when s,s−1 dΩs,s−2 (3.13) n,ξ,2 (X) ∩ Ωn+1,ξ,2 (X) = 0. One can see that if s = 2, and n arbitrary, or n = 1, and s arbitrary, the equality above becomes an identity. So in this cases one can speak of a globally defined Lagrange-Souriau form as in [13], [21] and [14] respectively.

4.

The Associated Poincar´e-Cartan n-Form

4.1.

The General Construction

Because the Euler-Lagrange expressions Tν are at most linear in the higher-order derivatives i h . yiσ1 ,...,is it is to be expected that they follow from a Lagrangian of minimal order r ≡ s+1 2 We prove this fact in this Section. The key observation is Proposition 4.1. Let α ∈ Ωs,s q+1,ξ (X) be closed. Then one can write it locally in the form α = dθ

(4.1)

where θ ∈ Ωsn (X) is a ρs,s−1 -projectable form and has the coordinates expression: n θ=

q X

X

k=0 |I1 |,...,|Ik |≤r−1

σk σ1 ik+1 k ∧ · · · ∧ dxiq LσI11,...,I ,...,σk ,ik+1 ,...,iq ωI1 ∧ · · · ∧ ωIk ∧ dx

(4.2)

k where LIσ11,...,I |I1 |, . . . , |Ik | ≤ r − 1 are smooth functions depending on the ,...,σk ,ik+1 ,...,iq , i σ σ σ variables (x , y , yj , . . . , yj1 ,...,js ) and verify the (anti)symmetry properties in the couples (Ip , σp ) p = 1, . . . k and in the indices ik+1 , . . . , iq .

Proof. Let us use the proposition 3.2 and write α ∈ Ωs,s q+1,ξ (X) in the form (3.7). It is easy to see that α0 has the following generic form: α0 =

q+1 X

X

k=0 |I1 |+···+|Ik |≤s−1

σk σ1 ik+1 k AIσ11,...,I ∧ · · · ∧ dxiq+1 ,...,σk ,ik+1 ,...,iq+1 dyI1 ∧ · · · ∧ dyIk ∧ dx

k where AIσ11,...,I |I1 |, . . . , |Ik | ≤ s − 1 are smooth functions depending on the ,...,σk ,ik+1 ,...,iq+1 , i σ σ σ variables (x , y , yj , . . . , yj1 ,...,js ) and verify the (anti)symmetry properties in the couples

On a Generalization of the Poincar´e-Cartan Form

67

(Ip , σp ) p = 1, . . . k and in the indices ik+1 , . . . , iq . In particular the differentials dyIσ , |I| ≥ r can appear at most once in every term of the preceding sum because if there would exists a term with at least two such differentials we would get a contradiction according to the obvious inequality 2r ≥ s. So, one can write α0 = β +

X

|I|≥r

dyIσ ∧ ασI

(4.3)

where β and ασI are (q + 1) (resp. q) forms which do not contain the differentials dyIσ , |I| ≥ r. But the form α0 is closed (see proposition 3.2) so one can write it, locally, as follows: α0 = dθ0

(4.4)

with θ0 having a structure similar to (4.3): θ0 = γ +

X

|I|≥r

dyIσ ∧ θσI ;

(4.5)

here γ and θσI are q (resp. (q − 1)) forms which do not contain the differentials dyIσ , |I| ≥ r. If we substitute (4.3) and (4.5) into (4.4) we easily obtain the consistency condition: ∂νJ θσI = ∂σI θνJ ,

∀|I|, |J| ≥ r.

Applying the usual Poincar´e lemma one gets from here that θσI , following expression θσI = ∂σI λ, |I| ≥ r

where λ is a (q − 1)-form which does not contain the differentials dyIσ , substitute this into the expression (4.5) above and gets: θ0 = γ + dλ −

X

|I|≤r

dyIσ ∧ ∂σI λ − dxi ∧

|I| ≥ r have the |I| ≥ r. Now one

∂λ . ∂xi

It follows that one can take in (4.4) θ0 = γ −

X

|I|≤r

dyIσ ∧ ∂σI λ − dxi ∧

∂λ ∂xi

without affecting it. But it is clear that this form has the structure θ0 =

q X

X

k=0 |I1 |+···+|Ik |≤r−1

σk σ1 ik+1 k BσI11,...,I ∧ · · · ∧ dxiq . ,...,σk ,ik+1 ,...,iq dyI1 ∧ · · · ∧ dyIk ∧ dx

If we define θ ≡ (ρs,s−1 )∗ θ0 then we have the equality from the statement. n If q = n then we have similarly to (3.8): Li1 ,...,in = εi1 ,...,in L

(4.6)

where L is a smooth real function called the (local) Lagrangian. Now we have a result similar to theorem 2 from [32], ch. 3.2, but as we can see the proof is much simpler and do not make use of the Young diagrams technique.

68

D.R. Grigore

Theorem 4.2. Let α ∈ Ωs,s n+1,ξ (X) be closed and verifying the Lepage condition (3.4). Suppose that we have written it as in proposition 4.1. Then the following formula is true: θ = θ0 + dλ + µ

(4.7)

where θ0 ≡ n!Ldx1 ∧ · · · ∧ dxn + n also λ ∈

Ωs,r−2 n−1,ξ,1

and µ ∈

(−1)|J| dJ ∂σI11Ji1 Li1 ,...,in ωIσ11 ∧ dxi2 ∧ · · · ∧ dxin ;

X

|J|≤r−1−|I1 |

(4.8)

Ωs,s−1 n,ξ,2 .

Proof. We start from the formula (4.2) obtained before and notice that the term corresponding to k = 1 can be written as follows: LIσ11 ,i2 ,...,in ωIσ11 ∧ dxi2 ∧ · · · ∧ dxin =n

X

(−1)|J| dJ ∂σI11Ji1 Li1 ,...,in ωIσ11 ∧ dxi2 ∧ · · · ∧ dxin + dλ

|J|≤r−1−|I1 |

where λ≡

X

|I1 |≤r−2

ΛIσ11 ,i2 ,...,in−1 ωIσ11 ∧ dxi2 ∧ · · · ∧ dxin−1 .

is a (n − 1)-form with the order of contactness equal to 1. Now we define the form µ to be the sum of the terms corresponding to the contributions k ≥ 2 in the expression (4.2); this gives us a n-form with the order of contactness equal to 2. The formula from the statement follows. We prove now that the Euler-Lagrange expressions Tσ are following from a Lagrangian of order r. Proposition 4.3. In the conditions from above the following result is true: Tσ =

X

(−1)|J| dJ ∂σJ L.

(4.9)

|J|≤r

Proof. We have by direct computation ∅ Tσ,i = ∂σ Li1 ,...,in + 1 ,...,in

n 1X (−1)p dip L∅σ,i ,...,iˆ ,...,i p n 1 n p=1

(4.10)

and L∅σ1 ,i2 ,...,in = n

X

(−1)|J| dJ ∂σJi1 1 Li1 ,...,in + (δΛ)∅σ1 ,i2 ,...,in .

(4.11)

|J|≤r−1

If we substitute the second relation into the first one, we obtain the formula from the statement.

On a Generalization of the Poincar´e-Cartan Form

69

Let us comment this result. First we can say that because the expressions Tσ have the usual Euler-Lagrange expression, they verify the generalized Helmholtz equations, so we have an alternative proof of the corollary 3.4. Next, we notice that in fact we have a sharper result, namely the expressions Tσ follow from a Lagrangian of order r which is the minimal possible order. Indeed, if Tσ would follow from a Lagrangian of order strictly smaller than r, the the Euler-Lagrange equations would have the order strictly smaller than s which would contradict the basic stating point of our analysis. So, we can say that we have obtained above a form of the conjecture regarding the reduction to the minimal order in the higher-order Lagrangian formalism. We close this Subsection with the following result. Proposition 4.4. In the conditions of the proposition 4.1, let us suppose that q = n and k moreover that the tensors LIσ11,...,I |I1 | = · · · = |Ik | = r − 1, k = 1, . . . , n ,...,σk ,ik+1 ,...,in , are traceless. The we have the following formula: k LIσ11,...,I ,...,σk ,ik+1 ,...,in

=

!

n k (r − 1)! ∂ i1 I1 · · · ∂σikkIk Li1 ,...,in , r k (k + r − 1)! σ1

|I1 | = · · · = |Ik | = r − 1,

k = 0, . . . , n

(4.12)

The proof goes by induction and is based on the condition (3.6). We do not give the details but we only mention that the preceding formula appears also in [31].

4.2.

Hyper-Jacobians and Variationally Trivial Lagrangians

By definition, the hyper-Jacobians of order s are the following expressions: σ ,...,σ ,ik+1 ,...,in

JI11,...,Ik k

≡ εi1 ,...,in

k Y

yIσllil ,

|I1 | = · · · = |Ik | = s − 1,

l=1

k = 0, . . . , n. (4.13)

Then we have a result of combinatorial nature. Proposition 4.5. In the conditions of proposition 4.1 we have for k ≥ 2 and |I1 | + · · · + |Ik | ≥ r the following formulas: – for q ≤ n: k LIσ11,...,I ,...,σk ,ik+1 ,...,iq

X

|Ik+1 |=···=|Il |=r−1

k(q+1)

= (−1)

σ

q

!

X l 1 n εi1 ,...,in n! q − k k l=k

!

,...,σ ,jl+1 ,...,jq ,i1 ,...,ik ,iq+1 ,...,in

k+1 l l LIσ11,...,I ,...,σl ,jl+1 ,...,jq JIk+1 ,...,Il

;

(4.14)

– for q ≥ n: k LIσ11,...,I ,...,σk ,ik+1 ,...,iq

X

|Ik+1 |=···=|Il |=r−1

k(n+1)

= (−1)

!

q

X l 1 n εiq−n+1 ,...,iq n! q − k k l=k σ

,...,σ ,jl+1 ,...,jq ,iq−n+1 ,...,ik

k+1 l l LIσ11,...,I ,...,σl ,jl+1 ,...,jq JIk+1 ,...,Il

!

(4.15)

70

D.R. Grigore σ

,...,σ ,j

,...,j

q k+1 l l+1 l where JIk+1 are the hyper-Jacobians of order r and LIσ11,...,I ,...,σl ,jl+1 ,...,jq , ,...,Il 1 σ |I1 |, . . . , |Ik | ≤ r − 1 are tensors depending on the variables (x , y , . . . , yjσ1 ,...,jr−1 ) and with the symmetry properties in the couples (Ip , σp ) p = 1, . . . k and in the indices ik+1 , . . . , iq . If s = 2r − 1 then the formulas above are valid for k = 1 also.

Based on the preceding two results we can give a new proof of an important result from [17] namely that a(local) Lagrangian L of order r which is variationally trivial (i.e. the associated Euler-Lagrange expressions are identically zero) must be a linear combination of hyper-Jacobians of order r (however the coefficients are not completely arbitrary).

5.

Two Particular Cases

In this Section we present two particular case which do to have physical relevance. We will obtain, essentially, known results but we think that it is profitable to see how they follow as particularizations of the main framework developed in this paper. There will also be some refinements of these old results.

5.1.

The Case s = 2 and n Arbitrary

If we particularize in this case the main theorem of Section 3, we get: Theorem 5.1. Let X be a differential manifold of dimension N > n and let Pn2 X be the second order Grassmann manifold associated to it. Let α ∈ Ω2,2 n+1 (X) be closed and verifying the Lepage condition (3.4) and the tracelessness condition (3.6). Then α has the following local expression: α=

n X

k=1

Tσi00 ,...,σk ,ik+1 ,...,in ωiσ00 ∧ ω σ1 ∧ · · · ω σk ∧ dxik+1 ∧ · · · ∧ dxin +

n X

k=1

Tσ0 ,...,σk ,ik+1 ,...,in ω σ0 ∧ · · · ω σk ∧ dxik+1 ∧ · · · ∧ dxin ,

(5.1)

where: – the coefficients Tσi00 ,...,σk ,ik+1 ,...,in and Tσ0 ,...,σk ,ik+1 ,...,in are smooth functions of the σ ); variables (xi , y σ , yjσ , yjl – they are antisymmetric in the indices σ1 , . . . , σk (resp. in σ0 , . . . , σk ) and in the indices ik+1 , . . . , in ; – the following traceless condition is valid: Tσj0 ,...,σk ,j,ik+2 ,...,in = 0.

(5.2)

The form α is globally defined by these conditions. The proof follows directly from Theorem 3.3 if we take note that the condition (3.13) is true in this particular case. Let us remark that the previous tracelessness condition is weaker than the global condition Kα = 0 introduced in [21], [13]. One can express all the coefficients of the form α in terms of the expressions Tσ ; as a consequence we have α = 0 ⇐⇒ Tσ = 0. This is a stronger form of the relation (3.12). Finally we have the analogue of Theorem 4.1:

On a Generalization of the Poincar´e-Cartan Form

71

Theorem 5.2. In the conditions of the preceding theorem one can write α locally as follows: α = dθ

(5.3)

n i1 θ= ∂σ1 · · · ∂σikk Li1 ,...,in ω σ1 ∧ · · · ω σk ∧ dxik+1 ∧ · · · dxin . k! k k=0

(5.4)

where n X 1

!

Here we have Li1 ,...,in = εi1 ,...,in L

(5.5)

where L is a smooth local function depending on the variables (xi , y σ , yjσ ). It is instructive to prove all these results directly in this particular case. We remark in the end that there are no restriction on the local Lagrangian L other that the independence on the second order derivatives.

5.2.

The Case n = 1 and s Arbitrary

This case was studied in [25], [33], [35] and [14] with minor differences. As before we have: Let X be a differential manifold of dimension N = n+1 and let P1s X be the associated Grassmann manifold of order s. We will denote the local coordinates on it as follows: σ σ σ σ σ x1 7→ t and y1, . . . , 1 7→ qk i.e. we have the coordinates (t, q0 , q1 , . . . , qs ); it is natural to ωσ

| {z } k−times

put also: 1, . . . , 1 7→ ωkσ . | {z } k−times

Theorem 5.3. In the conditions described above, let α ∈ Ωs,s 2 (X) closed and verifying the tracelessness and the Lepage conditions. Then one can write it locally as follows: α = Tσ ω σ ∧ dt + with Tσ ,

X

i+j≤s−1

ij σ Tσν ωi ∧ ωjν

(5.6)

ij smooth functions depending on the variables (t, q σ , q σ , . . . , q σ ) verifying: Tσν 0 1 k ij ji Tσν = −Tνσ .

(5.7)

In this case the form α is globally defined. We define now the total derivative operator by: dt ≡

dst

s−1 X ∂ ∂ σ ≡ + qj+1 ∂t j=0 qjσ

(5.8)

and assume that ij Tσν = 0,

i + j ≥ s.

(5.9)

72

D.R. Grigore The closeness conditions for this local description is: ∂Tσ s−1,0 = 0, + 2Tνσ ∂qsν ij ∂Tσν = 0, ∂qsρ

i, j = 0, . . . , s − 1.

1 ∂Tσ j−1,0 j,0 + Tνσ + dt Tνσ = 0, 2 ∂qjν ij i−1,j i,j−1 dt Tσν + Tσν + Tσν = 0, ij ∂Tσν + ∂qkρ

jk ∂Tνρ ∂qiσ

1 2

+

ki ∂Tρσ ∂qjν

= 0,

∂Tσ ∂Tν − σ ν ∂q0 ∂q0

(5.10)

(5.11)

j = 1, . . . , s − 1,

(5.12)

i, j = 1, . . . , s − 1,

(5.13)

i, j, k = 0, . . . , s − 1,

(5.14)

00 + dt Tνσ = 0.

(5.15)

One can relax the condition (5.9). Indeed one can accept that the form α is given by the expression (5.6) with the summations restricted only to i, j ≤ s − 1. In that case one obtains from the closeness condition, beside the relations (5.10) - (5.15) above, also: s−1,j Tσν = 0,

j, . . . , s − 1.

(5.16)

Now one uses (5.12) to prove by induction that we have in fact (5.9) [35]. Let us mention two more facts. First, we have from in our particular case directly from (5.10) - (5.13) + (5.15): ij Tσν =−

1 2

s−1−i−j X

(−1)k+j

k=0

j+k ∂Tν (dt )k σ k ∂qi+j+k+1

(5.17)

so in particular we have α = 0 ⇐⇒ Tσ = 0. From this expression one can obtain the Helmholtz equations as in Corollary 3.4. They are: ! s X ∂Tσ ∂Tν k k = (−1) (dt )k−j σ , j = 0, . . . , s. (5.18) ∂qjν j ∂qk k=j We also note that the expressions (5.17) from above verify identically the system (5.10) - (5.15). Indeed, only the equation (5.14) should be investigated because the others are used completely in the induction process to obtain (5.17). But it is not very hard to prove that (5.17) verify identically (5.14) [33], [35]. Finally we give the analogue of Theorem 4.1 in this case: Theorem 5.4. In the conditions of the theorem above one can write α locally in the form α = dθ where θ=

r−1 X j=0

Ljσ ωjσ .

(5.19)

(5.20)

On a Generalization of the Poincar´e-Cartan Form

73

Here r = [(s + 1)/2] as before, Ljσ ≡

r−1−j X i=0

(−1)i (dt )i

∂L , σ ∂qi+j+1

j = 0, . . . , r − 1

(5.21)

and L is a smooth function depending only on the variables: (t, q σ , qjσ , . . . , qrσ ) which remains arbitrary for s = 2r and is constrained to be at most linear in qrσ for s = 2r − 1. The proof is elementary. We provide finally the expressions of the coefficients of the form α in terms of L: r X ∂L Tσ = (−1)j (dt )j σ (5.22) ∂qj j=0 (i.e. the usual Euler-Lagrange expressions) and ij Tσν =

6.

∂Ljν ∂Liσ − , σ ∂qi ∂qiν

i, j = 0, . . . , s − 1.

(5.23)

Conclusions

We first mention that one can use the formalism developed in this paper to analyse higher order Lagrangian systems with Noetherian symmetries, as in [13], [14], [35], [36],[37] and [38]. Indeed, if φ is a diffeomorphisms of the manifold X then one can see that its lift J s φ to Pns X leaves invariant the subspace of forms appearing in the left hand side of (3.13). This means that we can define a Noetherian symmetry as a map φ such that J s φ leaves the Lagrange-Souriau class invariant. It is to be expected that the computations will be much more difficult than in the two particular cases from the last Section. Next, we mention that it is not clear if in the general case studied here, the only restriction on the Euler-Lagrange expressions are given by the generalized Helmholtz equations, but it is reasonable to conjecture that this is true. Last, we remark that the formalism above could be generalized, in principle, to the case when the Euler-Lagrange expressions are not restricted by the condition of linearity in the highest order derivatives, trying for instance to relax the condition (3.3) i.e to factorize α to a smaller subspace.

References [1] H. F. Ahner and A. E. Moose, Covariant Inverse Problem of the Calculus of Variation, Journ. Math. Phys. 18 (1977) 1367–1373. [2] I. M. Anderson, The Variational Bicomplex (Utah State Univ. preprint, 1989, Academic Press, Boston, to appear). [3] I. M. Anderson and T. Duchamp, On the Existence of Global Variational Principles, American Journ. Math. 102 (1980) 781–868.

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[4] R. W. Atherton and G. M. Homsy, On the Existence and Formulation of Variational Principles for Nonlinear Differential Equations, Studies in Appl. Math. LIV (1975) 31–60. [5] D. E. Betounes, Extensions of the Classical Cartan Form, Phys. Rev. D 29 (1984) 599–606. [6] D. E. Betounes, Differential Geometric Aspects of the Cartan Form: Symmetry Theory, J. Math. Phys. 28 (1987) 2347–2353. [7] J. M. Ball, J. C. Currie and P. J. Olver, Null Lagrangians, Weak Continuity, and Variational Problems of Arbitrary Order, Journ. Functional Anal. 41 (1981) 135–174. [8] E. Cartan, Lec¸ons sur les Invariants Integraux (Hermann, 1922). [9] A. Galindo and L. Mart´ınez Alonso, Kernels and Ranges in the Variational Formalism, Lett. Math. Phys. 2 (1978) 385–390. [10] P. L. Garcia, The Poincar´e-Cartan Invariant in the Calculus of Variations, Symp. Math. XIV (1974) 219–246. [11] H. Goldschmit and S. Sternberg, The Hamilton-Cartan Formalism in the Calculus of Variations, Ann. Inst. Fourier 23 (1973) 203–267. [12] M. J. Gotay, A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations. I. Covariant Hamiltonian Formalism, In: Mechanics, Analysis and Geometry: 200 Years after Lagrange ((M. Francaviglia and D. D. Holms, Eds.) North-Holland, Amsterdam, 1990) 203–235. [13] D. R. Grigore, A Generalized Lagrangian Formalism in Particle Mechanics and Classical Field Theory, Fortschr. der Phys. 41 (1993) 569–617. [14] D. R. Grigore, Higher-Order Lagrangian Theories and Noetherian Symmetries, Romanian Journ. Phys. 39 (1994) 11–35. [15] D. R. Grigore, On a Generalisation of the Poincar´e-Cartan Form to Classical Field Theory; arXiv:math.DG/9801073. [16] D. R. Grigore, The Variational Sequence on Finite Jet Bundle Extensions and the Lagrangian Formalism, Diff. Geom. Appl. 10 (1999) 43–77; arXiv:dg-ga/9702016. [17] D. R. Grigore, Variationally Trivial Lagrangians and Locally Variational Differential Equations of Arbitrary Order, Diff. Geom. Appl. 10 (1999) 79–105. [18] D. R. Grigore, Higher-Order Lagrangian Formalism on Grassmann Manifolds; arXiv:dg-ga/9709005. [19] D. R. Grigore, Lagrangian Formalism on Grassmann Manifolds, In: Handbook of Global Analysis ((D. Krupka and D. Saunders, Eds.) Elsevier, 2008) 327–373.

On a Generalization of the Poincar´e-Cartan Form

75

[20] D. R. Grigore and D. Krupka, Invariants of Velocities and Higher Order Grassmann Bundles, Journ. Geom. Phys. 24 (1997) 244–266; arXiv:dg-ga/9708013. [21] D. R. Grigore and O. T. Popp, On the Lagrange-Souriau Form in Classical Field Theory, Mathematica Bohemica 123 (1998) 73–86. [22] A. Hakov´a and O. Krupkov´a, Variational first-order partial differential equations, Journ. Differential Equations 191 (2003) 67–89. [23] P. Horv´athy, Variational Formalism for Spinning Particles, Journ. Math. Phys. 20 (1979) 49–52. [24] I. Kijowski, A Finite-Dimensional Canonical Formalism in Classical Field Theory, Comm. Math. Phys. 30 (1973) 99–128. [25] L. Klapka, Euler-Lagrange Expressions and Closed Two-Forms in Higher Order Mechanics, In: Geometrical Methods in Physics (Conf. on Differential Geometry and Applications, Czechoslovakia, 1983, Univ. Brno, (D. Krupka, Ed.)). [26] J. Klein, Espaces Variationels et M´ecanique, Ann. Inst. Fourier 12 (1962) 1–124. [27] D. Krupka, A Map Associated to the Lepagean Forms of the Calculus of Variations in Fibered Manifolds, Czech. Math. Journ. 27 (1977) 114–118. [28] D. Krupka, Lepagean Forms in Higher Order Variational Theory, In: Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics (Turin, 1982, Atti della Academia delle Scienze di Torino, Suppl. al Vol. 117, 1983) 198–238. [29] D. Krupka, Geometry of Lagrangian Structures, In: Proceedings of the 14th Winter School on Abstract Analysis (Srn´ı, 1986, Suppl. ai Rendiconti del Circolo Matematico di Palermo, Serie II, no.14, 1987) 187–224. [30] D. Krupka, Variational Sequence on Finite Order Jet Spaces, In: Proceedings of the Conference Differential Geometry and its Applications (August, 1989, World Scientific, Singapore, 1990) 236–254. [31] D. Krupka, Topics in the Calculus of Variation: Finite Order Variational Sequences, In: Differential Geometry and its Applications (Proceedings Conf. Opava, 1992, Open Univ. Press) 437–495. [32] D. Krupka, The Geometry of Lagrange Structures, Preprint Series in Global Analysis, GA 7/97, Dept. of Math., Opava Univ., Czech Rep. [33] O. Krupkov´a, Lepagean 2-forms in higher order Hamiltonian mechanics, I. Regularity, Arch. Math. (Brno) 22 (1986) 97–120. [34] O. Krupkov´a, Lepagean 2-forms in Higher Order Hamiltonian Mechanics. II Inverse Problem, Arch. Math. (Brno) 23 (1987) 155–170.

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[35] O. Krupkov´a, Variational Analysis on Fibered Manifolds over One-Dimensional Bases, Ph. D. Thesis, Opava Univ., 1992. [36] O. Krupkov´a, Liouville and Jacobi theorems for vector distributions, In: Differential Geometry and Its Applications (Proc. Conf., Opava, August 1992, (O. Kowalski and D. Krupka, Eds.) Mathematical Publications 1, Silesian University, Opava, Czechoslovakia, 1993) 75–88. [37] O. Krupkov´a, Symmetries and first integrals of time-dependent higher-order constrained systems, Journ. Geom. Phys. 18 (1996) 38–58. [38] O. Krupkov´a, The Geometry of Ordinary Variational Equations (Lecture Notes in Mathematics 1678, Springer, Berlin, 1997). [39] O. Krupkov´a, Hamiltonian field theory revisited: A geometric approach to regularity, In: Steps in Differential Geometry (Proc. Colloq. Diff. Geom., Debrecen, July 2000, (L. Kozma, P. T. Nagy and L. Tam´assy, Eds.) Debrecen University, Debrecen, 2001) 187–207. [40] O. Krupkov´a, Hamiltonian field theory, Journ. Geom. Phys. 43 (2002) 93–132. [41] P. J. Olver, Hyperjacobians, Determinant Ideals and the Weak Solutions to Variational Problems, Proc. Roy. Soc. Edinburgh 95A (1983) 317–340. [42] P. J. Olver, Applications of Lie Groups to Differential Equations (Springer, 1986). [43] H. Poincar´e, Lec¸ons sur les M´ethodes Nouvelles de la M´ecanique C´eleste (GauthierVillars, Paris, 1892). [44] H. Rund, A Cartan Form for the Field Theory of Charath´eodory in the Calculus of Variations of Multiple Integrals, Lect. Notes in Pure and Appl. Math. 100 (1985) 455– 469. [45] H. Rund, Integral Formulae Associated with the Euler-Lagrange Operator of Multiple Integral Problems in the Calculus of Variation, Æquationes Math. 11 (1974) 212–229. [46] D. J. Saunders, An Alternative Approach to the Cartan Form in the Lagrangian Field Theories, J. Phys. A 20 (1987) 339–349. [47] J. M. Souriau, Structure des Systemes Dynamique (Dunod, Paris, 1970).

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 77-84

Chapter 4

K RUPKA’ S F UNDAMENTAL L EPAGE E QUIVALENT AND THE E XCESS F UNCTION OF W ILKINS D.J. Saunders∗ Palack´y University, Olomouc, Czech Republic

Abstract We recall the features of Lepage equivalents of first-order Lagrangians on jet spaces of fibred manifolds, and the corresponding structures associated with homogeneous Lagrangians. We demonstrate the correspondence between Krupka’s fundamental Lepage equivalent and a variant of the Weierstrass excess function introduced by J. E. Wilkins.

1.

Introduction

I first met Demeter Krupka in 1989 at the DGA conference; although we have not published any joint work, there are many areas of our subject where we take similar approaches, and over the years we have held many useful discussions. In this short note (which I state at the outset contains no original material) I should like to make some remarks about one of Demeter’s constructions which has fascinated me for several years: this is the Fundamental Lepage equivalent of a Lagrangian [6]. This object also appears in a paper by David Betounes [1], published a few years later but as a result of independent work. It may be described as follows. Given a Lagrangian m-form λ = Lω

(ω = dx1 ∧ · · · ∧ dxm )

in n dependent variables y α and their derivatives yjα , consider the m-form min{m,n}

X r=0

∗

1 ∂rL θα1 ∧ · · · ∧ θαr ∧ ωi1 ···ir α (r!)2 ∂yj11 · · · ∂yjαrr

E-mail address: [email protected]

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D.J. Saunders

where θαs = dy αs − ykαs dxk are contact forms, and ωi1 ···ir = ∂/∂xir ωi1 ···ir−1 is defined recursively. This m-form has the property that it is closed precisely when the original Lagrangian m-form λ is null: that is, when the Euler-Lagrange equations of the Lagrangian vanish identically. In this note, I want to relate the circle of ideas surrounding this object — which arises, of course, in the ‘exterior differential forms’ approach to the calculus of variations — to those of an apparently different approach, associated with the Weierstrass excess function.

2.

Single-Integral Problems

Consider the variational problem with fixed endpoints given in classical notation by Z b δ L(x, y α , yxα )dx = 0 . a

If we modify the integrand L by adding to it a ‘total derivative’ the problem retains the same extremals: Z b d α α α δ L(x, y , yx ) + f (x, y ) dx = 0 , dx a because

Z b a

d α f (x, y ) dx = [f (x, y α )]ba dx

is independent of the path along which the integral is taken. We may rewrite this problem in modern notation, using the language of fibred manifolds and differential forms. Take a fibred manifold π : E → R with coordinate x on R, and fibred coordinates (x, y α ) (1 ≤ α ≤ n) on E. Given a Lagrangian 1-form λ = L dx on the first-order jet manifold J 1 π, the problem is to find local sections φ such that Z b (j 1 φ)∗ λ a

takes extreme values. For any f : E → R, adding the horizontal differential dh f = (df /dx)dx does not affect the extremals, because Z b Z b 1 ∗ (j φ) dh f = d(φ∗ f ) = [φ∗ f ]ba a

a

is independent of the section φ as the endpoints are fixed. Instead of adding a horizontal differential dh f to the Lagrangian, we could add a contact form, in other words some linear combination of the forms written locally as dy α − yxα dx where yxα are the jet coordinates. Once again, we would have (j 1 φ)∗ (L dx + gα (dy α − yxα dx)) = (j 1 φ)∗ (L dx) because (j 1 φ)∗ (dy α − yxα dx) = 0. Choosing in particular the coefficient functions g α to be gα =

∂L ∂yxα

Krupka’s Fundamental Lepage Equivalent and the Excess Function of Wilkins

79

gives the Cartan form Θλ = L dx +

∂L (dy α − yxα dx) . ∂yxα

The Cartan form Θλ is a ‘Lepage form’ [7]: that is, iZ dΘλ

is a contact form

whenever the vector field Z on J 1 π is vertical over E. Furthermore, if φ is an extremal section for λ then (j 1 φ)∗ iX 1 dλ = 0 for the prolongation X 1 of any variation field X on E; and then (j 1 φ)∗ iY dΘλ = 0 for any vector field Y on J 1 π by the Lepage property and the fixed endpoints. So j 1 φ is an extremal section for Θλ . A straightforward calculation in jet coordinates shows that the Euler-Lagrange form, ∗ dΘ : which is a 2-form on J 2 π, is the 1-contact part of π2,1 λ dΘλ =

∂L d ∂L − ∂y α dx ∂yxα

(dy α − yxα dx) ∧ dx + . . . .

Thus it is obvious that, when Θλ is closed, λ is automatically null. In this present situation, the converse also holds: we can see this because a null Lagrangian is necessarily of the form λ = L dx ehere df ∂f ∂f L= = + yxα α , dx ∂x ∂y and then Θλ =

3.

∂f ∂f + yxα α ∂x ∂y

dx +

∂f (dy α − yxα dx) = df . ∂y α

Multiple-Integral Problems

The theory for multiple-integral problems, unlike that for single-integral problems, is by no means as straightforward. Two quite distinct approaches are associated with De Donder [5] and Weyl [9], and with Carath´eodory [2]. We start with the De Donder-Weyl approach. Take a fibred manifold π : E → M where i x (1 ≤ i ≤ m) are coordinates on M and (xi , y α ) (1 ≤ α ≤ n) are fibred coordinates on E; and suppose given a Lagrangian m-form λ = Lω on J 1 π, where ω = dx1 ∧ . . . ∧ dxm is a fixed volume form on the m-dimensional base manifold M (and the same symbol is used for its pull-back to J 1 π). The variational problem is to find local sections φ such that Z (j 1 φ)∗ λ C

80

D.J. Saunders

takes extreme values, where C ⊂ M is a simply-connected compact m-dimensional submanifold with boundary and φ is given on ∂C. If µ = µi ωi is a horizontal (m − 1)-form on E, adding the horizontal differential dh µ = (dµi /dxi )ω does not affect the extremals: Z Z Z 1 ∗ ∗ (j φ) dh µ = d(φ µ) = φ∗ µ C

C

∂C

is independent of the section φ as the boundary ∂C is fixed. Alternatively, instead of adding a horizontal differential dh µ, we could add a 1-contact form. Put b λ = Lω + ∂L θα ∧ ωi , Θ ∂yiα

where θα = dy α − yiα dxi are local contact forms; then

b λ = (j 1 φ)∗ (Lω) (j 1 φ)∗ Θ

b λ is often called the Cartan form, by analogy with because (j 1 φ)∗ θα = 0. The m-form Θ the single-integral case, and it is a Lepage equivalent of λ; but, unlike the single-integral case, there are other possible Lepage equivalents. b λ (it is easy to see that this will be The Euler-Lagrange form is the 1-contact part of dΘ the case for any Lepage equivalent, not merely the one we have described): ∂L d ∂L b dΘλ = − θα ∧ ω + . . . . ∂y α dxi ∂yiα b λ is closed then clearly λ is null. And for a partial converse, if a null Lagrangian is If Θ b λ is closed. But a complete converse is not available: if µi are given by λ = dh µ then Θ functions on E then i dµ λ = det ω = dh µ1 ∧ · · · ∧ dh µm dxj

b λ is not closed. is also a null Lagrangian, and now Θ A somewhat different approach was taken by Carath´eodory . Again we consider local sections φ such that Z (j 1 φ)∗ λ

C

takes extreme values; but now, instead of adding a trace, we add a determinant. For any m functions µi on E, adding det dµi /dxj ω = dh µ1 ∧ · · · ∧ dh µm

does not affect the extremals: Z Z 1 ∗ i j (j φ) det dµ /dx ω = C

∂C

φ∗ µ1 dh µ2 ∧ · · · ∧ dh µm

Krupka’s Fundamental Lepage Equivalent and the Excess Function of Wilkins

81

is independent of the section φ as the boundary is fixed. Alternatively, instead of adding a i i horizontal form det dµ /dx ω, we could add a contact form in such a way that the result was a decomposable m-form. Put eλ = Θ

then

1 Lm−1

m ^

i=1

∂L L dx + α θα ∂yi i

;

e λ = (j 1 φ)∗ (Lω) (j 1 φ)∗ Θ

e λ is another Lepage equivalent of λ, and is called the because (j 1 φ)∗ θα = 0. The m-form Θ e λ , and Carath´eodory form. As before, the Euler-Lagrange form is the 1-contact part of dΘ e λ is closed then λ is null. Furthermore, if a null Lagrangian is given by the determinant if Θ e λ is closed. λ = det dµi /dxj ω then Θ e λ is not linear, so that taking the sum of two Now, however, we have the problem that Θ determinants we obtain i i dµ2 dµ1 ω + det ω λ = det j dx dxj e λ is not closed. as another null Lagrangian, where Θ The fundamental Lepage equivalent is designed to avoid these problems; in order to do this, it is necessary to use higher derivatives of the Lagrangian function. Put min{m,n}

Θλ =

X r=0

1 ∂rL θα1 ∧ · · · ∧ θαr ∧ ωi1 ···ir ; (r!)2 ∂yjα11 · · · ∂yjαrr

then, once again, Θλ is a Lepage equivalent of λ. If λ = det dµi /dxj ω then it may be e λ . But now Θλ is linear, and dΘλ = 0 for any null lagrangian λ. shown [4] that Θλ = Θ

4.

Homogeneity

We turn now to a different, but related, type of variational problem. A ‘parametric variational problem’ is one where the submanifolds we consider are given parametrically, and are not the images of sections. The basic example of such a problem arises in Finsler geometry, where we consider 1-dimensional parametric problems: Z b δ (j 1 σ)∗ L dt = 0 a

where E is a manifold with local coordinates ua (1 ≤ a ≤ 1 + n), σ : [a, b] → E is a curve, and the Lagrangian L is a function. In order for the extremals to be independent of parametrization (but nevertheless to have a particular orientation), it is necessary and sufficient for L to be positively homogeneous of degree 1: u˙ a

∂L = L. ∂ u˙ a

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D.J. Saunders

A homogeneous Lagrangian L gives rise to its Hilbert form ΘL =

∂L a du ; ∂ u˙ a

if σ is an extremal curve for L then j 1 σ is an extremal curve for ΘL . Any Lagrangian involving x explicitly (on a jet bundle) gives rise, in a standard way, to a homogeneous Lagrangian, and the Hilbert form of this homogeneous Lagrangian then projects to the Cartan form of the original Lagrangian. Parametric m-dimensional variational problems, on a manifold E with coordinates ua (1 ≤ a ≤ m + n) and its manifold of regular m-velocities FE with coordinates (ua , uai ) (1 ≤ i ≤ m) are considered in the same way: Z δ (j 1 σ)∗ L dm t = 0 C

where L now satisfies the homogeneity condition uaj

∂L = δji L . ∂uai

There are now m Hilbert 1-forms i θL =

∂L a du , ∂uai

and any Lagrangian involving xi explicitly (on a jet bundle) gives rise to a homogeneous Lagrangian. The Cartan form from the De Donder-Weyl theory makes no sense in this context. But we can construct m ^ i eL = 1 Θ θL Lm−1 i=1

and this projects to the Carath´eodory form of the original Lagrangian [3]. If we define the tensors ∂ S i = dua ⊗ a ∂ui

then we can construct the Hilbert forms by i θL = S i dL .

We can also construct the ‘fundamental form’ ΘL =

1 1 2 S dS d · · · S m dL m!

and this projects to the fundamental Lepage equivalent of the original Lagrangian [4]. The coordinate formula for ΘL is ΘL =

1 ∂mL dua1 ∧ · · · ∧ duamm . m! ∂ua11 · · · ∂uamm 1

(1)

Krupka’s Fundamental Lepage Equivalent and the Excess Function of Wilkins

5.

83

The Weierstrass Excess Function

We return to inhomogeneous variational problems, and consider the classical question of whether an extremal is a genuine local minimum of the action functional; a useful reference for this is [8]. For questions like these, we need to distinguish between weak and strong local minima, with the distinction arising from the topology on the space of sections: if we regard sections as close if their values are close then we obtain a strong local minimum, whereas if we also require their derivatives to be close then we obtain a weak local minimum. In order to consider this question, one approach for a single-integral problem is to define the Weierstrass excess function: in classical notation, this is given by E(x, y α , yxα , z α ) = L(x, y α , z α ) − L(x, y α , yxα ) −

∂L α (z − yxα ) , ∂yxα

or in modern notation, with π1,0 (ψ(x)) = φ(x), we would write E(jx1 φ, ψ(x)) = L(ψ(x)) − L(jx1 φ) − dF L(ψ(x) − jx1 φ) . The Weierstrass necessary condition is then that, if φ is a strong minimum of the variational problem, then E(jx1 φ, ψ(x)) ≥ 0 for any section ψ of J 1 π → R projecting to φ. For multiple-integral problems the question is, as we might expect, more complicated. The De Donder-Weyl theory suggests that we should use b i , y α , yiα , ziα ) = L(xi , y α , ziα ) − L(xi , y α , yiα ) − ∂L (ziα − yiα ) E(x ∂yiα

as an excess function, whereas the Carath´eodory theory suggests 1 ∂L α i i α α α i α α α e E(x , y , yi , zi ) = L(x , y , zi ) − p−1 det δj L + α (zj − yj ) L ∂yi

b ≥ 0 nor E e ≥ 0 is, in general, a necessary instead. It turns out, however, that neither E condition for a strong minimum. Similar difficulties arise for homogeneous problems. For these problems the De Donder-Weyl theory is not appropriate. The excess function for the Carath´eodory theory becomes 1 ∂L a a a a a a e v E(u , ui , vi ) = L(u , vi ) − p−1 det L ∂uai j e ≥ 0 is not a necessary condition for a strong miniin the homogeneous case, but again E mum. Consider, for example, L = u11 u22 − u12 u21 + u31 u42 − u32 u41 :

e 6= 0. this is a null Lagrangian, but E

84

D.J. Saunders

It is, however, possible to define an excess function for homogeneous problems which does not have this problem; a suitable definition was proposed in a paper by J. E. Wilkins in 1944 [10]. This paper defines an excess function E by the formula E(ua , uai , via ) = L(ua , via ) −

1 ∂mL j1 ···jm v α1 · · · vjαmm , ε (m!)2 i1 ···im ∂uαi11 · · · ∂uαimm j1

(2)

and demonstrates that E = 0 whenever the Lagrangian is null. The structure of this formula is interesting, as it involves higher derivatives of the Lagrangian. Indeed, by comparing equations (1) and (2) it may be seen that the derivatives of the Lagrangian, and their coefficients, are combined in exactly the same arrangement here as in the fundamental form ΘL of the homogeneous Lagrangian, a form directly related to Krupka’s fundamental Lepage equivalent. It seems, therefore, that this type of structure, discovered in its two different manifestations in 1944 and 1977, is of some importance in studying the properties of null Lagrangians.

Acknowledgements The author expresses his acknowledgments to the Czech Science Foundation (grant no. 201/06/0922 for Global Analysis and its Applications).

References [1] D. E. Betounes, Extensions of the classical Cartan form, Phys. Rev. D 29 (1984), 599– 606. ¨ [2] C. Carath´eodory, Uber die Variationsrechnung bei mehrfachen Integralen, Acta Szeged. Sect. Scient. Mathem. 4 (1929), 193–216. [3] M. Crampin and D. J. Saunders, The Hilbert-Carath´eodory form for parametric multiple integral problems in the calculus of variations, Acta Appl. Math. 76 (2003), 37–55. [4] M. Crampin and D. J. Saunders, On null Lagrangians, Diff. Geom. Appl. 22 (2005), 131–146. [5] Th. De Donder, Th´eorie invariantive du calcul des variations (nouvelle e´ dit.: Paris, Gauthier-Villars, 1935). [6] D. Krupka, A map associated to the Lepagean forms in the calculus of variations, Czech Math. J. 27 (1977), 114–118. [7] Th.-H.-J. Lepage, Sur les champs g´eod´esiques du calcul des variations, Bull. Acad. Roy. Belg. Cl. Sci. V S´er 22 (1936), 716–729, 1036–1046. [8] H. Rund, The Hamilton-Jacobi theory in the calculus of variations (London: Van Nostrand, 1966). [9] H. Weyl, Geodesic fields in the calculus of variations for multiple integrals, Ann. Math. (2nd Ser.) 36 (1935), 607–629. [10] J. E. Wilkins, Multiple integral problems in parametric form in the calculus of variations, Ann. Math. (2nd Ser.) 45 (1944), 312–334.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 85-97

Chapter 5

L EPAGE C ONGRUENCES IN D ISCRETE M ECHANICS∗ 1

Antonio Fern´andez1† and Pedro L. Garc´ıa2‡ Dpto. de Matem´atica Aplicada, Universidad de Salamanca. 2 Dpto. de Matem´aticas, Universidad de Salamanca

Abstract We introduce the concept of contact 1-form in Discrete Mechanics. In terms of this concept, we express the Poincar´e-Cartan form of a discrete Lagrangian by two formulas that generalizes to the Discrete Mechanics the classical Lepage congruences. The requirement for these congruences to have similar properties to those in the continuous case leads to a special class of mechanical systems, which interest is illustrated with some examples.

1.

Introduction

One of the most beautiful geometrical doctrines of the last century is, without any doubt, that known in the literature from the beginning of the 70’s as the Hamilton-Cartan formalism of the Variational Calculus. This doctrine starts in the 30’s with the works of De Donder [4] and Weyl [18] and its generalization by Lepage [15, 16] some years after, its modern formulation in terms of jet fiber bundles of bundled manifolds was stablished 30 years after. Two key facts of this formulation were in its starts the identification of the Poincar´e-Cartan form with the “boundary term” of the formula of variation of the integral of the action for first order variational problems and the characterization of this form from a natural globalization of the classical Lepage congruences (three basic references from this first stage are: Goldschmidt and Sternberg [10], Garc´ıa [7] and Krupka [12, 13]. The generalization of this setup to higher order problems was more problematic (Garc´ıa and Mu˜noz [8], ˇ ep´ankov´a [14], Hor´ak and Kol´aˇr [11] and refFerraris and Francaviglia [6], Krupka and Stˇ erences therein), and moreover, the recent treatment of the constrained problems and its ∗

Dedicated to Demeter Krupka E-mail address: [email protected] ‡ E-mail address: [email protected] †

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Antonio Fern´andez and Pedro L. Garc´ıa

relation with the Lagrangian reduction topic (Fernandez, Garc´ıa and Rodrigo [5], Bibbona, Fatibene and Francaviglia [1], Garc´ıa and Rodrigo [9], etc.). In its simplest case –Analytical Mechanics– the setup is well known: Given a fibration p : Q˙ = Q × R → R (Q : configuration manifold of a mechanical system and R the time line), the natural space were the Lagrangian density Ldt lives is ˙ → Q˙ of the 1-jets of the local sections of p which geometry the affine bundle π : J 1 (Q) ˙ (V (Q) ˙ : bundle of the is driven by a 1-form θ with values on the vector bundle π ∗ V (Q) ˙ p-vertical vector fields of Q) defined by the rule: ˙ θjt1 s (D) = (dvert. s)(π∗ D) = π∗ D − st∗ (p∗ ◦ π∗ D) D ∈ Tjt1 s (J 1 (Q)). t ˙ In the standard set of local coordinates (t, q i , q˙i ) of J 1 (Q): X ∂ θ= dq i − q˙i dt ⊗ i . ∂q i

This differential form, known in the literature as the contact 1-form of the 1-jet bundle, allows toR obtain an intrinsic expression of the differential δs L of the integral of the action L(s) = j 1 s Ldt on a section s in the following terms: Z Z δs L = LD1 Ldt = E(s) · θ(D1 ) dt + d (j 1 s)∗ iD1 Θ , j1s

j1s

where j 1 s and D1 are the 1-jet extensions of the section s and a vector field D on Q˙ respectively, E : s → E(s) ∈ s∗ (V E)∗ is a second order differential operator (Euler-Lagrange ˙ (Poincar´e-Cartan form) and · is the duality pairing. operator), Θ is a 1-form on J 1 (Q) Even more, it is possible to characterize the Poincar´e-Cartan form by the conditions: Θ = p · θ + Ldt,

¯ θ, dΘ = E ∧

˙ with values on π ∗ V (Q) ˙ ∗ where p and E are, respectively, a function and a 1-form on J 1 (Q) ¯ are taken with respect to the duality pairing. and where the products · and ∧ These are in the case of the mechanics the so-called Lepage congruences, from where it is possible to recover the variation of the integral of the action by restricting to j 1 s the Lie derivative with respect to D1 of this congruences bearing in mind that θ(j 1 s) = 0 and y (j 1 s)∗ L1D θ = 0. Established the problem in these terms, in the present work we are going to start the study of this question in Discrete Mechanics, which we believe that it has not been treated up to now, perhaps because for the non existence of a clear concept of tangency to a discrete curve in this field. After a brief outline of the basic principles of the Discrete Mechanics on section 2. (see [17, 2, 3]), in section 3. we introduce a general notion of contact 1-form characterizing intrinsically one of them in terms of which the usual discretization of the “derivative” as an “incremental quotient” in two nearby instants is codified (Theorem 3). From this concept, the Poincar´e-Cartan 1-form of a discrete Lagrangian is characterized by a set of conditions that extend to the discrete case the classical Lepage congruences (Theorem 4). The requirement for this congruences to have similar properties to those in the continuous case leads to a special class of mechanical systems, which interest is illustrated in section 5. with some examples.

Lepage Congruences in Discrete Mechanics

2.

87

Discrete Mechanics

Let Q˙ = R × Q be the configuration space.

Definition 1. A discrete section S d is a collection of points of Q˙ sd = {(t0 , q0 ), (t1 , q1 ), . . . , (tN , qN )} or, equivalently, a point of Q˙ N +1 . Definition 2. A discrete action is a differentiable application Sd : Q˙ N +1 → R From now onwards, we are only going to consider the discrete actions of the form d

S (t0 , q0 , t1 , q1 , . . . , tN , qN ) =

N −1 X k=0

Lk (tk , qk , tk+1 , qk+1 )(tk+1 − tk ),

(1)

where Lk : Q˙ × Q˙ → R is a differentiable function that will be called local discrete Lagrangian. Definition 3. A critical discrete section of the discrete action Sd is a discrete section sd such that dsd Sd (0, D1 , . . . , DN −1 , 0) = 0,

∀Dk ∈ T(tk ,qk ) Q˙

k = 1, . . . , N − 1.

If we calculate dSd we get: d

dS =d

NX −1 k=0

i Lk (tk , qki , tk+1 , qk+1 )(tk+1

− tk ) =

X ∂L0 ∂L0 (t1 − t0 )dq0i (t1 − t0 ) − L0 dt0 + = i ∂t0 ∂q0 i N −1 X ∂Lk−1 ∂Lk + (tk − tk−1 ) + Lk−1 + (tk+1 − tk ) − Lk dtk ∂tk ∂tk k=1 ! X ∂Lk−1 ∂Lk (tk − tk−1 ) + (tk+1 − tk ) dqki + i i ∂qk ∂qk i ∂LN −1 ∂LN −1 i + (tN − tN −1 ) + LN −1 )dtN + (tN − tN −1 dqN i ∂tN ∂qN

(2)

and, hence, the following theorem holds Theorem 1. A discrete section sd = (t0 , q0 , t1 , q1 , . . . , tN , qN ) is critical of the discrete action (1) if and only if satisfies the discrete Euler-Lagrange equations  ∂Lk−1 ∂Lk  0= (tk − tk−1 ) + (tk+1 − tk ), i = 1, . . . , n  ∂qki ∂qki k = 1, . . . , N − 1.  ∂Lk ∂Lk−1  0= (tk − tk−1 ) + Lk−1 + (tk+1 − tk ) − Lk  ∂tk ∂tk

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If the Euler-Lagrange equations allow us to define (tk+1 , qk+1 ) as functions of (tk−1 , qk−1 ) and (tk , qk ), we can define a set of maps φk : Q˙ × Q˙ → Q˙ × Q˙

((tk−1 , qk−1 ), (tk , qk )) 7→ ((tk , qk ), (tk+1 , qk+1 )) that is called the discrete flux of the critical section sd . In terms of the discrete flux, the Euler-Lagrange equations are ∂Lk−1 ∂Lk (t − t ) =− (tk − tk−1 ), φ∗k k+1 k ∂qki ∂qki ∂Lk ∂Lk−1 ∗ φk − (tk+1 − tk ) + Lk = (tk − tk−1 ) + Lk−1 ∂tk ∂tk

(3)

and, hence, a discrete flux φk defines a critical section sd if and only if equations (3) hold. On the other hand, if we look at the first and last terms of the differential of the action (2), we can define a pair of 1-forms: X ∂Lk ∂Lk i − (t − t )dq + (t − t ) + L Θ− = − k+1 k k+1 k k dtk , k i ∂tk ∂q k i X ∂Lk ∂Lk + i Θ = (tk+1 − tk )dqk+1 + (tk+1 − tk ) + Lk dtk+1 i ∂tk+1 ∂qk+1 i

and, then, a 1-form (Θ− , Θ+ ) on Q˙ × Q˙ that will be called Poincar´e-Cartan form. The two 1-forms Θ− and Θ+ are related by the formula Θ+ − Θ− = d Lk (tk , qk , tk+1 , qk+1 )(tk+1 − tk )

(4)

and, if is the discrete flux of a critical section, by (3), we have that Θ− and Θ+ in two consecutive stages are φk related, that is φ∗k Θ− = Θ+ . Finally, combining this two equations we have Theorem 2 (Discrete Cartan Equation). A discrete flux φk defines a critical section sd if and only if φ∗k Θ+ − Θ+ = d φ∗k Lk (tk+1 − tk ) . (5)

Finally, observe that if we differentiate the discrete Cartan Equation (5) we get the simplecticity of the discrete flux φk Corollary 1. If φk is the discrete flux of a critical section sd , then φ∗k dΘ+ = dΘ+ . As we have done in the continuous mechanics, we have obtained the Poincar´e-Cartan form from the discrete variational principle. The next two sections of the paper will be devoted to obtain (for a particular class of Lagrangians) a characterization of the Poincar´eCartan form on a stage Q˙ × Q˙ from the local discrete Lagrangian only.

Lepage Congruences in Discrete Mechanics

3.

89

Discrete Contact 1-Forms

In order to introduce the notion of contact 1-form in the discrete realm, we need two definitions in the direct product of a manifold. Definition 4. Let M be a differentiable manifold. The conjugation of M × M is the diffeomorphism of M × M that interchanges the factors: ϕ: M × M → M × M

(x1 , x2 ) 7→ (x2 , x1 ).

The conjugation diffeomorphism acts in a natural way on the vector fields of M × M , and thus defining a 1-1 tensor on M × M , T , which local expression is the following: If (x1 , . . . , xn ) is a coordinate system in M , let (x11 , . . . , xn1 , x12 , . . . , xn2 ) be the induced coordinate system in M × M . As ϕ(x11 , . . . , xn1 , x12 , . . . , xn2 ) = (x12 , . . . , xn2 , x11 , . . . , xn1 ), it is ϕ∗ (xi2 ) = xi1 and ϕ∗ (xi1 ) = xi2 , and, hence, ∂ ∂ ∂ ∂ T = , T = . ∂xi1 ∂xi2 ∂xi2 ∂xi1 ˙ (θ− , θ+ ) verifying Definition 5. A contact 1-form on Q˙ × Q˙ is a 1-1 tensor on Q˙ × Q, 1 (θ− , θ+ ) is a projector of Q˙ × Q˙ → R × R. 2 T ∗ θ+ = θ− Local expression: Let (q 1 , . . . , q n ) be a set of local coordinates on Q, and let ˙ Then, the general (t0 , q01 , . . . , q0n , t1 , q11 , . . . , q1n ) be the induced coordinates on Q˙ × Q. − + expression of θ and θ is: θ− =

X

(dq0i + f0i dt0 ) ⊗

∂ , ∂q0i

θ+ =

X

(dq1i + f1i dt1 ) ⊗

∂ ∂q1i

i

i

and the condition T ∗ θ+ = θ− implies that f1i = f0i . Hence, the local expression of the contact 1-form is θ− =

X

(dq0i − ui dt0 ) ⊗

X

(dq1i

i

+

θ =

i

∂ , ∂q0i

∂ − u dt1 ) ⊗ i . ∂q1

(6)

i

Definition 6. The contact distribution D will be the kernel of the contact 1-form (θ− , θ+ ) D = {(D0 , D1 ) : (θ− (D0 ), θ+ (D1 )) = (0, 0)}.

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Antonio Fern´andez and Pedro L. Garc´ıa

From the local expresion of the contact 1-form (6), it follows that the contact distribution is locally spanned by the vector fields D− and D+ given by X ∂ ∂ D− = + ui i , ∂t0 ∂q0 i X ∂ ∂ D+ = + ui i . ∂t1 ∂q1 i

Theorem 3. There exists a unique contact 1-form (θ− , θ+ ) which contact distribution D is integrable and its first integrals are ϕ-invariant. Proof. From the local expression of the local vector fields D− and D+ , it follows that the integrability condition is X − + − j ∂ + j ∂ [D , D ] = D (u ) j − D (u ) j = 0 ∂q1 ∂q0 j and, hence, D− (uj ) = D+ (uj ) = 0,

j = 1, . . . , n

ui

and the functions are first integrals of the distribution. From this set of first integrals we can construct a set of n additional first integrals v0j = q0j − uj t0 ,

j = 1, . . . , n.

Now, by imposing the condition of ϕ-invariance of the first integrals, we have that, being

it should be ϕv0j = v0j , that is

ϕv0j = q1j − uj t1

q0j − uj t0 = q1j − uj t1 ,

⇒ uj =

q1j − q0j , t1 − t0

j = 1, . . . n.

From now onwards we are going to consider only this canonically determined contact 1-form.

4.

Discrete Lepage Congruences

Let L : Q˙ × Q˙ → R a discrete Lagrangian, and let (Θ− , Θ+ ) be its Poincar´e-Cartan form as defined in Section 2.. In these conditions, we have: ˙ its Poincar´eTheorem 4. If L is a first integral of the contact distribution D on Q˙ × Q, − + Cartan form (Θ , Θ ) is univocally determined by the following conditions: Θ− = p− · θ− + Ldt0 ;

Θ+ = p+ · θ+ + Ldt1 , Θ+ − Θ− = d L(t1 − t0 ) ,

(7)

where p− and p+ are functions on Q˙ × Q˙ with values on T ∗ Q0 and T ∗ Q1 respectively. And, conversely, if the Poincar´e-Cartan form of a discrete Lagrangian L satisfies the conditions (7), then L is a first integral of the contact distribution D

Lepage Congruences in Discrete Mechanics

91

Proof. Let (q 1 , . . . , q n ) be local coordinates on Q, and (t0 , q01 , . . . , q0n , t1 , q11 , . . . , q1n ) the ˙ In these coordinates, the local expression of the Poincar´einduced coordinates on Q˙ × Q. Cartan form is X ∂L ∂L + i Θ = (t1 − t0 )dq1 + (t1 − t0 ) + L dt1 = ∂t1 ∂q1i i X ∂L (t1 − t0 ) dq1i − ui dt1 + Ldt1 = i ∂q1 i X ∂L i ∂L + (t1 − t0 ) u + i ∂t1 ∂q 1 i X ∂L i ∂L + + , =p · θ + Ldt1 + (t1 − t0 ) u + ∂t1 ∂q1i i

where p+ =

X ∂L (t − t0 )dq1i i 1 ∂q 1 i

and the local expression of Θ− is X ∂L i ∂L Θ = p · θ + Ldt0 + (t1 − t0 ) , u + ∂t0 ∂q0i i −

−

−

where p− = −

X ∂L (t − t0 )dq0i . i 1 ∂q 0 i

From these two local expressions, it follows that the discrete Lepage’s congruences (7) hold if and only if the Lagrangian function L is a first integral of D. On the other hand, if a 1-form (ω − , ω + ) satisfies the Lepage’s congruences (7) we have that X X i + i p− = p− dq , p = p+ 0 i i dq1 i

i

with p− i

=ω

−

∂ , ∂q0i

p+ i

=ω

+

∂ ∂q1i

and, hence, p+ i

=ω

+

∂ ∂q1i

+ p− i =−ω

−

∂ ∂L = d L(t1 − t0 ) = i (t1 − t0 ), i ∂q1 ∂q 1 ∂ ∂ ∂L = −d L(t1 − t0 ) = − i (t1 − t0 ) ∂q0i ∂q0i ∂q1

∂ ∂q1i

−ω ∂ + ω− ∂q0i

and, then, it is ω − = Θ− ,

ω + = Θ+ .

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Antonio Fern´andez and Pedro L. Garc´ıa

Remark 1. Despite its apparent restriction, there is a rather natural way of obtaining local discrete Lagrangians that fullfil the hypothesis of Theorem (4): Starting from a continuous Lagrangian L(t, q, q), ˙ we can construct a family of local discrete Lagrangians, simply by evaluating the Lagrangian in an intermediate point of the segment joining the points (tk , qk ) and (tk+1 , qk+1 ) i Ld (tk , qki , tk+1 , qk+1 , α)

i qk+1 − qki i i . = L (1 − α)tk + αtk+1 , (1 − α)qk + αqk+1 , tk+1 − tk

Then, the condition of being Ld a first integral of the contact distribution can be expressed as: ! ∂L X i ∂L − D Ld =(1 − α) + uk i = 0, ∂t ∂q i ! ∂L X i ∂L + uk i = 0. D+ Ld =α ∂t ∂q i

On the other hand, as X ∂L dLd ∂L = (tk+1 − tk ) + (q i − qki ) dα ∂t ∂q i k+1 i

it follows that Ld is a first integral of the contact distribution if and only if dLd /dα = 0, that is, if and only if Ld is independent of the parameter α.

5.

Examples

In this section we are going to deal with discrete local Lagrangians obtained from continuous Lagrangians of the form n

L(ui , v i ) =

1X i 2 (u ) − Ψ(v 1 , . . . , v n ), 2 i=1

where ui = q˙i and v i = q i − tq˙i . The Euler-Lagrange equations of the variational problem defined by this Lagrangians are: d ∂L ∂L − i = 0, (8) i dt ∂ q˙ ∂q where

∂L ∂Ψ = ui + i t, i ∂ q˙ ∂v

∂L ∂Ψ =− i i ∂q ∂v

and, hence, equations (8) are: dui ∂Ψ d ∂Ψ +2 i + = 0. dt ∂v dt ∂v i

Lepage Congruences in Discrete Mechanics But, given that i v only:

dv i dt

93

i

= −t du dt we can express the Euler-Lagrange equations in terms of

∂Ψ dv i d 2 ∂Ψ 1 dv i d ∂Ψ +2 i + ⇔ = t 0=− t dt ∂v dt ∂v i dt dt ∂v i

(9)

∂Ψ that is, the variables v i are related by the equations v i = t2 ∂v i + λ. Let us assume that this i equations allow us to determine v as functions of t. Given that v i = q − tq, ˙ we have that Z i vi v (t) q i − tq˙i q ′ , q(t) = −t = = − dt. 2 2 t t t t2

By Theorem (4), the Poincar´e-Cartan form (Θ− , Θ+ ) in each stage (t0 , q0 , t1 , q1 ), is given by the canonical momenta X ∂L ∂uj ∂L ∂L ∂v j ∂L ∂L + pi = i (t1 − t0 ) = (t1 − t0 ) + j i = − t0 i , i j i ∂u ∂v ∂u ∂v ∂q1 ∂q1 ∂q1 j X ∂L ∂uj ∂L ∂v j ∂L ∂L ∂L − + j i = − t1 i pi = − i (t1 − t0 ) = (t1 − t0 ) i j i ∂u ∂v ∂u ∂v ∂q1 ∂q0 ∂q0 j in our particular case, ∂Ψ , ∂v i ∂Ψ i p− . i = u + t1 ∂v i On the other hand, as we have seen, the Cartan equation (5) is equivalent to i p+ i = u + t0

(10)

φ∗k Θ− = Θ+ and, hence, by (7) to

φ∗k

X i

+ φ∗k p− i,k =pi,k , X i i p+ p− i,k uk − Lk = i,k−1 uk−1 − Lk−1 .

(11)

i

Example 1 (The P free particle). This mechanical system is given by the Lagrangian function L(ui ) = 21 i (ui )2 ; and its Euler-Lagrange equations are u ¨i = 0 ⇔ ui (t) = λi t + µi ,

i = 1, . . . , n.

(12)

On the other hand, the local discrete Lagrangians defined by this Lagrangian function are Lk (uik ) = and, hence, the momenta (10) are

1X i 2 (uk ) , 2

uik =

i

− i p+ i,k = uk = pi,k

i qk+1 − qki tk+1 − tk

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Antonio Fern´andez and Pedro L. Garc´ıa

and the discrete Cartan equations (11) are φ∗k uik = uik−1 ⇔

i i − qki qk+1 q i − qk−1 = k tk+1 − tk tk − tk−1

that is, the discrete Cartan equations do not impose conditions on all the discrete variables; in particular, every time discretization t0 , . . . , tN is compatible with them. If we choose a regular discretization, whith tk+1 − tk = tNN−t0 = h, we have the following discrete equations: i i qk+1 − 2qki + qk−1 =0 h that is the standard central-point discretization (multiplied by h) of the Euler-Lagrange equations (12). Example 2.

Let us now add a linear potential to the free particle Lagrangian: L(ui , v i ) =

1 1X i 2 X (u ) − αi v i = u · ut − A · v t , 2 2 i

i

where we have wrote u = (u1 , . . . , un ), v = (v 1 , . . . , v n ), A = (a1 , . . . , an ). The Euler-Lagrange equations are, then du + 2A = 0 ⇔ q¨ = −2A. dt The solutions of the Euler-Lagrange equations can be obtained by the general procedure previously introduced: Z i v (t) βi i 2 i v (t) = t αi + βi , q (t) = −t dt = −t tαi − + γi = −αi t2 − γi t + βi . t2 t The local discrete Lagrangians defined by this Lagrangian function can be written in vectorial form as 1 Lk (uik , vki ) = uk · utk − A · vkt 2 and, then, the canonical momenta are: p+ k = uk + Atk ,

p− k = uk + Atk+1 .

The first equation of (11) is, then φ∗k (uk + Atk+1 ) = uk−1 + Atk−1 . Let us now drop the φ+ k in order to simplify the explanation; and, then, we have that uk = uk−1 − A(hk + hk−1 ),

hk = tk+1 − tk , hk−1 = tk − tk−1 .

(13)

The second equation is 1 1 + t t t t t t p− k −Lk = pk−1 −Lk−1 ⇔ 2 uk ·uk +A·(uk tk+1 +qk+1 ) = 2 uk−1 ·uk−1 +A·(uk−1 tk−1 +qk−1 )

Lepage Congruences in Discrete Mechanics

95

and, given that vk = qk − tk uk = qk+1 − tk+1 uk+1 , this equation is 1 1 uk · utk + A · utk hk = uk−1 · utk−1 − A · utk−1 hk−1 . 2 2 If we substitute uk using (13), we get, after a straightforward computation, that 1 A · At (h2k − h2k−1 ) = 0 2 that is, the time discretization should be uniform: hk = hk−1 = h =

tN − t0 N

and, returning to the first equation, we have that qk+1 − qk qk − qk−1 qk+1 − 2qk + qk−1 − = −2Ah ⇔ = −2Ah h h h that is, once again, the central-point discretization of the Euler Lagrange equations. Example 3.

Finally, we are going to consider a quadratic potential: L(ui , v i ) =

1X i 2 1X i 1 1 (u ) − v αij v j = u · ut − v · A · v t , 2 2 2 2 i

i,j

where A is a symmetric matrix. Once again, the solutions of the Euler-Lagrange equations can be easily obtained: v(t) =t2 v(t) · A + λ ⇒ v(t) = λ(I − t2 A)−1 , Z λ(I − t2 A)−1 q(t) = − t dt, t2 where λ = (λ1 , . . . , λn ) is a vector of constants and we have assumed that the matrix I − t2 A is invertible. The local discrete Lagrangians defined by this Lagrangian function can be written in vectorial form as 1 1 Lk (uk , vk ) = uk · utk − v · A · vkt 2 2 and, then, the canonical momenta are: p+ k = uk + tk vk · A,

p− k = uk + tk+1 vk · A.

Using the same conventions that in the previous example, the first equation of (11) is, then uk + tk+1 vk · A = uk−1 + tk−1 vk−1 · A and, hence uk = uk−1 Bk−1 · Bk−1 + (tk−1 − tk+1 )qk · A · Bk−1 ,

(14)

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Antonio Fern´andez and Pedro L. Garc´ıa

where we have denoted by Bk = I − tk+1 tk A. From this expression, it is easy to obtain that vk = vk−1 Bk−1 · Bk−1 . (15) The second equation is 1 1 + t t t t t p− k −Lk = pk−1 −Lk−1 ⇔ 2 uk ·uk +vk ·A·(uk tk+1 +vk ) = 2 uk−1 ·uk−1 +vk−1 ·A·qk−1 and, by substituting uk and vk using (14) and (15) it is possible to obtain tk+1 in terms of (tk−1 , qk−1 , tk , qk ); however, in this case the time distribution it is not the regular one tk = t0 + kh in general, as it is easy to see in the simplest case of being A a diagonal matrix.

Acknowledgements This work has been partially supported by the Spanish Ministerio de Ciencia y Tecnolog´ıa, project number MTM2004-01683.

References [1] E. Bibbona, L. Fatibene and M. Francaviglia, Gauge-natural parameterized variational problems, vakonomic field theories and relativistic hydrodynamics of a charged fluid, Int. J. Geom. Methods Mod. Phys. 3 (8) (2006) 1573–1608. [2] J.-B. Chen, H.-Y. Guo and K. Wu, Total variation and variational symplectic-energymomentum integrators, (2001); arXiv:hep-th/0109178. [3] J.-B. Chen, H.-Y. Guo and K. Wu, Discrete total variation calculus and Lee’s discrete mechanics, Appl. Math. Comput. 177 (1) (2006) 226–234. [4] Th. De Donder, Th´eorie invariantive du calcul des variations (Nuov. e´ d. GauthierVillars, Paris, 1935). [5] A. Fern´andez, P. L. Garc´ıa and C. Rodrigo, Lagrangian reduction and constrained variational calculus, In: Proceedings of the IX Fall Workshop on Geometry and Physics (Vilanova i la Geltr´u, 2000, Publ. R. Soc. Mat. Esp., vol. 3, R. Soc. Mat. Esp., Madrid, 2001) 53–64. [6] M. Ferraris and M. Francaviglia, On the global structure of Lagrangian and Hamiltonian formalisms in higher order calculus of variations, In: Proceedings of the International Meeting on Geometry and Physics (Florence, 1982, Bologna, Pitagora, 1983) 43–70. [7] P. L. Garc´ıa, The Poincar´e-Cartan invariant in the calculus of variations, In: Symposia Mathematica (Vol. XIV, Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973, Academic Press, London, 1974) 219–246.

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[8] P. L. Garc´ıa and J. Mu˜noz, On the geometrical structure of higher order variational calculus, In: Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics (Vol. I, Torino, 1982, vol. 117, 1983) 127–147. [9] P. L. Garc´ıa and C. Rodrigo, The momentum map in vakonomic mechanics, In: Proceedings of the XII Fall Workshop on Geometry and Physics (Publ. R. Soc. Mat. Esp., vol. 7, R. Soc. Mat. Esp., Madrid, 2004) 111–123. [10] H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble) 23 (1) (1973) 203–267. [11] M. Hor´ak and I. Kol´arˇ, On the higher order Poincar´e-Cartan forms, Czechoslovak Math. J. 33 (108) (3) (1983) 467–475. [12] D. Krupka, A geometric theory of ordinary first order variational problems in fibered manifolds. I. Critical sections, J. Math. Anal. Appl. 49 (1975) 180–206. [13] D. Krupka, A geometric theory of ordinary first order variational problems in fibered manifolds. II. Invariance, J. Math. Anal. Appl. 49 (1975) 469–476. ˇ ep´ankov´a, On the Hamilton form in second order calculus of [14] D. Krupka and O. Stˇ variations, In: Proceedings of the International Meeting on Geometry and Physics (Florence, 1982, Bologna, Pitagora, 1983) 85–101. [15] Th. Lepage, Sur les champs g´eod´esiques des int´egrales multiples, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 27 (1941) 27–46. [16] Th. Lepage, Champs stationnaires, champs g´eod´esiques et formes int´egrables. II, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 28 (1942) 247–265. [17] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer. 10 (2001) 357–514. [18] H. Weyl, Geodesic fields in the calculus of variations for multiple integrals, Ann. of Math. (2) 36 (3) (1935) 607–629.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 99-115

Chapter 6

F INITE O RDER VARIATIONAL S EQUENCES : A S HORT R EVIEW Raffaele Vitolo∗ Department of Mathematics “E. De Giorgi”, University of Lecce, Via per Arnesano, 73100 Lecce, Italy

Abstract Variational sequences are complexes of modules or sheaf sequences in which one of the maps is the Euler–Lagrange operator, i.e., the differential operator taking a Lagrangian into its Euler–Lagrange form. In this review paper we discuss variational sequences on finite order jets, with special emphasis on Krupka’s approach. We also discuss recent results on this topic as well as possible research directions.

2000 Mathematics Subject Classification. Primary 58J10, secondary 58A12, 58A20. Key words and phrases. Jet spaces, variational sequence, variational bicomplex.

Introduction In the Seventies, during a process of geometrization of the calculus of variations, it was realized that operations like passing from a Lagrangian to its Euler–Lagrange form were part of a complex, namely, the variational sequence. Foundational contributions to variational sequences are in the papers [3, 7, 12, 39, 40, 41, 42, 43, 44, 45]. Among the problems which were solved by the variational sequence was the so-called global inverse problem of the calculus of variations: given a set of Euler–Lagrange equations, the vanishing of Helmholtz conditions is a necessary and sufficient condition for the existence of a local Lagrangian for the given equations; does there exist a global Lagrangian? It was proved that the answer is in the cohomology of the variational sequence. More precisely, the cohomological obstruction for always having a global Lagrangian is the n + 1-st de Rham cohomology of the space of independent and dependent variables. ∗

E-mail address: [email protected]

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The geometric framework for variational sequences is that of jet spaces. Infinite order jet spaces were used as a rule, with the exception of [3]. There are some technical reasons for that choice: the first and most important is that on infinite order jet spaces the contact distribution is integrable and admits an intrinsic direct summand. This fact leads to much simpler computations. On the other hand, using infinite order jets one simply drops any information on the order of the objects involved in the computations. In this sense, the use of finite order jets can lead to finer results. A first approach in this sense was in [3]. In that paper the finite order variational sequence was truncated after the space of Euler–Lagrange forms. Moreover, in order to obtain the solution of the global inverse problem the authors resorted to infinite order jets. Another approach was through C-spectral sequences in [8, 9]. But it used one conjecture about the structure of contact forms (see Theorem 1.3). In [23] Krupka proved the above conjecture and was able to give the first formulation of the (long) variational sequence on finite order jets. The formulation was different from both the so-called variational bicomplex [2, 37] and the C-spectral sequence [7, 44]. The idea is rather simple: consider the de Rham complex on jets of order r. Then a subsequence of forms which yield trivial contribution to action-like functionals is defined. The quotient of the former sequence with the latter one yields the finite order variational sequence. In this paper, after a preliminary section on jet spaces and contact forms, we describe Krupka’s finite order variational sequence. In the final section we discuss the state of the research on this topic.

1.

Jet Spaces

Manifolds and maps between manifolds are C ∞ . All morphisms of fibred manifolds (and hence bundles) will be morphisms over the identity of the base manifold, unless otherwise specified. In particular, when speaking of ‘forms’ we will always mean ‘C ∞ differential forms’. We recall some basic facts on jet spaces. Our framework is a fibred manifold π : Y → X, with dim X = n, dim Y = n + m, and n, m ≥ 1. We have the vector subbundle def VY = ker T π of T Y , which is made by vectors which are tangent to the fibres of Y . For 1 ≤ r, we are concerned with the r-th jet space J r π; we also set J 0 π ≡ Y . For 0 ≤ s < r we recall the natural fibrings πr,s : J r π → J s π,

πr : J r π → X,

and the affine bundle πr,r−1 : J r π → J r−1 π associated with the vector bundle ⊙r T ∗ X ⊗J r−1 π V Y → J r−1 π. Charts on Y adapted to the fibring are denoted by (xi , y σ ). Latin indices i, j, . . . run from 1 to n and label base coordinates, Greek indices σ, τ , . . . run from 1 to m and label fibre coordinates, unless otherwise specified. We denote by (∂/∂xi , ∂/∂y σ ) and (dxi , dy σ ), respectively, the local bases of vector fields and 1-forms on Y induced by an adapted chart.

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We denoteP(symmetrized) multi-indices by capital letters: I = (i1 , . . . , in ) ∈ Nn . We def def also set |I| = k ik and I! = σ1 ! · · · σn !. The sum of a multiindex with a Latin index I + i will denote the sum of I and the multiindex (0, . . . , i, 0, . . . , 0), where 1 is at the i-th entry. def σ The charts induced on J r π are denoted by (xi , yIσ ), where 0 ≤ |I| ≤ r and y0σ = y . The local vector fields and forms of J r π induced by the fibre coordinates are denoted by (∂/∂yIσ ) and (dyIσ ), 0 ≤ |I| ≤ r, 1 ≤ i ≤ m, respectively. An r-th order (ordinary or partial) differential equation is, by definition, a submanifold S ⊂ J r π. We denote by jr s :X → J r π the jet prolongation of a section s : X → Y and by r J f : J r π → J r π the jet prolongation of a fibred morphism f : Y → Y over a diffeomorphism f¯: X → X. Any vector field ξ : Y → T Y which projects onto a vector field ξ : X → T X can be prolonged to a vector field ξ r : J r π → T J r π by prolonging its flow; its coordinate expression is well-known (see, e.g., [5, 37]). The fundamental geometric structure on jets is the contact distribution (or Cartan distribution) C r ⊂ T J r π. It is the distribution on J r π generated by all vectors which are tangent to the image jr s(X) ⊂ J r π of a prolonged section jr s. It is locally generated by the vector fields ∂ ∂ ∂ σ Di = + yI+i , , (1) σ i ∂x ∂yI ∂yJσ with 0 ≤ |I| ≤ r − 1, |J| = r. It is easy to show that this distribution is not involutive and does not admit any natural direct summand that complement it to T J r π. While the contact distribution has an essential importance in the symmetry analysis of PDE [5], in this context the dual concept of contact differential forms plays a central role. Let us denote by Fr the sheaf of smooth functions on J r π. We denote by Ωkr the sheaf of k-forms on J r π. We denote by Ω∗r the sheaf of forms of any degree on J r π. Definition 1.1. We say that a form α ∈ Ωkr is a contact k-form if (jr s)∗ α = 0 for all sections s of π. We denote by C 1 Ωkr the sheaf of contact k-forms on J r π. We denote by C 1 Ω∗r the sheaf of contact forms of any degree on J r π. Note that if k > n then every form is contact, i.e., C 1 Ωkr = Ωkr . It is obvious from the commutation of d and pull-back that dC 1 Ωkr ⊂ C 1 Ωk+1 . Morer 1 ∗ over, it is obvious that C Ωr is a sheaf of ideals (with respect to the exterior product) in Ω∗r . Unfortunately, C 1 Ω∗r does not coincide with the ideal generated by 1-forms which annihilate the contact distribution (for this would contradict the non-integrability). More precisely, the following lemma can be easily proved (see, e.g., [23]). Lemma 1.2. The sheaf C 1 Ω1r is locally generated (on Fr ) by the 1-forms σ ωIσ = dyIσ − yI+i dxi , def

0 ≤ |I| ≤ r − 1.

The above differential forms generate an ideal of Ω∗r . However, such an ideal is not differential, hence it does not coincide with C 1 Ω∗r . To realize it, the following formula can be easily proved σ dωIσ = −ωI+i ∧ dxi , (2)

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from which it follows that, when |I| = r − 1, then dωIσ , which is a contact 2-form, cannot σ contains derivatives of order be expressed through the 1-forms of lemma 1.2 because ωI+i r + 1. The following theorem is an important achievement by Krupka. It has been first conjectured in [9] (C 1 Ω-hypothesis), then proved in [23, 24]. Theorem 1.3. Let k ≥ 2. The sheaf C 1 Ωkr is locally generated (on Fr ) by the forms ωIσ ,

dωJσ ,

0 ≤ |I| ≤ r − 1,

|J| = r − 1.

We can consider forms which are generated by p-th exterior powers of contact forms. More precisely, we have the following definition. Definition 1.4. Let p ≥ 1. We say that a form α ∈ Ωkr is a p-contact k-form if it is generated by p-th exterior powers of contact forms. We denote by C p Ωkr the sheaf of p-contact k-forms on J r π. We denote by C p Ω∗r the sheaf of p-contact forms of any degree on J r π. def Finally, we set C 0 Ω∗r = Ω∗r . In other words, C p Ω∗r is the p-th power of the ideal C 1 Ω∗r in Ω∗r . Of course, a 1-contact form is just a contact form. We have the obvious inclusion C p+1 Ω∗r ⊂ C p Ω∗r . It follows that C p+1 Ω∗r is a sheaf of ideals of C p Ω∗r , hence of Ω∗r . Moreover, dC p+1 Ω∗r ⊂ C p+1 Ω∗r . Now, we would like to introduce a tool to extract from a form α ∈ Ωkr the non-trivial part (to the purposes of calculus of variations). In other words, we would like to introduce a map whose kernel is precisely the set of contact forms. Such forms yield no contribution to action-like functionals (see Remark 1.10). First of all, we observe that eq. (2) and Theorem 1.3 suggest that such a map can be constructed if we allow it to increase the jet order by 1. More precisely, it can be easily proved that the contact 1-forms ωIσ , with 0 ≤ |I| ≤ r − 1 generate a natural subbundle Cr∗ ⊂ J r π ×J r−1 π T ∗ J r−1 π ⊂ T ∗ J r π [46]. We have the following lemma (see [32, 37]). Lemma 1.5. We have the splitting J

r+1

r+1 ∗ ∗ π × T J π= J π×T X ⊕ Cr+1 , ∗ r

Jrπ

X

J r+1 π

with projections Dr+1 : J r+1 π → T ∗ X ⊗ T J r π, X

ω r+1 : J r+1 π → T ∗ J r π ⊗ V J r π, Jrπ

and coordinate expression ∂ ∂ σ + y , I+i ∂xi ∂yIσ ∂ ∂ σ = ωIσ ⊗ σ = (dyIσ − yI+i dxi ) ⊗ σ . ∂yI ∂yI

Dr+1 = dxi ⊗ Di = dxi ⊗ ω r+1

(3)

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103

Note that the above construction makes sense through the natural inclusions V J r π ⊂ T J r π and J r+1 π ×X T ∗ X ⊂ J r+1 π ×J r π T ∗ J r π, the latter being provided by T ∗ πr . From elementary multilinear algebra it turns out that we have the splitting M r+1 k ∗ r r+1 q ∗ ∗ J π ×J r π ∧ T J π = J π×∧ T X ⊕ ∧p Cr+1 . p+q=k

J r+1 π

X

Now, we observe that a form α ∈ Ωkr fulfills ∗ πr+1,r (α) : J r+1 π → ∧k T ∗ J r π ⊂ ∧k T ∗ J r+1 π, ∗ (α) can be split where the inclusion is realized through the map T ∗ πr+1,r . Hence, πr+1,r into k + 1 factors which, respectively, have 0 contact factors, 1 contact factor, . . . , k contact factors. More precisely, let us denote by Hrq the set of q-forms of the type

α : J r π → ∧q T ∗ X. We have the following proposition (for a proof, see [23, 46, 48]). Proposition 1.6. We have the natural decomposition M q ∗ πr+1,r (Ωkr ) ⊂ C p Ωpr+1 ∧ Hr+1 , p+q=k

q with splitting projections prp,q : Ωkr → C p Ωpr+1 ∧ Hr+1 defined by p+q p,q p q ∗ pr (α) = ⊙ iDr+1 ⊙ ⊙ iωr+1 ◦ πr+1,r , q

where iDr+1 , iωr+1 stand for contractions followed by a wedge product. Note that the above maps prp,q are not surjective. See [46] for more details. Definition 1.7. We say the horizontalization to be the map q hp,q : C p Ωp+q → C p Ωpr+1 ∧ Hr+1 , r

We denote by

α 7→ prp,q (α).

p,q

def p,q Ωr = h (C p Ωp+q r )

(4) 0,q

the image of the horizontalization; we say an element α ¯ ∈ Ωr to be a horizontal form. Probably the first occurrence of horizontalization is in [22]. Of course, horizontalization is just the projection on forms which have no contact factors. Note that, if q > n, then horizontalization is the zero map. In coordinates, if 0 < q ≤ n, then σh σ1 ih+1 h α = ασI11···I ∧ · · · ∧ dxiq ...σh ih+1 ···iq dyI1 ∧ · · · ∧ dyIh ∧ dx

and the coordinate expression of the horizontalization is i1 iq h h0,q (α) = yIσ11+i1 · · · yIσhh+ih ασI11···I ...σh ih+1 ···iq dx ∧ · · · ∧ dx ,

(5)

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Raffaele Vitolo

where 0 ≤ h ≤ q. The coordinate expressions of hp,q can be obtained in a similar way (see [3, 23, 24, 46]). Note that if n > 1 then the above form is not the most general polynomial in (r + 1)-st derivatives, even if q = 1. For q > 1 the skew-symmetrization in the indexes i1 ,. . . , ih yields a peculiar structure in the polynomial, in which the sums of all terms of the same degree are said to be hyperjacobians. Finally, we observe that if n = 1 then the horizontalization is surjective on the space of forms with affine coefficients with respect to r + 1-st derivatives [25]. The technical importance of horizontalization is in the next two results. Lemma 1.8. Let α ∈ Ωqr , with 0 ≤ q ≤ n, and s : X → Y be a section. Then (jr s)∗ (α) = (jr+1 s)∗ (h0,q (α))

Proposition 1.9. Let p ≥ 0. The kernel of hp,q coincides with p + 1-contact q-forms, i.e., C p+1 Ωq = ker hp,q .

For a proof of both results, see, for example, [48]. The above decomposition also affects the exterior differential. Namely, the pull-back of the differential can be split in two operators, one of which raises the contact degree by one, and the other raises the horizontal degree by one. More precisely, in view of proposition 1.6 and following [37], we introduce the maps iH : Ωkr → Ωkr+1 , iV :

Ωkr

→

Ωkr+1 ,

∗ iH = iDr+1 ◦ πr+1,r ,

iV = iωr+1 ◦

∗ πr+1,r .

(6a) (6b)

The maps iH and iV are two derivations along πr+1,r of degree 0. Together with the exterior differential d they yield two derivations along πr+1,r of degree 1, the horizontal and vertical differential def dH = iH ◦ d − d ◦ iH : Ωkr → Ωkr+1 , def dV = iV ◦ d − d ◦ iV : Ωkr → Ωkr+1 ,

It can be proved (see [37]) that dH and dV fulfill the properties d2H = d2V = 0, dH + ∗

(jr+1 s) ◦ dV = 0,

dH ◦ dV + dV ◦ dH = 0,

dV = (πrr+1 )∗ d ◦ (jr s)∗

◦ d,

∗

= (jr+1 s) ◦ dH .

(7a) (7b) (7c)

The action of dH and dV on functions f : J r Y → R and one–forms on J r Y uniquely characterizes dH and dV . We have the coordinate expressions ∂f ∂f σ i + yI+i σ dxi , (8a) dH f = Di f dx = ∂xi ∂yI dH dxi = 0,

σ dH dyIσ = −dyI+i ∧ dxi , ∂f dV f = σ ωIσ , ∂yI

dV dxi = 0 , We note that dH dyIσ = dH ωIσ .

σ dV dyIσ = dyI+i ∧ dxi ,

σ dH ωIσ = −ωI+i ∧ dxi ,

(8b) (8c)

dV ωIσ = 0.

(8d)

Finite Order Variational Sequences: A Short Review Remark 1.10. A form α ∈ Ωnr defines an action functional Z def A(s, U ) = (jr s)∗ α,

105

(9)

U

where U ⊂ X is any oriented n-dimensional submanifold of X with regular boundary. This is slightly more general than the usual notion, where a horizontal form of the type λ : J r π → ∧n T ∗ X is used (see, e.g., [37]). It follows that contact forms yield no contribution to action-like functionals. The definition (9) is a first motivation for the computations of the above section.

2.

Finite Order Variational Sequence

The first statement of a partial version of finite order variational sequence was in [3]. This finite order variational sequence stopped with a trivial projection to 0 just after the space of finite order source forms (see below). The local exactness of this sequence was proved, together with an original solution of the global inverse problem (despite the fact that in order to do that the authors used infinite order jets). For more detailed comments about that variational sequence see remark 2.8. The first formulation of a (long) variational sequence on finite order jet spaces is due to Krupka [23] (see [25] for the case n = 1). Below we will describe the main points of the approach of [23], and compare it with other approaches. In [23] a natural exact subsequence of the de Rham sequence on J r π is defined. This subsequence is made by contact forms and their differentials. Then we define the r–th order variational sequence to be the quotient of the de Rham sequence on J r π by means of the above exact subsequence. Local and global results about the variational sequence are proved using the fact that the above subsequence is globally exact and using the abstract de Rham theorem. Let us consider the sheaf of 1-contact forms C 1 Ω∗r , and denote by (dC p Ωkr )˜the sheaf generated by the presheaf dC p Ωkr . We set def 1 q Θqr = C Ωr + (dC 1 Ωq−1 r )˜ 0 ≤ q ≤ n,

def p p+n p p+n−1 Θp+n )˜ 1 ≤ p ≤ dim J r π. = C Ωr + (dC Ωr r

(10)

We observe that dC 1 Ωrq−1 ⊂ C 1 Ωqr , so that the second summand of the above first equation yields no contribution to C 1 Ωqr . The sheaves Θp+n become trivial when p + n > P , where r the value of P is computed in [23] using Theorem 1.3. Moreover, we have the following property (proved in [23]). Lemma 2.1. Let 0 ≤ k ≤ dim J r π. Then the sheaves Θkr are soft sheaves. We have the following natural soft subsequence of the de Rham sequence on J r π 0

Θ1r

d

Θ2r

d

...

d

ΘPr

d

0

Definition 2.2. The sheaf sequence (11) is said to be the contact sequence.

(11)

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Raffaele Vitolo

Theorem 2.3. The contact sequence is an exact soft resolution of C 1 Ω1r , hence the cohomology of the associated cochain complex of sections on any open subset of J r π vanishes. The above theorem is proved in [23] by first proving the local exactness of the contact sequence and then using standard results from sheaf theory (for which an adequate source is [50]). Standard arguments of homological algebra prove that the diagram in Figure 1 (p.106) is commutative, and its rows and columns are exact.

0

0

0

R

Ω0r

d

0

0

Θ1r

d

Θ2r

d

...

d

ΘPr

d

Ω1r

d

Ω2r

d

...

d

ΩPr

d EI

Ω1r /Θ1r 0

E1

Ω2r /Θ2r 0

E2

...

0

ΩPr +1

d

···0

EP −1

ΩPr /ΘPr 0

Figure 1. The r-th order variational bicomplex. Definition 2.4. The diagram in Figure 1 is said to be the r-th order variational bicomplex associated with the fibred manifold π : Y → X. We say the bottom row of the above diagram to be the r-th order variational sequence associated with the fibred manifold π : Y → X. Due to theorem 2.3 the finite order variational sequence is an exact sheaf sequence (this means that the sequence is locally exact, [50]). Hence both the de Rham sequence and the variational sequence are acyclic resolutions of the constant sheaf R (‘acyclic’ means that the sequences are locally exact with the exception of the first sheaf R). Next corollary follows by the abstract de Rham theorem. Corollary 2.5. The cohomology of the variational sequence is naturally isomorphic to the de Rham cohomology of J r π. The above finite order diagram yields a variational sequence which can be proved to be equal to the finite order variational sequence obtained from a finite order analogue of the C-spectral sequence [49]. Moreover, as one could expect, for 0 ≤ s < r pull-back via πr,s yields a natural inclusion of the s-th order variational bicomplex into the r-th order variational bicomplex. More precisely, we have the following lemma (see [23]).

Finite Order Variational Sequences: A Short Review

107

Lemma 2.6. Let 0 ≤ s < r. Then we have the injective sheaf morphism ∗ χrs : Ωks /Θks → Ωkr /Θkr , [α] 7→ [πr,s α].

Hence, there is an inclusion of the s–th order variational bicomplex into the r–th order variational bicomplex. The inclusion commutes with the operators of the variational bicomplexes of orders s and r.

Having already dealt with local and global properties of the r-th order variational sequence, we are left with the problem of representing the quotient sheaves. This problem has been independently solved by many authors in the infinite order case. We recognize two different approaches to the problem: with differential forms (see for example [41, 42]) and with differential operators [43, 44]. The restriction to finite order jets of the former approach has been developed in [46] for p = 1, p = 2, and in [20, 21] for all p. See [49] for a finite order differential operator approach. We will describe the differential forms approach. First of all, it is obvious that, for 0 ≤ q ≤ n, horizontalization provides such a representation (see [23, 46]). Proposition 2.7. Let 0 ≤ q ≤ n. Then we have the isomorphism 0,q

Hq : Ωqr /Θqr → Ωr ,

[α] 7→ h0,q (α).

The quotient differential Eq reads through the above isomorphism as Hq+1 (Eq ([α])) = Hq+1 ([dα]) = h0,q+1 (dα) = dH h0,q (α). The last equality of the above equation is the least obvious, and was first proved in [3]. σ σ The proof depends on the fact that Di yI+j = yI+j+i , and that the indexes i, j are skew0,q symmetrized in the coefficients of dH h (α) (see the coordinate expression of h0,q ). Remark 2.8. In [3] the finite order variational sequence is developed starting from the idea of finding a subsequence of forms whose order do not change under dH . The authors prove that the above property characterizes the forms which are in the image of h0,q (see also [2]). Conversely, in [23] the idea is to start with forms on finite order jets, but the result is the same up to the degree q = n. When the degree of forms is greater than n we are able to provide isomorphisms of the quotient sheaves with other quotient sheaves made with proper subsheaves. This helps both to the purpose of representing quotient sheaves and to the purpose of comparing the current approach with others, as we will see. Proposition 2.9. Let p ≥ 1. The horizontalization hp,n induces the natural sheaf isomorphism p,n

Hp+n : Ωp+n /Θp+n → Ωr /hp,n ((dC p Ωp+n−1 )˜), r r r

[α] 7→ [hp,n (α)].

The quotient differential Ep+n reads through the above isomorphism as Hp+1+n (Ep+n ([α])) = Hp+1+n ([dα]) = [hp+1,n (dα)].

108

Raffaele Vitolo For a proof, see [46, 48]. Following [2, 41, 42], let us introduce the map 1 Ip (α) = ω σ ∧ (−1)|I| DI i∂/∂uσI α p

n Ip : C p Ωpr ∧ Hrn → C p Ωp2r ∧ H2r ,

(12)

where DI stands for the iterated Lie derivative (LD1 )i1 · · · (LDn )in . We say the map Ip to be the interior Euler operator. It can be proved [2, 20, 42] that the following properties of Ip hold - Ip is a natural map, i.e., LX 2r (Ip (α)) = Ip (LX r (α)), hence Ip is a global map; n , which is of the - if α ∈ C p Ωpr ∧ Hrn then there exists a unique form β ∈ C p Ωp2r ∧ H2r p n−1 type β = dH γ with γ ∈ C p Ω2r−1 ∧ H2r−1 , such that

α = I(α) + β.

(13)

Remark 2.10. The above form γ is not uniquely defined, in general. For p = 1, if the order of α is 1 it is easily proved that γ is uniquely defined; if the order of α is 2 then there exists a unique γ fulfilling a certain intrinsic property; if the order is 3 it is proved in [16, 17] that no natural γ of the above type exists. However, suitable linear connections on M and on the fibres of π : E → M can be used to determine a unique γ. See [1, 2] for the case of p > 1. n−1 It follows from the above theorem that if γ ∈ C p Ωp2r−1 ∧ H2r−1 then Ip (dH γ) = 0, so 2 that Ip = Ip .

Theorem 2.11. We have the isomorphism Ωp+n /Θp+n → Vrp , r r

[α] 7→ Ip (Hp+n ([α])),

n is a suitable subspace (see [46] for a characterization for where Vrp ⊂ C p Ωp2r+1 ∧ H2r+1 p = 1, p = 2).

For a proof, see [46] (p = 1, p = 2) and [20, 21] for any p. The above theorem also mean that, despite the fact that the denominator in proposition 2.9 is made by forms which are locally total divergences, only global divergences really matter. We say the elements of V p to be the p-th degree variational forms; for p = 1 they are also known as source forms. The map Ip+1 allows us to represent the differentials Ep+n through forms: Ip+1 (Hp+1+n (Ep+n ([α]))) = Ip+1 (Hp+1+n ([dα])).

(14)

From the coordinate expression of Ip it follows that En is just the Euler–Lagrange opdef erator and E1+n is just the Helmholtz operator. In fact, let ν = dx1 ∧ · · · ∧ dxn . Then, if 0,n λ ∈ Ωr , then λ = h0,n (α) = Lν, where L is a function with polynomial structure in r + 1-st order derivatives as in (5). Now we can use (14) on α, but if α is not known the computational problem of finding it can be technically difficult in principle. On the other hand, we can use the commutativity of the inclusion of Lemma 2.6 with the operators Ep+n and consider λ ∈ Ωnr+1 . Then h0,n (λ) = λ and En (λ) is the standard Euler–Lagrange

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operator on the r + 1-st order Lagrangian λ. A similar reasoning proves that E1+n coincide with the Helmholtz operator. A different, computational approach to the problem of the representation of quotients is presented in [13, 14]. A further approach to the problem of representation appeared in [30] for the case n = 1. Here the concept of Lepagean equivalent is introduced in full generality (older version of this concept can be found e.g., in [22], with references to older foundational works). Namely, let α ∈ Ωp+n . Then a Lepage equivalent of [α] ∈ Ωp+n /Θp+n is a differential r r r p+n form β ∈ Ωr such that hp,n (β) = hp,n (α),

hp+1,n (dβ) = Ip+1 (hp+1,n (dα)).

The most important example of a Lepagean equivalent is the Poincar´e–Cartan form of a Lagrangian (see, e.g., [22]).

3.

Some Related Problems

In this section we will briefly describe what are the most recent results which involve finite order variational sequences.

3.1.

Variationally Trivial Lagrangians

A variationally trivial Lagrangian is an element [α] ∈ Ωnr /Θnr such that En ([α]) = 0. If [α] is a variationally trivial Lagrangian, then by the local exactness of the variational sequence we have h0,n (α) = dH (h0,n−1 (β)) with [β] ∈ Ωn−1 /Θn−1 a local form. A r r n−1 n−1 global horizontal n − 1-form [β] ∈ Ωr /Θr such that [α] = dH [β] exists if and only if [α] induces the zero cohomology class in the variational sequence. A refinement of this result is the following theorem. Theorem 3.1. Let λ : J r π → ∧n T ∗ X induce a variationally trivial Lagrangian [λ]. Then, locally, λ = dH µ, where µ = h0,n−1 (α) and α ∈ Ωn−1 r−1 . In other words, λ = h0,n (dα), hence λ is the representative of a class En−1 ([α]) = r−1 [dα] ∈ Ωn−1 r−1 /Θn−1 . This means that λ depends on r-th order derivatives through hyperjacobians. This result has been proved in [3], [4] (here the proof is for the special case when the Lagrangian does not depend on (xi )), [14, 29] (here the proof uses the finite order variational sequence). See also [27] for another approach to the problem. Of course, the result is sharp: the order cannot be further lowered.

3.2.

Locally Variational Source Forms

such that E1+n ([α]) = 0. /Θn+1 A locally variational source form is an element [α] ∈ Ωn+1 r r If [α] is a locally variational source form, then by the local exactness of the variational sequence [α] is the Euler–Lagrange expression of a local Lagrangian, i.e., [α] = En ([β]) with [β] ∈ Ωnr /Θnr . A global Lagrangian [β] ∈ Ωnr /Θnr such that [α] = En ([β]) exists if and only if [α] = 0 ∈ H n+1 (Y ).

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The previous result is sharp with respect to the order [23, 46]. However, it can be very difficult to check that a source form is in the space Ωn+1 /Θn+1 . A result proved in [2] is r r (r) helpful in this sense. Let y denote all derivative coordinates of order r on a jet space. Let f ∈ C ∞ (J 2r π), and suppose that f (xi , y (0) , . . . , y (r) , ty (r+1) , t2 y (r+2) , . . . , tr y (2r) ) is a polynomial of degree less than or equal to r in y (s) , with r + 1 ≤ s ≤ 2r. Then f is said to be a weighted polynomial of degree r in the derivative coordinates of order r + 1 ≤ s ≤ 2r. Theorem 3.2. Let [∆] be a locally variational source form, with ∆ : J 2r π → C0∗ ∧∧n T ∗ X. Suppose that the coefficients of ∆ are weighted polynomials of degree less than or equal to r. Then ∆ = E(λ), where λ : J r π → ∧n T ∗ X. Again, the result is sharp with respect to the order of the jet space where the Lagrangian is defined. The above theorem is complemented in [2] by a rather complex algorithm for building the lowest order Lagrangian. This algorithm is an improvement of the well-known Volterra Lagrangian Z 1

L=

0

y σ ∆σ (xi , tyIτ )dt

for a locally variational source form ∆. In fact, the above Lagrangian is defined on the same jet space as ∆. The finite order variational sequence yields another method for computing lower order Lagrangians, provided we know that ∆ = [α] ∈ Ωn+1 /Θn+1 . Namely, we r r n+2 apply the contact homotopy operator to the closed form dα ∈ Θr , finding β ∈ Θn+1 r such that dβ = dα. Using the (standard) homotopy operator we find γ ∈ Ωnr such that def 0,n dγ = β − α, and λ = h (γ) is the required Lagrangian. Of course, the most difficult point is to invert the representation of quotients in the variational sequence, i.e., to find a least order α such that ∆ = [α]. The above theorem does not exhaust the finite order inverse problem. A locally variational source form ∆ on J 2r π seems to have a definite form of the coefficients with respect to its derivatives of order s, with r + 1 ≤ s ≤ 2r. It is an open problem to determine such a structure, e.g. prove that such forms always lie in Ωn+1 /Θn+1 for a minimal value of s; a s s least order Lagrangian would follow from the local exactness of the variational sequence. Finally, we recall that recently some geometric results on variational first-order partial differential equations have been obtained in [15]. Such equations arise in multisymplectic field theories.

3.3.

Contact Elements

Let Y be an n + m-dimensional manifold, and x ∈ Y . We say that two n-dimensional submanifolds L1 , L2 such that x ∈ L1 ∩ L2 are r-equivalent if they have a contact of order r at x. It is possible to choose a chart of Y at x of the form (xi , y σ ), 1 ≤ i ≤ n, 1 ≤ i ≤ m, where both L1 and L2 can be expressed as graphs y σ = f1σ (xi ), y σ = f2σ (xi ). Then the contact condition is the equality of the derivatives of the above functions f1 , f2 at x up to the order r. This is an equivalence relation whose quotient set is J r (Y, n), the manifold of r-th order n-dimensional contact elements. This construction was first formalized in [10], and is also known as r-th order jet space of n-dimensional submanifolds of Y [45] or extended jet bundle [34]). If Y is endowed with a fibring π, then J r π is the open and dense subspace of J r (Y, n) which is made by submanifolds which are transverse to the fibring at

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a point (which, of course, can be locally identified with the images of sections, hence with local sections themselves). Of course, manifolds of contact elements have a contact distribution, hence a variational sequence can be formulated through the C-spectral sequence [7, 43, 44]. Manifolds of contact elements can also be seen as jets of parametrizations of submanifolds (i.e., jets of local n-dimensional immersions) up to the action of the reparametrization group [18]. In this setting another approach to the variational sequence is [38]. In [33] the finite-order C-spectral sequence on the manifold of contact elements is computed. Research based on Krupka’s approach on a variational sequence on finite order contact elements is in progress [31]. Another interesting research topic is the development of finite order variational structures on differential equations, i.e. submanifolds of jet spaces. This would possibly lead to a classification of their conservation laws of a certain order [35].

3.4.

Variational Sequence and Symmetries

The Lie derivative of variational forms is interesting for the determination of symmetries of Lagrangians and source forms. However, the result of a Lie derivative with respect to a prolonged vector field is a form which, in general, contains dH -exact terms. For this reason it is natural to use a new operator, the variational Lie derivative, which is defined up to dH -exact terms. Such a formula first appeared in [45] (‘infinitesimal Stokes’formula’) in the infinite order formalism. The finite order case has been dealt with in [11, 28]. See also [6] for symmetries of source forms which are locally but not globally variational. This topic has clear connections with Noether’s theorem, for which we invite the reader to consult the above literature.

3.5.

Further Topics

We already mentioned that other approaches to variational sequences exist in literature, mostly on infinite order jets. It can be proved that there exists an infinite order analogue of Krupka’s r-th order variational bicomplex [47]. This is defined in view of Lemma 2.6 via a direct limit of the injective family of r-th order variational bicomplexes. Nonetheless the direct limit infinite order bicomplex will be a bicomplex of presheaves, because gluing forms defined on jets of increasing order provides ‘forms’ which are only locally of finite order. The C-spectral sequence on jets of fibrings yields an infinite order variational sequence [7, 43, 44]. See [26, 48] for a comparison with Krupka’s approach and [49] for some finite order C-spectral sequence computations. In [36] the relationship between a part of the finite order variational sequence and the Spencer sequence are stressed. This relationship was already explored in [43, 44] in the case of infinite order jet spaces.

Acknowledgements It is a pleasure for me to acknowledge Professor Demeter Krupka. His hospitality and his inspiring seminars during my Ph.D. studies have been an invaluable contribution to my

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mathematical education. This research has been supported by PRIN 2005/2007 “Simmetrie e supersimmetrie classiche e quantistiche”, by the section GNSAGA of the Istituto Nazionale di Alta Matematica http://www.altamatematica.it, and by the Dipartimento di Matematica “E. De Giorgi” of the University of Salento.

References [1] R. Alonso Blanco, On the Green-Vinogradov formula, Acta Appl. Math. 72 (1-2) (2002) 19–32. [2] I. M. Anderson, The Variational Bicomplex (book preprint, technical report of the Utah State University, 1989); available at http://www.math.usu.edu/˜fg mp/). [3] I. M. Anderson and T. Duchamp, On the existence of global variational principles, American J. of Math. 102 (1980) 781–868; http://www.jstor.org/. [4] J. M. Ball, J. C. Currie and P. J. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal. 41 (1981) 135–174. [5] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor′ kova, I. S. Krasil′ shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky and A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics ((I. S. Krasil′ shchik and A. M. Vinogradov, Eds.) Amer. Math. Soc., 1999). [6] J. Brajerˇc´ık and D. Krupka, Variational principles for locally variational forms, J. Math. Phys. 46 (2005) 052903. [7] P. Dedecker, On applications of homological algebra to calculus of variations and mathematical physics, In: Proceedings of the IV international colloquium on differential geometry (Santiago de Compostela, Universidad de Santiago de Compostela, 1978, Cursos y Congresos de la Universidad de Santiago de Compostela 15) 285– 294. [8] S. V. Duzhin, C-spectral sequence on the manifold J 1 M , Uspekhi Math. Nauk 38 (1983) 165–166 (in Russian); English translation: Russian Math. Surveys 38 (1) (1983) 179–181. [9] S. V. Duzhin, C-spectral sequence on manifolds of jets of finite order, preprint VINITI UDK 514.763.8 (1982) (in Russian). [10] C. Ehresmann, Les prolongements d’une variete differentiable. IV. Elements de contact et elements d’enveloppe, C. R. Acad. Sci. Paris 234 (1952) 1028–1030. [11] M. Francaviglia, M. Palese and R. Vitolo, Symmetries in finite order variational sequences, Czech. Math. J. 52 (127) (2002), 197–213; http://poincare.unile.it/vitolo.

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[12] I. M. Gel′ fand and L. A. Dikii, Asymptotic behaviour of the resolvent of Sturm– Liouville equations and the algebra of the Kortexeg–De Vries equations, Uspekhi Mat. Nauk 30 (5) (1975) 67–100 (in Russian); Russian Math. Surveys 30 (5) (1975) 77–113 (English translation). [13] D. R. Grigore, The Variational Sequence on Finite Jet Bundle Extensions and the Lagrangian Formalism, Diff. Geom. Appl. 10 (1999) 43–77. [14] D. R. Grigore, Variationally trivial Lagrangians and locally variational differential equations of arbitrary order, Diff. Geom. Appl. 10 (1999) 79–105. [15] A. Hakov´a and O. Krupkov´a, Variational first-order partial differential equations, J. Diff. Equat. 191 (2003) 67–89. [16] I. Kol´aˇr, A geometrical version of the higher order Hamilton formalism in fibred manifolds, J. Geom. Phys. 1 (2) (1984) 127–137. [17] I. Kol´aˇr, Natural operators related with the variational calculus, In: Proc. Conf. Diff. Geom. Appl. 1992 (Silesian Univ. Opava, 1993) 461–472. [18] I. Kol´aˇr, P. Michor and J. Slov´ak, Natural Operations in Differential Geometry (Springer-Verlag, 1993); http://www.emis.de/monographs/index.html. [19] I. S. Krasil′ shchik and A. M. Verbovetsky, Homological methods in equations of mathematical physics, Open Education and Sciences, Opava (Czech Rep.) 1998; arXiv:math.DG/9808130. [20] M. Krbek and J. Musilov´a, Representation of the variational sequence by differential forms, Rep. Math. Phys. 51 (2/3) (2003) 251–258. [21] M. Krbek and J. Musilov´a, Representation of the variational sequence by differential forms, Acta Appl. Math. 88 (2) (2005) 177–199. [22] D. Krupka, Some geometric aspects of the calculus of variations in fibred manifolds, Folia Fac. Sci. Nat. UJEP Brunensis, Brno University, 14 (1973); arXiv:mathph/0110005. [23] D. Krupka, Variational sequences on finite order jet spaces, In: Proc. Conf. Diff. Geom. Appl. 1989 (World Scientific, New York, 1990) 236–254. [24] D. Krupka, The contact ideal, Diff. Geom. Appl. 5 (1995) 257–276. [25] D. Krupka, Variational sequences in mechanics, Calc. Var. 5 (1997) 557–583. [26] D. Krupka, Variational sequences and variational bicomplexes, In: Differential Geometry and Applications (Proc. Conf., (I. Kol´aˇr, O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Brno, August 1998, Masaryk Univ., Brno, Czech Republic, 1999) 525– 531; http://www.emis.de/proceedings. [27] D. Krupka, The total divergence equation, Lobachevskii Journal of Mathematics 23 (2006) 71–93.

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[28] D. Krupka, O. Krupkov´a, G. Prince and W. Sarlet, Contact symmetries and variational sequences, In: Differential Geometry and Its Applications (Proc. Conf., (J. Bureˇs, O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Prague, August 2004, Charles University in Prague, Czech Republic, 2005) 605–615. [29] D. Krupka and J. Musilov´a, Trivial Lagrangians in Field Theory, Diff. Geom. and Appl. 9 (1998) 293–505. ˇ enkov´a, Variational sequences and Lepage forms, In: Differen[30] D. Krupka and J. Sedˇ tial Geometry and Its Applications (Proc. Conf., (J. Bureˇs, O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Prague, August 2004, Charles University in Prague, Czech Republic, 2005) 617–627. [31] D. Krupka and Z. Urban, Differential invariants and higher order Grassmann bundles, In: Differential Geometry and its Applications (Proc. 10th Int. Conf. on Diff. Geom. and Appl., Olomouc 2007, World Scientific, Singapore, 2008) 463–473. [32] L. Mangiarotti and M. Modugno, Fibered Spaces, Jet Spaces and Connections for Field Theories, In: Proc. of the Int. Meeting on Geometry and Physics (Pitagora Editrice, Bologna, 1983) 135–165. [33] G. Manno and R. Vitolo, Variational sequences on finite order jets of submanifolds, In: Proc. of the VIII Conf. on Diff. Geom. and Appl. (Opava 2001, Czech Republic, http://www.emis.de/proceedings); see also the longer preprint arXiv:math.DG/0602127. [34] P. J. Olver, Applications of Lie Groups to Differential Equations (GTM 107, 2nd edition, Springer 1993). [35] P.J. Olver, private communication (2007). [36] J. F. Pommaret, Spencer sequence and variational sequence, Acta Appl. Math. 41 (1995) 285–296. [37] D. J. Saunders, The Geometry of Jet Bundles (Cambridge Univ. Press, 1989). [38] D. J. Saunders, Homogeneous variational complexes and bicomplexes; arXiv:math.DG/0512383. [39] F. Takens, A global version of the inverse problem of the calculus of variations, J. Diff. Geom. 14 (1979) 543–562. [40] W. M. Tulczyjew, Sur la diff´erentielle de Lagrange, C. R. Acad. Sc. Paris, s´erie A 280 (1975) 1295–1298. [41] W. M. Tulczyjew, The Lagrange Complex, Bull. Soc. Math. France 105 (1977) 419– 431. [42] W. M. Tulczyjew, The Euler–Lagrange Resolution, In: Internat. Coll. on Diff. Geom. Methods in Math. Phys. (Aix–en–Provence, 1979; Lecture Notes in Mathematics, n. 836, Springer–Verlag, Berlin, 1980) 22–48.

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[43] A. M. Vinogradov, On the algebro–geometric foundations of Lagrangian field theory, Soviet Math. Dokl. 18 (1977) 1200–1204; http://diffiety.ac.ru/. [44] A. M. Vinogradov, A spectral sequence associated with a non–linear differential equation, and algebro–geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978) 144–148; http://diffiety.ac.ru/. [45] A. M. Vinogradov, The C–Spectral Sequence, Lagrangian Formalism and Conservation Laws I and II, J. Math. Anal. Appl. 100 (1) (1984); http://diffiety.ac.ru/. [46] R. Vitolo, Finite order variational bicomplexes, Math. Proc. of the Camb. Phil. Soc. 125 (1998) 321–333; http://poincare.unile.it/vitolo; a more detailed version is available at arXiv:math-ph/0001009. [47] R. Vitolo, A new infinite order formulation of variational sequences, Arch. Math. Un. Brunensis 4 (34) (1998) 483–504; http://poincare.unile.it/vitolo. [48] R. Vitolo, On different geometric formulations of Lagrangian formalism, Diff. Geom. and its Appl. 10 (3) (1999) 225–255; updated version at http://poincare.unile.it/vitolo. [49] R. Vitolo, Finite order formulation of Vinogradov C–spectral sequence, Acta Appl. Math. 70 (1-2) (2002) 133–154; updated version at http://poincare.unile.it/vitolo. [50] R. O. Wells, Differential Analysis on Complex Manifolds (GTM, n. 65, Springer– Verlag, Berlin, 1980).

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 117-128

Chapter 7

C ONCATENATING VARIATIONAL P RINCIPLES AND THE K INETIC S TRESS -E NERGY-M OMENTUM T ENSOR Marco Castrill´on L´opez1∗, Mark J. Gotay2† and Jerrold E. Marsden3‡ Departamento de Geometr´ıa y Topolog´ıa, Facultad de Ciencias Matem´aticas, Universidad Complutense de Madrid, 28040 Madrid, Spain, 2 Department of Mathematics, University of Hawai‘i, Honolulu, Hawai‘i 96822, USA, 3 Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, California 91125, USA

1

Abstract We show how to “concatenate” variational principles over different bases into one over a single base, thereby providing a unified Lagrangian treatment of interacting systems. As an example we study a Klein–Gordon field interacting with a mesically charged particle. We employ our method to give a novel group-theoretic derivation of the kinetic stress-energy-momentum tensor density corresponding to the particle.

1.

Introduction and Setup

Let us recall the geometric setting of a classical variational principle [3]: We are given a fibration Y → X, with dim X = n + 1, and we wish to extremize an action of the form Z S(ψ) = L(j 1 ψ) X

where ψ : X → Y is a section and L : J 1 Y → Λn+1 X is a specified Lagrangian density.1 ∗

E-mail address: [email protected] E-mail address: [email protected] ‡ E-mail address: [email protected] 1 For simplicity we consider only first order theories. We also ignore technical issues and proceed formally. †

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One commonly encounters several (say K) such variational principles simultaneously, for instance when one studies the Newtonian dynamics of a swarm of charged particles (in a background electromagnetic field), or the interaction between Dirac and Yang–Mills fields. In the cases cited, the relevant fibrations have the form Yi → X for the ith variational principle; the key point being that each fibration has the same base X. To combine these variational principles into a single principle is a straightforward matter: one builds the fiber product Y1 ×X · · · ×X YK → X, and then on the first jet of this bundle one takes as the Lagrangian density an expression of the form L1 + · · · + LK + Lint for some interaction terms Lint . It is less clear how to deal with variational principles with disparate bases, that is, fibrations Yi → Xi in which the Xi are all different. A simple example is a nucleon moving in a dynamic Klein–Gordon field. (Here the configuration bundle for the nucleon is X ×R → R, where X is 4-dimensional spacetime and R is the material world line of the nucleon. The fibration for the Klein–Gordon field is R × X → X, sections of which are scalar fields on spacetime.) Even if the bases are identical, it may be desirable to distinguish them. This is the case, for instance, in relativistic multiparticle systems (cf. [1]), when one wants to parametrize each particle’s trajectory by its own proper time, as opposed to a single “universal” time. In this context of disparate bases, one standard way to proceed is as follows. Construct an action functional using sections ψi : Xi → Yi for the ith bundle by setting S(ψi , . . . , ψK ) =

K Z X i=1

Xi

1

Li (j ψi ) +

Z

X1 ×···×XK

Lint (j 1 ψ1 , . . . , j 1 ψK ).

(1.1)

Then varying these fields ψi , one obtains the Euler–Lagrange equations for the problem. See equation (2.1) for a specific example. However, while producing the Euler–Lagrange equations, this approach has the unsatisfactory feature of not yielding a field theory in the usual sense, in which the fields are sections of a single bundle and which has a well-defined Lagrangian density. This or some other formalism is needed if one wishes to tap into the machinery of multisymplectic geometry, multimomentum maps, stress-energy-momentum (“SEM”) tensors, and constraint theory, etc. To concatenate variational principles with disparate bases in such a way as to recapture a genuine field theory, we proceed as follows. To begin, construct the product bundle Y1 × · · · × YK → X1 × · · · × XK , which we denote Y → X for short. In agreement with experience we restrict attention to product sections of this bundle of the form ψ = (ψ1 , . . . , ψK ), where each ψi is a section of Yi → Xi . With ψ = (ψ1 , . . . , ψK ) such a section, j 1 ψ(x) = j 1 ψ1 (x1 ), . . . , j 1 ψK (xK ) where x = (x1 , . . . , xK ). Denote by J¯1 Y the subbundle of J 1 Y consisting of all such jets; equivalently, J¯1 Y = J 1 Y1 × · · · × J 1 YK . Given Lagrangian densities Li on the jet bundles J 1 Yi , it is simple enough to lift them to maps, still denoted by Li , on the concatenated jet bundle J¯1 Y by composing with projections: j 1 ψ(x) 7→ Li j 1 ψi (xi ) .

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But how do we concatenate these Li into a single Lagrangian density? Even ignoring interaction terms, we cannot just add the Li as they take values in different spaces, viz. Λni +1 Xi and so need not be forms of equal rank. The trick is to “suspend” the Li : J¯1 Y → Λni +1 Xi to maps J¯1 Y → ΛN +K X, where N = n1 +· · ·+nK , by inserting suitable tensor densities in the Li to “even out” their ranks in the target. First, we pull Li back via the projection X → Xi to an (ni + 1)-form on X. Secci · · · × XK . Now in ond, for each i choose scalar densities Di of weight 1 on X1 × · · · X n +1 Li = Li d i xi the coefficient Li transforms as a scalar density of weight 1 on Xi , so the coefficient in ni +1x ∧ · · · ∧ d nK +1x = ±L D d N +Kx L¯i := Li d ni +1xi ∧ Di d n1 +1x1 ∧ · · · ∧ d\ i i i K

will also transform as a scalar density of weight 1 on X under the subgroup Diff(X1 ) × · · · × Diff(XK ) ⊂ Diff(X) (which is sufficient for our purposes). The densities Di are to be chosen by hand, depending on the precise structure of the system; see the examples in §§2. and 3.. Thus modified, we may assemble L¯1 + · · · + L¯K into a map L¯ : J¯1 Y → ΛN +K X. Interaction terms, which are typically defined over several of the bases Xi (again, see the following examples) are treated similarly. Finally, it is straightforward to deal with composite situations in which some of the bases are identical and others are not. Ultimately, the specific choice of the Di will not matter as long as Z Z N +K Li d ni +1x Li Di d x= Xi

X

for each i, that is, the concatenated action reduces to the original action. Specifically, this means that Z ¯ 1 ψ) = S(ψ) L(j X

where the right hand side is given by (1.1). In particular, the Euler–Lagrange equations remain unaltered when the Lagrangian L¯ is used in place of the action functional (1.1). Once we have a total Lagrangian density in hand (albeit possibly a distributional one), we may proceed in the usual fashion. Thus we may compute the equations of motion and various geometric objects, such as SEM tensors. To extract physical information from these objects, however, it will normally be necessary to “project” them from X to some Xi or products thereof; this projection is accomplished by integration over the remaining Xj . Rather than continuing to try to describe the procedure in generality, it is more instructive to illustrate it via a simple example. (It really is easier done than said!) In §2. we apply this method to a system consisting of a Klein–Gordon field interacting with a mesically charged particle. (Think of a pion field interacting with a nucleon.) Beyond illustrating concatenation, this example has interesting features which are worth elucidating. In particular, we study the SEM tensor density of this system. Its computation, following

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[5], is interesting in that it naturally produces the Minkowski, or kinetic, SEM tensor for a moving particle as a matter of course. To our knowledge, this SEM tensor has never been derived via a Lagrangian from first principles; it has always been inserted into the formalism in an ad hoc manner. An important point therefore is that our method is not merely a ‘tidy’ means of packaging variational principles; it is capable of providing, in an entirely straightforward fashion, quantities which otherwise cannot be obtained except in makeshift ways. Finally in §3. we briefly indicate some other contexts in which our results should be useful.

2.

Motion of a Mesically Charged Particle in a Klein–Gordon Field

Let X be an oriented spacetime with metric G. We consider a real Klein–Gordon field φ : X → R of mass M interacting with a particle of mass m and mesic charge ε. The particle’s trajectory in spacetime (or “placement field”) is z : R → X. The base for the system is thus X × R, the second factor being thought of as a time axis,2 and the configuration bundle Y is then (R × X) × (X × R) → X × R with coordinates (φ, X a ) on the fiber and (xµ , λ) on the base. We set z a = X a ◦ z. Our presentation is based upon the excellent exposition in Chapter 8 of [1], to which we refer the reader for further information. The action (1.1) for the system in this case is usually written Z p 1 µν S(φ, z) = G (x)φ,µ (x)φ,ν (x) − M 2 φ(x)2 −G(x) d 4x 2 X Z Z 4 4 εφ(x)kz(λ)k − ˙ δ (x − z(λ)) d x dλ − mkz(λ)k ˙ dλ, (2.1) X×R

R

p where the dot denotes differentiation with respect to λ and kzk ˙ = −Gab z˙ a z˙ b . Observe that the bases for the free Klein–Gordon term and the free particle term are different, and that the interaction term in the middle lives on the product of these. Before proceeding, there are two technical issues that need to be resolved, stemming from the presence of the two factors of X in the configuration bundle. First, note that in the leading term of S, G is regarded as living on the X in the base, while in the last term it evidently resides on the X in the fiber. It is necessary to know precisely where G lives, as this has an effect on the subsequent analysis: if on the base, then G is treated as a field, while if on the fiber it is simply thought of as a geometric object. We reconcile these two interpretations by taking G to be anchored to the base, and then pulling it back to the fiber by means of the following construction.3,4 Introduce yet another factor of X in the 2

Not necessarily proper time. This is a variant of the Kuchaˇr method of parametrizing a classical field theory; see [6] and [2] for details. 4 At the end of this section we will briefly examine what happens if instead we anchor G to the fiber.

3

Concatenating Variational Principles

121

fiber along with diffeomorphisms η : X → X, viewed as sections of X × X → X, with corresponding configuration and multivelocity variables η a = X a ◦ η and η a µ = ∂(X a ◦ η)/∂xµ , respectively. (We can, and do, regard the two copies of X in the fiber as identical.) We use these auxiliary nondynamic fields, the diffeomorphism fields (or “diffields” for short, to (i) identify the copies of X in the fiber with that in the base, and (ii) endow the new copy of X in the fiber with the metric g = η∗ G with components gab = Gµν κµ a κν b , where (κµ a ) = (η a µ )−1 . All this is summarized in the following figure.

The general set up for the introduction of diffeomorphism fields.

Second, the delta function δ 4 (x − z(λ)) must be modified, as it compares elements x in the base with elements z(λ) in the fiber. As just indicated we can use the diffields fields to remedy this problem as well: we need only write δ 4 x − η −1 (z(λ)) instead. It is sometimes convenient to replace δ 4 x − η −1 (z(λ)) = δ 4 (η(x) − z(λ))(det η∗ )

(2.2)

using the properties of delta functions (cf. the Appendix), where η∗ is the Jacobian of η. From this we see that δ 4 x − η −1 (z(λ)) (i) is a scalar density on X (again, see the Appendix), and (ii) depends upon the spacetime derivatives of η, even though this is not obvious at first glance. The reason we do not insist on a fixed identification of the base X with the fiber X, and instead allow a variable identification by means of the diffields, is to allow some gauge freedom in the fields; see also footnote 8 below. Remark. Analogous fields η, called “covariance fields,”, are introduced in [6] and [2], but there they have a different purpose, namely, to make a field theory on a given background generally covariant and in doing so, they are introduced as dynamic fields.

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In addition to the diffields η, we introduce a (positive-definite) metric K on R as a nondynamic field. We suppose that K is chosen so that R has metric volume 1. With these fixes we may concatenate the three action terms over the composite R now 4 ¯ base X × R as S(φ, z) = L d x dλ, with the Lagrangian √ √ ¯ µ , λ, φ, φ,µ , z b , z˙ b ; η a , η a µ , Gρσ , K) = 1 Gµν φ,µ φ,ν − M 2 φ2 −G K L(x 2 4 − m + εφ kzk ˙ δ (η − z)(det η∗ ). (2.3)

Notice that the interaction term itself needs no essential modification, as the corresponding term in (2.1) is already an integral over X × R,but informs our choice of scalar density in the free particle term, viz. δ 4 x − η −1 (z(λ)) , when we suspend the latter to X × R. ¯ is defined We have also written this delta function in the form (2.2) to make it clear that L pointwise. √ Remark. The choice of R scalar density D = K in the Klein–Gordon term is hardly unique; all we require is that R D dλ = 1. For instance, we could instead take δ(λ) for D with no essential difference. As evident from (2.3), the modified configuration bundle is taken to be Y ′ = Y ×X X 2 ×X Lor(X) ×R Riem(R), where we abbreviate the bundle X × X → X by X 2 , Lor(X) is the bundle whose sections are Lorentz metrics on X and, similarly, Riem(R) is the bundle whose sections are Riemannian metrics on R. However, in our approach φ and z are variational, while η, G and K are nondynamic fields. As per the above, we now regard q kzk ˙ = −Gµν κµ a κν b z˙ a z˙ b . Remark. Occasionally, as in [9], one encounters what one might call “noncovariant concatenations.” In the current example, this amounts to writing the terms in the action as integrals over X alone and is effectively accomplished by imposing the coordinate condition x0 = λ. As this procedure is not covariant, it can lead to problems [10]. We compute the Euler–Lagrange equations. Varying with respect to φ and employing (2.2), we obtain p p −M 2 φ(x) −G(x) K(λ) − εkz(λ)k ˙ δ 4 x − η −1 (z(λ)) p √ − ∂µ Gµν φ,ν −G (x) K(λ) = 0.

Integrating with respect to λ, using the fact that volK (R) = 1, and rearranging, this reduces to the Klein–Gordon equation ∇µ ∇µ φ + M 2 φ = −ρ where ∇ denotes the G-covariant derivative and Z − 12 ρ(x) = ε(−G) kz(λ)k ˙ δ 4 x − η −1 (z(λ)) dλ R

(2.4)

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123

is the source density. Similarly, varying with respect to z and employing (2.2) yield i ∂ h 4 −1 m + εφ(x) k z(λ)k ˙ δ x − η (z(λ)) ∂z a gab (z(λ))z˙ b (λ) ∂ −1 m + εφ(x) x − η (z(λ)) = 0. + ∂λ kz(λ)k ˙

Carrying out the differentiation and then integrating over X, some manipulations give " # b (λ) d g (z(λ)) z ˙ ab m + εφ η −1 (z(λ)) dλ kz(λ)k ˙ = −εκµ a φ,µ η −1 (z(λ)) kz(λ)k ˙ gbc,a (z(λ))z˙ b (λ)z˙ c (λ) −1 + m + εφ η (z(λ)) . (2.5) 2kz(λ)k ˙ To give insight into these equations, note that in the special case when (X, G) is Minkowski spacetime, η = IdX , and λ is taken to be proper time along the particle’s world line, these equations simplify in a global Lorentz frame to i d h m + εφ(z(λ)) z˙a (λ) = −εφ,a (z(λ)). dλ

This is the mesic analogue of the Lorentz force law in electrodynamics. Neither K, the Gµν , nor the η a have field equations, since they are not variational. Thus one is free to assign them whatever values one wishes in (2.4) and (2.5). Often, however, one has specific values of G and η in mind, e.g., the given spacetime metric for G and IdX for η. Turning now to the SEM tensor, let Diffc (X) × Diffc (R) (that is, the group of diffeomorphisms that are the identity outside a compact set) act on the modified configuration bundle Y ′ according to (σ × f ) · (x, λ, φ, z; η, G, K) = σ(x), f (λ), φ, z; η, σ∗ G, f∗ K . (We assume that all diffeomorphisms are positively oriented.) The Lagrangian density L¯ = ¯ d 4x dλ is then visibly equivariant with respect to the induced action on J¯1 Y ′ , that is,5 L L¯ (σ × f ) · j 1 (φ, z; η, G, K = (σ × f )∗ L¯ j 1 (φ, z; η, G, K) . We may thus use equation (3.12) in [5] to compute the 5-dimensional SEM tensor den-

sity T = 5

T µν T 4ν T µ4 T 44

Even though the pointwise action of Diff(X) on the fiber of the“diffeomorphism bundle” X × X → X is taken to be trivial, its action on sections thereof is not: σ · η = η ◦ σ −1 . Thus the identification of the factor of X in the base with that in the fiber can fluctuate, which is one of the reasons we allow η to be variable in the first place.

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of the interacting system (where x4 = λ).6 Integrating over λ and raising an index, we project out the spacetime SEM tensor density: Z µν T = T µν dλ = tµν + (m + ǫφ)Θµν , (2.6) R

where

√ 1 (2Gµα Gνβ − Gµν Gαβ )φ,α φ,β + Gµν M 2 φ2 −G 2 is the canonical SEM tensor density of the (free) Klein–Gordon field and tµν = −

µν

µ

ν

Θ (x) = κ a κ

b

z˙ a (λ)z˙ b (λ) 4 δ x − η −1 (z(λ)) dλ kz(λ)k ˙

Z

R

is the Minkowski tensor density. (mΘµν is then the kinetic SEM tensor density). As well, we compute T 4 ν = 0 = T µ 4 . Finally, we find that when integrated over X, T 4 4 is effectively the Klein–Gordon action: Z √ √ 1 4 4 µν 2 2 −G d x K. T 4= G φ,µ φ,ν − M φ 2 X Remark. The kinetic SEM tensor density is a familiar object in microscopic continuum mechanics, cf. Chapter 8 of [1] and §33 of [9]. Minkowski [11] originally introduced it in flat-spacetime electrodynamics in order to recover the continuity equation Tµν ,ν = 0 in view of the fact that Tµν EM ,ν 6= 0 when currents are present. In the continuum limit of a noninteracting clutch of particles, Θµν goes over to the SEM tensor density for a perfect fluid as in §§9.1-2 of [1]. It is interesting that in this limit, the infinite time integrals in the kinetic SEM tensor density disappear and one is left with a local tensor density. To our knowledge, ours is the first genuine derivation of the Minkowski tensor density from first principles in a variational context, once again illustrating the power of multisymplectic geometry in classical field theory and in particular, the usefulness of having a concatenated theory for which one can make use of concepts such as the SEM tensor. As we have defined it, the Minkowski tensor density depends upon the diffields as well as the particle placement field. However, note that when η = IdX , Θ reduces to the more familiar expression ab

Θ (x) =

Z

R

z˙ a (λ)z˙ b (λ) 4 δ (x−z(λ)) dλ. kz(λ)k ˙

Remark. Suppose we focus solely on the particle dynamics so that the (original) configuration bundle is X × R → R. The corresponding Lagrangian density −mkz(λ)kdλ ˙ is Diffc (R)-covariant, and so we may compute the corresponding SEM scalar density as in Example a, Interlude II of [4]. We obtain T = −E, the energy of the particle, which vanishes as the Lagrangian is time reparametrization-invariant. (This is reflected by the 6

Using the product metric G ⊕ K on X × R, one could also compute T via the Hilbert formula (4.2) in [5]. See also [10].

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125

vanishing of T 4 4 in the 5-dimensional context when there is no Klein–Gordon field.) Thus only when the spacetime X is part of the base of the variational principle do we encounter the kinetic SEM tensor density; it does not appear in standard particle dynamics per se. To reiterate, even in the absence of other fields, our technique yields yet another (5dimensional!) treatment of the relativistic free particle that has the advantage of automatically incorporating the Minkowski tensor. Remark. Note that the term εφΘµν in (2.6) arises from the interaction of φ with the mesically charged particle. This term has no analogue in the electrodynamics of particles; there we get simply µν Tµν = Tµν EM + mΘ . Charged strings behave similarly, as we show in §3A (cf. equation (3.1)). This can be traced to the fact that the electromagnetic field is a covector, while the Klein–Gordon field is a scalar. The SEM tensor density T is symmetric. It is also divergence-free, as can be seen from general principles (cf. Proposition 5 in [5]). One may verify this directly, via a long calculation. We end with a discussion of an alternate treatment of this system. Suppose we consider the physical metric as a geometric object g on the fiber as opposed to a field on spacetime. Then we would define G = η ∗ g with components Gµν = η a µ η b ν gab . Proceeding as in the above, the Lagrangian density would be ¯ µ , λ, φ, φ,µ , z b , z˙ b ; η a , η a µ , K) L(x √ √ 1 µ ν ab κ a κ b g φ,µ φ,ν − M 2 φ2 −g (det η∗ ) K = 2 − m + εφ kzk ˙ δ 4 (η − z)(det η∗ )

p √ √ ˙ = −gab (z)z˙ a z˙ b . where −G = −g (det η∗ ) and kzk Computing the SEM tensor density in this formulation, we obtain T µ ν ≡ 0 and the other components as before. That the spacetime components vanish is actually a consequence of the generalized Hilbert formula (3.13) in [5], since the nondynamic fields η and K do not transform under Diff(X). (In the original formulation, the nondynamic metric G on X does transform under the spacetime diffeomorphism group with the result that (2.6) is nonzero.) The difference between this SEM tensor density and the previous one stems from: (i) the spacetime metric no longer being regarded as a field, so that it cannot contribute to the energy, momentum, and stress content of the system, and (ii) the subtly different manners in which the diffields appear in the two formulations. That one can encounter several SEM tensor densities for the ‘same’ system may seem surprising, but is unavoidable and can also be regarded as different “packaging” of the same information. What the SEM tensor density turns out to be depends upon what the fields are, whether they are dynamic, and precisely how they appear in the Lagrangian. And even the size of the SEM tensor density depends upon how the system is formulated! For instance, for something as simple as a relativistic free particle, we can have a 1 × 1 SEM tensor density (which vanishes identically)—as noted in a previous remark, or a 5 × 5 SEM tensor

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Marko Castrill´on L´opez, Mark J. Gotay and Jerrold E. Marsden

density (which does or doesn’t, depending on where the spacetime metric is anchored). And in the latter case, the 5 × 5 object reduces to the 4 × 4 Minkowski tensor density! Thus how the system is formulated plays a substantial role insofar as how various quantities, and in particular the SEM tensor density, are to be understood.

3.

Further Examples and Outlook

To conclude we briefly mention some other systems for which our techniques should prove helpful. We begin by upping the dimension of the matter from 1 to 2, that is, we replace the particle by a string. For variety, we also replace the mesic interaction by an electromagnetic one.

Charged Strings. We closely follow the exposition in §2. Let (X, G) and (W = R × B, H = −HR ⊕ HB ) be 4- and 2-dimensional Lorentzian spacetimes, respectively. We consider a string worldsheet in X, this being a map z : W → X. We use coordinates xµ , λA = (τ, σ) as coordinates on X × W . The configuration bundle Y is correspondingly Λ1 X ×W (X × W ) ×X Lor(X) ×W Lor(W ) → X × W. Assume that the string carries a charge density ρ : B → R and interacts with a dynamic electromagnetic field described by a potential 1-form A on X. We also take the metric H on W to be dynamic; thus we adopt the Polyakov approach as in [7]. The action for this system is 1 S(A, z, H) = − 4

Z

ZX

p F µν (x)Fµν (x) −G(x) d 4x

p ∂z µ (τ, σ) δ 4 (x − z(τ, σ)) HB (σ) d 4x dτ dσ ∂τ X×W Z p T − H AB (λ) Gµν (z(λ))z µ (λ),A z ν (λ),B −H(λ) d 2λ 2 W +

Aµ (x)ρ(σ)

where T is the tension. As in the case of the meson, we see that z takes values in X, which is the base for the electromagnetic field. So we need to introduce diffields as before. As well, we take the spacetime metric to reside on the factor of X in the base. Finally, let K be a nondynamic Riemannian metric on W with total volume 1. The modified configuration bundle for the concatenated variational principle is then Y ×X X 2 ×W Riem(W ) → X × W

Concatenating Variational Principles

127

and the Lagrangian reads ¯ µ , τ, σ, Aµ , Aµ,ν , z a , z a ,A , HAB ; η a , η a ν , Gµν , KAB ) L(x √ √ 1 = − Gµα Gνβ Fµν Fαβ −G K 4 p ∂z a 4 + Aµ κµ a ρ δ η − z)(det η∗ ) HB ∂τ √ T AB − H Gµν κµ a κν b z a ,A z b ,B δ 4 (η − z)(det η∗ ) −H. 2 Now build the product metric G ⊕ K on X × W . Using the Hilbert formula and then integrating as before, we compute the 6-dimensional SEM tensor density as follows: the spacetime components are µν Tµν = Tµν (3.1) EM + T Θ where Tµν EM

√ 1 µν αβ νβ αµ = − G Fαβ F + G F Fβα −G 4

is the free electromagnetic SEM tensor density, and Z √ µν µ ν Θ = κ aκ b H AB z a ,A z b ,B δ 4 x − η −1 (z(σ)) −H d 2σ W

is the analogue of the Minkowski tensor density for strings. The ‘extra’ T µA and T Aµ components are zero and, after integrating over X, the T AB subblock reduces to Z √ √ 1 µα νβ AB 4 G G Fµν Fαβ −G d x K AB K. T =− 4 X Continua. Another intriguing example that we intend to pursue in future works is a charged elastic body, fluid, or plasma, in which one concatenates a continuum with electromagnetism on a given background metric spacetime. Such theories will likely have significant differences with the particle and string examples presented above. Amongst these differences, we expect that, unlike mesically or electrically charged particles, continua should have well-defined initial value problems (see also the discussion of this point in [1]). Evidence for this can be found in works such as [8] and [12]. One other interesting aspect of a charged elastic body is the following. If B is the body manifold, then its motion in spacetime is determined by a map z : R × B → X. The main difference from our previous two examples is that rather than the delta functions 4 −1 δ (η(x) − z(λ)) we must now use characteristic functions χ η (z(R × B)) . We expect that examples such as this will be key players in the future development of the point of view given in this paper.

Appendix Let M be a manifold with coordinates x = (x1 , . . . , xm ). Here we prove that the delta function δ m (x − x0 ) transforms as a scalar density of weight 1.

128

Marko Castrill´on L´opez, Mark J. Gotay and Jerrold E. Marsden Let η : M → M be a diffeomorphism and f ∈ C ∞ (X). On the one hand, Z (f ◦ η)(x0 ) = (f ◦ η)(x) δ m (x − x0 ) d m x. X

On the other hand, by the change of variables theorem with y = η(x), Z f (η(x0 )) = f (y) δ m (y − η(x0 )) d m y X Z = f (η(x)) δ m (η(x) − η(x0 )) |J(x)| d m x. X

where J is the Jacobian determinant of η. Since f is arbitrary the desired result follows upon comparing these two formulæ.

References [1] J. L. Anderson, Principles of Relativity Physics (Academic Press, New York, 1967). [2] M. Castrill´on L´opez, M. J. Gotay and J. E. Marsden, Parametrization and stressenergy-momentum tensors in metric theories, J. Phys. A:Math. Theor. (2008); to appear. [3] M. J. Gotay, J. A. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical fields, I: Covariant field theory, (1998); arXiv:physics/9801019. [4] M. J. Gotay, J. A. Isenberg and J. E. Marsden, Momentum maps and classical fields, II: Canonical analysis of field theories, (2004); arXiv:math-ph/0411032. [5] M. J. Gotay and J. E. Marsden, Stress-energy-momentum tensors and the Belinfante– Rosenfeld formula, Contemp. Math. 132 (1992) 367–391. [6] M. J. Gotay and J. E. Marsden, Parametrization theory, (2008); in preparation. [7] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Volume I: Introduction (Cambridge Univ. Press, Cambridge, 1987). [8] M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov–Klein–Gordon system, Electronic J. Differential Equations (1) (2005) pp. 17. [9] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Fourth revised English ed., Permagon Press, Aberdeen, 1979). [10] M. Leclerc, Canonical and gravitational stress-energy tensors, (2006); arXiv:grqc/0510044. [11] H. Minkowski, Die Grundgleichungen f¨ur die elektromagnetischen Vorg¨ange in bewegten K¨orpern, Nach. Ges. Wiss. G¨ottingen (1908) 53–111. [12] G. Rein, Generic global solutions of the relativistic Vlasov–Maxwell system of plasma physics, Comm. Math. Phys. 135 (1990) 41–78.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 129-140

Chapter 8

A G EOMETRIC H AMILTON -JACOBI T HEORY FOR C LASSICAL F IELD T HEORIES∗ Manuel de Le´on1†, Juan Carlos Marrero2‡ and David Mart´ın de Diego3§ 1 Instituto de Ciencias Matem´aticas, CSIC-UAM-UC3M-UCM, Serrano 123, 28006 Madrid, Spain 2 Departamento de Matem´atica Fundamental, Universidad de La Laguna, La Laguna, Canary Islands, Spain, 3 Instituto de Ciencias Matem´aticas, CSIC-UAM-UC3M-UCM, Serrano 123, 28006 Madrid, Spain

Abstract In this paper we extend the geometric formalism of the Hamilton-Jacobi theory for hamiltonian mechanics to the case of classical field theories in the framework of multisymplectic geometry and Ehresmann connections.

2000 Mathematics Subject Classification. 70S05, 49L99. Key words and phrases. Multisymplectic field theory, Hamilton-Jacobi equations.

1.

Introduction

The standard formulation of the Hamilton-Jacobi problem is to find a function S(t, q A ) (called the principal function) such that ∂S A ∂S + H q , A = 0. (1.1) ∂t ∂q ∗

To Prof. Demeter Krupka in his 65th birthday E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: [email protected] †

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M. de Le´on, J.C. Marrero and D. Mart´ın de Diego

If we put S(t, q A ) = W (q A ) − tE, where E is a constant, then W satisfies A ∂W H q , A = E; ∂q

(1.2)

W is called the characteristic function. Equations (1.1) and (1.2) are indistinctly referred as the Hamilton-Jacobi equation. There are some recent attempts to extend this theory for classical field theories in the framework of the so-called multisymplectic formalism [15, 16]. For a classical field theory the hamiltonian is a function H = H(xµ , y i , pµi ), where (xµ ) are coordinates in the spacetime, (y i ) represent the field coordinates, and (pµi ) are the conjugate momenta. In this context, the Hamilton-Jacobi equation is [17] µ ∂S µ ν i ∂S =0 + H x ,y , ∂xµ ∂y i

(1.3)

where S µ = S µ (xν , y j ). In this paper we introduce a geometric version for the Hamilton-Jacobi theory based in two facts: (1) the recent geometric description for Hamiltonian mechanics developed in [6] (see [8] for the case of nonholonomic mechanics); (2) the multisymplectic formalism for classical field theories [3, 4, 5, 7] in terms of Ehresmann connections [9, 10, 11, 12]. We shall also adopt the convention that a repeated index implies summation over the range of the index.

2.

A Geometric Hamilton-Jacobi Theory for Hamiltonian Mechanics

First of all, we give a geometric version of the standard Hamilton-Jacobi theory which will be useful in the sequel. Let Q be the configuration manifold, and T ∗ Q its cotangent bundle equipped with the canonical symplectic form ωQ = dq A ∧ dpA

where (q A ) are coordinates in Q and (q A , pA ) are the induced ones in T ∗ Q.

Let H : T ∗ Q −→ R a hamiltonian function and XH the corresponding hamiltonian vector field: iXH ωQ = dH The integral curves of XH , (q A (t), pA (t)), satisfy the Hamilton equations: ∂H dpA ∂H dq A = , =− A. dt ∂pA dt ∂q Theorem 2.1 (Hamilton-Jacobi Theorem). Let λ be a closed 1-form on Q (that is, dλ = 0 and, locally λ = dW ). Then, the following conditions are equivalent:

Hamilton-Jacobi Theory for Classical Field Theories

131

(i) If σ : I → Q satisfies the equation ∂H dq A = dt ∂pA then λ ◦ σ is a solution of the Hamilton equations; (ii) d(H ◦ λ) = 0. To go further in this analysis, define a vector field on Q: λ XH = T π Q ◦ XH ◦ λ

as we can see in the following diagram: XH

T ∗Q

T (T ∗ Q)

πQ

λ

T πQ λ XH

Q

TQ

Notice that the following conditions are equivalent: (i) If σ : I → Q satisfies the equation ∂H dq A = dt ∂pA then λ ◦ σ is a solution of the Hamilton equations; λ , then λ ◦ σ is an integral curve of X ; (i)’ If σ : I → Q is an integral curve of XH H λ are λ-related, i.e. (i)” XH and XH λ T λ(XH ) = XH ◦ λ

so that the above theorem can be stated as follows: Theorem 2.2 (Hamilton-Jacobi Theorem). Let λ be a closed 1-form on Q. Then, the following conditions are equivalent: λ and X are λ-related; (i) XH H

(ii) d(H ◦ λ) = 0.

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M. de Le´on, J.C. Marrero and D. Mart´ın de Diego

3.

The Multisymplectic Formalism

3.1.

Multisymplectic Bundles

The configuration manifold in Mechanics is substituted by a fibred manifold π : E −→ M such that (i) dim M = n, dim E = n + m (ii) M is endowed with a volume form η. We can choose fibred coordinates (xµ , y i ) such that η = dx1 ∧ · · · ∧ dxn . We will use the following useful notations: dn x = dx1 ∧ · · · ∧ dxn , dn−1 xµ = i

∂ ∂xµ

dn x .

Denote by V π = ker T π the vertical bundle of π, that is, their elements are the tangent vectors to E which are π-vertical. Denote by Π : Λn E −→ E the vector bundle of n-forms on E. The total space Λn E is equipped with a canonical n-form Θ: Θ(α)(X1 , . . . , Xn ) = α(e)(T Π(X1 ), . . . , T Π(Xn )), where X1 , . . . , Xn ∈ Tα (Λn E) and α is an n-form at e ∈ E. The (n + 1)-form

Ω = −dΘ ,

is called the canonical multisymplectic form on Λn E. Denote by Λnr E the bundle of r-semibasic n-forms on E, say Λnr E = {α ∈ Λn E | iv1 ∧···∧vr α = 0, whenever v1 , . . . , vr are π-vertical}. Since Λnr E is a submanifold of Λn E it is equipped with a multisymplectic form Ωr , which is just the restriction of Ω. Two bundles of semibasic forms play an special role: Λn1 E and Λn2 E. The elements of these spaces have the following local expressions: Λn1 E

:

p0 dn x,

Λn2 E

: p0 dn x + pµi dy i ∧ dn−1 xµ

Hamilton-Jacobi Theory for Classical Field Theories

133

which permits to introduce local coordinates (xµ , y i , p0 ) and (xµ , y i , p0 , pµi ) in Λn1 E and Λn2 E, respectively. Since Λn1 E is a vector subbundle of Λn2 E over E, we can obtain the quotient vector space denoted by J 1 π ∗ which completes the following exact sequence of vector bundles: 0 −→ Λn1 E −→ Λn2 E −→ J 1 π ∗ −→ 0 . We denote by π1,0 : J 1 π ∗ −→ E and π1 : J 1 π ∗ −→ M the induced fibrations.

3.2.

Ehresmann Connections in the Fibration π1 : J 1 π ∗ −→ M

A connection (in the sense of Ehresmann) in π1 is a horizontal subbundle H which is complementary to V π1 ; namely, T (J 1 π ∗ ) = H ⊕ V π1 where V π1 = ker T π1 is the vertical bundle of π1 . Thus, we have: (i) there exists a (unique) horizontal lift of every tangent vector to M ; (ii) in fibred coordinates (xµ , y i , pµi ) on J 1 π ∗ , then ∂ ∂ V π1 = span , , H = span {Hµ } , ∂y i ∂pµi where Hµ is the horizontal lift of

∂ ∂xµ .

(iii) there is a horizontal projector h : T J 1 π ∗ −→ H.

3.3.

Hamiltonian Sections

Consider a hamiltonian section h : J 1 π ∗ −→ Λn2 E of the canonical projection µ : Λn2 E −→ J 1 π ∗ which in local coordinates read as h(xµ , y i , pµi ) = (xµ , y i , −H(x, y, p), pµi ) . Denote by Ωh = h∗ Ω2 , where Ω2 is the multisymplectic form on Λn2 E. The field equations can be written as follows: ih Ωh = (n − 1) Ωh ,

(3.1)

where h denotes the horizontal projection of an Ehresmann connection in the fibred manifold π1 : J 1 π ∗ −→ M . The local expressions of Ω2 and Ωh are:

Ω2 = −d(p0 dn x + pµi dy i ∧ dn−1 xµ ),

Ωh = −d(−H dn x + pµi dy i ∧ dn−1 xµ ) .

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3.4.

The Field Equations

Next, we go back to the Equation (3.1). The horizontal subspaces are locally spanned by the local vector fields ∂ ∂ ∂ ∂ Hµ = h = + Γiµ i + (Γµ )νj , ∂xµ ∂xµ ∂y ∂pνj where Γiµ and (Γµ )νj are the Christoffel components of the connection. Assume that τ is an integral section of h; this means that τ : M −→ J 1 π ∗ is a local section of the canonical projection π1 : J 1 π ∗ −→ M such that T τ (x)(Tx M ) = Hτ (x) , for all x ∈ M . If τ (xµ ) = (xµ , τ i (x), τiµ (x)) then the above conditions becomes ∂τ i ∂H ∂τiµ ∂H = µ , =− i µ µ ∂x ∂y ∂pi ∂x which are the Hamilton equations.

4.

The Hamilton-Jacobi Theory

Let λ be a 2-semibasic n-form on E; in local coordinates we have λ = λ0 (x, y) dn x + λµi (x, y) dy i ∧ dn−1 xµ . Alternatively, we can see it as a section λ : E −→ Λn2 E, and then we have λ(xµ , y i ) = (xµ , y i , λ0 (x, y), λµi (x, y)) . A direct computation shows that ∂λ0 ∂λµi − µ dy i ∧ dn x dλ = ∂y i ∂x µ ∂λi + dy j ∧ dy i ∧ dn−1 xµ . ∂y j Therefore, dλ = 0 if and only if ∂λ0 ∂y i

=

∂λµi ∂y j

=

∂λµi , ∂xµ ∂λµj . ∂y i

(4.1) (4.2)

Using λ and h we construct an induced connection in the fibred manifold π : E −→ M by defining its horizontal projector as follows: ˜e h

: Te E −→ Te E, ˜ he (X) = T π1,0 ◦ h(µ◦λ)(e) ◦ ǫ(X)

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135

where ǫ(X) ∈ T(µ◦λ)(e) (J 1 π ∗ ) is an arbitrary tangent vector which projects onto X. From the above definition we immediately proves that

˜ is a well-defined connection in the fibration π : E −→ M . (i) h (ii) The corresponding horizontal subspaces are locally spanned by ∂ ∂ ˜ ∂ ˜µ = h H = + Γiµ ((µ ◦ λ)(x, y)) i . ∂xµ ∂xµ ∂y The following theorem is the main result of this paper. ˜ is a flat conTheorem 4.1. Assume that λ is a closed 2-semibasic form on E and that h nection on π : E −→ M . Then the following conditions are equivalent: ˜ then µ ◦ λ ◦ σ is a solution of the Hamilton equations. (i) If σ is an integral section of h (ii) The n-form h ◦ µ ◦ λ is closed. Before to begin with the proof, let us consider some preliminary results. We have (h ◦ µ ◦ λ)(xµ , y i ) = (xµ , y i , −H(xµ , y i , λµi (x, y)), λµi (x, y)) , that is h ◦ µ ◦ λ = −H(xµ , y i , λµi (x, y)) dn x + λµi dy i ∧ dn−1 xµ . Notice that h ◦ µ ◦ λ is again a 2-semibasic n-form on E.

A direct computation shows that d(h ◦ µ ◦ λ) = −

∂λµi ∂H ∂λνj ∂H + + ∂y i ∂pνj ∂y i ∂xµ

!

dy i ∧ dn x +

Therefore, we have the following result. Lemma 4.2. Assume dλ = 0; then d(h ◦ µ ◦ λ) = 0 if and only if

∂λµi ∂H ∂H ∂λνj + + =0. ∂y i ∂pνj ∂y i ∂xµ

Proof of the Theorem. (i) ⇒ (ii)

It should be remarked the meaning of (i). Assume that σ(xµ ) = (xµ , σ i (x))

∂λµi dy j ∧ dy i ∧ dn−1 xµ . ∂y j

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˜ then is an integral section of h;

∂σ i ∂H = µ . ∂xµ ∂pi

(i) states that in the above conditions, (µ ◦ λ ◦ σ)(xµ ) = (xµ , σ i (x), σ ¯jν = λνj (σ(x))) is a solution of the Hamilton equations, that is, ∂σ ¯iµ ∂λµi ∂λµi ∂σ j ∂H = + = − i. ∂xµ ∂xµ ∂y j ∂xµ ∂y Assume (i). Then ∂λµi ∂H ∂H ∂λνj + + ∂y i ∂pνj ∂y i ∂xµ =

∂λµi ∂H ∂H ∂λνi + + , ∂y i ∂pνj ∂y j ∂xµ

∂λµi ∂H ∂σ j ∂λνi + + , ∂y i ∂xν ∂y j ∂xµ =0 (since (i))

=

(since dλ = 0) (since the first Hamilton equation)

which implies (ii) by Lemma 4.2. (ii) ⇒ (i)

Assume that d(h ◦ µ ◦ λ) = 0. ˜ is a flat connection, we may consider an integral section σ of h. ˜ Suppose that Since h σ(xµ ) = (xµ , σ i (x)). Then, we have that

∂σ i ∂H = µ. ∂xµ ∂pi

Thus, ∂σ ¯jµ ∂xµ

∂λµj

∂λµj ∂σ i = + , ∂xµ ∂y i ∂xµ ∂λµj ∂λµi ∂σ i = + , (since dλ = 0) ∂xµ ∂y j ∂xµ ∂λµj ∂λµi ∂H = + , (since the first Hamilton equation) ∂xµ ∂y j ∂pµi ∂H =− j , (since (ii)). 2 ∂y

Assume that λ = dS, where S is a 1-semibasic (n − 1)-form, say S = S µ dn−1 xµ

Hamilton-Jacobi Theory for Classical Field Theories Therefore, we have λ0 =

137

∂S µ ∂S µ µ , λ = i ∂xµ ∂y i

and the Hamilton-Jacobi equation has the form µ µ ∂S ∂ ν i ∂S + H(x , y , ) =0. ∂y i ∂xµ ∂y i The above equations mean that µ ∂S µ ν i ∂S = f (xµ ) + H x , y , ∂xµ ∂y i ˜ = H − f we deduce the standard form of the Hamilton-Jacobi equation so that if we put H ˜ (since H and H give the same Hamilton equations): µ ∂S µ ν i ∂S ˜ + H x ,y , =0. ∂xµ ∂y i An alternative geometric approach of the Hamilton-Jacobi theory for Classical Field Theories in a multisymplectic setting was discussed in [15, 16].

5.

Time-dependent Mechanics

A hamiltonian time-dependent mechanical system corresponds to a classical field theory when the base is M = R. We have the following identification Λ12 E = T ∗ E and we have local coordinates (t, y i , p0 , pi ) and (t, y i , pi ) on T ∗ E and J 1 π ∗ , respectively. The hamiltonian section is given by h(t, y i , pi ) = (t, y i , −H(t, y, p), pi ) , and therefore we obtain Ωh = dH ∧ dt − dpi ∧ dy i . If we denote by η = dt the different pull-backs of dt to the fibred manifolds over M , we have the following result. The pair (Ωh , dt) is a cosymplectic structure on E, that is, Ωh and dt are closed forms and dt ∧ Ωnh = dt ∧ Ωh ∧ · · · ∧ Ωh is a volume form, where dimE = 2n + 1. The Reeb vector field Rh of the structure (Ωh , dt) satisfies iRh Ωh = 0 , iRh dt = 1. The integral curves of Rh are just the solutions of the Hamilton equations for H. The relation with the multisymplectic approach is the following: h = Rh ⊗ dt ,

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or, equivalently, ∂ = Rh . h ∂t A closed 1-form λ on E is locally represented by λ = λ0 dt + λi dy i . Using λ we obtain a vector field on E: (Rh )λ = T π1,0 ◦ Rh ◦ µ ◦ λ such that the induced connection is ˜ = (Rh )λ ⊗ dt h Therefore, we have the following result. Theorem 5.1. The following conditions are equivalent: (i) (Rh )λ and Rh are (µ ◦ λ)-related. (ii) The 1-form h ◦ µ ◦ λ is closed. Remark 5.2. An equivalent result to Theorem 5.1 was proved in [14] (see Corollary 5 in [14]). ⋄ Now, if λ = dS =

∂S ∂S dt + i dy i , ∂t ∂y

then we obtain the Hamilton-Jacobi equation ∂ ∂S i ∂S + H t, y , =0. ∂y i ∂t ∂y i

Acknowledgement This work has been partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2007-62478, project “Ingenio Mathematica” (i-MATH) No. CSD 2006-00032 (ConsoliderIngenio 2010) and S-0505/ESP/0158 of the CAM.

References [1] R. Abraham and J. E. Marsden, Foundations of Mechanics (2nd edition, BenjaminCumming, Reading, 1978). [2] V. I. Arnold, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics 60, Springer-Verlag, Berlin, 1978).

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[3] E. Binz, J. Sniatycki and H. Fischer, Geometry of Classical Fields (North-Holland Mathematics Studies, 154, North-Holland Publishing Co., Amsterdam, 1988). [4] F. Cantrijn, A. Ibort and M. de Le´on, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. (Series A) 66 (1999) 303–330. [5] F. Cantrijn, A. Ibort and M. de Le´on, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino 54 (3) (1996) 225–236. [6] J. F. Cari˜nena, X. Gracia, G. Marmo, E. Mart´ınez, M. Mu˜noz-Lecanda and N. Rom´anRoy, Geometric Hamilton-Jacobi theory, Int. J. Geom. Meth. Mod. Phys. 3 (7) (2006) 1417–1458. [7] M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory; arXiv:physics/9801019v2. [8] M. de Le´on, D. Iglesias-Ponte and D. Mart´ın de Diego, Hamilton-Jacobi Theory for Nonholonomic Mechanical Systems, J. Phys. A: Math. Theor. 41 (2008) 015205, pp. 14. [9] M. de Le´on, J. C. Marrero and D. Mar´ın, A geometrical approach to Classical Field Theories: a constraint algorithm for singular theories, In: New Developments in Differential Geometry (Debrecen, 1994, Mat. Appl. 350, Kluwer, Dordrecht, 1996) 291– 312. [10] M. de Le´on, J. C. Marrero and D. Mar´ın, Ehresmann connections in Classical Field Theories, In: Differential Geometry and its Applications (Granada, 1994, Anales de F´ısica, Monograf´ıas 2, 1995) 73–89. [11] M. de Le´on, D. Mart´ın de Diego and A. Santamar´ıa-Merino, Tulczyjew’ s triples and lagrangian submanifolds in classical field theory, In: Applied Differential Geometry and Mechanics ((W. Sarlet and F. Cantrijn, Eds.) Universiteit Gent, 2003) 21–47. [12] M. de Le´on, D. Mart´ın de Diego and A. Santamar´ıa-Merino, Symmetries in Field Theory, Int. J. Geom. Meth. Mod. Phys. 1 (5) (2004) 651–710. [13] M. de Le´on, P. R. Rodrigues, Generalized Classical Mechanics and Field Theory (North-Holland, Amsterdam, 1985). [14] J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie Affgebroids, Int. J. Geom. Meth. Mod. Phys. 3 (3) (2006) 605–622. [15] C. Paufler and H. Romer, De Donder-Weyl equations and multisymplectic geometry, In: XXXIII Symposium on Mathematical Physics (Tor´un, 2001, Rep. Math. Phys. 49, 2002) 325–334. [16] C. Paufler and H. Romer, Geometry of Hamiltonean n-vector fields in multisymplectic field theory, J. Geom. Phys. 44 (2002) 52–69.

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[17] H. Rund, The Hamilton-Jacobi theory in the Calculus of Variations (Robert E. Krieger Publ. Co., Nuntington, N.Y. 1973). [18] D. J. Saunders, The Geometry of Jet Bundles (London Mathematical Society Lecture Notes Ser. 142, Cambridge Univ. Press, Cambridge, 1989).

Part II

N ATURAL B UNDLES AND D IFFERENTIAL I NVARIANTS

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In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 143-166

Chapter 9

N ATURAL L AGRANGIAN S TRUCTURES Josef Janyˇska∗ Department of Mathematics and Statistics Masaryk University, Jan´acˇ kovo n´am 2a, 602 00 Brno Czech Republic

Abstract We use the theory of natural and gauge-natural bundles and natural differential operators to give a general description of invariant and gauge invariant Lagrangian structures on natural and gauge natural bundles.

2000 Mathematics Subject Classification. 58A32, 53C05, 53C80, 70G45, 70G50, 70S05, 70S15 Key words and phrases. Natural bundle, gauge-natural bundle, natural differential operator, invariant Lagrangian, infinitesimal symmetr

1.

Introduction

Let π : Y → M be a fibered manifold and π r : J r Y → M its r-jet prolongation, Saunders [43]. An r-order Lagrangian is a fibered morphism, over M , L : J r Y → ∧m T ∗ M .

(1.1)

A vector field η on Y projectable on a vector field ξ on M is said to be an infinitesimal symmetry of L if the Lie derivative of L with respect to the r-jet lift J r η of η vanishes, Krupka and Trautman [35], i.e. LJ r η L(y r ) = 0 , where y r ∈ J r Y . But from the general theory of Lie derivatives LJ r η L(y r ) := L(J r η,∧m T ∗ ξ) L(y r ) = (T L ◦ J r η)(y r ) − ∧m T ∗ ξ(L(y r )) , ∗

E-mail address: [email protected]

(1.2)

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Josef Janyˇska

where ∧m T ∗ ξ, m = dim M , is the vector field on ∧m T ∗ M obtained by the flow prolongation of ξ, see Subsection 2.13.. So the condition for infinitesimal symmetries LJ r η L = 0 reads as T L(J r η)(y r ) = ∧m T ∗ ξ(L(y r )) .

(1.3)

Usually, in physical theories, Y is a bundle (tensor bundle, bundle of connections, gauge bundle, ...) with a geometrical structure given by an action of a Lie group. Then, if η is a vector field preserving the given structure, a Lagrangian satisfying (1.3) is said to be invariant with respect to the given structure. There are two main geometrical structures of Y : 1. Y is a natural bundle (in the sense of Nijenhuis [40]) with a geometrical structure given by the differential group Grm = invJ0r (Rm , Rm )0 . Examples of such bundles are tensor bundles (r = 1) and the bundle of classical (linear) connections (r = 2). Invariant Lagrangians on tensor bundles were studied by many authors, see for instance Krupka [26, 27, 28, 30, 31] and Novotn´y [41]. 2. Y is a gauge-natural bundle (in the sense of Eck [6]) with a geometrical structure induced by a gauge group G. In this case a special attention is devoted to invariant Lagrangians on the bundle of principal connections, see for instance Betounes [1], Eck [6], Horndeski [10], Utiyama [51] and others. In this paper we will study natural (invariant and gauge invariant) Lagrangian structures on natural and gauge-natural bundles. To describe invariant Lagrangians on natural or gauge-natural bundles we can use general properties of natural differential operators on natural bundles or gauge-natural bundles. The paper is organized as follows. In Section 2. we recall basic properties of natural bundles and natural differential operators on natural bundles. In Section 3. we study invariant (natural) Lagrangians on natural bundles, especially on the natural bundle of classical connections. In Section 4. we recall basic properties of gauge-natural bundles and natural differential operators on gauge-natural bundles and finally, in Section 5., we study gauge invariant (natural) Lagrangians on gauge-natural bundles, especially on the gaugenatural bundle of general linear and principal connections by using higher order versions of Utiyama’s reduction method, Janyˇska [15, 17, 18]. In what follows we will use the following notations. M is the category of all smooth manifolds and smooth mappings, Mm is the category of all m-dimensional smooth manifolds and local diffeomorphisms, F M m is the category of all fibred manifolds with mdimensional bases and smooth fibred morphisms covering local diffeomorphisms of bases, V B m (A B m ) is the category of all vector (affine) bundles with m-dimensional bases and smooth linear (affine) fibred morphisms covering local diffeomorphisms of bases and, finally, PB m (G) is the category of all principal G-bundles with m-dimensional bases and smooth principal fibred morphisms covering local diffeomorphisms of bases. In what follows all manifolds and maps are supposed to be smooth.

Natural Lagrangian Structures

2.

145

Natural Bundles and Operators

We recall here definitions and basic properties of the theory of natural bundles and natural differential operations, for details see [9, 11, 19, 24, 33, 40, 48, 53]. As examples we mention functors and operators which will be used later. Natural differential operators on natural bundles are closely related with the term geometric invariant (or concomitant) which has been used in differential geometry since the end of the 19th century. In the 1930’s Schouten and his collaborators, [45], used the notion of geometric object and invariant operations with geometric objects. A modern functorial approach to the theory of geometrical objects and invariant operations with geometric objects was introduced by Nijenhuis [39] in the 1950’s. Starting from the famous paper by Nijenhuis [40] geometrical objects and invariant operations with geometrical objects have been very intensively studied by using the concepts of natural bundles and natural differential operators. Nijenhuis defined natural bundles as lifting functors on the category Mm . Lifting functors are supposed to satisfy three conditions: prolongation, localization and regularity (continuity). Kol´aˇr [22] generalized lift functors to the category M of all differentiable manifolds and their smooth mappings. Such functors are called prolongation functors. Later various prolongation functors on subcategories of M were studied. Main problem of the theory of natural differential operators is to give a complete classification of them for concrete underlying geometric structures. Such classification is based on the one-to-one correspondence of natural differential operators and equivariant maps between standard fibres. To classify equivariant maps we can use several methods. Formerly the method of Lie equations was used, Krupka and Janyˇska [33], recently we use the algebraic method by Kol´aˇr, Michor and Slov´ak [24]. In literature it is possible to find many examples and applications of natural operations used in geometry and physics. For wide list of references we recommend to see Fatibene and Francaviglia [9], Kol´aˇr, Michor and Slov´ak [24] and Krupka and Janyˇska [33].

2.1.

Natural Bundles

Natural bundles were introduced by Nijenhuis [40] over the category Mm . We use more general definition by Kol´aˇr, Michor and Slov´ak [24]. Definition 2.1. A natural bundle functor on a subcategory C of M is a covariant functor F from C to the category F M satisfying (i) (prolongation) for each manifold M ∈ Ob C , pM : F M → M is a fibred manifold over M , (ii) (localization) for each f ∈ Mor C , F f is a fibred manifold map covering f such that F ι(U ) = ι(F U ) for any open subset ι : U ֒→ M . A natural bundle functor on the subcategory Mm of M , for a certain m, is a natural lift functor, Nijenhuis [40]. In literature natural bundle functors on M are also called prolongation functors, Kol´aˇr [22]. In original definitions of natural lift and prolongation functors there is the regularity condition saying that a smoothly parameterized family of diffeomorphisms is prolonged

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Josef Janyˇska

into a smoothly parameterized family of diffeomorphisms. But this condition turns out to be a consequence of remaining prolongation and localization properties. This was proved by Epstein and Thurston [7] for natural lift functors and by Kol´aˇr and Slov´ak [25] for natural prolongation functors. A natural bundle is then a triplet (F M, pM , M ). Later (Theorem 2.2) we will see that pM : F M → M is indeed a bundle.

2.2.

Geometrical Object

A geometrical object on a manifold M is now an element from F M , where F is a natural bundle functor. A section σ : M → F M is a field of geometrical objects on M .

2.3.

Order of Natural Bundle Functors

We say that a natural lift functor F is of finite order r if r is the smallest number such that jxr f = jxr g ⇒ F f |Fx M = F g|Fx M ¯ ) ∈ Mor Mm and any x ∈ M . for any (f, g : M → M

Palais and Terng [42] proved that the order r of a natural lift functor is finite r < 2n+1 where n is the dimension of the fiber F0 Rm . Later Epstein and Thurston [7] gave much better bound. They proved r ≤ 2n + 1 and that this bound is sharp for m = 1. Finally Zajtz n n [54] proved r ≤ max{ n−1 ;m + 1}. Mikulski [38] has shown that a natural prolongation functor with infinite order exists.

2.4.

Differential Group

Let us denote by Grm the Lie group Grm = invJ0r (Rm , Rm )0 of invertible r-jets (with source and target 0) of diffeomorphisms of Rm which preserve 0. The group multiplication is given by the jet composition. The canonical coordinates on Grm will be denoted by (aλµ , . . . , aλµ1 ...µr ) and tilde will refer to the inverse element.

2.5.

Standard Fiber

Let F be an r-order natural lift functor and let F0 = F0 Rm . Because of (ii) of Definition 2.1 F0 is diffeomorphic with Fx M for any x ∈ M , M ∈ ObMm . F0 will be called the standard fiber of F . Applying F on origin-preserving diffeomorphisms of Rm we get a left action of Grm on F0 which defines on F0 a structure of a left smooth Grm -manifold, Krupka [29] and Terng [48].

2.6.

Natural Fibred Coordinate Chart

Local coordinate charts (xλ ) on M and (y p ) on F0 induce a fibred coordinate chart (xλ , y p ) on F M , which is said to be the natural fibred coordinate chart.

Natural Lagrangian Structures

2.7.

147

Examples

1. The tangent functor T is a natural bundle functor of order one on the category M with values in the category V B. In dimension m the corresponding standard fiber is Rm on which G1m = Gl(m, R) acts in the standard way by the matrix multiplication. The natural fibred coordinate chart on T M will be denoted by (xλ , x˙ λ ). 2. The cotangent functor T ∗ is a natural lift functor of order one with values in the category V B m . The standard fiber is Rm∗ with the standard action of G1m . 3. The functor ∧p T ∗ of p–forms is a natural lift functor of order one with values in the category V B m . The standard fiber is ∧p Rm∗ on which G1m acts in the standard tensor way. The natural fibred coordinate chart on ∧p T ∗ M will be denoted by (xλ , ωλ1 ...λp ), 1 ≤ λ1 < · · · < λp ≤ m. Especially, for p = m, we obtain the natural lift functor of volume forms. 4. The functor of pseudo–Riemannian metrics pRm is a natural lift functor of order one such that pRm(M ) are subbundles of bundles from the category V B m . The standard fiber (pRm)0 is the subspace in ⊙2 Rm∗ of non-degenerate symmetric matrices with the tensor action of G1m . The natural fibred coordinate chart on pRm(M ) will be denoted by (xλ , gλµ ), gλµ = gµλ , det(gλµ ) 6= 0. 5. The functor of k r -velocities Tkr is a natural bundle functor of order r on the category M . For any M ∈ Ob M , we define Tkr M = J0r (Rk , M ) and, for any f ∈ Mor M , ¯ , we define T r f (j r α) = j r (f ◦ α), where j r α ∈ T r M . The standard fiber of f :M →M 0 0 0 k k r Tk , in dimension m, is J0r (Rk , Rm )0 and the action of Grm on the standard fiber is given by the jet composition. 6. The functor of r-order frames F r is a natural lift functor of order r. For any M ∈ Ob Mm , we define F r M = invJ0r (Rm , M ) and, for any f ∈ Mor Mm , F r f is defined as in Example 2.7..5. The values of the functor F r are in the category PB m (Grm ). 7. The functor Cla of classical (linear) connections on a given manifold is a natural lift functor of order two with values in the category A B m . Its standard fiber is R∗m ⊗ Rm ⊗ Rm∗ on which G2m acts via the well known transformation relations of the Christoffel symbols, Christoffel [3], ¯ µ λ ν = aλρ (Λσ ρ τ a Λ ˜σµ a ˜τν − a ˜ρµν ) . The natural fibred coordinate chart on Cla M will be denoted by (xλ , Λµ λ ν ). By Claτ will be denoted the functor of torsion free linear connections. In natural fibred coordinates Claτ is characterized by Λµ λ ν = Λν λ µ . 8. Let F be a natural lift functor of order r and J s be the functor of s-jet prolongation, Saunders [43]. Then J s F ≡ J s ◦ F is a natural lift functor of order (r + s). If F0 is the standard fiber of F , then the standard fiber of J s F is (J s F )0 = Tns F0 and the action of s r Gr+s m on (J F )0 is obtained by the jet prolongation of the action of Gm on F0 .

2.8.

The Bundle Structure

In the theory of natural lift functors the functor of r-order frames, defined in Example 2.7..6, plays a fundamental role. Namely, we have the following theorem, Krupka [29] and Terng [48].

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Theorem 2.2. Any natural lift functor F of order r, with the standard fiber F0 , is canonically represented by F M = [F r M, F0 ],

F f = [F r f, idF0 ],

where M ∈ Ob Mm , f ∈ Mor Mm , and [F r M, F0 ] = (F r M, F0 )/Grm is the bundle associated with F r M . This theorem implies that there is the one-to-one correspondence between r-order natural lift functors and left Grm -manifolds.

2.9.

Natural Differential Operators

¯ be a mapping in MorMm and σ : M → F M Let F be a natural lift functor, f : M → M ∗ ¯ → FM ¯ by f ∗ σ = F f ◦ σ ◦ f −1 . be a section. Then we define the section f σ : M Definition 2.3. A natural differential operator D from a natural lift functor F1 to a natural lift functor F2 is a family of differential operators ¯ )}M ∈ObM {D(M ) : C ∞ (F1 M ) → C ∞ (F2 M m such that ¯ )(f ∗ σ) = f ∗ D(M )(σ) for every section σ ∈ C ∞ (F1 M ) and every f : M → (i) D(M ¯ M in MorMm , (ii) DU (σ|U ) = (DM σ)|U for every section σ ∈ C ∞ (F1 M ) and every open submanifold U ⊂ M , (iii) every smoothly parameterized family of sections of F1 M is transformed into a smoothly parametrized family of sections of F2 M .

2.10.

Order of Natural Differential Operator

A natural differential operator is of order k, 0 ≤ k ≤ ∞, if all D(M ), M ∈ ObMm , are of order k. Thus, a k-order natural differential operator D from F1 to F2 is characterized by the associated fibred manifold morphisms D(M ) : J k F1 M → F2 M , over M , according to the formula D(M )(jxk σ) = D(M )(σ)(x). The family D = {D(M )}M ∈ObMm defines a natural transformation of the functors J k F1 and F2 . In what follows we will identify korder natural differential operators with the corresponding natural transformations and use the same symbol.

2.11.

Equivariant Mappings Given by Natural Operators

Coordinate independent geometrical constructions are in fact natural differential operators between natural lift functors. The study of natural differential operators is based on relations between natural differential operators and equivariant mappings. The basic tool is the following theorem, Terng [48]. Theorem 2.4. There is a bijective correspondence between the set of k-order natural differential operators from a natural lift functor F1 to a natural lift functor F2 and equivariant mappings from the standard fiber of J k F1 to the standard fiber of F2 .

Natural Lagrangian Structures

2.12.

149

Examples

1. The exterior differential d is a first order natural differential operator from ∧p T ∗ , p ≥ 0, to ∧p+1 T ∗ . The corresponding G2n -equivariant mapping from J 1 (∧p T ∗ )0 = Tn1 (∧p Rm∗ ) to (∧p+1 T ∗ )0 = ∧p+1 Rm∗ is given, in the canonical coordinate chart (ωλ1 ...λp ), 1 ≤ λ1 < · · · < λp ≤ m, on (∧p Rm∗ ), by ωλ1 ...λp+1 ◦ d = ω[λ1 ...λp ,λp+1 ] , where [...] denotes the anti-symmetrization. For p ≥ 1 the naturality determines d up to a constant multiple while in classical proofs the linearity was supposed, see for instance Kol´aˇr [23], Krupka and Janyˇska [33] and Krupka and Mikol´asˇov´a [34]. 2. The Levi-Civita connection is a first order natural differential operator from pRm to Claτ . The corresponding G2m -equivariant mapping from J 1 (pRm)0 to Cla0 is given by the formal Christoffel symbols, Christoffel [3], Λµ λ ν = − 12 g λρ (gρµ,ν + gρν,µ − gµν,ρ ), where (g λµ ) is the inverse matrix of (gλµ ). The uniqueness of the Levi–Civita connection is the classical geometrical problem. The proof of the uniqueness by using natural techniques can be found in Krupka and Janyˇska [33], Krupka and Mikol´asˇov´a [34] and Slov´ak [47]. 3. The curvature tensor of a classical connection is a first order natural differential operator from Cla to T ∗ ⊗ T ⊗ (∧2 T ∗ ). The corresponding G3m -equivariant mapping from J 1 Cla0 to (T ∗ ⊗ T ⊗ (∧2 T ∗ ))0 = R∗m ⊗ Rm ⊗ (∧2 Rm∗ ) is given by wν κ λµ = Λν κ λ,µ − Λν κ µ,λ + Λρ κµ Λν ρλ − Λρ κ λ Λν ρ µ . The curvature tensor is not unique operator of this type and plays an important role in classification of natural operators defined on classical connections, see for instance Kol´aˇr [23], Kol´aˇr, Michor and Slov´ak [24] and Schouten [44].

2.13.

Infinitesimal Properties of Natural Lift Functors

The regularity property of lift functors allows us to lift vector fields on a manifold M to projectable vector fields on a natural bundle F M by using flows. Namely if exp(tξ) is the flow of a vector field ξ on M then F (exp(tξ)) = exp(tFξ) is the flow of the vector field Fξ on F M which is said the flow lift of ξ. Moreover, if F is of order r, then Fξ depends on r-jets of ξ. For instance the flow lift of a vector field ξ = ξ λ ∂x∂ λ with respect to the tangent functor is the vector field ∂ ∂ξ λ ∂ T ξ = ξ λ λ + ρ x˙ ρ λ ∂x ∂x ∂ x˙

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on T M and the flow lift of ξ with respect to the natural lift functor of classical connections, see Example 2.4..7, is the vector field ∂ξ λ ∂ Cla ξ = ξ λ λ + Λµ ρ ν (2.1) ∂x ∂xρ ρ ∂ξ ρ ∂2ξλ ∂ λ ∂ξ − Λρ λ ν − Λ + µ ρ ∂xµ ∂xν ∂xµ ∂xν ∂Λµ λ ν on Cla M .

2.14.

Infinitesimal Properties of Natural Operators

If σ : M → F M is a section of an r-th order natural bundle (a field of geometrical objects) then we can define the Lie derivative of σ with respect to a vector field ξ by the formula d |0 {exp(−tξ)∗ σ} . (2.2) dt Lξ σ is a section of V F M . Natural differential operators D from a natural lift functor F1 to a natural lift functor F2 are infinitesimally characterized by the commutativity with the Lie derivatives, Kirillov [21] and Janyˇska and Modugno [19], in the following sense. Lξ σ =

Theorem 2.5. A k-order differential operator D(M ) from a natural bundle F1 M to a natural bundle F2 M is natural if and only if Lξ D(M )(σ) = T D(M )(Lξ σ)

(2.3)

for all σ ∈ C ∞ (F1 M ) and all vector fields ξ on M . If we identify D with the corresponding natural transformation D, then the above condition (2.3) is equivalent with T D(M )(J k F1 ξ)(j k σ) = F2 ξ(D(M )(j k σ)) .

3.

(2.4)

Natural Lagrangian Structures on Natural Bundles

If we assume in (2.4) the natural bundle of volume forms as the target bundle, i.e. F2 = ∧m T ∗ , the condition (2.4) means that natural Lagrangians on the natural bundle F1 M are just invariant Lagrangians on F1 M (in the sense of (1.2)) and infinitesimal symmetries of such Lagrangians are vector fields F1 ξ, for all vector fields ξ on M . So a k-order Lagrangian L : J k F1 M → ∧m T ∗ M is invariant (natural) if and only if LJ k F1 ξ L = 0

(3.1)

for any vector field ξ on M . The above infinitesimal characterization of invariant Lagrangians on natural bundles leads in natural fibered local coordinates to a system of partial differential equations which is generally difficult to solve. Much more simple is to use algebraical method of classification of equivariant mappings corresponding to natural operators. Special situation we obtain in the case of invariant Lagrangians on the natural bundle of classical symmetric connections. In this case we can use the reduction theorems by Schouten [44].

Natural Lagrangian Structures

3.1.

151

The First and the Second Reduction Theorems

Let us assume the natural bundle of classical symmetric connections on manifolds given by the lift functor Claτ , see Example 2.7..7. Schouten [44] proved that all finite order polynomial concomitants on Claτ M and a bundle of geometrical object of order one with values in an other bundle of geometrical objects of order one factorize through the curvature tensor, a given field of geometrical objects and their covariant differentials. By using the theory of natural differential operators on natural bundles the results of Schouten were generalized by Kol´aˇr, Michor and Slov´ak [24] and we obtain the first reduction theorem in the form. Theorem 3.1. All natural differential operators of order r with values in a natural bundle of order one of a classical symmetric connection are natural differential operators of order zero of the curvature tensor and its covariant differentials up to the order (r − 1), i.e. e (r−1) R[Λ]) , D(j r Λ) = D(∇

where ∇(r−1) = (id, ∇, . . . , ∇r−1 ).

The second reduction theorem can be formulated as follows. Theorem 3.2. Let Φ be a field of geometrical objects of order one and s ≥ r − 1. All natural differential operators with values in a natural bundle of order one of a classical symmetric connection (in order s) and Φ (in order r) are natural differential operators of the curvature tensor, Φ and their covariant differentials up to the orders (s − 1) and r, respectively, i.e. e (s−1) R[Λ], ∇(r) Φ) . D(j s Λ, j r Φ) = D(∇

Remark 3.3. If Λ is a classical non-symmetric connection on M , then there exists its unique e + T , where Λ e is the classical symmetric connection obtained by the symsplitting Λ = Λ metrization of Λ and T is the torsion tensor of Λ. Then all natural operators of Φ and Λ are of the form e j s T, j r Φ) = D( e ∇ e (s−1) R[Λ], e ∇ e (s) T, ∇ e (r) Φ) , D(j s Λ, j r Φ) = D(j s Λ,

e refers to the connection Λ. e where ∇

3.2.

Invariant Lagrangians of Classical Symmetric Connections

Now, as a simple consequence of Theorem 3.1, we have Theorem 3.4. All invariant (natural) Lagrangians of order r on the natural bundle of classical symmetric connections are of the type e (r−1) R[Λ]) . L(j r Λ) = L(∇

Remark 3.5. r-order invariant Lagrangians on classical symmetric connections are infinitesimally given as solutions of the system of partial differential equations given by LJ r Cla ξ L = 0

(3.2)

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for all vector fields ξ on M and Cla ξ given by (2.1). From Theorem 3.4 it follows that all solutions of (3.2) factorize through the curvature tensor and its covariant differentials up to the order (r − 1). As a consequence of Theorem 3.2, we have Theorem 3.6. All invariant Lagrangians on the natural bundle of classical symmetric connections (in order s) and a natural bundle of order one (in order r) are of the type

3.3.

e (s−1) R[Λ], ∇(r) Φ) . L(j s Λ, j r Φ) = L(∇

Invariant Lagrangians of Metric Fields

In physical theories M is usually an oriented spacetime with a metric g. In this case we have the Levi Civita connection Λ(g) and the canonical volume form p ω(g) = |g| dx1 ∧ . . . ∧ dxm .

Any invariant Lagrangian is of the form L(j r g, j r Φ) = l(j r g, j r Φ) ω(g) where l(j r g, j r Φ) is an invariant Lagrangian function. Then, as a consequence of Theorem 3.6, invariant Lagrangian functions are of the type l(j r g, j r Φ) = e l(g, ∇(r−2) R[Λ(g)], ∇(r) Φ) .

Invariant Lagrangians on the natural bundle of metrics were studied for instance by Krupka [27, 28] and Novotn´y [41].

4.

Gauge-Natural Bundles

Natural differential operators on natural bundles describe the invariance of geometrical or physical theories with respect to changes of local coordinates. But in physical theories another sort of invariance plays an important role, the so called gauge invariance. Invariant gauge theory has been introduced in the book by H. Weyl [52] in 1918 as a generalization of the Einstein’s general relativity. Weyl considered operators on a spacetime invariant not only with respect to isomorphisms of spacetime but also with respect to gauge transformations g 7→ eα g (the term ”gauge” was used for the first time by H. Weyl). The original invariant physical gauge theories was related with the gauge group U (1) acting on wave functions and electromagnetic potentials. In early 1950’s the concept of gauge invariance was generalized first for the spin group, see for instance Yang and Mills [55], and then for any Lie group G playing the role of the gauge group. The first geometrical interpretation of gauge invariance with respect to a general gauge group can be found in the famous paper by Utiyama [51]. Gauge invariant theories can be described geometrically by using the concepts of gauge-natural bundle functors and natural differential operators between gaugenatural bundles. So in Section 4. we recall basic definitions and properties of gauge-natural bundle functors, see Eck [6], Fatibene and Francaviglia [9], Kol´aˇr [23] and Kol´aˇr, Michor and Slov´ak [24].

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4.1.

153

Gauge-natural Bundle Functors

Gauge-natural bundle functors was introduced by Eck [6]. We recall here the definition of Kol´aˇr, Michor and Slov´ak [24]. Let us recall that B is the base functor from the category F M to the category M . Definition 4.1. A gauge-natural bundle functor over m-dimensional manifolds is a functor F : PB m (G) → F M m such that (i) every PB m (G)-object π : P → BP is transformed into a fibered manifold qP : F P → BP over BP , (ii) every PB m (G)-morphism f : P → P¯ is transformed into a fibered morphism F f : F P → F P¯ over Bf , (iii) for every open subset U ⊂ BP , the inclusion ι : π −1 (U ) ֒→ P is transformed into the inclusion F ι : qP−1 (U ) ֒→ F P . A gauge-natural bundle is then a quadruple (F P, πP , M, π : P → BP ). Later (Theorem 3.3.) we will see that F P is actually a bundle. In the original definition by Eck [6] there is one more regularity (continuity) condition which says that a smoothly parametrized family of diffeomorphisms of P is ”transformed” into a smoothly parameterized family of isomorphisms of F P . But this condition is a consequence of (i), (ii) and (iii), Kol´aˇr, Michor and Slov´ak [24].

4.2.

Functor W r

r Let (π : P → M ) ∈ Ob PB m (G), let W r P be the space of all r-jets j(0,e) ϕ, where m m ϕ : R × G → P is in Mor PB m (G), 0∈ R and e is the unity in G. The space W r P is r G = Jr m m a principal fibre bundle over M with the structure group Wm (0,e) (R × G, R × G) of all r-jets of principal fibre bundle isomorphisms Ψ : Rm × G → Rm × G covering the diffeomorphisms ψ : Rm → Rm such that ψ(0) = 0. The group Wnr G is the semidirect r G of Gr and T r G with respect to the action of Gr on T r G given by product Grm ⋊ Tm m m m m the jet composition. Let (ϕ : P → P¯ ) ∈ Mor PB m (G), then we can define the principal bundle morphism W r ϕ : W r P → W r P¯ by the jet composition. The rule transforming r G) and any ϕ ∈ Mor PB (G) into any P ∈ Ob PB m (G) into W r P ∈ Ob PB m (Wm m r r W ϕ ∈ Mor PB m (Wm G) is a gauge-natural bundle functor, Kol´aˇr [23].

4.3.

Bundle Structure

The gauge-natural bundle functor W r described in Subsection 4.2. plays a fundamental role in the theory of gauge-natural bundle functors. We have, Eck [6], Theorem 4.2. Every gauge-natural bundle F P is a fibred bundle associated with the gauge-natural bundle W r P for a certain order r.

4.4.

Order of Gauge-natural Bundle Functors

The number r from Theorem 4.2 is called order of the gauge-natural bundle functor F . So if F is an r-order gauge-natural bundle functor then F P = [W r P, F0 ],

F ϕ = [W r ϕ, idF0 ],

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r G-manifold called the standard fibre of F . where F0 is a left Wm

4.5.

Gauge and Total Order of Gauge-natural Functors

Let F be an s-order gauge-natural bundle functor and let r ≤ s be the minimal number s G = Gs ⋊ T s G on F can be factorized through the canonical such that the action of Wm 0 m m s G → T r G, s ≥ r. Then s is said to be the total order of F , r is projection πrs : Tm m the gauge order and we say that F is of order (s, r). In what follows we shall denote by (s,r) r G and by W (s,r) P the corresponding principal bundle. Wm G = Gsm ⋊ Tm

4.6.

Gauge-natural Fibred Coordinate Chart

A local fibred coordinate chart (xλ , pσ ) on P and a coordinate chart (y p ) on F0 induce a fibred coordinate chart (xλ , y p ) on F P which is said to be the gauge-natural fibred coordinate chart.

4.7.

Examples

1. Any r-order natural lift functor in the sense of Definition 2.1 is the (r,0)-order gaugenatural bundle functor with the trivial gauge action, i.e. the action (Grm × G) × F0 −→ F0 does not depend on G. 2. Let (π : P → M ) ∈ Ob PB m (G) and let us denote by Pri P → M the bundle of principal connections on P . Then Pri is a (1,1)-order gauge-natural bundle functor with 1 G given by, Kol´ the standard fibre G ⊗ Rm∗ and with the action of Wm aˇr [23], (X, g, Z)(Y ) = ad(g)(Y + Z)X −1 .

In particular, let G = Grn , then Pri P can be viewed as the bundle Lin E of linear connections on an associated vector bundle E → M with n-dimensional fibres. The standard fibre of Lin is Lin0 = R∗n ⊗ Rn ⊗ Rm∗ with coordinates (Kj i λ ), i, j = 1, ..., n, (1,1) 1 G1 on Lin is given, in the canonical λ = 1, ..., m, and the action of Wm G1n = G1m ⋊ Tm 0 n λ i i 1 1 1 coordinates (aµ , aj , ajλ ) on Gm ⋊ Tm Gn , by ¯ j i λ = aip Kq p ρ a K ˜qj a ˜ρλ + aipρ a ˜pj a ˜ρλ , where tilde refers to the inverse element. 3. Let F0 be a left G–manifold. The associated gauge-natural bundle functor is defined by assF0 (P ) = [P, F0 ], assF0 (ϕ) = [ϕ, idF0 ] , where P ∈ ObPB m (G), ϕ ∈ MorPB m (G). assF0 (P ) is a 0-order gauge-natural bundle. Especially the adjoint bundle ad P is the 0-order gauge-natural bundle given by the adjoint action of G on its Lie algebra G . 4. If F is a gauge-natural bundle functor of order (s, r) then J k F is a gauge-natural bundle functor of order at most (s + k, r + k). The number (s + k) is exact, but (r + k) may be too big. For instance if F is an s-order natural lift functor, i.e. an (s,0)-order gaugenatural bundle functor, then J k F is an (s+k)-order natural lift functor, i.e. an (s+k,0)-order gauge-natural bundle functor. 5. ad P ⊗ (∧p T ∗ M ) is a (1,0)-order gauge-natural vector bundle.

Natural Lagrangian Structures

4.8.

155

Natural Operators

¯ , F be a gauge-natural bundle Let (ϕ, f ) ∈ Mor PB m (G), ϕ : P → P¯ over f : M → M ¯ → F P¯ by functor and σ : M → F P be a section. Then we define the section ϕ∗ σ : M ∗ −1 ϕ σ = Fϕ ◦ σ ◦ f . Definition 4.3. A natural differential operator D from a gauge-natural bundle functor F1 to a gauge-natural bundle functor F2 is a family of differential operators {D(P ) : C ∞ (F1 P ) → C ∞ (F2 P )}P ∈ObPBm (G) such that (i) D(P¯ )(ϕ∗ σ) = ϕ∗ D(P )(σ) for every sec∞ tion σ ∈ C (F1 P ) and every (ϕ, f ) ∈ MorP ¯, Bm (G), ϕ : P → P¯ over f : M → M −1 (ii) D(π (U ))(σ|U ) = (D(P )(σ))|U for every section σ ∈ C ∞ (F1 P ) and every open subset U ⊂ M , (iii) every smoothly parameterized family of sections of F1 P is transformed into a smoothly parameterized family of sections of F2 P .

4.9.

Order of Natural Differential Operators

A natural differential operator D from F1 to F2 is of a finite order k if all D(P ), (π : P → M ) ∈ObPB m (G), depend on k-order jets of sections of F1 P . Thus, a k-order natural differential operator from F1 to F2 is characterized by the associated fibred manifold morphism D(P ) : J k F1 P → F2 P , over M , such that the family D = {D(P )}P ∈ObPBm (G) is a natural transformation of J k F1 to F2 . In what follows we will identify k-order natural differential operators with the corresponding natural transformations and use the same symbol. Theorem 4.4. Let F1 and F2 be gauge-natural bundle functors of order ≤ r. Then we have a one-to-one correspondence between natural differential operators of order k from F1 to r+k G-equivariant mappings from (J k F ) to (F ) . F2 and Wm 1 0 2 0 This theorem is due to Eck [6], see also Kol´aˇr, Michor and Slov´ak [24].

4.10.

Curvature Operator

The curvature operator of principal connections is a 1-order natural differential operator (2,2) from Pri to ad ⊗ (∧2 T ∗ ) with the associated Wm G-equivariant morphism (ua λµ ) ◦ R = Γa λ,µ − Γa µ,λ + cabd Γb λ Γd µ , where c = (cabd ) are the structure constants of G.

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Josef Janyˇska

Infinitesimal Properties of Gauge-natural Bundle Functors

The regularity (continuity) property of gauge-natural bundle functors allows to transform right G-invariant vector fields on a principal G-bundle P to projectable vector fields on a gauge-natural bundle F P by using flows. Namely if exp(tΞ) is the flow of a right Ginvariant vector field Ξ on P , projectable on the vector field ξ on M , then F (exp(tΞ)) = exp(tFΞ) is the flow of the vector field FΞ on F P which is said the flow transformation of Ξ. Moreover, if F is of order r, then FΞ depends on r-jets of Ξ. ea ) is the base of vertical right G-invariant vector fields on P given by For instance if (B a base (Ba ) of the Lie algebra G and Ξ is a right G-invariant vector field Ξ = ξ λ (x) ∂x∂ λ + ea on P . Then the flow transformation of Ξ with respect to the gauge-natural bundle Ξa (x)B functor of principal connections, see Example 4.7..2, is the vector field ∂ ∂ξ ρ ∂Ξa ∂ P ri Ξ = ξ λ λ + cabd Ξb Γd λ − Γa ρ λ + (4.1) ∂x ∂x ∂xλ ∂Γa λ on Pri P .

4.12.

Infinitesimal Properties of Natural Differential Operators

If σ : M → F P is a section of an r-th order gauge-natural bundle then we can define the Lie derivative of σ with respect to a right G-invariant vector field Ξ on P , over the vector field ξ on M , by the formula d |0 {exp(−tΞ)∗ σ} . dt LΞ σ is a section of V F P . Natural differential operators D from a gauge-natural bundle functor F1 to a gauge-natural functor functor F2 can be infinitesimally characterized by the commutativity with the Lie derivatives, Janyˇska and Modugno [19], in the following sense. LΞ σ =

Theorem 4.5. A k-order differential operator D(P ) from a gauge-natural bundle F1 P to a gauge-natural bundle F2 P is natural if and only if LΞ D(P )(σ) = T D(P )(LΞ σ) ,

(4.2)

for all right G-invariant vector fields Ξ on P and all sections σ ∈ C ∞ (F1 P ). If we identify D with the corresponding natural transformation D, then the above condition (4.2) is equivalent with T D(P )(J k F1 Ξ)(j k σ) = F2 Ξ(D(P )(j k σ)) .

5.

(4.3)

Natural Lagrangians on Gauge-Natural Bundles

Gauge invariant theories can be very efficiently formulated by using the theory of gaugenatural bundles and natural differential operators and have wide applications in gauge field theories, see Fatibene and Francaviglia [9]. As applications of natural differential operators on gauge-natural bundles we will generalize the Utiyama’s reduction method for the linear group Gl(n, R) as a gauge group, Janyˇska [15], and for a general Lie group G as a gauge group, Janyˇska [17, 18].

Natural Lagrangian Structures

5.1.

157

Gauge Invariant Lagrangians on Electromagnetic Potentials

Originally gauge invariant Lagrangians were studied on a complex wave function (a particle field) φ and a gauge field A = (Aλ (x)). The gauge invariance was considered with respect to the gauge transformation of φ given in the form φ(x) 7→ eiα(x) φ(x). In order to obtain a gauge invariant Lagrangian of φ of order one it is necessary to consider also a µ gauge field A with the gauge transformation Aλ 7→ (Aµ + ∂µ α) ∂x . Then a gauge invariant ∂x ¯λ Lagrangian of order one is of the form e ∇φ) , L(j 1 φ, A) = L(φ,

(5.1)

e ). L(j 1 A) = L(F

(5.2)

where ∇λ φ = ∂λ φ − i Aλ φ. In physical theories A is an electromagnetic potential and the above result is known as the minimal coupling principle. Further, a gauge invariant Lagrangian of order one of the gauge field A only factorizes through the 2-form (electromagnetic 2-form) Fλµ = ∂λ Aµ − ∂µ Aλ , i.e.

The above classical gauge invariance corresponds to the invariance with respect to the gauge group U (1). The above results can be interpreted geometrically as follows: let Q → M be a complex line bundle (a quantum bundle) with a Hermitian product h and local fibered coordinates (xλ , z). Then φ : M → Q is a section and A = dxλ ⊗ (

∂ ∂ + iAλ (x) ) λ ∂x ∂z

is a linear connection such that ∇h = 0 (a Hermitian connection). Then ∇φ is the standard covariant differential and the 2-form F is given by the curvature tensor of A.

5.2.

Utiyama’s Reduction Theorems for Principal Connections

In [51] Utiyama generalized the results of Subsection 5.1. for a general Lie group G as a gauge group. A particle field Φ is now a section of a vector bundle associated with a principle G-bundle P and a gauge field is a principal connection Γ on P . Then any gauge invariant first order Lagrangian on Γ is given by a gauge invariant Lagrangian of the curvature tensor R[Γ], i.e. e L(j 1 Γ) = L(R[Γ]) .

(5.3)

e ∇Φ) . L(Γ, j 1 Φ) = L(Φ,

(5.4)

This result is in literature cited as the Utiyama’s theorem. Further, Utiyama generalized the minimal coupling principle as the invariant interaction of the particle field Φ and the gauge field Γ in the form

In his original paper [51] Utiyama considered his theorems only locally with gauge transformations described in coordinates. Later the Utiyama’s theorem was reproved by many authors also globally, see for instance Castrill´on, Mu˜noz and Ratiu [2], Eck [6] and Mangiarotti and Modugno [36]. The Utiyama’s results can be very simply generalized

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for operators with values in a gauge-natural bundle of order (1, 0). In this case we shall use the term Utiyama-like theorem instead of the Utiyama’s theorem. The Utiyama-like theorem was proved (in order 1) in Kol´aˇr, Michor and Slov´ak [24]. A generalization of the invariant interaction (5.4) was proved globally by Betounes [1] who proved L(j 1 Γ, j 1 Φ) = e L(R[Γ], Φ, ∇Φ).

5.3.

First Reduction Theorem for General Linear Connections

In Janyˇska and Modugno [20], see also Janyˇska [12, 13, 14], we have studied second order natural quantum Lagrangians and second order natural Schr¨odinger operators on the quantum bundle, i.e. we studied second order operators on a gravitational field (a classical symmetric connection on spacetime), an electromagnetic potential (a quantum connection, i.e. a general linear connection on the quantum bundle preserving the Hermitian product) and a wave function (a section of the quantum bundle). In both situations such operators are factorized through the covariant differentials of sections of the quantum bundle where the first order covariant differentials are given by the quantum connection only but the second order covariant differentials are given by both quantum and spacetime connections. This fact was a motivation how to generalize the reduction theorems, see Subsection 3.1., for general linear connections on vector bundles. Let E → M be a vector bundle with a m-dimensional base and n-dimensional fibres. Local linear fiber coordinate charts on E will be denoted by (xλ , y i ). We define a linear connection on E to be a linear splitting K : E → J 1 E . Considering the contact morphism J 1 E → T ∗ M ⊗ T E over the identity of T M , a linear connection can be regarded as a T E-valued 1-form K : E → T ∗ M ⊗ T E projecting onto the identity of T M . The coordinate expression of a linear connection K is of the type K = dλ ⊗ ∂λ + Kj i λ y j ∂i , with Kj i λ ∈ C ∞ (M, R) .

Linear connections can be regarded as sections of a (1,1)-order G = Gl(n, R)-gaugenatural bundle Lin E → M described in Example 4.7..2. The curvature of a linear connection K on E turns out to be the vertical valued 2–form R[K] = −[K, K] : E → V E ⊗ ∧2 T ∗ M , where [, ] is the Froelicher-Nijenhuis bracket. If we consider the identification V E = E × E and linearity of R[K], the curvature R[K] can M

be considered as the curvature tensor field R[K] : M → E ∗ ⊗ E ⊗ ∧2 T ∗ M and R[K] : C ∞ (LinE) → C ∞ (E ∗ ⊗ E ⊗ ∧2 T ∗ M )

is a natural differential operator which is of order one. p,r Let us set Eq,s = ⊗p E ⊗ ⊗q E ∗ ⊗ ⊗r T M ⊗ ⊗s T ∗ M . Then a classical connection Λ on M and a linear connection K on E induce the linear tensor product connection Kqp ⊗ Λrs = p,r ⊗p K ⊗ ⊗q K ∗ ⊗ ⊗r Λ ⊗ ⊗s Λ∗ on Eq,s p,r p,r Kqp ⊗ Γrs : Eq,s → T ∗ M ⊗ T Eq,s M p Kq ⊗

p,r p,r → J 1 Eq,s . Let Φ ∈ which can be considered as a linear splitting Λrs : Eq,s p,r ∞ C (Eq,s ). We define the covariant differential of Φ with respect to the pair of connecp,r tions (K, Λ) as a section of Eq,s ⊗ T ∗ M defined by, Janyˇska [16],

∇(K,Λ) Φ = j 1 Φ − (Kqp ⊗ Λrs ) ◦ Φ .

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159

The iterated rth order covariant differential applied on the curvature tensor of the linear connection K is a natural differential operator which is of order (r − 1) with respect to the classical connection Λ and of order (r + 1) with respect to the linear connection K. Let (s) (r) us denote by CLr E the image of this operator and by CC M ×M CL E the (s, r)-order curvature bundle of classical and linear connections given as the image of the pair of the operators (∇(s+1) R[Λ], ∇(r+1) R[K]), s ≥ r − 2, ∇(s) = (id, ∇, . . . , ∇s ), defined on Claτ M × Lin E. Let us assume a (1, 0)-order Gl(n, R)-gauge natural bundle F E, then the first reduction theorem for linear and classical connections can be formulated as follows, Janyˇska [15]. Theorem 5.1. Let s ≥ r − 2, r ≥ 0. All natural differential operators D : C ∞ (Claτ M × Lin E) → C ∞ (F E) M

which are of order s with respect to classical symmetric connections and of order r with respect to general linear connections are of the form e (s−1) R[Λ], ∇(r−1) R[K]) D(j s Λ, j r K) = D(∇

e is a zero order natural operator where D

e : C ∞ (C (s−1) M × C (r−1) E) → C ∞ (F E) . D C L M

5.4.

Second Reduction Theorem for General Linear Connections

Let us assume the kth order covariant differential of sections of Eqp11,q,p22 (particle fields). It is a natural operator of order k with respect to sections of Eqp11,q,p22 and of order (k − 1) with respect to classical and linear connections. Let us define the k-th order Ricci bundle Z (k) E as the image of the triplet of the operators (∇(k−2) R[Λ], ∇(k−2) R[K], ∇(k) Φ) defined on Claτ M × Lin E × Eqp11,q,p22 . Then the second reduction theorem for linear and classical connections can be formulated as follows, Janyˇska [15]. Theorem 5.2. Let s, r ≥ k − 1, s ≥ r − 2. All natural differential operators D : C ∞ (Claτ M × Lin E × Eqp11,q,p22 ) → C ∞ (F E) M

M

which are of order s with respect to classical symmetric connections, of order r with respect to general linear connections and of order k with respect to sections of Eqp11,q,p22 are of the form e (s−1) R[Λ], ∇(r−1) R[K], ∇(k) Φ) D(j s Λ, j r K, j k Φ) = D(∇

e is a zero order natural operator where D e : C ∞ ((C (s−1) M × C (r−1) E) D C L M

×

(k−2) (k−2) CC M × CL E M

Z (k) E) → C ∞ (F E) .

160

5.5.

Josef Janyˇska

Higher Order Utiyama’s Theorem

Higher order local version of the Utiyama-like theorem was studied by Horndeski [10] who generalized the replacement theorem of Thomas and his collaborators, [49, 50], for gauge fields. The results obtained in [10] are local and not complete since only concomitants obtained from the covariant differentials of the curvature tensor of the gauge field are assumed, while concomitants obtained from classical connections on the base only are not considered. By using the methods of gauge-natural bundles we obtain complete and global coordinate free description of higher order Utiyama-like theorem. Let G be an n-dimensional Lie group, P ∈ Ob PBm (G), Γ be a principal connection on P and ad P is the adjoint vector bundle associated with the principal bundle P . Then we have the induced adjoint linear connection ad(Γ) on ad P . If Γ has the coordinate expression Γ = dλ ⊗ (

∂ ea ) , + Γa λ (x) B ∂xλ

(5.5)

then ad(Γ) has, in the induced fibered coordinates (xλ , ua ) on ad P , the coordinate expression ∂ λ a b d ∂ ad(Γ) = d ⊗ + cbd Γ λ (x) u . (5.6) ∂xλ ∂ua The curvature tensors of principal connections are given by a 1-order natural operator from Pri P into ad P ⊗ ∧2 T ∗ M . The covariant differential of the curvature tensor R[Γ] with respect to Γ and a classical connection Λ on the base M is then defined as the covariant differential with respect to ad(Γ) and Λ, see Subsection 5.2. and Janyˇska [17]. Then the iterated rth order covariant differential ∇r R[Γ] is a natural operator on Cla M × Pri P which is of order (r − 1) with respect to classical connections and of order (r + 1) with (s) (r) respect to principal connections. Let us denote by CC M × CP P , s ≥ r − 2, (s, r)order curvature bundle for classical and principal connections obtained as the image of the pair of the operators (∇(s) R[Λ], ∇(r) R[Γ]) defined on Claτ × Pri P . Then higher order Utiyama-like theorem for principal and classical connections can be formulated as follows, Janyˇska [17]. Theorem 5.3. Let s ≥ r − 2, r ≥ 0, and let F be a (1, 0)-order G-gauge-natural bundle functor. All natural differential operators D : C ∞ (Claτ M × Pri P ) → C ∞ (F P ) M

which are of order s with respect to classical symmetric connections and of order r with respect to principal connections are of the form e (s−1) R[Λ], ∇(r−1) R[Γ]) D(j s Λ, j r Γ) = D(∇

e is a zero order natural operator where D

e : C ∞ (C (s−1) M × C (r−1) P ) → C ∞ (F P ) . D C P M

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Remark 5.4. The curvature bundle of classical symmetric and principal connections is given by identities depending on the structure constants of the group G. So all natural operators defined on the curvature bundle depend also on the structure constants, i.e. e ∇(s−1) R[Λ], ∇(r−1) R[Γ]) . D(j s Λ, j r Γ) = D(c,

For instance cbab ∇ρr . . . ∇ρ1 Ra λµ is an example of a natural tensor field of the type (0, r+2) on M given by Λ (in order (r − 1)) and Γ (in order (r + 1)). In the case of (general) linear connections the structure constants are given by the Kronecker deltas and they contract with the curvature tensor fields, i.e. they are not ”visible”.

5.6.

Higher Order Utiyama’s Invariant Interaction

Let E → M be a vector bundle obtained as the bundle associated with P with respect to a linear representation ℓ : G → Gl(n, R), ℓ = (ℓij (g)), and let us denote by ℓ′ the set ∂ℓi

of constants ℓija = ∂gja |e, g a beeing the local coordinates on G and e ∈ G is the unit element. Then any principal connection Γ on P induces the linear connection ℓ(Γ) on E by ℓ(Γ)j i λ = ℓijb Γb λ . Let us assume Eqp11,q,p22 defined as in Subsection 5.3., then the covariant differential of sections of Eqp11,q,p22 (particle fields) with respect to the pair of connection (Γ, Λ) is defined as the covariant differential with respect to the pair of connections (ℓ(Γ), Λ), see Subsection 5.3.. The k-order covariant differential is then a natural differential operator of order k with respect to sections of Eqp11,q,p22 and of order (k − 1) with respect to classical and principal connections. Let us define the k-th order Ricci bundle Z (k) P as the image of the triplet of the operators (∇(k−2) R[Λ], ∇(k−2) R[K], ∇(k) Φ) defined on Claτ M × Lin E × Eqp11,q,p22 . Then the higher order invariant interaction of particle fields, principal and classical symmetric connections can be formulated as follows, Janyˇska [18]. Theorem 5.5. Let s, r ≥ k − 1, s ≥ r − 2. All natural differential operators D : C ∞ (Claτ M × Pri P × Eqp11,q,p22 ) → C ∞ (F P ) M

M

which are of order s with respect to classical symmetric connections, of order r with respect to principal connections and of order k with respect to sections of Eqp11,q,p22 are of the form e (s−1) R[Λ], ∇(r−1) R[Γ], ∇(k) Φ) D(j s Λ, j r Γ, j k Φ) = D(∇

e is a zero order natural operator where D e : C ∞ ((C (s−1) M × C (r−1) P ) D C P M

×

(k−2) (k−2) CC M × CP P M

Z (k) P ) → C ∞ (F P ) .

Remark 5.6. The Ricci identities and the identities on the curvature bundle of classical symmetric and principal connections depend on the structure constants of the group G and the constants ℓ′ . So all natural differential operators defined on the Ricci bundle depend also on the structure constants and ℓ′ , i.e. e ℓ′ , ∇(s−1) R[Λ], ∇(r−1) R[Γ]) . D(j s Λ, j r Γ) = D(c,

162

5.7.

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Gauge Invariant Lagrangians

As direct consequences of Theorems 5.3 and 5.5 we obtain higher order gauge invariant Lagrangians in the form. Theorem 5.7. All natural Lagrangians of orders s in Λ, r in Γ and k in Φ are of the form e ∇(s−1) R[Λ], ∇(r−1) R[Γ]) , L(j s Λ, j r Γ) = L(c, e ′ , c, ∇(s−1) R[Λ], ∇(r−1) R[Γ], ∇(k) Φ) , L(j s Λ, j r Γ, j k Φ) = L(ℓ

where Le is a unique zero order gauge invariant Lagrangian.

Remark 5.8. Gauge invariant Lagrangians on classical symmetric connections (in order s) and principal connections (in order r) are infinitesimally given as solutions of the system of partial differential equations given by LJ s Cla ξ+J r P ri Ξ L = 0

(5.7)

for all right G-invariant vector fields Ξ on P over ξ on M , where Cla ξ is given by (2.1) and P ri Ξ is given by (4.1). From Theorem 5.3 it follows that all solutions of (5.7) factorize through the curvature tensors of both connections and their covariant differentials up to the orders (s − 1) and (r − 1). Example 5.9. Let (M, g) be a (pseudo-)Riemannian oriented manifold. Any natural Lagrangian of g, Γ and Φ is given by an gauge invariant Lagrangian function e l(ℓ′ , c, ∇(s−2) R[Λ(g)], ∇(r−1) R[Γ], g, ∇(k) Φ) .

All natural Lagrangians used in fields theories are of this type (for r, s ≤ 1), Fatibene and Francaviglia [9]. It is easy to see that the invariant Lagrangian function e l = g λ1 µ1 g λ2 µ2 g ρ1 σ1 . . . g ρr σr cdae cebd Ra λ1 λ2 ;ρ1 ;...;ρr Rb µ1 µ2 ;σ1 ;...;σr

defines a higher order Yang-Mills Lagrangian which is of order (r+1) with respect to Γ and of order r with respect to g. For r = 0 we obtain just the classical Yang-Mills Lagrangian.

5.8.

Final Remarks

Let us note that in literature as gauge invariant operators are sometimes considered operators invariant with respect to principal bundle morphisms over the identity of base manifolds, see for instance Etayo, Garc´ıa, Mu˜noz and P´erez [8] and Manno, Pohjanpelto and Vitolo [37]. In this case the corresponding mapping between standard fibres is equivariant with r G ⊂ Gs ⋊ T r G. Then for r-order (r ≥ 2) operators respect to the subgroup {e} × Tm m m on the gauge-natural bundle of principal connections it is not necessary to use auxiliary classical connections on base manifolds and all r-order natural differential operators (with values in a gauge-natural bundle of order (r + 1, 0)) factorize through gauge covariant differentials of the curvature tensor up to the order r. The gauge covariant differentiation was defined intrinsically by Eck [6], for local formulas see also normal gauge concomitants

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by Horndeski [10]. For instance the gauge covariant differential of the curvature tensor R[Γ] is given in coordinates by Γ

∇ν R[Γ]a λµ = R[Γ]a λµ;ν = ∂ν R[Γ]a λµ − cabd Γb ν R[Γ]a λµ

and it is a (2, 0)-order field. Then we have Theorem 5.10. All Lagrangians of orders r in Γ and k in Φ which are invariant with respect to principal bundle morphisms over the identity of the base manifold are of the form e Γ ∇(r−1) R[Γ]) , L(j r Γ) = L(c, e ′ , c, Γ ∇(r−1) R[Γ], Γ ∇(k) Φ) , L(j r Γ, j k Φ) = L(ℓ

where Le is a unique zero order Lagrangian.

For instance, if in Example 5.9 we replace covariant differentials with respect to (Γ, Λ) by gauge covariant differentials with respect to Γ only, then the corresponding higher-order Yang-Mills Lagrangian is invariant with respect to principal bundle morphisms over the identity of the base manifold.

Acknowledgement This research has been supported by the Ministry of Education of the Czech Republic under the project MSM0021622409, by the Grant agency of the Czech Republic under the project GA201/05/0523.

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[41] J. Novotn´y, On the generally invariant Lagrangians for the metric field and other tensor fields, Internat. J. Theoret. Phys. 17 (1978) 677–684. [42] R. S. Palais and C. L. Terng, Natural bundles have finite order, Topology 16 (1977) 271–277. [43] D. J. Saunders, The Geometry of Jet Bundles (London Math. Soc., Lecture Note Series 142, Cambridge University Press, 1986). [44] J. A. Schouten, Ricci Calculus (Berlin-G¨ottingen, 1954). [45] J. A. Schounten and J. Haantjes, On the theory of the geometric object, Proc. London Math. Soc. 42 (1937) 356–376. [46] J. Slov´ak, On finite order of some operators, In: Proc. Conf. Diff. Geom. and Its Appl. (Communications, Brno 1986) J. E. Purkynˇe University, 1987 283–294. [47] J. Slov´ak, On natural connections on Riemannian manifolds, Comment. Math. Univ. Carolinae 30 (1989) 389–393. [48] C. L. Terng, Natural vector bundles and natural differential operators, Am. J. Math. 100 (1978) 775–828. [49] T. Y. Thomas, Differential Invariants of Generalized Spaces (Cambridge University Press, Cambridge, 1934). [50] T. Y. Thomas and A. D. Michal, Differential invariants of affinely connected manifolds, Ann. Math. 28 (1927) 196–236. [51] R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956) 1597–1607. [52] H. Weyl, Raum - Zeit - Materie (Space - Time - Matter) (1918, translated from the 4th German Edition, London: Methuen 1922). [53] A. Zajtz, Foundations of Differential Geometry on Natural Bundles (Caracas, 1983). [54] A. Zajtz, The sharp upper bound on the order of natural bundles of given dimensions, Bull. Soc. Math. Belgique 3 (1987) 347–357. [55] C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Physical Rev. (2) 96 (1954) 191–195.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 167-188

Chapter 10

C ONNECTIONS ON H IGHER O RDER F RAME B UNDLES AND T HEIR G AUGE A NALOGIES∗ Ivan Kol´arˇ† Institute of Mathematics and Statistics Faculty of Science, Masaryk University Jan´acˇ kovo n´am. 2a, 602 00 Brno Czech Republic

Abstract In the first part of the paper, we present a survey of the basic properties of connections on the r-th order frame bundle of a manifold. Special attention is paid to the torsion and torsion-free connections. In the second part, connections on the r-th principal prolongation of a principal bundle are treated from similar points of view. The case of the first principal prolongation is discussed in detail.

2000 Mathematics Subject Classification. 58A20, 53C05, 58A32 Key words and phrases. r-th order frame bundle, connection, torsion, natural operator, semiholonomic 2-jet, r-th principal prolongation of principal bundle, gauge-natural operator In the present paper, connection means a principal connection on a principal bundle unless otherwise specified. Several properties of connections on the r-th order frame bundle P r M of a manifold M appear in the framework of the general theory of natural bundles and operators. This is described in the book by D. Krupka and J. Janyˇska, [21], and in the monograph [18]. So the first part of the present paper is devoted to a survey of some more specific properties of connections on P r M that are mostly related with the idea of torsion. In Section 1 we underline that the Lie algebroid version of a connection on the principal bundle P r M (M, Grm ) is a linear r-th order connection on T M . So we have two different approaches to the concept of torsion. Proposition 2 reads that both approaches are equivalent. ∗ †

To Demeter Krupka, on the occasion of his 65th birthday E-mail address: [email protected]

Ivan Kol´aˇr

168

In Section 3 we clarify that the torsion-free connections on P r M are in bijection with the reductions of P r+1 M to the canonical injection of G1m into Gr+1 m . This enables us to define the r-th exponential operator transforming every torsion-free connection Λ on P 1 M into a torsion-free connection on P r M . In particular, this implies that Λ determines a general connection on every natural bundle over m-manifolds. In Section 5, the Lie algebroid construction of the exponential operator is based on two interesting lemmas concerning the r-jet of the commutator of vector fields on M . Then we deduce that the torsion-free connections on P r M are in bijection with the splittings from the cotangent bundle T ∗ M into the bundle of (1, r + 1)-covelocities on M . In Section 7 we discuss a connection on P r M from the viewpoint of the theory of higher order G-structures and we characterize its integrability in this sense. In Section 8 we present a recent result by W. Mikulski, [27], who determined all natural operators transforming a torsion-free connection on P 1 M into a connection on P r M . Section 9 is devoted to the basic properties of semiholonomic 2-jets, that represent a useful tool for several problems of the present paper. The principal prolongation W r P of an arbitrary principal bundle P (M, G) is defined in Section 10 in a formally slightly different way to [18]. We hope this could be useful in applications. Then we summarize the basic properties of connections on W r P and their torsions. In Section 11 we clarify that every connection Φ on W 1 P is canonically identified with the triple (Γ, Λ, D) of a connection Γ on P , a connection Λ on P 1 M and a section D 2 N

of L0 P ⊗ T ∗ M , where L0 P is the adjoint bundle of P . In Section 12 we present the list of all gauge-natural operators transforming every pair (Γ, Λ) of a connection Γ on P and a torsion-free connection Λ on P 1 M into a connection on W 1 P . Finally we outline how the semiholonomic 2-jets can be used in the theory of connections on W 1 P . In particular, ˜ to every connection Φ on W 1 P by using the we introduce the conjugate connection Φ canonical involution of semiholonomic 2-jets. All manifolds and maps are assumed to be infinitely differentable. Unless otherwise specified, we use the terminology and notations from the book [18].

1.

The Algebroid form of Connections on P r M

First we recall that the Lie algebroid LP → M of an arbitrary principal bundle P (M, G) is defined by LP = T P/G. So the elements of LP are the right invariant families of tangent vectors along the individual fibers of P , every section σ : M → LP is identified with a right invariant vector field σ : P → T P and the bracket [σ 1 , σ 2 ] of right invariant vector fields on P induces the bracket [[σ1 , σ2 ]] of LP . The canonical projection q : LP → T M is called the anchor map. Clearly, a connection Γ on P can be interpreted as a linear morphism γ : T M → LP satisfying q ◦ γ = id T M , [26]. We write P r M for the r-th order frame bundle of an m-dimensional manifold M . This is a principal bundle over M with structure group Grm = inv J0r (Rm , Rm )0 . Every local diffeomorphism f : M1 → M2 induces a principal bundle morphism P r f : P r M1 → P r M2 , so that P r is a bundle functor on the category Mfm of m-dimensional manifolds and their local diffeomorphisms, [18]. For every vector field X : M → T M , its flow

Connections on Higher Order Frame Bundles and Their Gauge Analogies

169

prolongation ∂ r P (F ltX ) (1) ∂t 0 is a right invariant vector field on P r M . This follows directly from the fact that the values of P r are in the category PBm (Grm ) of principal Grm -bundles over m-manifolds and their local principal bundle isomorphisms. Since P r is an r-th order bundle functor, the restriction P r X | Pxr M depends on jxr X only, x ∈ M . P r X :=

Proposition 1. The rule identifies J r T M with LP r M .

r IM (jxr X) = P r X | Pxr M

(2)

r is a diffeomorphism. But P r M = reg T r M is an open Proof. We have to prove that IM m r M of all (m, r)-velocities on M and P r X is the restriction of subset of the bundle Tm r follows from the the flow prolongation Tmr X to this subset. Hence the bijectivity of IM r r existence of an exchange isomorphism κM : Tm T M → T Tm M such that Tmr X = κM ◦ r M , [16], [18]. Tm

A linear r-th order connection on T M is a linear morphism T M → J r T M that splits the target jet projection. According to Proposition 1, every connection Γ : P r M → J 1 P r M is identified with a linear splitting γ : T M → J r T M . We say that γ is the algebroid form of Γ.

2.

Two Approaches to the Torsion on P r M

r m The canonical (Rm ×gr−1 m )-valued 1-form ϕr on P M is defined as follows. We have R × r−1 r−1 r−1 m r m gm = Ter−1 P R , where er−1 = j0 id Rm . Every u = j0 f , f : R → M , induces P r−1 f : P r−1 Rm → P r−1 M . The tangent map u ˜ := Ter−1 P r−1 f : Ter−1 P r−1 Rm → r (u), depends on u only. Then one defines Tur−1 P r−1 M , ur−1 = πr−1 r (A) , ϕr (A) = u ˜−1 T πr−1

A ∈ Tu P r M .

P. C. Yuen introduced the torsion of a connection Γ on P r M as the exterior covariant differential DΓ ϕr of ϕr , [33]. Since DΓ ϕr is a horizontal 2-form on P r M , it can be 2 ∗ interpreted as a map P r M → (Rm ×gr−1 m )⊗Λ T M . Taking into account the identification m r u ˜1 : R → Tx M , u1 = π1 (u), we construct 2 m∗ DΓ ϕr : P r M → (Rm × gr−1 . m )⊗Λ R

(3)

On the other hand, the (r − 1)-jet at x ∈ M of the bracket [ξ, η] of two vector fields ξ, η on M depends on the r-jets jxr ξ and jxr η. This defines a map [ , ]r−1 : J r T M ×M J r T M → J r−1 T M . Let γ : T M → J r T M be the algebroid form of Γ. According to A. Zajtz, [28], the torsion of γ is a map τ γ : T M ×M T M → J r−1 T M defined by

τ γ(A, B) = γ(A), γ(B)

r−1

,

A, B ∈ Tx M .

(4)

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Clearly, τ γ can be interpreted as a section of J r−1 T M ⊗ Λ2 T ∗ M . This is a fiber bundle 2 m∗ . So the frame form of τ γ is associated to P r M with standard fiber (Rm × gr−1 m )⊗Λ R a map 2 m∗ τ γ : P r M → (Rm × gr−1 . m )⊗Λ R In [14], we deduced Proposition 2. We have DΓ ϕr = 12 τ γ. Let Γ and ∆ be two connections on P r M over the same connection on P r−1 M . Conr sider their algebroid forms γ, δ : T M → J r T M . Since the kernel of πr−1 : J rT M → J r−1 T M is T M ⊗ S r T ∗ M , [18], the difference of γ and δ is a section γ − δ : M → T M ⊗ Sr T ∗M ⊗ T ∗M .

(5)

Proposition 3. If both γ and δ are torsion-free, then the values of γ − δ lie in T M ⊗ S r+1 T ∗ M . Proof. If xi are some local coordinates on M , X i = dxi are the induced coordinates on T M and Xαi are the jet coordinates on J r T M , then the equations of γ or δ are Xαi = Γiαj (x)X j

or

Xαi = ∆iαj (x)X j ,

1 ≤ |α| ≤ r ,

where α is a multi-index of range m. The difference γ − δ can be interpreted as a map T M ×M T M → T M ⊗ S r−1 T ∗ M of the form Γiβjk − ∆iβjk ξ j η k ,

|β| = r − 1 .

The only r-th order terms in j r−1 [ξ, η] are ξj

r i ∂ r ηi j ∂ ξ − η , ∂xj ∂β x ∂xj ∂β x

|β| = r − 1 .

(6)

If γ is torsion-free, then (6) yields Γiβjk = Γiβkj . If δ is also torsion-free, (5) is symmetric in the last two subscripts. From the proof one sees directly that γ − δ is an arbitrary section of T M ⊗ S r+1 T ∗ M .

3.

Torsion-Free Connections on P r M as Reductions of P r+1 M

Every a ∈ G1m is a matrix, which defines a linear map l(a) : Rm → Rm . This induces a group homomorphism lr−1 : G1m → Grm ,

lr−1 (a) = j0r l(a) .

S. Kobayashi proved, [8], that the torsion-free connections on P 1 M are in bijection with the reductions of P 2 M to the subgroup l1 (G1m ) ⊂ G2m . We deduce an analogous result for arbitrary order r. This is based on the following injection irM : P r+1 M ֒→ J 1 P r M . Every u = j0r+1 f ∈ P r+1 M determines a local section ψ of P r M → M ψ(y) = j0r f ◦ tf −1 (y) ,

(7)

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where y lies in a neighbourhood of f (0) ∈ M and tf −1 (y) : Rm → Rm is the translation x 7→ x + f −1 (y). Then we set irM (u) = jf1(0) ψ. If xi , xij , . . . , xiji ...jr are the standard coordinates on P r Rm , xij,k , . . . , xij1 ...jr ,k are the induced coordinates on J 1 P r Rm and xij1 ...jr+1 are the additional coordinates on P r+1 Rm , then (7) implies directly the following coordinate form of ir xij,k = xijl x ˜lk , . . . , xij1 ...jr ,k = xij1 ...jr l x ˜lk , (8) where x ˜ij is the inverse matrix to xij . Every X ∈ J 1 P r M over βX ∈ P r M is identified with an m-plane in the tangent space TβX P r M , which will be denoted by the same symbol X. Hence we can consider the restriction dϕr | X of the exterior differential of ϕr to X. Denote by X1 ∈ J 1 P r−1 M the underlying element of X. The following lemma from [14] is close to a result by Yuen, [33]. r r+1 M ) if and only Lemma 1. Let X ∈ J 1 P r M satisfy X1 = ir−1 M (βX). Then X ∈ iM (P if dϕr | X = 0.

For r = 1 we have no X1 and the claim dϕ1 | X = 0 if and only if X ∈ i1M (P 2 M ) was used in [8]. For every torsion-free connection Γ on P r M we define a map µ(Γ) : P 1 M → P r+1 M by the following induction. Consider a connection Γ : P r M → J 1 P r M such that the underlying connection Γ1 : P r−1 M → J 1 P r−1 M is torsion-free, so that Γ1 determines a map µ(Γ1 ) : P 1 M → P r M be the induction hypothesis. Proposition 4. Γ is torsion-free, if and only if the values of Γ ◦ µ(Γ1 ) lie in irM (P r+1 M ). Then we define µ(Γ) = (irM )−1 ◦ Γ ◦ µ(Γ1 ) : P 1 M → P r+1 M . Proof. By Lemma 1, we have dϕr | X = 0 for all X ∈ Γ µ(Γ1 )(P 1 M ) . But ϕr is a pseudotensorial form, [18], so that dϕr | A = 0 holds for every A ∈ Γ(P r M ). This is equivalent to DΓ ϕr = 0.

For every principal bundle P (M, G), we have an induced right action of G on J 1 P , jx1 s(y), g 7→ jx1 s(y)g , where s is a local section of P on a neighbourhood of x ∈ M and g ∈ G. This action will be denoted by (X, g) 7→ X̺(g). Lemma 2. For every v ∈ P r+1 M and a ∈ G1m , we have irM vlr (a) = irM (v)̺ lr−1 (a) .

Proof. If v = j0r+1 f , then irM (v)̺ lr−1 (a) = jx1 j0r f ◦ tf −1 (y) ◦ l(a) . On the other hand, irM vlr (a) = jx1 j0r f ◦ l(a) ◦ tl(a)−1 (f −1 (y)) . But tz ◦ l(a) = l(a) ◦ tl(a)−1 (z) , z ∈ Rm , is a well known relation from the affine geometry.

By Lemma 2, µ(Γ)(P 1 M ) is a reduction of P r+1 M to the subgroup lr (G1m ) ⊂ Gr+1 m . Indeed, using induction we obtain µ(Γ)(ua) = (irM )−1 Γ µ(Γ1 )(u) ̺ lr−1 (a)

= µ(Γ)(u)lr (a) .

On the other hand, every reduction Q ⊂ P r+1 M to the subgroup lr (G1m ) induces a map (denoted by the same symbol) Q : P 1 M → P r+1 M as follows. For every v ∈ Q we

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construct u = π1r+1 (v) and we set Q(u) = v. Any other v¯ in the same fiber of Q → M is of the form v¯ = vlr (a), a ∈ G1m . This implies π1r+1 (¯ v ) = ua, so that our definition is correct. Proposition 5. Proposition 4 establishes a bijection between torsion-free connections on P r M and reductions of P r+1 M to lr (G1m ). Proof. First we deduce that µ(Γ) : P 1 M → P r+1 M is a reduction to lr (G1m ). For every u ∈ P 1 M and a ∈ G1m we have µ(Γ)(ua) = (irM )−1 Γ µ(Γ1 )(ua)

= (irM )−1 Γ µ(Γ1 )(u)lr−1 (a)

= (irM )−1 Γ µ(Γ1 )(u) ̺ lr−1 (a)

= (irM )−1 (irM ) µ(Γ)(u) lr (a)

by definition, by the induction hypothesis, by right-invariance of Γ and by Lemma 2. Conversely, if Q : P 1 M → P r+1 M is a reduction to lr (G1m ), then Q1 = πrr+1 ◦ Q : P 1 M → P r M is a reduction to lr−1 (G1m ). We define Γ : Q1 (P 1 M ) → J 1 P r M by Γ Q1 (u) = irM Q(u). By Lemma 2, it holds Γ Q1 (ua) = irM Q(ua) = irM Q(u)lr (a) = r iM Q(u) ̺ lr−1 (a) = Γ Q1 (u) ̺ lr−1 (a) . Hence Γ is a right-invariant map, which is canonically extended into a connection on P r M .

4.

The r-th Order Exponential Prolongation

The following construction represents an interesting application of Proposition 5. Consider a torsion-free connection Λ on P 1 M . For every x ∈ M , Λ determines the exponential map expΛ x : Ux → M , where Ux ⊂ Tx M is a neighbourhood of the origin. Then we define a map Er (Λ) : P 1 M → P r+1 M by Er (Λ)(u) = j0r+1 (expΛ x ◦u) ,

u ∈ Px1 M ,

(9)

where u is interpreted as a map Rm → Tx M . Proposition 6. Er (Λ)(P r M ) is a reduction of P r+1 M to lr (G1m ). Proof. For all u ∈ P 1 M and a ∈ G1m , we have Er (Λ)(ua) = j0r+1 expΛ x ◦u ◦ l(a) = Er (Λ)(u)lr (a).

By Proposition 5, Er (Λ) is a torsion-free connection on P r M , that is called the r-th exponential prolongation of Λ. The rule Λ 7→ Er (Λ) is said to be the r-th order exponential operator on the bundle Qτ P 1 M of torsion-free connections on P 1 M . W. Mikulski invented another construction of the exponential prolongation, [27]. Every ˜ on Tx M by means of translations. The expoX ∈ Tx M is extended into a vector field X Λ ˜ locally into a vector field (expΛ ˜ nential map expx transforms X x )∗ (X) on M . Then we can construct r ˜ εr (Λ)(X) = j0r (expΛ (10) x )∗ (X) ∈ Jx T M .

In Section 5 we deduce that εr (Λ) is the algebroid form of Er (Λ).

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This result enables us to describe another geometrically interesting construction of r ˜ Er (Λ). The flow prolongation P r (expΛ x )∗ (X) is a vector field on P M . By Section 1, r r the lifting map P M ×M T M → T P M of Er (Λ) is ˜ Er (Λ)(u, X) = P r (expΛ x )∗ (X) (u) ,

u ∈ Pxr M .

Further, consider an r-th order natural bundle F over m-manifolds, [18]. So F M is a fiber bundle associated to P r M with standard fiber F0 Rm . Every principal connection Γ on P r M induces a general connection ΓF on P r M . We shall use the construction of ΓF by means of lifting vector fields. In general, every right-invariant vector field Z on P r M r induces a vector field ZF on F M as follows. If Z(u) = dc(0) dt , c : R → P M , u = c(0), then d ZF {u, a} = 0 c(t), a , a ∈ F0 Rm . dt Since Z is right-invariant, this definition is correct. Then the ΓF -lift of a vector field X on M is prescribed by ΓF (X) = (ΓX)F . Clearly, the flow prolongation FX of a vector field X on M with respect to F satisfies FX = (P r X)F . Thus we have deduced Proposition 7. For every r-th order natural bundle F , the rule ˜ Er (Λ)F (v, X) = F (expΛ x )∗ (X) (v) ,

v ∈ Fx M, X ∈ Tx M

transforms every torsion-free connection Λ on P 1 M into general connection Er (Λ)F on FM.

5.

The Exponential Prolongation in the Algebroid Form

Consider an arbitrary linear splitting γ : T M → J r T M . For a linear frame u ∈ Px1 M , u = (A1 , . . . , Am ), Ai ∈ Tx M , we take vector fields Xi satisfying jxr Xi = γ(Ai ), i = 1, . . . , m. Then F ltX1 1 ◦ · · · ◦ F ltXmm (x)

is a local map Rm → M and we define

σ(γ)(u) = j0r+1 F ltX1 1 ◦ · · · ◦ F ltXmm (x) ∈ Pxr+1 M .

(11)

One verifies easily that σ(γ)(u) depends on u and γ only. Proposition 8. If γ is torsion-free, then σ(γ)(P 1 M ) is a reduction of P r+1 M to lr (G1m ). Proof is based, in a very instructive way, on the definition (4) of τ γ. We shall use the following two lemmas from [15]. Consider two vector fields X and Y on M . Then F ltX ◦ F lτY (x) is a local map r+1 R2 → M , so that j0,0 F ltX ◦ F lτY (x) ∈ (T2r+1 M ) is a (2, r + 1)-velocity on M . Lemma 3. If jxr−1 [X, Y ] = 0, then

r+1 r+1 j0,0 F ltX ◦ F lτY (x) = j0,0 F lτY ◦ F ltX (x) .

(12)

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Further, F ltX ◦ F lt (x) is a local map R → M , so that j0r+1 F ltX ◦ F ltY (x) ∈ (T1r+1 M )x is a (1, r + 1)-velocity on M . Y

Lemma 4. If jxr−1 [X, Y ] = 0, then j0r+1 F ltX ◦ F ltY (x) = j0r+1 F ltX+Y (x) .

We shall also apply the well known formula X F lct = F ltcX ,

(13)

c ∈ R.

(14)

Take a = (aij ) ∈ G1m and consider ua = (aji Aj ). Since γ is torsion-free, by (13), (14) and (12) we obtain gradually a1 X1 +···+am 1 Xm

σ(γ)(ua) = j0r+1 F lt11

a 1 X1

= j0r+1 F lt11

a1 X1 +···+am m Xm

◦ · · · ◦ F ltmm

a m Xm

◦ · · · ◦ F lt11

a 1 X1

◦ · · · ◦ F ltmm

a m Xm

◦ · · · ◦ F ltmm

m = j0r+1 F laX11t1 ◦ · · · ◦ F laXmmt1 ◦ · · · ◦ F laX11 tm ◦ · · · ◦ F laXm m mt

=

m

1

1

j0r+1

F laX11t1 +···+a1 tm m 1

◦ ··· ◦

F laXmmt1 +···+am tm . m 1

This proves Proposition 8. To clarify the relation of σ(γ) to the reduction µ(Γ) from Section 3, we need the folr M . Clearly, lowing form of the injection irM : P r+1 M → J 1 P r M . We have P r M ⊂ Tm r r m j0 f ∈ Tm M , f : R → M , can be expressed in the form r j0r f = (Tm f )(er ) ,

er = j0r id Rm .

(15)

∂ r i r Rm , where τ i : Rm → Rm is the translation t¯1 = j τ ∈ Ter Tm Write Ei = ∂t t 0 0 t t1 , . . . , t¯i = ti + t, . . . , t¯m = tm . If we consider j0r+1 ψ ∈ P r+1 M , then

r (T Tm ψ)(Ei )

(16)

is an m-tuple of tangent vectors at j0r ψ ∈ P r M . The linear span of these vectors defines r+1 r iM j0 ψ ∈ J 1 P r M .

Proposition 9. If γ is torsion-free and Γ is the corresponding connection on P r M , then σ(γ) = µ(Γ). Proof. We proceed by introduction. If γ1 and Γ1 are the underlying connections in the order r − 1, then σ(γ1 ) = µ(Γ1 ) by the induction hypothesis. Consider u = (A1 , . . . , Am ) ∈ Px1 M and write v = σ(γ1 )(u) = µ(Γ1 )(u) . By (16), irM j0r+1 F ltX1 1 ◦ · · · ◦ F ltXmm (x) is the linear span of the vectors

r T Tm F ltX1 1 ◦ · · · ◦ F ltXmm (Ei ) ,

i = 1, . . . , m .

(17)

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Using the basic properties of flows, Lemma 3 and (15), we deduce that (17) is equal to ∂ r Xi Xm Tm F ltX1 1 ◦ · · · ◦ F lt+t (er ) i ◦ · · · ◦ F ltm 0 ∂t ∂ T rX T rX T rX = 0 F lt m i ◦ F lt1m 1 ◦ · · · ◦ F ltmm m (er ) ∂t r = Tmr Xi Tm (F ltX1 1 ◦ · · · ◦ F ltXmm )(er ) = Tmr Xi (v) .

By (2) and by the induction hypothesis, this m-tuple spans µ(Γ)(v).

From the proof of Proposition 8 we obtain easily that the construction (10) of εr (Λ) by W. Mikulski is the algebroid form of the exponential prolongation Er (Λ) introduced in Section 4.

6.

Splittings T ∗ M → T r+1∗ M

The space T r+1∗ M = J r+1 (M, R)0 of all (1, r + 1)-covelocities on M is a vector bundle, [18]. By a splitting s : T ∗ M → T r+1∗ M we mean a linear morphism satisfying π1r+1 ◦ s = id T ∗ M . We remark that such splittings play an interesting role in the construction of Poincar´e-Cartan morphisms in the higher order variational calculus, [11]. Proposition 10. There is a canonical bijection between reductions Q ⊂ P r+1 M to subgroup lr (G1m ) and splittings s : T ∗ M → T r+1∗ M . Proof. Every b ∈ Tx∗ M determines a linear map λ(b) : Tx M → R. Let v = j0r+1 f ∈ Qx , so that u = π1r+1 (v) ∈ Px1 M can be interpreted as a map u : Rm → Tx M . Then we set s(b) = jxr+1 λ(b) ◦ u ◦ f −1 ∈ Txr+1∗ M .

(18)

Since Q is a reduction to lr (G1m ), (18) does not depend on the choice of v ∈ Qx , The fact that s is a splitting follows directly from (18). Conversely, let s : T ∗ M → T r+1∗ M be a splitting. A frame u ∈ Px1 M is a basis (e1 , . . . , em ) of Tx M . Consider the dual basis u∗ = (e1 , . . . , em ) of Tx∗ M . Then s(e1 ), . . . s(em ) are the components of an (r + 1)-jet −1 s(u∗ ) ∈ Jxr+1 (M, R)0 . Write Q(u) = s(u∗ ) ∈ Pxr+1 M for the inverse jet. If we j ∗ i j i take ua = (ai ej ), then (ua) = (˜ aj e ), where a ˜j is the inverse matrix to aij . Hence s (ua)∗ = a ˜ij s(ej ) = lr (a−1 ) ◦ s(u∗ ), which implies Q(ua) = Q(u)lr (a). Finally, one verifies easily that the maps Q 7→ s and s 7→ Q are inverse each other. Taking into account Proposition 5, we obtain a canonical bijection between torsion-free connections on P r M and splittings T ∗ M → T r+1∗ M . We remark that Proposition 10 yields another proof of Proposition 3.

7.

The Viewpoint of Higher Order G-Structures

A connection on P r M can be viewed as a kind of higher order G-structure on M . We recall that a k-th order G-structure on M is said to be integrable, if it is locally isomorphic to the

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product Rm ×G, where G ⊂ Gkm is the structure group. We are going to apply this approach to the algebroid form γ : T M → J r T M . (We remark that this kind of integrability plays an important role in our theory of the flow prolongation of some tangent valued forms, [1].) On Rm , there is a distinguished connection Cr : T Rm → J r T Rm defined by ˜, Cr (X) = jxr X

X ∈ Tx Rm ,

˜ is the constant vector field on Rm constructed from X by means of translations. where X Definition 1. We say that γ : T M → J r T M is integrable, if for every x ∈ M there exists a neighbourhood U and a diffeomorphism f : U → Rm satisfying Cr ◦T f = J r T f ◦(γ | U ). Clearly, every integrable connection γ is torsion-free. According to a classical result, a connection Λ on P 1 M is integrable, iff it is both torsion-free and curvature-free. Then every exponential prolongation εk (Λ) is also integrable. Thus, the torsion of γ is the first obstruction to its integrability. Consider the underlying connections γk = πkr ◦ γ, k = 1, . . . , r. Clearly, if γ is torsion-free or integrable, then each γk is so. Assume that γ is torsion-free. Then the curvature of γ1 is the second obstruction to the integrability of γ. If this curvature vanishes, each connection εk (γ1 ) is integrable. The difference γ2 − ε2 (γ1 ) is a tensor field of type T M ⊗ S 3 T ∗ M that is the third obstruction to the integrability of γ. Assume by induction that the first up to (k + 1)-st obstruction to the integrability of γ vanish. Then γk = εk (γ1 ) and the tensor field of type T M ⊗ S k+2 T ∗ M γk+1 − εk+1 (γ1 ) is the next obstruction to the integrability of γ. If all these r + 1 obstructions vanish, then γ = εr (γ1 ) is integrable. Thus, we have proved Proposition 11. γ is integrable if and only if all the following conditions are satisfied a) γ is torsion-free, b) γ1 is curvature-free, c) all the gradually defined tensor fields γk − εk (γ1 ), k = 2, . . . , r, vanish.

8.

Natural Operators C ∞ Qτ P 1 M → C ∞ QP r M

We write QP for the connection bundle of an arbitrary principal bundle P (M, G), [18]. The connections on P form the space C ∞ QP of all sections of QP → M . Further, we write Qτ P r M for the bundle of all torsion-free connections on P r M , [14]. So the r-th exponential operator on M is a natural operator C ∞ Qτ P 1 M → C ∞ Qτ P r M . Using Er , W. Mikulski solved a rather sophisticated problem of finding all natural operators C ∞ Qτ P 1 M → C ∞ QP r M and C ∞ Qτ P 1 M → C ∞ Qτ P r M , [27]. Every torsion-free connection Λ on P 1 M defines a vector bundle isomorphism ψΛ : J r T M →

r M

k=0

T M ⊗ Sk T ∗M

(19)

Connections on Higher Order Frame Bundles and Their Gauge Analogies as follows. Write I : J0r T Rm →

r M

k=0

177

T0 Rm ⊗ S k T0∗ Rm

for the standard identification. Let ϕ be a Λ-normal coordinate system on M with center x and B ∈ Jxr T M . We define ψΛ (B) =

r M

k=0

(T0 ϕ−1 ⊗ S k T0∗−1 ϕ−1 ) I J r T ϕ(B)

.

(20)

Since the identification I is G1m -equivariant, (20) is a correct definition. Proposition 12. Let D : C ∞ Qτ P 1 M → C ∞ QP r M be a natural operator. Then there exist uniquely determined natural operators Ak : C ∞ Qτ P 1 M → C ∞ (T M ⊗ S k T ∗ M ⊗ T ∗ M ) , k = 0, . . . , r, such that A0 = 0, A1 = 0 and

D(Λ) = Er (Λ) + 0, 0, A2 (Λ), . . . , Ar (Λ) in the sense of the identification (19).

Proof. The difference D(Λ) − Er (Λ) is decomposed into r + 1 natural operators by (19). The natural operators A0 and A1 vanish according to 25.3 and Lemma 33.4 in [18]. Now Proposition 3 yields directly Proposition 13. Let D : C ∞ Qτ P 1 M → C ∞ Qτ P r M be a natural operator. Then there exist uniquely determined natural operators Ak : C ∞ Qτ P 1 M → C ∞ T M ⊗ S k+1 T ∗ M , k = 0, . . . , r, such that A0 = 0, A1 = 0 and

D(Λ) = Er (Λ) + 0, 0, A2 (Λ), . . . , Ar (Λ) . The natural operators A2 , . . . , Ar in Proposition 12 or Proposition 13 can be prescribed arbitrarily. According to Lemma 33.4 of [18], all natural operators C ∞ Qτ P 1M → C ∞ T M ⊗ Nk ∗ T M are R-linearly generated by – the curvature tensor and its covariant derivatives, – constructing tensor products (including tensor products with invariant tensors) and contractions. In the case of some prescribed symmetries in the covariant part we add the corresponding symmetrizations of the operators in question.

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i i Example 1. Write Rjkl = −Rjlk for the curvature tensor of Λ. If we look for all natu∞ 1 ral operators D : C Qτ P M → C ∞ Qτ P 2 M , we have to determine all natural operators C ∞ Qτ P 1 M → C ∞ (T M ⊗ S 3 T ∗ M ). By the above mentioned procedure, all these operm Ri ators are the constant multiples of δ(j kl)m . Hence all D’s form the one-parameter family m i D(Λ) = E2 (Λ) + c(δ(j Rkl)m ) ,

c ∈ R.

Example 2. J. Janyˇska and the author determined all natural operators C ∞ Qτ P 1 M → C ∞ QP 2 M by using a direct approach, [7], [12]. The list of them is rather long. Using Proposition 12, we can interpret that list in a very clear geometric way.

9.

Semiholonomic 2-Jets

Some aspects of our problems are properly related with the theory of semiholonomic 2-jets. First we describe the general ideas, [3], [23]. Consider a fibered manifold p : Y → M . Its second nonholonomic prolongation J˜2 Y is defined by the iteration J˜2 Y = J 1 (J 1 Y → M ) . If xi , y p are some fiber coordinates on Y , the induced coordinates on J 1 Y are yrp = ∂i y p (x) and the coordinates further induced on J˜2 Y are p y0i = ∂i y p (x) and

p yij = ∂j yip (x) .

There are two canonical projections J˜2 Y → J 1 Y , namely the target jet projection β1 : J˜2 Y → J 1 Y and the jet prolongation J 1 β : J˜2 Y → J 1 Y of the target jet projection β : J 1 Y → Y . The second semiholonomic prolongation J¯2 Y is the set of all A ∈ J˜2 Y satisfying β1 (A) = (J 1 β)(A) . In coordinates, this condition means p y0i = yip .

(21)

The injection J 2 Y ֒→ J˜2 Y is defined by jx2 s 7→ jx1 (j 1 s) . So the subset J 2 Y ⊂ J˜2 Y is characterized by p yip = y0i

p p and yij = yji .

(22)

Hence J 2 Y ⊂ J¯2 Y . According to the general theory, both β1 : J˜2 Y → J 1 Y and J 1 β : J˜2 Y → J 1 Y are affine bundles. For two manifolds M and N , the space J˜2 (M, N ) or J¯2 (M, N ) of nonholonomic or semiholonomic 2-jets of M into N is the second nonholonomic or semiholonomic

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prolongation of the product fibered manifold M × N → M , respectively. C. Ehresmann introduced the composition of nonholonomic jets, [3]. Consider another manifold Q and A ∈ J˜x2 (M, N )y , B ∈ J˜y2 (N, Q)z . So A = jx1 ϕ and B = jy1 ψ, where ϕ : M → J 1 (M, N ) and ψ : N → J 1 (N, Q) are sections of the source jet projection α. Hence αψ βϕ(u) = βϕ(u), u ∈ M , so that the composition of 1-jets ψ βϕ(u) and ϕ(u) is defined. Then we set B ◦ A = jx1 ψ βϕ(u) ◦ ϕ(u) ∈ J˜x2 (M, Q)z .

(23)

a , z a be the induced coordinates on J˜2 (N, Q) Let z a be some local coordinates on Q, zpa , z0p pq a a a and wi , w0i , wij be the induced coordinates on J˜2 (M, Q). Evaluating (23), we obtain the coordinate formula for the composition of nonholonomic 2-jets

wia = zpa yip ,

a a p w0i = z0p y0i ,

p a a p q wij = zpq yi y0j + zpa yij .

(24)

Clearly, the composition of two semiholonomic or holonomic 2-jets is semiholonomic or holonomic as well. A jet A ∈ J˜x2 (M, N )y is called regular, if there exists B ∈ J˜y2 (N, M )x such that B ◦ A = jx2 id M . By (24), A is regular iff both β1 (A) and (J 1 β)(A) are regular. In p coordinates this means that both yip and y0i are regular matrices. If dim M = dim N , then B ◦ A = jx2 id M implies A ◦ B = jy2 id N . In this case, regular is equivalent to invertible. Every ϕ(u) defines a linear map Tu M → T N , η p = yip (u)ξ i . This yields a local map T M → T N , whose tangent map at each point of Tx M is determined by jx1 ϕ. So the nonholonomic 2-jet A = jx1 ϕ(u) ∈ J˜x2 (M, N )y can be interpreted as a map (T T M )x → (T T N )y of the form η p = yip ξ i ,

p dy p = y0i dxi ,

p i dη p = yij ξ dxj + yip dξ i .

(25)

Consider the canonical involution ι of the iterated tangent functor, ιM (ξ i , dxi , dξ i ) = p (dxi , ξ i , dξ i ), and A ∈ J¯x2 (M, N )y in the form (25) with y0i = yip . Then the map ιN ◦A◦ιM is of the form η p = yip ξ i ,

dy p = yip dxi ,

p i dη p = yji ξ dxj + yip dξ i .

(26)

This map corresponds to another semiholonomic 2-jet κ(A) ∈ J¯x2 (M, N )y ,

p p κ(yip , yij ) = (yip , yji ).

(27)

Definition 2. The map κ is called the canonical involution of semiholonomic 2-jets. Since J¯2 (M, N ) → J 1 (M, N ) is an affine bundle, A and κ(A) determine a tensor ∆(A) = A − κ(A) ∈ Ty (N ) ⊗ Λ2 Tx∗ M

(28)

called the difference tensor of semiholonomic 2-jet A. (J. Pradines uses the name “dissym´etrie”, [30].) Clearly, A is holonomic, iff ∆(A) = 0.

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Example 3. We present the first remarkable application of this concept. Consider a general connection Γ : Y → J 1 Y on an arbitrary fibered manifold Y → M , [18]. If Γ is viewed as a morphism over M , we can construct J 1 Γ : J 1 Y → J˜2 Y . Clearly, the values of the composition Γ′ = J 1 Γ ◦ Γ lie in J¯2 Y . The difference tensor ∆ ◦ Γ′ : Y → V Y ⊗ Λ 2 T ∗ M coincides with the curvature of Γ. The second order semiholonomic frame bundle P¯ 2 M of M is defined by P¯ 2 M = ¯ 2m ), where G ¯ 2m = reg J¯02 (Rm , M ). This is a principal bundle P¯ 2 M (M, G 2 m m inv J¯0 (R , R )0 is the second order semiholonomic jet group in dimension m. The in¯ 2m defines an injection (denoted by the same symbol) l1 : G1m → G ¯ 2m . One clusion G2m ⊂ G 1 1 1 2 ¯ verifies easily that (8) with r = 1 defines an identification iM : J P M ≈ P M . Consider a principal connection Γ : P 1 M → J 1 P 1 M with the coordinate expression xij,k = Γilk xlj .

(29)

Using (8), we verify directly the following result by P. Libermann, [23]. Proposition 14. The rule Γ 7→ i1M ◦ Γ defines a bijection between the connections on P 1 M ¯ 2m . and the reductions of P¯ 2 M to the subgroup l1 (G1m ) ⊂ G It is remarkable that the canonical involution κ yields a simple construction of the ˜ conjugate to Γ. Indeed, we have connection Γ ˜ = κ ◦ (i1M ◦ Γ) , i1M ◦ Γ

(30)

˜ i.e. κ transforms the reduction determined by Γ into the reduction determined by Γ. Example 4. Another interesting application of the concept of difference tensor is in the theory of G-structures. We recall that a (first order) G-structure on M is a reduction P of P 1 M to a subgroup G ⊂ G1m . Then J 1 P ⊂ J 1 P 1 M ≈ P¯ 2 M and ∆ suggests a very conceptual way to the construction of the structure function of P , [19], [23]. In particular, this approach clarifies, in an instructive way, the difference between the prolongability and the flatness of P , [19]. Finally we remark that the theory of the covariant differentation with respect to connections on P 2 M is systematically developed in [6].

10.

W r P as a Generalization of P r M

Consider a principal bundle π : P → M with structure group G, dim M = m. Its rr ϕ of local principal th order principal prolongation W r P is the bundle of all r-jets j(0,e) bundle isomorphisms ϕ : Rm × G → P ,

0 ∈ Rm ,

e = the unit of G.

(31)

r G := W r (Rm × G), whose This is a principal bundle over M with structure group Wm 0 action on W r P is given by the jet composition, [2], [18].

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r P , we write Given A = j0r f ∈ Tm r πA = j0r (π ◦ f ) ∈ Tm M.

Further, we introduce r r reg π Tm P = {A ∈ Tm P;

r πA ∈ reg Tm M} .

Clearly, the local PB-isomorphism (31) is determined by its restriction ϕ | Rm × {e} : Rm → P . Hence r W r P = reg π Tm P. (32) ¯ , G) be another principal G-bundle, m = dim M ¯ . For every local principal bunLet P¯ (M r r r ¯ ¯ dle isomorphism f : P → P , Tm f : Tm P → Tm P restricts and corestricts into a map r P → reg T r P r r r¯ ¯ reg π Tm π m . This defines W f : W P → W P . r P , then πA ∈ T r M is invertible, so that A ◦ (πA)−1 satisfies π A ◦ If A∈ reg π Tm m −1 r (πA) = jx id M , x = π f (0) . This implies A ◦ (πA)−1 ∈ J r P . Hence W r P = P r M ×M J r P .

(33)

¯ is the base map of f . In Clearly, W r f is identified with P r f ×f J r f , where f : M → M particular, the structure group r r W0r (Rm × G) =: Wm G = Grm ⋊ Tm G

(34)

is the group semidirect product with the group composition

(g1 , C1 )(g2 , C2 ) = g1 ◦ g2 , (C1 ◦ g2 ) • C2 , r G, [18]. The first product projection where • denotes the induced group composition in Tm W r P → P r M is a principal bundle morphism with the associated group homomorphism r G → Gr determined by (34). Wm m W r P is a fundamental structure for the gauge theories of mathematical physics, [5]. In differential geometry, the main role of W r P is based on the fact that forevery associated bundle P [S], where S is a left G-space, the r-th jet prolongation J r P [S] is a fiber bundle associated to W r P , [18]. Further, we have a canonical injection P r M ֒→ W 1 P r−1 M , 1 P r−1 f , f : Rm → M . So W 1 P can play the role of a suitable recurrence j0r f 7→ j(0,e r−1 ) model for several geometric problems, [18]. In particular, the reductions of the principal bundle W 1 P are called generalized G-structures, [10]. Several properties of higher order G-structures are well reflected in the framework of this more general theory. If G = {e} is the one-element group, then M × {e} is identified with M and W r (M × {e}) = P r M . Hence many properties of W r P can be viewed as a generalization of the case r of P r M . In particular, we have the canonical one-form ϕr : T P r M → Rm ×gr−1 m on P M . On W r P , we introduce analogously a canonical one-form θr : T W r P → Rm × wr−1 m G= r−1 G, wr−1 G = Lie(W r−1 G). Consider T(0,Er−1 ) W r−1 (Rm × G), Er−1 = the unit of Wm m m r (u) ∈ W r−1 P , where π r u = j(0,e) ψ ∈ W r P and write u1 = πr−1 r−1 is the jet projection. The tangent map r−1 u ˜ = T(0,Er−1 ) W r−1 ψ : Rm × wr−1 P m G → Tu 1 W

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is a linear isomorphism depending on u only. For every Z ∈ Tu W r P , we define θr (Z) = u ˜−1 T πr−1 (Z) .

Clearly, the following diagram commutes T W rP

T P rM

θr

Rm × wr−1 m G

ϕr

Rm × gr−1 m

Analogously to Section 2, we introduced in [9] Definition 3. The torsion of a connection Φ on W r P is the covariant exterior differential DΦ θ r . The Lie algebroid LW r P of W r P coincides with the r-th jet prolongation J r (LP → M ), [17], [22]. Let ϕ : T M → J r LP be the algebroid form of Φ. Analogously to the case of J r T M , we have the truncated bracket [[ , ]]r−1 : J r LP ×M J r LP → J r−1 LP . The torsion τ ϕ of ϕ can be introduced as a morphism τ ϕ : T M ×M T M → J r−1 LP defined by (τ ϕ)(Z1 , Z2 ) = [[ϕ(Z1 ), ϕ(Z2 )]]r−1 ,

(Z1 , Z2 ) ∈ T M ×M T M .

In [17] it is deduced that DΦ θr and τ ϕ are naturally equivalent. The r-jets j0r gˆ, g ∈ G, of the constant maps gˆ : Rm → G, x 7→ g, define an injection r r G. Clearly, the direct group product l 1 r ν m : G → Tm r−1 (Gm ) × νm (G) is a subgroup of r Wm G. The following assertion, that is an analogy of Proposition 5, is proved in [17]. Proposition 15. The torsion-free connections on W r P are in bijection with the reductions r+1 (G) ⊂ W r+1 G. of W r+1 P to the subgroup lr (G1m ) × νm m

11.

Connections on W 1 P

We are going to discuss the connections on W 1 P in more details. By (33), W 1 P = P 1 M ×M J 1 P .

(35)

Write p1 : W 1 P → P 1 M and p2 : W 1 P → J 1 P for the product projections. Since p1 : W 1 P → P 1 M and the target jet projection β : W 1 P → P are principal bundle morphisms, every connection Φ : W 1 P → J 1 (W 1 P ) = J 1 P 1 M ×M J˜2 P

(36)

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induces a pair of connections p1∗ Φ on P 1 M and β∗ Φ on P . Conversely, consider two connections Γ : P → J 1 P and Λ : P 1 M → J 1 P 1 M . Define W 1 P ⊃ R(Γ) =

u, Γ(v) ; (u, v) ∈ P 1 M ×M P .

(37)

Using the action ̺ of G on P from Section 3, one finds easily that R(Γ) is a reduction of 1 (G) ⊂ W 1 G. Since (37) identifies R(Γ) with P 1 M × P , W 1 P to the subgroup G1m × νm M m the product connection Λ × Γ on P 1 M ×M P is identified with a connection on R(Γ) and the latter connection is uniquely extended into a connection p(Γ, Λ) on W 1 P . Clearly, β∗ p(Γ, Λ) = Γ and p1∗ p(Γ, Λ) = Λ. Write L0 P for the adjoint bundle of P . (Our notation is motivated by the fact that L0 P is the subset of the Lie algebroid LP of all elements A satisfying q(A) = 0.) The projections β and p1 give rise to projections L0 W 1 P ⊗T ∗ M → L0 P ⊗T ∗ M and L0 W 1 P ⊗ T ∗ M → L0 P 1 M ⊗ T ∗ M . The common kernel of these projections is L0 P ⊗ [12].

2 N

T ∗M ,

Proposition 16. Connections on W 1 P are in bijection with triples (Γ, Λ, D), where Γ ∈

C ∞ (QP ), Λ ∈ C ∞ (QP 1 M ) and D ∈ C ∞ (L0 P ⊗

2 N

T ∗ M ).

Proof. For Φ ∈ C ∞ (QW 1 P ) we set Γ = β∗ Φ, Λ = p1∗ Φ and D = Φ−p(β∗ Φ, p1∗ Φ). 2 N

The factor T ∗ M ⊗ T ∗ M gives rise to an exchange map ex : L0 P ⊗

2 N

T ∗ M → L0 P ⊗

˜ and D by ex◦D, we T ∗ M . Thus, if we replace Λ by the classical conjugate connection Λ ˜ obtain a connection Φ said to be conjugate to Φ. In Section 13 we present a more geometric ˜ by using the canonical involution of semiholonomic 2-jets. construction of Φ There is another construction transforming the pair (Γ, Λ) into a connection on W 1 P . It is based on the general idea of flow prolongation of connections, [18]. Consider Γ in the lifting form Γ : P ×M T M → T P . For every vector field X on M , we first construct its Γ-lift ΓX : P → T P and then the flow prolongation W 1 (ΓX) : W 1 P → T W 1 P . This defines a map W 1 Γ : W 1 P ×M J 1 T M → T W 1 P . If we add Λ in its algebroid form T M → J 1 T M , we obtain the lifting map W 1 (Γ, Λ) : W 1 P ×M T M → T W 1 P of a principal connection on W 1 P . In [20] we deduced β∗ W 1 (Γ, Λ) = Γ, p1∗ W 1 (Γ, Λ) = 2

˜ So the difference p(Γ, Λ) − W 1 (Γ, Λ) ˜ is a section of L0 P ⊗ N T ∗ M . We recall that the Λ. curvature C(Γ) of Γ is a section of L0 P ⊗ Λ2 T ∗ M . In [20], we proved ˜ = C(Γ). Proposition 17. We have p(Γ, Λ) − W 1 (Γ, Λ)

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12.

Gauge-Natural Operators on Connections

Analogously to Section 8 one can pose the question of finding all geometric operators transforming a pair (Γ, Λ) of a connection Γ on P and a connection Λ on P 1 M into a connection on W 1 P . The precise meaning of “geometric” is “gauge-natural” in the sense of [2]. Roughly speaking, when passing from the classical natural operators to the gauge-natural ones, we meet the higher order principal prolongations W r P in the former role of the higher order frame bundles P r M , see [18] for a complete theory. In [20] we deduced the following list of all gauge-natural operators A : C ∞ (QP ) × C ∞ (Qτ P 1 M ) → C ∞ (QW 1 P ). (The assumption Λ is torsion-free is of technical character. If we replace Qτ P 1 M by QP 1 M , the list will be much longer with many further terms of less geometric interest.) First of all one proves that the underlying connections of A(Γ, Λ) are β∗ A(Γ, Λ) = Γ and p1∗ A(Γ, Λ) = Λ. So the difference A(Γ, Λ) − p(Γ, Λ) is 2 N

a section of L0 P ⊗ T ∗ M . We already know that the curvature C(Γ) is a section of that bundle. Let Z ⊂ Lin(g, g) be the subspace of all linear maps commuting with the adjoint action of G. Since every z ∈ Z is an equivariant map between the standard fibers, it induces a vector bundle morphism zP : L0 P → L0 P . Hence one can construct the modified curvature operator C(Γ)(z) = (zP ⊗ id ) ◦ C(Γ). On the other hand, by Example 28.7 of [18] all natural operators C ∞ (Qτ P 1 M ) → C ∞ (T ∗ M ⊗ T ∗ M ) are linearly generated by the contractions k ) and R (Λ) = (Rk ) of the curvature tensor (Ri ) of Λ. Let S ⊂ g be R1 (Λ) = (Rkij 2 ikj jkl the subspace of all vectors invariant with respect to the adjoint action. Since every B ∈ S is an invariant element of the standard fiber, it determines a section BP of L0 P . Our result from [20] reads Proposition 18. All gauge-natural operators C ∞ (QP ) × C ∞ (Qτ P 1 M ) → C ∞ (QW 1 P ) are of the form p(Γ, Λ) + C(Γ)(z) + B1P ⊗ R1 (Λ) + B2P ⊗ R2 (Λ) for all z ∈ Z and all B1 , B2 ∈ S. We underline that there exist many interesting open problems concerning the gaugenatural operators related with connections on W r P .

13.

¯ 2P Connections on W 1 P as Reductions of W

Finally we outline how the semiholonomic 2-jets can be used in the theory of connections on W 1 P . In general, the bundle of nonholonomic (n, 2)-velocities on a manifold M is defined by T˜n2 M = J˜02 (Rn , M ) . We shall frequently use a natural identification T˜n2 M ≈ Tn1 (Tn1 M ) .

(38)

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Write tu : Rn → Rn for the translation x 7→ x + u. If ψ : Rn → J 1 (Rn , M ) is a section, then u 7→ ψ(u) ◦ j01 tu is a map Rn → Tn1 M . Passing to 1-jets defines (38). One verifies easily that (38) identifies reg T˜n2 M with reg Tn1 (reg Tn1 M ). The second order nonholonomic frame bundle of M is defined by P˜ 2 M = inv J˜02 (Rm , M ) . This is a principal bundle over M with structure group ˜ 2m = inv J˜02 (Rm , Rm )0 , G the right action of which on P˜ 2 M is determined by the composition of nonholonomic 2-jets. On the other hand, W 1 (P 1 M ) = P 1 M ×M J 1 P 1 M . Using (32) and (38), we obtain P˜ 2 M = W 1 (P 1 M ) .

(39)

1 G1 . ˜ 2m = G1m ⋊ Tm In particular, G m Further we introduce the second nonholonomic prolongation of P (M, G) by the iteration ˜ 2 P = W 1 (W 1 P ) . W (40) 2 G := W 1 (W 1 G). We have ˜m This is a principal bundle over M with structure group W m m

W 1 (W 1 P ) = P 1 M ×M J 1 P 1 M ×M J˜2 P = P˜ 2 M ×M J˜2 P and 1 1 1 1 1 1 2 ˜ 2m ⋊ T˜m Wm (Wm G) = G1m × Tm G m × Tm Tm G ≈ G G,

where the group semidirect product has an analogous meaning to (34). In the semiholonomic case, we have ˜ 2P ⊃ W ¯ 2 P := P¯ 2 M ×M J¯2 P . W (41)

2G = G 2 G. ¯ 2 P is W ¯m ¯ 2m ⋊ T¯m The structure group of W 1 1 1 Consider a connection Φ : W P → J (W P ) = J 1 P 1 M ×M J˜2 P . Using the identification J 1 P 1 M ≈ P¯ 2 M from Section 9 and the inclusion J¯2 P ⊂ J˜2 P , we obtain an ¯ 2 P ⊂ J 1 W 1 P . Let Γ and Λ be the underlying connections of Φ. In Secinclusion W 1 (G). One tion 11 we constructed the reduction R(Γ) ⊂ W 1 P to the subgroup G1m × νm 2 (G) ⊂ ¯ 2 P is a reduction to the subgroup l1 (G1m ) × νm verifies easily that Φ R(Γ) ⊂ W 2 2 2 2 ¯ ¯ Gm ⋊ Tm G ⊂ Gm ⋊ Tm G. The following assertion generalizes the result by P. Libermann mentioned in Section 9.

¯ 2 P to Proposition 19. The connections on W 1 P are in bijection with the reductions of W 2 2 2 1 ¯ m ⋊ T¯m G. the subgroup l1 (Gm ) × νm (G) ⊂ G

2 (G). Hence its projection Q into ¯ 2 P to l1 (G1m ) × νm Proof. Let Q be a reduction of W 1 1 1 1 W P is a reduction to the subgroup Gm × νm (G). Then Q can be interpreted as a map (denoted by the same symbol) Q : Q1 → J 1 W 1 P . This map is equivariant, so that Q can be uniquely extended into a connection on W 1 P .

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˜ conjugate to Φ by using the canonical inNow we can construct the connection Φ volution κ of semiholonomic 2-jets. Using Proposition 3 of [20], one proves that if ¯ 2 P is the reduction corresponding to Φ, then red(Φ) ⊂ W ˜ κ ◦ red(Φ) = red(Φ)

(42)

˜ is the reduction corresponding to Φ.

Acknowledgement The author was supported by the Ministry of Education of the Czech Republic under the project MSM 0021622409 and the grant GACR No. 201/05/0523.

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In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 189-207

Chapter 11

N ATURAL L IFTS IN R IEMANNIAN G EOMETRY Oldˇrich Kowalski1∗and Masami Sekizawa2† 1 Faculty of Mathematics and Physics Charles University in Prague, Sokolovsk´a 83, 186 75 Praha 8 Czech Republic 2 Department of Mathematics, Tokyo Gakugei University Koganei-shi Nukuikita-machi 4-1-1, Tokyo 184-8501, Japan

Abstract Using the concepts and the methods developed by D. Krupka, J. Janyˇska and V. Mikol´asˇov´a, we fully classified (during 1986-88) naturally lifted metrics to tangent bundles, linear frame bundles and cotangent bundles. All classical constructions of metrics on such bundles are special examples of our lifted metrics. We shall survey our own earlier work and also the later development in the geometrical study of natural metrics.

2000 Mathematics Subject Classification. 53A55, 53C20, 53C25 Key words and phrases. Riemannian metric, natural lift, tangent bundle, linear frame bundle, cotangent bundle, sectional curvature, scalar curvature.

Introduction There are well-known classical examples of “lifted metrics” on the tangent bundle T M and on the linear frame bundle LM over a Riemannian manifold (M, g), and also of a lifted metric on the the cotangent bundle T ∗ M over an affine manifold (M, ∇). Namely, they are the Sasaki metric, the horizontal lift and the vertical lift on T M ; then the diagonal lift, the horizontal lift and the vertical lift on LM ; and finally the Riemann extension on T ∗ M . All these constructions have been studied extensively (see e.g. [12, 38, 30, 11, 14, 37, 43, 44, 47].) As we shall see, the classical constructions are examples of “natural transformations ∗ †

E-mail address: [email protected] E-mail address: [email protected]

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Oldˇrich Kowalski and Masami Sekizawa

of the second order”. We shall survey our work [22, 21, 39] on the full classifications of (possibly degenerate) naturally lifted metrics on T M , LM and T ∗ M . We have proved that the complete family of naturally lifted metrics on T M and LM (for a fixed base metric) is a module over real functions generated by some generalizations of known classical lifts. In the case of T ∗ M , just two arbitrary parameters appear. Our idea of naturality is closely related to that of A. Nijenhuis [33], D. B. A. Epstein [13], P. Stredder [42] and others. Yet, we shall use for our purposes the concepts and methods developed by D. Krupka [26, 27] and D. Krupka and V. Mikol´asˇov´a [28]. See also I. Kol´aˇr, P. W. Michor and J. Slov´ak [18, pp. 227–280], and D. Krupka and J. Janyˇska [29, pp. 160–166] for other presentations of our study and for the concept of naturality in general. “Rigidity” and “heredity” of general natural metrics on tangent bundles and linear frame bundles have been extensively studied. This study is usually a hard work. We shall survey our own work from the 1980s, and the later development in the geometrical study of natural metrics.

1.

Natural Transformations

Let us recall the general theory of natural transformations due to D. Krupka. We refer to [26, 27, 28, 29] for more details. Let r be any non-negative integer. Then the r-th order differential group Lrn of the n-dimensional Euclidean space Rn , n ≥ 2, is the Lie group of all r-jets of local diffeomorphisms of Rn with source and target at the origin o ∈ Rn . Let P and Q be smooth manifolds on which the group Lrn acts to the left. Then an r-th order differential invariant f : P −→ Q is an Lrn -equivariant map of the left Lrn -space P to the left Lrn -space Q. Further, let F r M denote the bundle of all frames of r-th order over M , which carries a natural structure of a principal Lrn -bundle F r (M, Lrn , πnr ). We get a natural functor from the category Dn of smooth n-manifolds and injective immersions into the category of principal Lrn -bundles and Lrn -bundle morphisms. For a left Lrn -space P , let FPr M denote the fiber bundle with fiber P , associated to the principal Lrn -bundle F r M . We obtain a natural functor FPr from the category Dn into the category of fiber bundles and their morphisms. For each manifold M and each differential invariant f : P −→ Q, we can define a morphism fM : FPr M −→ FQr M over the identity map id : M −→ M by fM ([y, p]) = [y, f (p)] for all [y, p] ∈ FPr M . This morphism fM is called the realization of a differential invariant f on the manifold M . An r-th order natural transformation T of the functor Fpr into the functor FQr is a collection of bundle morphisms TM : FPr M −→ FQr M over the identity map id : M −→ M , M ∈ Dn , such that FQr ϕ◦TM1 = TM2 ◦FPr ϕ holds for every morphism ϕ : M1 −→ M2 of Dn . The following Theorem by D. Krupka in [26] says that any concrete classification of all r-th order natural transformations of FPr to FQr can be reduced to a classification of all r-th order differential invariants f from P to Q. Theorem 1.1 ([26]). Let f : P −→ Q be an r-th order differential invariant. Then the correspondence Tf : M −→ fM , where M is an object of Dn , is a natural transformation

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of the functor FPr to the functor FQr . Moreover, the correspondence f −→ Tf is a bijection between the set of all r-th order differential invariants from P to Q and the set of all r-th order natural transformations of FPr to FQr .

2.

Tangent Bundle

After recalling classical examples of lifts of a given metric on the base manifold to its tangent bundle, we shall present our full classification of naturally lifted metrics and related results. Then, we shall survey some geometric properties of such lifted metrics.

2.1.

Classical Examples of Metrics on the Tangent Bundle

The tangent bundle T M over an n-dimensional smooth manifold M , n ≥ 2, consists of all pairs (x, u), where x is a point of M and u is a vector from the tangent space Mx of M at x. We denote by p the natural projection of T M to M defined by p(x, u) = x. Let g be a Riemannian metric on the manifold M and ∇ its Levi-Civita connection. Then the tangent space (T M )(x,u) of T M at (x, u) ∈ T M splits into the horizontal and vertical subspace H(x,u) and V(x,u) with respect to ∇: (T M )(x,u) = H(x,u) ⊕ V(x,u) . If a point (x, u) ∈ T M and a vector X ∈ Mx are given, then there exists a unique vector X h ∈ H(x,u) such that p∗ (X h ) = X. We call X h the horizontal lift of X to T M at (x, u). The vertical lift of X to (x, u) is a unique vector X v ∈ V(x,u) such that X v (df ) = Xf for all smooth functions f on M . Here we consider a one-form df on M as a function on T M defined by (df )(x, u) = uf for all (x, u) ∈ T M . The map X 7−→ X h is an isomorphism between Mx and H(x,u) , and the map X 7−→ X v is an isomorphism between Mx and V(x,u) . In an obvious way we can define horizontal and vertical lifts of vector fields on M . These are uniquely defined vector fields on T M . The three classical constructions of metrics on tangent bundles T M which are derived from a Riemannian metric g on M are given as follows: (a) The metric g s constructed by Sasaki [38] is a (positive definite) Riemannian metric on T M given by s (X h , Y h ) = gx (X, Y ), g(x,u)

s (X h , Y v ) = 0, g(x,u)

s g(x,u) (X v , Y h ) = 0,

s (X v , Y v ) = gx (X, Y ) g(x,u)

for all X, Y ∈ Mx . (b) The horizontal lift g h of g is a pseudo-Riemannian metric on T M with signature (n, n) which is given by h g(x,u) (X h , Y h ) = 0,

h g(x,u) (X h , Y v ) = gx (X, Y ),

h g(x,u) (X v , Y h ) = gx (X, Y ),

h g(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx .

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Oldˇrich Kowalski and Masami Sekizawa (c) The vertical lift g v of g is a degenerate metric of rank n on T M given by v g(x,u) (X h , Y h ) = gx (X, Y ),

v g(x,u) (X h , Y v ) = 0,

v g(x,u) (X v , Y h ) = 0,

v g(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx .

2.2.

Naturally Lifted Metrics on the Tangent Bundle

Let us consider the symmetric tensor product E = Rn∗ ⊙ Rn∗ , i.e., the vector space of symmetric bilinear forms on Rn and let E+ ⊂ E denote the open subset of all positive inner products. Put P = Rn × Tn1 E+ and denote by (uk , gij , gij,k ), 1 ≤ i ≤ j ≤ n; k = 1, 2, . . . , n, the system of canonical coordinates of P . Also, put Q = Rn ⊕ (R2n∗ ⊙ R2n∗ ) and denote by (v k , GAB ), k = 1, 2, . . . , n; A, B = 1, 2, . . . , 2n, the system of canonical coordinates on Q. This can be also written in the form (v k , Gij , Gik∗ , Gi∗j∗ ), 1 ≤ i ≤ j ≤ n; k = 1, 2, . . . , n, where i∗ stands for n + i. We define the actions of L2n on P and Q, respectively, having in mind the transformation rules on Riemannian objects under changes of systems of natural local coordinates. We see easily that the corresponding associated L2n -bundles FP2 M and FQ2 M over a manifold M always have canonical bundle projections FP2 M −→ T M,

FQ2 M −→ T M.

(2.1)

We define the problem to find all second order natural transformations of a Riemannian metric on a manifold to a metric on its tangent bundle as the problem to find all those natural transformations of FP2 M to FQ2 M which, for each M and via the projections (2.1), induce the identity map id : T M −→ T M . Hence, by Theorem 1.1, this reduces to the problem to find all second order differential invariants f : (uk , gij , gij,k ) −→ (v k , GAB (uk , gij , gij,k )) from P to Q such that v k ◦ f = uk , k = 1, 2, . . . , n. We use the computational method proposed by Krupka and Mikol´asˇov´a in [28]. We obtain Theorem 2.1 ([22]). Let αij , βij and γij be functions on P which are solutions of the following system of differential equations: ∂ζij = 0, ∂gpq,r 2

n X

a=1

gap

(2.2)

∂ζij ∂ζij − uq p = ζip δjq + ζpj δiq ∂gaq ∂u

(2.3)

for i, j, p, q, r = 1, 2, . . . , n. Then a necessary condition for a map f : P −→ Q such that v k ◦ f = uk , k = 1, 2, . . . , n, to be a differential invariant is that its representation by the canonical coordinates is of the form Gij =

n X

b,c,s,t=1

ub uc Γsbi Γtcj αst +

n X

b,s=1

ub Γsbi βsj +

n X

b,s=1

ub Γsbj βsi + γij ,

(2.4)

Natural Lifts in Riemannian Geometry Gik∗ =

n X

ub Γsbi αsk + βki ,

193 (2.5)

b,s=1

Gi∗j∗ = αij

(2.6)

for 1 ≤ i ≤ j ≤ n and k = 1, 2, . . . , n, where Γkij ’s are the formal Christoffel symbols derived from gij and gij,k . We have given in [22] a geometric meaning to the system of differential equations (2.3) and its solution. We have shown first which type of differential invariants of the form ζij = ζij (uk , gkl ) (and which type of natural transformations) belongs to the system of differential equations. Consider the left L1n -spaces V = Rn ×E+ and W = Rn∗ ⊗Rn∗ with the natural actions of L1n defined, again, having in mind transformation rules under changes of systems of natural local coordinates. Let (uk , gij ), 1 ≤ i ≤ j ≤ n; k = 1, 2, . . . , n, and (ζij ), i, j = 1, 2, . . . , n, be the systems of canonical coordinates of V and W , respectively. We can check easily that the system of differential equations (2.3) gives a necessary condition for a map ζij = ζij (uk , gkl ) to be (first order) differential invariant from V to W . Long but routine calculations show that the formulas (2.4)–(2.6) provide a differential invariant f from P to Q if and only if the functions αij , βij and γij defined on V describe some differential invariants α, β and γ from V to W . Thus our problem reduces to the 1 . study of the first order natural transformations of FV1 into FW We say that a bundle morphism of the form ζ : T M ⊕ T M ⊕ T M −→ M × R is an F-metric on M if it is linear in the second and the third argument (and smooth in the first argument). We also say that ζ is symmetric or skew-symmetric if it is symmetric or skewsymmetric with respect to the second and third argument, respectively. (We use here the letter “F” to recall the Finsler geometry.) Any Riemannian metric g on M is a symmetric F-metric which is independent on u. In our special case, letting g be a given Riemannian metric on M , we speak about natural F-metrics derived from g which are F-metrics ζ, for a fixed u ∈ T M , whose components ζ(u)ij = ζ(u, ∂/∂xi , ∂/∂xj ) with respect to a system of local coordinates (x1 , x2 , . . . , xn ) in M are solutions of the system of differential equations (2.3). We obtain that Theorem 2.2 ([22]). Let (M, g) be an n-dimensional oriented Riemannian manifold. Then all natural F-metrics ζ on M derived from g are given as follows: (1) For n = 2, all symmetric natural F-metrics are of the form ζ(u; X, Y ) = α(kuk2 )g(X, Y ) + β(kuk2 )g(X, u)g(Y, u) + γ(kuk2 ){g(X, u)g(Y, Ju) + g(X, Ju)g(Y, u)}, and all skew-symmetric natural F-metrics are of the form ζ(u; X, Y ) = δ(kuk2 ){g(X, u)g(Y, Ju) − g(X, Ju)g(Y, u)}, where α, β, γ and δ are arbitrary functions of kuk2 = g(u, u) and J is one of the two canonical almost complex structures of (M, g) (for which (M, g, J) is a K¨ahler manifold).

194

Oldˇrich Kowalski and Masami Sekizawa (2) For n = 3, all symmetric natural F-metrics are of the form ζ(u; X, Y ) = α(kuk2 )g(X, Y ) + β(kuk2 )g(X, u)g(Y, u)

(2.7)

and all skew-symmetric natural F-metrics are of the form ζ(u; X, Y ) = ϕ(kuk2 )g(X × Y, u),

where α, β and ϕ are arbitrary functions of kuk2 = g(u, u), and X × Y is the usual vector product of X and Y . (3) For n > 3, all natural F-metrics are symmetric and of the form (2.7). I. Kol´aˇr, P. W. Michor and J. Slov´ak have given in [18, Proposition 33.22] a new and elegant proof of the classification of natural F-metrics on non-oriented Riemannian manifolds. Theorem 2.3 ([18]). Let (M, g) be an n-dimensional non-oriented Riemannian manifold, n ≥ 1. Then all natural F-metrics ζ on M derived from g are symmetric and given by (2.7), where α and β are arbitrary smooth functions defined on the interval (0, ∞). In particular, β = 0 if n = 1. M. T. K. Abbassi has proved in [1] explicitly that all basic functions from Theorems 2.2 and 2.3 can be prolonged to smooth functions on the set of all non-negative real numbers. This result found many applications in the techniques used for the thorough investigation of g-natural metrics by M. T. K. Abbassi and M. Sarih. For a given Riemannian metric g on M , we define the classical lifts of F-metrics from M to T M , with respect to g, as follows: (a) The Sasaki lift ζ s,g of a symmetric F-metric ζ with respect to g is defined by s,g ζ(x,u) (X h , Y h ) = ζx (u; X, Y ),

s,g ζ(x,u) (X h , Y v ) = 0,

s,g ζ(x,u) (X v , Y h ) = 0,

s,g ζ(x,u) (X v , Y v ) = ζx (u; X, Y )

for all X, Y ∈ Mx . (b) The horizontal lift ζ h,g of an arbitrary F-metric ζ with respect to g is defined by h,g ζ(x,u) (X h , Y h ) = 0,

h,g ζ(x,u) (X h , Y v ) = ζx (u; Y, X),

h,g ζ(x,u) (X v , Y h ) = ζx (u; X, Y ),

h,g ζ(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx . (c) The vertical lift ζ v of a symmetric F-metric ζ with respect to g is defined by v (X h , Y h ) = ζx (u; X, Y ), ζ(x,u)

v ζ(x,u) (X h , Y v ) = 0,

v ζ(x,u) (X v , Y h ) = 0,

v ζ(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx . Obviously, the vertical lift does not depend on the choice of g. We note that ζ s,g , ζ h,g and ζ v are (not necessarily regular) pseudo-Riemannian metrics on T M . If we take ζ = g, then ζ s,g , ζ h,g and ζ v are just the classical lifts g s , g h and g v , respectively. Thus we have all metrics on T M which come from a second order natural transformation of a given Riemannian metric on M .

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Theorem 2.4 ([22]). Let g be a Riemannian metric on an n-dimensional smooth manifold M , n ≥ 2, and let G be a (possibly degenerate) pseudo-Riemannian metric on the tangent bundle T M which comes from a second order natural transformation of g. Then there are natural F-metrics ζ1 , ζ2 and ζ3 derived from g, where ζ1 and ζ3 are symmetric, such that G = ζ1 s,g + ζ2 h,g + ζ3 v . Moreover, all natural F-metrics derived from g are given by Theorem 2.2. If n = 2, then the family of all natural metrics G on T M depends on 10 arbitrary functions of one variable, for n = 3 it depends on seven arbitrary functions of one variable, and for n > 3 on six arbitrary functions of one variable. Our concept of F-metric coincides with that of “M-tensor of type (0, 2)” introduced by Y. C. Wong and K. P. Mok in [46]. Further, our Theorem 2.2 is strongly related to Theorem 5.1 of the paper by K. P. Mok, E. M. Patterson and Y. C. Wong in [31]. Nevertheless, the authors above do not mention the naturality problem in their investigations, and they also do not give any nontrivial examples of M-tensor. In fact, Theorem 5.1 mentioned above only says that each metric on the tangent bundle T M is a sum of three classical lifts of three independent M-tensors of type (0, 2) with respect to a generalized connection (called M-connection). The metrics given by Theorem 2.4 are called g-natural metrics by M. T. K. Abbassi in [1]. He has formulated these metrics in the form: Theorem 2.5 ([1, 4]). Let (M, g) be an n-dimensional Riemannian manifold and G be a g-natural metric on the tangent bundle T M . Then there are real valued functions αi and βi , i = 1, 2, 3, defined on [0, ∞) such that G(x,u) (X h , Y h ) = (α1 + α3 )(r2 )gx (X, Y )

+ (β1 + β3 )(r2 )gx (X, u)gx (Y, u), G(x,u) (X h , Y v ) = α2 (r2 )gx (X, Y ) + β2 (r2 )gx (X, u)gx (Y, u), G(x,u) (X v , Y h ) = α2 (r2 )gx (X, Y ) + β2 (r2 )gx (X, u)gx (Y, u), G(x,u) (X v , Y v ) = α1 (r2 )gx (X, Y ) + β1 (r2 )gx (X, u)gx (Y, u) hold at each point (x, u) ∈ T M for all u, X, Y ∈ Mx , where r2 = gx (u, u). For n = 1, the same holds with βi = 0, i = 1, 2, 3.

2.3.

Riemannian Geometry of the Tangent Bundle

Let φ be a (local) transformation of a manifold M . Then we define a transformation Φ of T M by Φ(x, u) = (φx, φ∗x u) for all (x, u) ∈ T M . If φ is a (local) affine transformation with respect to the Levi-Civita connection ∇ of (M, g), then we have Φ∗ (X h ) = (φ∗ X)h ,

Φ∗ (X v ) = (φ∗ X)v

for all X ∈ X(M ). Using this fact we can easily see that all g-natural metrics are invariant:

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Oldˇrich Kowalski and Masami Sekizawa

Theorem 2.6 ([24]). Let φ be a (local) isometry of a Riemannian manifold (M, g). Then every g-natural metric G on the tangent bundle T M over (M, g) is invariant by the lift Φ of φ. In other words, Φ is a local isometry of (T M, G) whose projection on (M, g) is φ. Riemannian geometry of the tangent bundle T M with the metric g s defined by S. Sasaki in [38] has been studied by many authors (see for example, [12, 19, 32, 47]). The first author of the present paper has proved in [19] that (T M, g s ) is never locally symmetric unless the base metric g on M is locally Euclidean. Further, E. Musso and F. Tricerri have proved in [32] that g s is “extremely rigid” in the following sense: Theorem 2.7 ([32]). The tangent bundle (T M, g s ) with the Sasaki metric has constant scalar curvature if and only if the base manifold (M, g) is locally Euclidean. It seems that another metric nicely fitted to the tangent bundle is the so-called CheegerGromoll metric. Its construction has been suggested by J. Cheeger and D. Gromoll in [8] and expressed more explicitly by E. Musso and F. Tricerri in [32]. It is a “nonclassical” natural metric on the tangent bundle T M over a Riemannian manifold (M, g) defined at each point (x, u) ∈ T M by cg g(x,u) =

1 kuk2 gxv − kxv + gxs,g + kxs,g , 2 1 + kuk

where k is the natural F-metric given by k(u; X, Y ) = g(X, u)g(Y, u). In the other form, g cg is given by cg g(x,u) (X h , Y h ) = gx (X, Y ), cg g(x,u) (X v , Y v ) =

cg cg g(x,u) (X h , Y v ) = g(x,u) (X v , Y h ) = 0,

1 + g (X, u)g (Y, u) g (X, Y ) x x x 1 + kuk2

for all X, Y ∈ Mx . The curvatures of g cg have been studied in detail by the second author of the present paper in [41]. In particular, it has been proved that the scalar curvature of g cg is never constant if g has constant sectional curvature. (Unfortunately there are computational errors in the paper [41], which have been pointed and corrected precisely by S. Gudmandsson and E. Kappose in [16]). It should be interesting to find “non-rigid” metrics on T M over a Riemannian manifold (M, g). The first example of such metrics has been given by V. Oproiu in [36]. The family of Oproiu metrics depends on two arbitrary functions of one variable, and belong to a family of metrics on T M which come from a second order natural transformation of g given by Theorem 2.4. In terms of Theorem 2.5, The Oproiu metrics are g-natural metrics on T M such that (α1 + α3 )(t) = v(t/2), 1 , v(t/2) α2 = β2 = 0, α1 (t) =

(β1 + β3 )(t) = w(t/2), β1 (t) =

w(t/2) , v(t/2) v(t/2) + tw(t/2)

where v and w are real valued smooth functions defined on (0, ∞) such that v > 0 and v(t) + 2tw(t) > 0 for all t ∈ (0, ∞). (see [5]). The main result of [36] is the following

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Theorem 2.8. If (M, g) is an n-dimensional space of negative constant sectional curvature, n > 2, then T M equipped with any Oproiu metric is a K¨ahler Einstein manifold with positive constant scalar curvature. M. T. K. Abbassi and M. Sarih have proved in [6] the following (highly nontrivial) “heredity” theorem: Theorem 2.9. Let G be any Riemannian g-natural metric on T M , where (M, g) is an arbitrary Riemannian manifold. Then, if (T M, G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M, g) possesses the same property, respectively. As an application of study on the Oproiu metric, they have proved that Theorem 2.10 ([6]). Let (M, g) be an n-dimensional space of negative constant sectional curvature, n ≥ 3. Then there is a one-parameter family F of g-natural metrics on T M with nonconstant defining functions αi and βi , i = 1, 2, 3, such that, for every G ∈ F, (T M, G) is a space of positive constant scalar curvature. Moreover, for each (M, g) as above, and each prescribed positive constant S, there is a metric G ∈ F with the constant scalar curvature S. Moreover they have obtained examples of Riemannian g-natural metrics with constant defining functions αi and βi , i = 1, 2, 3, which can have an arbitrary constant scalar curvature (not necessarily positive as in the above Theorem). Theorem 2.11 ([6]). Let (M, g) be a space of constant sectional curvature K 6= 0, and let G = a g s + b g h + c g v a metric on the √ tangent bundle T M , where a, b and c are constant such that a > 0 and b2 = (−1 + 13 )a(a + c)/6. Then (T M, G) is a space of nonzero constant scalar curvature with the same sign as K. Furthermore, we can choose the constants a, b and c so that the scalar curvature of G has any prescribed nonzero constant value with the same sign as K.

3.

Linear Frame Bundle

After recalling classical examples of lifts of a given metric on the base manifold to its linear frame bundle, we shall present our full classification of naturally lifted metrics. Then, we shall remind some rigidity results for special cases of these metrics.

3.1.

Classical Examples of Metrics on the Linear Frame Bundle

The linear frame bundle LM over an n-dimensional smooth manifold M , n ≥ 2, consists of all pairs (x, u), where x is a point of M and u is a basis for the tangent space Mx of M at x. We denote by p the natural projection of LM to M defined by p(x, u) = x. Let g be a Riemannian metric on the manifold M and ∇ its Levi-Civita connection. Then the tangent space (LM )(x,u) of LM at (x, u) ∈ LM splits into the horizontal and vertical subspace H(x,u) and V(x,u) with respect to ∇: (LM )(x,u) = H(x,u) ⊕ V(x,u) .

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If a point (x, u) ∈ LM and a vector X ∈ Mx are given, then there exists a unique vector X h ∈ H(x,u) such that p∗ (X h ) = X. We call X h the horizontal lift of X to LM at (x, u). We define naturally n different vertical lifts of X ∈ Mx . If ω is a one-form on M , then ιµ ω, µ = 1, 2, . . . , n, are functions on LM defined by (ιµ ω)(x, u) = ω(uµ ) for all (x, u) = (x, u1 , u2 , . . . , un ) ∈ LM . The vertical lifts X v,λ , λ = 1, 2, . . . , n, of X ∈ Mx to LM at (x, u) are the n vectors such that X v,λ (ιµ ω) = ω(X)δµλ , λ, µ = 1, 2, . . . , n, hold for all one-forms ω on M , where δµλ denotes the Kronecker’s delta. The n vertical lifts are always uniquely determined, and they are linearly independent if X 6= 0. In an obvious way we can define horizontal and vertical lifts of vector fields on M . These are uniquely defined vector fields on T M . The three classical constructions of metrics on linear frame bundles LM which are derived from a Riemannian metric g on M are given as follows: (a) The diagonal lift g d of g defined by K. P. Mok in [30] is a (positive definite) Riemannian metric on LM given by d g(x,u) (X h , Y h ) = gx (X, Y ),

d g(x,u) (X h , Y v,µ ) = 0,

d g(x,u) (X v,λ , Y h ) = 0,

d g(x,u) (X v,λ , Y v,µ ) = gx (X, Y )δ λµ

for all X, Y ∈ Mx and λ, µ = 1, 2, . . . , n. This metric is called also the Sasaki-Mok metric because it resembles the Sasaki metric of the tangent bundle over a Riemannian manifold. (b) The horizontal lift g h of g is a degenerate metric on LM of rank 2n and signature (n, n) which is given by h (X h , Y h ) = 0, g(x,u)

h (X h , Y v,µ ) = gx (X, Y ), g(x,u)

h (X v,λ , Y h ) = gx (X, Y ), g(x,u)

h (X v,λ , Y v,µ ) = 0 g(x,u)

for all X, Y ∈ Mx and λ, µ = 1, 2, . . . , n. (c) The vertical lift g v of g is a degenerate metric of rank n on LM given by v g(x,u) (X h , Y h ) = gx (X, Y ),

v g(x,u) (X h , Y v,µ ) = 0,

v g(x,u) (X v,λ , Y h ) = 0,

v g(x,u) (X v,λ , Y v,µ ) = 0

for all X, Y ∈ Mx and λ, µ = 1, 2, . . . , n.

3.2.

Naturally Lifted Metrics on the Linear Frame Bundle

Put P = GL(n, R) × Tn1 E+ and denote by (ukλ , gij , gij,k ), 1 ≤ i ≤ j ≤ n; k = 1, 2, . . . , n, the system of canonical coordinates of P . Also, put Q = GL(n, R) × {(Rn∗ ⊙ Rn∗ ) ⊕ (Rn∗ ⊗ Rn∗ ⊗ Rn ) ⊕ [(Rn∗ ⊗ Rn ) ⊙ (Rn∗ ⊗ Rn )]} and denote by (vνk , Gij , Gi νk , Gλµ ij ) the system of canonical coordinates on Q. Here 1 ≤ i ≤ j ≤ n; k, ν = 1, 2, . . . n and (1, 1) ≤ (i, λ) ≤ (j, µ) ≤ (n, n) in the lexicographic arrangement. We define the actions of L2n on P and Q, respectively, having in mind the transformation rules on Riemannian objects under changes of systems of natural local coordinates. Discussing as in the previous section, we have

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λµ Theorem 3.1 ([21]). Let αij , βi µj and γij be functions on P which are solutions of the following system of differential equations:

∂ζij = 0, ∂gpq,r 2

n X

a=1

gap

(3.1)

n X ∂ζij ∂ζij − uqα p = ζip δjq + ζpj δiq ∂gaq α=1 ∂uα

(3.2)

for i, j, p, q, r = 1, 2, . . . , n. Then a necessary condition for a map f : P −→ Q such that v k ◦ f = uk , k = 1, 2, . . . , n, to be a differential invariant is that its representation by the canonical coordinates is of the form Gij =

n X

νω uaν ubω Γsai Γtbj αst +

a,b,s,t,ν,ω=1

n X

ubω Γsbi βs ωj +

b,s,ω=1

Gi νk =

n X

n X

ubω Γsbj βs ωi + γij , (3.3)

b,s,ω=1

µν uaµ Γsai αsk + βi νk ,

(3.4)

a,s,µ=1 λµ Gλµ ij = αij

(3.5)

for 1 ≤ i ≤ j ≤ n; k, ν = 1, 2, . . . n and (1, 1) ≤ (i, λ) ≤ (j, µ) ≤ (n, n) in the lexicographic arrangement, where Γkij ’s are the formal Christoffel symbols derived from gij and gij,k . We have given in [21] a geometric meaning to the system of differential equations (3.2) and its solution. Consider the left L1n -spaces V = GL(n, R) × E+ and W = Rn∗ ⊗ Rn∗ with the natural actions of L1n defined, again, having in mind transformation rules under changes of systems of natural local coordinates. Let (ukλ , gij ), 1 ≤ i ≤ j ≤ n; k, λ = 1, 2, . . . , n, and (ζij ), i, j = 1, 2, . . . , n, be the systems of canonical coordinates of V and W , respectively. We can check easily that the system of differential equations (3.2) gives a necessary condition for a map ζij = ζij (uk , gkl ) to be (the first order) differential invariant from V to W . Long but routine calculations show that the formulas (3.3)–(3.5) provide a differential λµ invariant f from P to Q if and only if the functions αij , βi µj and γij defined on V describe some differential invariants α, β and γ from V to W . We say that a bundle morphism of the form ζ : LM ⊕ T M ⊕ T M −→ M × R is an L-metric on M if it is linear in the second and the third argument (and smooth in the first argument). We also say that ζ is symmetric or skew-symmetric if it is symmetric or skew-symmetric with respect to the second and third argument, respectively. Any Riemannian metric g on M is a symmetric L-metric which is independent on u. In our special case, letting g be a given Riemannian metric on M , we speak about natural L-metrics derived from g which are L-metrics ζ, for a fixed u ∈ LM , whose components ζ(u)ij = ζ(u, ∂/∂xi , ∂/∂xj ) with respect to a system of local coordinates (x1 , x2 , . . . , xn ) in M are solutions of the system of differential equations (3.2). Solving this system of differential equations, we obtain

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Theorem 3.2 ([21]). Let (M, g) be an n-dimensional Riemannian manifold. Then all natural L-metrics ζ on M derived from g are given by ζ(u; X, Y ) =

n X

ϕαβ ω α (X)ω β (Y )

(3.6)

α,β=1

where {ω 1 , ω 2 , . . . , ω n } are the dual frame to a linear frame u = {u1 , u2 , . . . , un } and ϕαβ , α, β = 1, 2, . . . , n, are arbitrary smooth functions of n(n + 1)/2 variables wλµ = g(uλ , uµ ), 1 ≤ λ ≤ µ ≤ n. For a given Riemannian metric g on M , we define the classical lifts of L-metrics from M to LM with respect to g as symmetric (0, 2)-tensor fields on LM which are constructed as follows: (a) Let ζ be a symmetric L-metric and (ζ λµ ), 1 ≤ λ ≤ µ ≤ n, a family of arbitrary L-metrics. The diagonal lift ξ d,g of the family ξ = (ζ, ζ λµ ) with with respect to g is defined by d,g ξ(x,u) (X h , Y h ) = ζx (u; X, Y ), d,g ζ(x,u) (X h , Y v,µ ) = 0,

µ = 1, 2, . . . , n,

d,g ξ(x,u) (X v,λ , Y v,µ ) = ζxλµ (u; X, Y ),

1 ≤ λ ≤ µ ≤ n,

for all X, Y ∈ Mx . (b) The horizontal lift ξ h,g of an n-tuple ξ = (ζ µ ) of L-metrics with respect to g is defined by h,g ξ(x,u) (X h , Y h ) = 0, h,g ξ(x,u) (X h , Y v,µ ) = ζxµ (u; Y, X),

µ = 1, 2, . . . , n,

h,g ξ(x,u) (X v,λ , Y v,µ ) = 0,

λ, µ = 1, 2, . . . , n,

for all X, Y ∈ Mx . If we take ζ = g and ζ λµ = g δ λµ in (a), and ζ µ = g in (b), then ξ d,g and ξ h,g are just the classical lifts g d and g h , respectively. Also, if we take ζ = g and ζ λµ = 0 in (a), then ξ d,g is the vertical lift g v of g. Thus we have all metrics on LM which come from a second order natural transformation of a given Riemannian metric on M : Theorem 3.3 ([21]). Let g be a Riemannian metric on an n-dimensional smooth manifold M , n ≥ 2, and let G be a (possibly degenerate) pseudo-Riemannian metric on the linear frame bundle LM which comes from a second order natural transformation of g. Then there are families ξ1 = (ζ, ζ λµ ) and ξ2 = (ζ ν ) of natural L-metrics derived from g, where 1 ≤ λ ≤ µ ≤ n, ν = 1, 2, . . . , n and ζ is symmetric, such that G = ξ1 d,g + ξ2 h,g . Moreover, all natural L-metrics derived from g are given by Theorem 3.2.

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The family of all natural metrics G on LM over an n-dimensional Riemannian manifold (M, g) depends on n(n3 + 3n2 + n + 1)/2 arbitrary functions of n(n + 1)/2 variables. Equivalently, the metrics G in Theorem 3.3 can be expressed in the following form: G(x,u) (X h , Y h ) =

n X

ϕαβ ωxα (X)ωxβ (Y ),

α,β=1

G(x,u) (X h , Y v,µ ) =

n X

ϕα µβ ωxα (X)ωxβ (Y ),

(3.7)

α,β=1

G(x,u) (X v,λ , Y v,µ ) =

n X

α β ϕλµ αβ ωx (X)ωx (Y )

α,β=1

for all X, Y ∈ Mx , where ϕαβ , ϕα µβ and ϕλµ αβ , α, β, λ, µ = 1, 2, . . . , n, are arbitrary smooth functions of n(n + 1)/2 variables wρσ = g(uρ , uσ ), 1 ≤ ρ ≤ σ ≤ n. In the following we shall call G a g-natural metric on LM . Remark. The case of symmetric affine connection given on the base manifold gives a completely analogous classifications of metrics on LM as described for the metric case in Theorems 3.2 and 3.3, where just two lifts of “metric type” of L-metrics are replaced by the lifts of “affine type”. (See [40].)

3.3.

Riemannian Geometry of the Linear Frame Bundle

Let φ be a (local) transformation of a manifold M . Then we define a transformation Φ of LM by Φ(x, u) = (φx, φ∗x u1 , φ∗x u2 , . . . , φ∗x un ) for all (x, u) = (x, u1 , u2 , . . . , un ) ∈ LM . If φ is a (local) affine transformation with respect to the Levi-Civita connection ∇ of (M, g), then we have Φ∗ (X h ) = (φ∗ X)h ,

Φ∗ (X v,λ ) = (φ∗ X)v,λ

for all X ∈ X(M ) and λ = 1, 2, . . . , n. Using this fact we can easily see that all g-natural metrics are invariant: Theorem 3.4 ([25]). Let φ be a (local) isometry of a Riemannian manifold (M, g). Then every g-natural metric G on the linear frame bundle LM over (M, g) is invariant by the lift Φ of φ. In other words, Φ is a local isometry of (LM, G) whose projection on (M, g) is φ. Let (M, g) be an n-dimensional Riemannian manifold, n ≥ 2, and LM its linear frame bundle. It seems that phenomenon similar to geometry of T M with the Sasaki metric g s happens about geometry of LM with the diagonal lift g d . L. A. Cordero and M. de Le´on have shown in [10] that g d is also “rigid” in some sense. Namely, they have proved Theorem 3.5 ([10]). The Riemannian manifold (LM, g d ) is never locally symmetric unless (M, g) is locally Euclidean.

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Theorem 3.6 ([10]). If (LM, g d ) is an Einstein manifold, then (M, g) is flat. Theorem 3.7 ([10]). The Riemannian manifolds (LM, g d ) and (M, g) have the same constant scalar curvature if and only if both are flat. It should be interesting to find “non-rigid” metrics on LM . We study in this section specific Riemannian metrics from our list given by Theorem 3.3, which are a bit more general ones than the diagonal lift g d . The horizontal and vertical distributions are no more orthogonal in general with respect to the metrics g¯ treated in this section. Usually, calculations for getting geometric objects of LM are not short. They are complicated sometimes. We have given in [23] some simpler procedure in order to calculate them. The metrics which we pick in the following are still rigid when the base manifold is of constant curvature. Let g¯ be a metric on LM defined by taking the coefficient functions in (3.7) as ϕα µβ = cµ wαβ

ϕαβ = wαβ ,

λµ and ϕλµ αβ = c wαβ ,

where cµ and cλµ = cµλ are constants. That is, g¯ is given by g¯(x,u) (X h , Y h ) = gx (X, Y ), g¯(x,u) (X h , Y v,µ ) = cµ gx (X, Y ),

(3.8)

g¯(x,u) (X v,λ , Y v,µ ) = cλµ gx (X, Y ) for all X, Y ∈ Mx and λ, µ = 1, 2, . . . , n. The metric g¯ is positive definite if and only if all principal minor determinants of 1

c1

c2

 1 c   2 c   

c11

c12



c21 ...

...

cn



. . . c1n    22 2n c ... c    ... ...  

cn cn1 cn2 . . . cnn

are positive. In particular, the matrix [cλµ ] is positive definite. If cλ = 0 and cλµ = δ λµ for λ, µ = 1, 2, . . . , n, then the metric g¯ is the diagonal lift g d of g. We have shown in [23] that the metric g¯ is rigid in the following sense: Theorem 3.8 ([23]). Let (M, g) be a space of constant sectional curvature K, and g¯ the positive definite metric on LM defined by (3.8). If the scalar curvature of the frame bundle (LM, g¯) is constant then both manifolds are flat. This can be applied to the diagonal metric and hence we obtain a new result in the spirit of Theorems 3.5–3.7.

4.

Cotangent Bundle

If the base manifold M has a Riemannian metric g, then its cotangent bundle T ∗ M is dual to the tangent bundle T M , and hence all natural lifts of g to T ∗ M are settled from the those

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of g to T M through this duality. But there is a situation when only T ∗ M comes in the game, namely if the base manifold is an affine manifold (M, ∇). Here a new operation, so-called Riemann extension is known [37]. After recalling this notion, we shall present the full classification of naturally lifted (pseudo-Riemannian) metrics from a given symmetric affine connection ∇ on M to T ∗ M .

4.1.

Classical Examples of Metrics on the Cotangent Bundle

The cotangent bundle T ∗ M over an n-dimensional smooth manifold M , n ≥ 2, consists of all pairs (x, w), where x is a point of M and w is a covector from the cotangent space Mx∗ of M at x. We denote by p the natural projection of T ∗ M to M defined by p(x, w) = x. Let p∗ : T (T ∗ M ) −→ T M be the differential of the natural projection p, and q : T (T ∗ M ) −→ T ∗ M be the natural projection of the tangent bundle T (T ∗ M ) (over T ∗ M ) to T ∗ M . The canonical one-form on T ∗ M is a one-form θ defined by θ(X) = q(X)(p∗ X) for all X ∈ T (T ∗ M ), and the canonical two-form on T ∗ M is the exterior derivative dθ of θ. Let ∇ be an affine connection on the base manifold M . Then the tangent space ∗ (T M )(x,w) of T ∗ M at (x, w) ∈ T ∗ M splits into the horizontal and vertical subspace H(x,w) and V(x,w) with respect to ∇: (T ∗ M )(x,w) = H(x,w) ⊕ V(x,w) . Let X = hX + vX be the decomposition of a vector filed X on T ∗ M into the horizontal and vertical part. The Riemann extension g¯ of the affine connection ∇ on M to T ∗ M is a pseudo-Riemannian metric of the signature (n, n) defined by g¯(X, Y) = (dθ)(vX, hY) + (dθ)(vY, hX) for all X, Y ∈ T (T ∗ M ). This metric does not depend on the skew-symmetric part of ∇.

4.2.

Naturally Lifted Metrics on the Cotangent Bundle

Let us consider the space P = Rn∗ ⊕ Rn ⊗ (Rn∗ ⊙ Rn∗ ) , and denote by (wh , Γhij ), h = 1, 2, . . . , n; 1 ≤ i ≤ j ≤ n, the system of canonical coordinates of P . Also, put Q = Rn∗ ⊕ (Rn ⊕ Rn∗ ) ⊙ (Rn ⊕ Rn∗ ) and denote by (zh , Gij , Ghi , Gij ), h = 1, 2, . . . , n; 1 ≤ i ≤ j ≤ n, the system of canonical coordinates on Q. We define the actions of L2n on P and Q, respectively, having in mind the transformation rules on Riemannian objects under changes of systems of natural local coordinates. Discussing as in the previous section, we have

Theorem 4.1 ([39]). All differential invariants f : P −→ Q such that, in the canonical coordinates, zh ◦ f = wh , h = 1, 2, . . . , n, are given by Gij = −2a

n X

ws Γsij + b wi wj ,

s=1

Ghi = a δih , Gij = 0 for 1 ≤ i ≤ j ≤ n and h = 1, 2, . . . , n, where a and b are constants.

204

Oldˇrich Kowalski and Masami Sekizawa Thus we have

Theorem 4.2 ([39]). Let ∇ be a symmetric affine connection on an n-dimensional manifold M , n ≥ 2. Then a (possibly degenerate) pseudo-Riemannian metric G on the cotangent bundle T ∗ M over M comes from a second order natural transformation of ∇ if and only if G = a g¯ + b θ2 , where g¯ is the Riemann extension of ∇, θ2 is the tensor square of the canonical one-form θ of T ∗ M , and a, b are constants. This metric G is non-degenerate (of signature (n, n)) if and only if a 6= 0.

Acknowledgement ˇ 201/05/2707 and by the project The first author was supported by the grant GA CR MSM 0021620839.

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[41] M. Sekizawa, Curvatures of tangent bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14 (1991) 407–417. [42] P. Stredder, Natural differential operators on Riemannian manifolds, Tˆohoku Math. J. 10 (1958) 338–354. [43] M. Toomanian, Killing vectorfields and infinitesimal affine transformations on a generalised Riemann extension, Tensor (N.S.) 32 (1978) 335–338. [44] L. Vanhecke and T. J. Willmore, Riemann extensions of D’Atri spaces, Tensor (N.S.) 38 (1982) 154–158. [45] J. A. Wolf, Elliptic spaces in Grassmann manifolds, Illinois J. Math. 7 (1963) 447– 462. [46] Y. C. Wong and K. P. Mok, Connections and M-tensors on the tangent bundle T M , In: Topics in Differential Geometry ((H. Rund and W. F. Forbes, Eds.) Academic Press, New York, 1976) 157–172. [47] K. Yano and S. Ishihara, Tangent and Cotangent Bundles (Marcel Dekker, Inc., New York, 1973).

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 209-235

Chapter 12

I NVARIANT VARIATIONAL P ROBLEMS AND I NVARIANT F LOWS VIA M OVING F RAMES Peter J. Olver∗ School of Mathematics, University of Minnesota Minneapolis 55455, USA

Abstract This paper reviews the moving frame approach to the construction of the invariant variational bicomplex. Applications include explicit formulae for the Euler-Lagrange equations of an invariant variational problem, and for the equations governing the evolution of differential invariants under invariant submanifold flows.

1.

Introduction

This survey paper describes some aspects of the author’s recent research, done partly in collaboration with Irina Kogan, [31, 53], into moving frames, the invariant variational bicomplex, and invariant submanifold flows. These results are based on combining two powerful ideas in the modern, geometric approach to differential equations and the variational calculus. The first is the variational bicomplex, which is of fundamental importance in the study of the geometry of jet bundles, differential equations and the calculus of variations. Its origins can be found in the work of Dedecker, [15], then developed in full detail by Tulczyjew, [67], and Vinogradov, [68, 69]. Later contributions of Tsujishita, [66], Anderson, [2, 3], and Krupka and Janyˇska, [32, 33], have amply demonstrated the power of the bicomplex formalism for both local and global problems in the geometric theory of differential equations and the calculus of variations. The second ingredient is a reformulation of Cartan’s method of moving frames, [17, 51]. For a general finite-dimensional transformation group G, a moving frame is defined as an equivariant map from an open subset of jet space to the Lie group G. Moving frames are constructed by the process of normalization based on the choice of cross-section to the ∗

E-mail address: [email protected],www.math.umn.edu/ olver

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Peter J. Olver

group orbits. The moving frame then provides a canonical mechanism, called invariantization, that allows us to systematically construct the invariant counterparts of all objects of interest in the usual variational bicomplex, including differential invariants, invariant differential forms, invariant differential operators, etc. The key recurrence formulae relate the differentials of ordinary functions and forms to the invariant differentials of invariant functions and forms, and thereby lead to the complete structure of the algebra1 of differential invariants, including the syzygies and commutation formulae, [26, 27, 52]. The equivariant moving frame method has impacted a remarkable range of subjects, including symmetry methods for partial differential equations, the calculus of variations, classical invariant theory, computer vision, numerical analysis, Hamiltonian systems, integrable soliton equations, materials and micromagnetics, joint invariants, relativity, quantum mechanics, invariants of Lie algebras, Lie pseudo-groups, symbolic methods, and (non-commutative) differential algebra; see [50, 51] for recent surveys of developments in the field. A key application of the invariant variational bicomplex is the general solution to an outstanding problem in the calculus of variations. Every group-invariant variational problem can be written in terms of the differential invariants. The associated Euler-Lagrange equations inherit the symmetry group, and so can also be written in terms of the differential invariants. The problem is to directly construct the invariant form of the Euler–Lagrange equations from the invariant form of the variational problem. Before the general solution to this problem appeared in [31], only a few specific examples were known, [3, 22]. A striking recent application of these techniques is the work of Starostin and van der Heijden, [65], on equilibrium configurations of flexible M¨obius bands. A second application is to the evolution of differential invariants under invariant submanifold flows. Invariant curve flows and surface flows arise in an impressive range of applications, including geometric optics, [7], elastodynamics, [37], computer vision, [55, 56, 60, 62, 64], visual tracking and control, [45], vortex dynamics, [25, 36], interface motions, [64], thermal grooving, [9], and elsewhere. A celebrated example is the Euclidean invariant curve shortening flow, [18, 20], in which a plane curve moves in its normal direction in proportion to its curvature. In computer vision, Euclidean curve shortening and its equi-affine counterpart have been successfully applied to image denoising and segmentation, [55, 61, 62]. In three dimensional space, Euclidean-invariant curve flows include the integrable vortex filament flow, [25, 36], while mean curvature and Willmore flows of surfaces have been the subject of extensive analysis and applications, [6, 14]. Given an invariant submanifold flow, a key issue is to track the induced evolution of its basic geometric invariants — curvature, torsion and the like. While a number of particular examples have been worked out by direct computation, e.g., in [18, 43], many cases of interest have yet to appear in the literature, owing to their computational complexity. Therefore, it is worth developing general, practical tools to ameliorate this often tedious task. Mansfield and van der Kamp, [39], have developed a method based on the differential invariant syzygies. Here we present a direct approach, applying the invariant variational bicomplex calculus discussed above. As we will see, the same basic invariant differential operators appearing in the construction of invariant Euler–Lagrange equations also play a 1

Technically, because differential invariants may only be locally defined, we should speak of the “sheaf of differential invariants”. However, as we work locally on suitable open subsets, this extra level of abstraction is not required; moreover, experts can readily translate our constructions into sheaf-theoretic language, [70].

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key role in this context.

2.

The Invariant Variational Bicomplex

In this section, we review the basics of prolonged group actions on submanifold jets, moving frames, and the induced invariant variational bicomplex. Basic references include [48, 49] for jets, contact forms, and prolonged Lie group actions, [3, 66] for the variational bicomplex, [17, 51, 52] for the equivariant approach to moving frames, and [31] for the moving frame construction of the invariant variational bicomplex. For simplicity, we will only deal with finite-dimensional Lie group actions in this paper, although the general ideas can be straightforwardly adapted to infinite-dimensional pseudo-group actions using more recent extensions of the moving frame technology, [54]. Let G be an r-dimensional Lie group, acting smoothly on a m-dimensional manifold M . We will study the induced action on p-dimensional submanifolds S ⊂ M . For 0 ≤ n ≤ ∞, let Jn = Jn (M, p) denote the n-th order (extended) jet bundle for such submanifolds, [49]. The action of G on M naturally prolongs to an action on Jn . Since the prolonged group actions are all mutually compatible under projection Jn → Jk , we will avoid explicit reference to the order of prolongation, and just use g · z (n) for the action of g ∈ G on the jet z (n) ∈ Jn , rather than the more traditional notation g (n) · z (n) . By definition, a moving frame is right-equivariant2 map3 ρ : Jn → G, meaning that ρ(g · z (n) ) = ρ(z (n) ) · g −1 for all g ∈ G and all z (n) ∈ Jn where defined. The existence of a moving frame requires that the prolonged group action be free, meaning the isotropy subgroups of each individual jet are trivial, and regular, meaning the prolonged group orbits form a regular foliation, on an open subset V ⊂ Jn . Under these conditions, a moving frame can be algorithmically constructed by a normalization process based on the choice of a compatible cross-section K n ⊂ Jn to the group orbits. Specifically, given z (n) ∈ Jn , we set g = ρ(z (n) ) to be the unique group element such that g · z (n) ∈ K n , when defined. Compatibility of moving frames under the jet space projections allows us to also suppress the order in the notation of ρ. We use ι to denote the invariantization process induced by the moving frame. The invariantization of a differential form Ω is the unique invariant differential form ι(Ω) that agrees with Ω when restricted to the cross-section. In particular, if Ω is an invariant differential form or function, then ι(Ω) = Ω. Invariantization defines an (exterior) algebra morphism that projects differential functions and forms on Jn to invariant differential functions and forms. Let (x, u) = (x1 , . . . , xp , u1 , . . . , uq ) be local coordinates on M . Viewing the x’s as independent variables and the u’s as dependent variables, we let uαJ = ∂ #J u/∂xJ be the usual induced local coordinates on Jn . Separating the local coordinates (x, u) on M into independent and dependent variables naturally splits the differential one-forms on J∞ into horizontal forms, spanned by dx1 , . . . , dxp , and vertical forms, spanned by the basic 2 All classical moving frames, [23], are left-equivariant, and can be obtained by composing ρ with the group inversion g 7→ g −1 . We choose to concentrate on the right-equivariant version to (slightly) simplify some of the calculations. 3 All maps, differential forms, differential functions, etc., need only be locally defined; thus, the domain of ρ is typically a suitable open subset of Jn .

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Peter J. Olver

contact one-forms θJα = duαJ −

p X

uαJ,i dxi ,

α = 1, . . . , q,

i=1

#J ≥ 0.

(1)

Let πH and πV denote the projections mapping one-forms on J∞ to their horizontal and vertical (contact) components, respectively. The induced splitting d = dH + dV of the differential into horizontal and vertical components results in the variational bicomplex4 . In particular, if F (x, u(n) ) is any differential function, its horizontal and vertical differentials are dH F =

p X

(Di F ) dxi ,

dV F = DF (θ) =

i=1

X ∂F X ∂F DJ θ α = θα , α ∂uJ ∂uαJ J

(2)

α,J

α,J

in which Di = Dxi denote the total derivative operators with respect to the independent variables, DJ = Dj1 · · · Djk are the higher order total derivatives, θ = (θ1 , . . . , θq )T is the column vector containing the order zero contact forms, while DF = (DF,1 , . . . , DF,q ) is the Fr´echet derivative or formal linearization of the differential function F . We will employ our moving frame to invariantize the variational bicomplex as follows. First, invariantization of the jet coordinate functions produces the fundamental differential invariants: H i = ι(xi ),

IJα = ι(uαJ ),

α = 1, . . . , q,

#J ≥ 0.

(3)

These naturally split into two classes: The r = dim G combinations defining the crosssection equations will be constant, and are known as the phantom differential invariants. The remainder, called the basic differential invariants, form a complete system of functionally independent differential invariants. Next, let ̟i = ω i + η i = ι(dxi ),

where

ω i = πH (̟i ), η i = πV (̟i ),

(4)

denote the invariantized horizontal one-forms. Their horizontal components ω 1 , . . . , ω p form, in the language of [49], a contact-invariant coframe for the prolonged group action, while η 1 , . . . , η p supply “contact corrections” that make the one-forms ̟1 , . . . , ̟p fully G-invariant. The corresponding dual invariant total differential operators D1 , . . . , Dp are defined so that H i = ι(xi ),

IJα = ι(uαJ ),

α = 1, . . . , q,

#J ≥ 0.

(5)

for any differential function F and, more generally, differential form Ω, on which the Di act via Lie differentiation. Finally, let ϑαJ = ι(θJα ),

α = 1, . . . , q,

#J ≥ 0.

(6)

4 Since the splitting depends on a choice of independent variables on M , the variational bicomplex is not intrinsic. A more refined version of this construction, known as the C spectral sequence, [68, 69], relies on the contact filtration of the algebra of differential forms. However, since all our calculations take place in local coordinates, we will avoid all the extra complications inherent in this more sophisticated machinery. Experts will be able to readily translate our results as desired.

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be the invariantized basis contact forms. As in the usual, non-invariant bicomplex construction, the decomposition of invariant one-forms on J∞ into invariant horizontal and invariant contact components induces a decomposition of the differential. However, now d = dH + dV + dW splits into three constituents, where dH adds an invariant horizontal form, dV adds a invariant contact form, while dW replaces an invariant horizontal one-form with a combination of wedge e r,s denote the space products of two invariant contact forms. In other words, if we let Ω of differential forms of degree r + s spanned by wedge products of r invariant horizontal one-forms (4) and s invariant contact one-forms (6), then e r,s −→ Ω e r+1,s , dH : Ω

e r,s −→ Ω e r,s+1 , dV : Ω

e r,s −→ Ω e r−1,s+2 . (7) dW : Ω

The resulting invariant variational quasi-tricomplex is characterized by the formulae d2H = 0,

dH dV + dV dH = 0,

d2W

dV dW + dW dV = 0,

= 0,

d2V + dH dW + dW dH = 0.

(8)

Fortunately, the third, anomalous component dW plays no role in the applications; in particular, dW F = 0 for any differential function F . The most important fact underlying the moving frame construction is that the invariantization map ι does not respect the exterior derivative operator. Thus, in general, d ι(Ω) 6= ι(dΩ). The recurrence formulae, [17, 31], which we now review, provide the missing “correction terms” dι(Ω) − ι(dΩ). Remarkably, these formulas can be explicitly and algorithmically constructed using only linear differential algebra — without knowing the explicit formulas for either the differential invariants or invariant differential forms, the invariant differential operators, or even the moving frame! The only required ingredients are the cross-section equations and the formulae for the prolonged infinitesimal generators of the group action. Let v1 , . . . , vr be a basis for the infinitesimal generators of our transformation group. We prolong each infinitesimal generator to Jn . For conciseness, we will retain the same notation vκ for the prolonged vector fields on any Jn which, in local coordinates, take the form p q n X X X ∂ ∂ i ξκ (x, u) i + vκ = ϕαJ,κ (x, u(j) ) α , κ = 1, . . . , r. (9) ∂x ∂uJ α=1 j=#J=0

i=1

The coefficients ϕαJ,κ = vκ (uαJ ) can be successively constructed by Lie’s recursive prolongation formula, [48, 49]: ϕαJi,κ = Di ϕαJ,κ −

p X

uαJj Di ξκj .

(10)

j=1

A straightforward induction establishes the explicit prolongation formula, first written down by the author in [47]: ϕαJ,κ

=

DJ Qακ

+

p X

ξκi uαJ,i ,

where

i=1

are the components of the characteristic of vκ .

Qακ

=

ϕακ

−

p X i=1

ξκi uαi

(11)

214

Peter J. Olver

Strikingly, all the recurrence relations are consequences of a single universal recurrence formula that prescribes the differential of an invariantized differential function or form. Theorem 1. If Ω is any differential form on J∞ , then d ι(Ω) = ι(d Ω) +

r X κ=1

ν κ ∧ ι [vκ (Ω)],

(12)

where ν 1 , . . . , ν r

are the invariantized Maurer–Cartan forms dual to the infinitesimal generators v1 , . . . , vr , while vκ (Ω) denotes the Lie derivative of Ω with respect to the prolonged infinitesimal generator vκ . The invariantized Maurer–Cartan forms ν 1 , . . . , ν r are obtained by pulling back the usual dual Maurer–Cartan forms µ1 , . . . , µr on G by the moving frame map: ν κ = ρ∗ µκ . Details would take us too far afield, [31], but, fortunately, are not required thanks to the following marvelous result that allows us to compute them directly without reference to their underlying definition: Lemma 2. Let I1 = ι(z1 ), . . . , Ir = ι(zr ) be the phantom differential invariants stemming from our cross-section. Then the corresponding phantom recurrence formulae 0 = dIς = d ι(zς ) = ι(dzς ) +

r X κ=1

ν κ ∧ ι [vκ (zς )],

ς = 1, . . . , r,

(13)

can be uniquely solved for the invariantized Maurer–Cartan forms ν 1 , . . . , ν r . Having solved the linear system (13) for ν 1 , . . . , ν r , we then decompose the resulting invariantized Maurer–Cartan forms into their invariant horizontal and contact components: κ

κ

κ

ν =γ +ε ,

κ

γ =

where

p X

Riκ ̟i ,

εκ =

i=1

X

Sακ,J ϑαJ ,

(14)

α,J

where Riκ , Sακ,J are certain differential invariants. The Riκ will be called the Maurer–Cartan invariants, [26, 27, 52]. In the case of curves, the Riκ appear as the entries of the Frenet– Serret matrix Dρ(x, u(n) ) · ρ(x, u(n) )−1 , in the case G ⊂ GL(N ) is a matrix Lie group, [23]. Substituting (14) back into the universal formula (12) produces a complete system of explicit recurrence relations for all the differentiated invariants and invariant differential forms. In particular, taking Ω to be any one of the individual jet coordinate functions xi , uαJ , results in the recurrence formulae for the fundamental differential invariants (3): dH i = ι(dxi ) + dIJα

=

ι(duαJ )

=

p X

r X

κ=1 r X

+

ν κ ι [vκ (xi )] = ̟i + ν

κ

ι [vκ (uαJ )]

κ=1

i=1

α IJi ̟i + ϑαJ +

=ι

r X

κ=1 p X

uαJi dxi

i=1

r X κ=1

ι(ξκi ) ν κ ,

ι(ϕαJ,κ ) ν κ .

+

θJα

!

+

r X κ=1

ι(ϕαJ,κ ) ν κ

(15)

Invariant Variational Problems and Invariant Flows via Moving Frames

215

In view of (14), the coefficient of ̟i in (15) yields the recurrence relations j

Di H =

δij

+

r X

Riκ ι(ξκi ),

Di IJα

κ=1

=

α IJi

+

r X

Riκ ι(ϕαJ,κ ),

(16)

κ=1

where δij is the usual Kronecker delta. Owing to the functional independence of the basic (non-phantom) differential invariants, these formulae, in fact, serve to completely characterize the structure of the non-commutative differential invariant algebra, [17, 26, 52]. Similarly, the contact components in (15) yield the vertical recurrence formulae dV H i =

r X

ι(ξκi ) εκ ,

dV IJα = ϑαJ +

κ=1

r X

ι(ϕαKκ ) εκ ,

(17)

κ=1

while, as noted earlier, dW H i = dW IJα = 0. The recurrence formulae (12) for the derivatives of the invariant horizontal forms are i

i

2 i

d̟ = d[ι(dx )] = ι(d x ) +

r X κ=1

p X

ν κ ∧ ι [vκ (dxi )]

! q i X ∂ξ κ α = νκ ∧ ι Dk ξκi dxk + θ ∂uα κ=1 α=1 k=1 i p q r X r X X X κ ∂ξκ i k = ι Dk ξκ ν ∧ ̟ + ν κ ∧ ϑα . ι ∂uα r X

κ=1 k=1

(18)

κ=1 α=1

The resulting two-form can be decomposed into three basic constituents, belonging, ree 2,0 ⊕ Ω e 1,1 ⊕ Ω e 0,2 . In view of (14), the terms in (18) spectively, to the invariant summands Ω e 2,0 , yield involving wedge products of two horizontal forms, i.e., in Ω X i dH ̟ i = − Yjk ̟j ∧ ̟k , j
where i Yjk =

p r X X κ=1 j=1

Rjκ ι(Dj ξκi ) − Rkκ ι(Dk ξκi )

(19)

are called the commutator invariants, since combining (19) with (5) produces the commutation formulae for the invariant differential operators: [ Dj , Dk ] =

p X i=1

i Yjk

Di = −

p X i=1

i Ykj Di .

(20)

Next, the terms in (18) involving wedge products of a horizontal and a contact form yield " q # p r X X X ∂ξκi i κ α i κ k dV ̟ = ι γ ∧ϑ + ι(Dk ξκ ) ε ∧ ̟ . (21) ∂uα κ=1

α=1

k=1

216

Peter J. Olver

Finally, the remaining terms, involving wedge products of two contact forms, provide the formulas for the anomalous third component of the differential: i q r X X ∂ξκ i dW ̟ = ι εκ ∧ ϑ α . (22) ∂uα κ=1 α=1

In a similar fashion, we derive the recurrence formulae (12) for the differentiated invariant contact forms: ! p r r X X X α α α κ α i α α dϑJ = d[ι(θJ )] = ι(dθJ ) + ν ∧ ι [vκ (θJ )] = ι dx ∧ θJi + ν κ ∧ ι(ψJκ ), κ=1

κ=1

i=1

(23)

where α ψJκ

=

vκ (θJα )

=

dϕαJκ

−

p X

ϕαJiκ dxi

+

uαJi dξκi

i=1

=

dV ϕαJκ

−

p X

uαJi dV ξκi

(24)

i=1

is known as the vertical prolongation coefficient of the vector field vκ . For our purposes, we only require the component of (23) that involves invariant horizontal forms: dH ϑαJ

=

p X i=1

Since5

i

̟ ∧

dH ϑ =

ϑαJi

+

r X κ=1

p X i=1

α γ κ ∧ ι(ψJκ ).

̟i ∧ Di ϑ

(25)

(26)

for any contact form ϑ, we deduce the recurrence formulae Di ϑαJ = ϑαJi +

r X

α Riκ ι(ψJκ )

(27)

κ=1

for the invariant (Lie) derivatives of the invariant contact forms. The latter can inductively be solved to express the higher order invariantized contact forms as certain invariant derivatives of those of order 0: q X α α ϑJ = EJ,β (ϑβ ) = EJα (ϑ), (28) β=1

in which ϑ = (ϑ1 , . . . , ϑq )T denotes the column vector containing the order zero invariantized contact forms, while EJα = (EJα , . . . , EJα ) are certain invariant differential operators. In view of (17, 28), if K = K(. . . H i . . . IJα . . .) is any differential invariant, we can write its invariant vertical derivative in the form p q X X ∂K X ∂K i α dV K = d H + d I = A (ϑ) = AK,α (ϑα ), (29) V V J K ∂H i ∂IJα i=1

α,J

α=1

in which AK = (AK,1 , . . . , AK,q ) is a row vector of invariant differential operators. We view (29) as the invariant version of the vertical differentiation formula dV F = DF (θ), cf. (2), which motivates the following terminology. 5

Warning: The analogous formula is not valid for horizontal forms.

Invariant Variational Problems and Invariant Flows via Moving Frames

217

Definition 3. The invariant linearization of a differential invariant K is the invariant differential operator AK that satisfies dV K = AK (ϑ). Remark. In [31], AK was called the Eulerian operator associated with K owing to its appearance in the differential invariant form of the Euler–Lagrange equations for an invariant variational problem; see Theorem 5 below. Similarly, we combine (14), (21), and (28), to produce formulae dV ̟ i =

p X q X

j=1 α=1

i Bjα (ϑα ) ∧ ̟j =

p X j=1

Bji (ϑ) ∧ ̟j

(30)

i , . . . , Bi ) for the vertical differentials of the invariant horizontal forms, in which Bji = (Bj1 jq 2 is a family of p row-vector-valued invariant differential operators, known, collectively, as the invariant Hamiltonian operator complex, cf. [58], again stemming from its role in the invariant calculus of variations.

Example 4. The Euclidean geometry of plane curves is governed by the standard action y = x cos φ − u sin φ + a, v = x sin φ + u cos φ + b, of the proper Euclidean group g = (φ, a, b) ∈ SE(2) on M = R 2 . The prolonged group transformations are constructed by applying the implicit differentiation operator Dy = (cos φ − ux sin φ)−1 Dx to v, and so vy =

sin φ + ux cos φ , cos φ − ux sin φ

vyy =

uxx , (cos φ − ux sin φ)3

etc.

Solving the normalization equations y = v = vy = 0 for the group parameters produces the right moving frame φ = − tan−1 ux ,

x + uux a=−p , 1 + u2x

xux − u b= p . 1 + u2x

(31)

(The classical moving frame, [23], is the left counterpart obtained by inverting the group element given in (31).) Invariantization of the coordinate functions, which is done by substituting the moving frame formulae into the prolonged group transformations, produces the fundamental normalized differential invariants ι(x) = H = 0,

ι(u) = I0 = 0,

ι(ux ) = I1 = 0,

ι(uxx ) = I2 = κ,

ι(uxxx ) = I3 = κs ,

ι(uxxxx ) = I4 = κss + 3κ3 ,

and so on. The first three, arising from the normalizations, are called phantom invariants. The lowest order non-trivial differential invariant is the Euclidean curvature I2 = κ = uxx (1 + u2xp )3/2 , while κs , κss , . . . denote the derivatives of κ with respect to the arc-length form ω = 1 + u2x dx. The invariant horizontal one-form dx + ux du p ux ̟ = ι(dx) = p = 1 + u2x dx + p θ 2 1 + ux 1 + u2x

(32)

is a sum of the contact-invariant arc length form along with a contact correction. In the same manner we obtain the basis invariant contact forms θ ϑ = ι(θ) = p , 1 + u2x

ϑ1 = ι(θx ) =

(1 + u2x ) θx − ux uxx θ , (1 + u2x )2

...

. (33)

218

Peter J. Olver

To obtain the explicit recurrence formulae, we begin with the prolonged infinitesimal generators of SE(2): v1 = ∂x ,

v2 = ∂u ,

v3 = −u ∂x + x ∂u + (1 + u2x ) ∂ux + 3ux uxx ∂uxx + · · · .

The one-forms γ κ , εκ governing the correction terms are found by applying the recurrence formulae (12) to the phantom invariants. From the first equation in (12), we obtain 0 = dH H = ι(dH x) + ι(v1 (x)) γ 1 + ι(v2 (x)) γ 2 + ι(v3 (x)) γ 3 = ̟ + γ 1 , 0 = dH I0 = ι(dH u) + ι(v1 (u)) γ 1 + ι(v2 (u)) γ 2 + ι(v3 (u)) γ 3 = γ 2 , 0 = dH I1 = ι(dH ux ) + ι(v1 (ux )) γ 1 + ι(v2 (ux )) γ 2 + ι(v3 (ux )) γ 3 = κ ̟ + γ 3 , and hence γ 1 = −̟, γ 2 = 0, γ 3 = −κ ̟. Similarly, applying dV to the phantom invariants and using the second equation in (12) yields ε1 = 0, ε2 = −ϑ, ε3 = −ϑ1 . We are now ready to substitute the non-phantom invariants into (12). The horizontal differentials dH Ik of the normalized differential invariants In = ι(un ) are used to produce the explicit recurrence formulae κ = I2 ,

κs = DI2 = I3 ,

κss = DI3 = I4 − 3I23 ,

...

relating them to the differentiated invariants Dm κ. Similarly, the second equation in (12) gives the vertical differential dV I2 = dV κ = ι(θ2 ) + ι(v3 (uxx )) ε3 = ϑ2 = (D2 + κ2 ) ϑ,

(34)

where the final equation follows from the invariant contact form recurrence formulae Dϑ = ϑ1 , Dϑ1 = ϑ2 − κ2 ϑ, which are found by applying dH to the invariant contact forms and using the first equation in (12). Thus, we deduce the following invariant linearization operators: Aκ = D2 + κ2 , Aκs = D3 + κ2 D + 3 κ κs , (35) Aκss = D4 + κ2 D2 + 5 κ κs D + 4 κ κss + 3 κ2s , etc. In fact, one can recursively construct the higher order operators starting with Aκ via Aκn = D · Aκn−1 + κ κn ,

(36)

where κn = Dn κ. Finally, applying the second formula in (12) to ̟ yields dV ̟ = − κ ϑ ∧ ̟, and hence the invariant Hamiltonian operator is B = − κ.

3.

(37)

Invariant Variational Problems

We now apply our construction to derive the formulae for the Euler-Lagrange equations associated with an invariant variational problem. Let us recall the variational bicomplex construction of the Euler-Lagrange equations.

Invariant Variational Problems and Invariant Flows via Moving Frames 219 R A variational problem I[ u ] = L[ u ] dx is determined by the Lagrangian form λ = L[ u ] dx ∈ Ωp,0 . Its differential dλ = dV λ ∈ Ωp,1 defines a form of type (p, 1). We introduce an equivalence relation on such forms, so that Θ ∼ Ω if and only if Θ = Ω+ dH Ψ for some Ψ ∈ Ωp−1,1 . The quotient space F 1 = Ωp,1 / ∼ is known as the space of source forms. Integration by parts proves that every source form has a canonical representative Pq (n) ) θ α ∧ dx, and so can be identified with a q-tuple of differential functions ∆ α (x, u α=1 ∆ = (∆1 , . . . , ∆q ). In applications, a source form is regarded as defining a system of q differential equations ∆1 = · · · = ∆q = 0 for the q dependent variables u = (u1 , . . . , uq ). Composing the differential d : Ωp,0 → Ωp,1 with the projection π∗ : Ωp,1 → F 1 produces the variational differential δ = π∗ ◦ d that takes a Lagrangian form λ = L[ u ] dx to its variational derivative source form δλ ≃

q X

α=1

Eα (L) θα ∧ dx,

where

Eα (L) =

X

(−D)J

J

∂L ∂uαJ

(38)

are the classical Euler-Lagrange expressions for the Lagrangian L. According to Lie, [38, 49], as long as we work on the open subset V ⊂ Jn where G acts regularly and freely, any G-invariant variational problem is given by an invariant e ω, where ω = ω 1 ∧ · · · ∧ ω p is the contact-invariant volume form, Lagrangian form λ = L e is an arbitrary differential invariant, and hence a function and the invariant Lagrangian L of the fundamental differential invariants I 1 , . . . , I l and their invariant derivatives DJ I α . The associated Euler-Lagrange equations E(L) = 0 admit G as a symmetry group, and so, under suitable nondegeneracy hypotheses, [49, Theorem 6.25], can themselves be written in terms of the differential invariants. The main problem is to go directly from the differential invariant formula for the variational problem to the differential invariant formula for the Euler-Lagrange equations. Not surprisingly, the required calculations rely on an invariant version of integration by parts. For this purpose, given invariant differential forms α, β, let us write α ≡ β whenever α = β + dH γ. If F is a differential function and ψ is a contact one-form, then6 dH F =

p X

(Di F ) ̟

i

dH ψ =

i=1

p X i=1

̟i ∧ Di ψ.

(39)

e ω by the To effect the computation, we begin by replacing our Lagrangian λ = L e e fully invariant version λ = L ̟, noting that, since they differ by contact forms, they have identical Euler–Lagrange equations. We then compute e= dV λ

Introduce the (p − 1)–forms ̟ (i) = Di 6

X ∂L e α e α dV I,K ∧ ̟ + L dV ̟, ∂I,K α,K

e p−1,0 . ̟ = (−1)i−1 ̟1 ∧ · · · ∧ ̟i−1 ∧ ̟i+1 ∧ · · · ∧ ̟p ∈ Ω

Warning: The second identity is not true for a general one-form.

(40)

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If F is any differential function and ψ any contact one-form, then dH (F ψ ∧ ̟ (i) ) = dH F ∧ ψ ∧ ̟ (i) + F dH ψ ∧ ̟ (i) − F ψ ∧ dH ̟ (i) .

(41)

e p,0 , it must be a multiple of the invariant volume form, and we write Since dH ̟ (i) ∈ Ω dH ̟ (i) = Zi ̟, where Z1 , . . . , Zp are certain differential invariants, which we will call the twist invariants. Using (39) we can rewrite (41) as F dH ψ ∧ ̟ (i) = F (Di ψ) ∧ ̟ ≡ − (Di + Zi )F ψ ∧ ̟ = (Di† F ) ψ ∧ ̟, (42)

where Di† = − (Di + Zi ) is called the twisted invariant adjoint of the invariant differential operator Di . If we choose ψ = dV H where H is a differential function, then (42) results in the multivariate invariant integration by parts formula F d(Di H) ∧ ̟ = (Di† F ) dV H ∧ ̟ −

p X j=1

F (Dj H) dV ̟j ∧ ̟ (i) .

(43)

We use (43) repeatedly to integrate the first term of (40) by parts, leading to

where e = Eα (L)

e≡ dV λ X K

q X

α=1

DK†

e dV I α ∧ ̟ − Eα (L)

e ∂L α , ∂I,K

p X i=1

e = −L e δji + Hji (L)

e dV ̟j ∧ ̟ (i) , Hji (L) q X X

α=1 J,K

α I,J,j DK†

e ∂L , α ∂I,J,i,K

(44)

(45)

are, respectively, the invariant Eulerian and invariant Hamiltonian tensor of the invariant e In (45), we use the twisted adjoints Lagrangian L. DK† = Dk†1 · · · Dk†m = (−1)m (Dk1 + Zk1 ) · · · (Dkm + Zkm ),

K = (k1 , . . . , km ),

of the repeated invariant differential operators. The second phase of the computation requires the vertical differentiation formulae α

dV I =

q X

β=1

Aαβ (ϑβ ),

j

dV ̟ =

q X

β=1

j Bi,β (ϑβ ) ∧ ̟i ,

(46)

where A = Aαβ denotes the Eulerian operator, which is an m × q matrix of invariant differential operators whose rows are the invariant linearizations of the fundamental differj j 1 l 2 ential invariants I , . . . , I , while the p row vectors Bi = Bi,β of invariant differential operators form the invariant Hamiltonian operator complex. This allows us to write (44) in the vectorial form e ≡ E(L) e A(ϑ) ∧ ̟ − dV λ

p X

i,j=1

e B j (ϑ) ∧ ̟. Hji (L) i

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We now apply (42) to further integrate both terms by parts. The final result is written in terms of twisted adjoints of the Eulerlian and Hamiltonian operators,   p X e ≡ δλ e =  A † E(L) e − e  ϑ ∧ ̟. dV λ (Bij ) † Hji (L) i,j=1

e = Theorem 5. The Euler-Lagrange equations of the invariant Lagrangian form λ (n) e L(I ) ̟ have the following invariant form: †

e − A E(L)

p X

i,j=1

e = 0. (Bij ) † Hji (L)

(47)

In the case of curves, when p = 1, there are no twist invariants, and so the general formula (47) reduces to e − B ∗ H(L) e = 0, A∗ E(L) (48) where A∗ and B ∗ are the ordinary formal adjoints of the invariant Eulerian and Hamiltonian operators, respectively.

Example 6. In the context of the Euclidean group acting on plane curves in Example 4, any Euclidean-invariant variational problem corresponds to a contact invariant Lagrangian e κs , κss , . . .) ω. Both the Eulerian operator (35) and the Hamiltonian operator λ = L(κ, (37) are invariantly self-adjoint: A = A∗ and B = B ∗ . Thus, the invariant Euler-Lagrange formula (48) reduces to the known formula, [3, 22], e + κ H(L) e =0 (D2 + κ2 ) E(L)

for the Euclidean-invariant Euler-Lagrange equation.

Example 7. Consider the standard action of the Euclidean group SE(3) on surfaces S ⊂ R 3 . We assume that the surface is parametrized by z = (x, y, u(x, y)), noting that the final formulae are, in fact, parameter-independent. The classical (local) left moving frame ρ(x, u(2) ) = (R, a) ∈ SE(3) consists of the point on the curve defining the translation component a = z, while the columns of the rotation matrix R contain the unit tangent vectors forming the Frenet frame along with the unit normal to the surface. The fundamental differential invariants are the principal curvatures κ1 = ι(uxx ), κ2 = ι(uyy ). The mean and Gaussian curvature invariants H = 21 (κ1 + κ2 ), K = κ1 κ2 , are often used as convenient alternatives, since they eliminate some of the residual discrete ambiguities in the moving frame. Higher order differential invariants are obtained by repeatedly applying the dual invariant differential operators D1 , D2 associated with the diagonalizing Frenet coframe ̟1 = ι(dx1 ), ̟2 = ι(dx2 ). The resulting differentiated invariants are not functionally independent, owing to the Codazzi identity

κ1,22 − κ2,11 +

κ1,1 κ2,1 + κ1,2 κ2,2 − 2(κ2,1 )2 − 2(κ1,2 )2 − κ1 κ2 (κ1 − κ2 ) = 0. κ1 − κ2

(49)

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The Codazzi syzygy can, in fact, be directly deduced from our infinitesimal moving frame computations by comparing the recurrence formulae for κ1,22 and κ2,11 with the normalized invariant ι(uxxyy ). Any Euclidean-invariant variational problem has the form Z e (n) ) ω 1 ∧ ω 2 , L(κ where ω 1 ∧ ω 2 = π2,0 (̟1 ∧ ̟2 )

e is an arbitrary differis the usual intrinsic surface area 2-form. The invariant Lagrangian L ential invariant, and so can be rewritten in terms of the principal curvature invariants and their derivatives, or, equivalently, in terms of the Gaussian and mean curvatures. The former representation leads to simpler formulae and will be retained. Since dH ̟ (1)

κ2,1 = dH ̟ = 1 ̟, κ − κ2 2

dH ̟ (2)

κ1,2 = − dH ̟ = 2 ̟, κ − κ1 1

the twist invariants are Z1 =

κ2,1 , κ1 − κ2

Z2 =

κ1,2 . κ2 − κ1

These invariants appear in Guggenheimer’s proof of the fundamental existence theorem for Euclidean surfaces, [23, p. 234]. The denominator vanishes at umbilic points on the surface, where the moving frame is not valid. The Codazzi syzygy (49) can be written compactly as K = κ1 κ2 = D1† (Z1 ) + D2† (Z2 ) = − (D1 + Z1 )Z1 − (D2 + Z2 )Z2 , which expresses the Gaussian curvature K as an invariant divergence. This fact lies at the heart of the Gauss–Bonnet Theorem. The invariant vertical derivatives of the principal curvatures are straightforwardly determined from (12), dV κ1 = ι(θxx ) = D12 + Z2 D2 + (κ1 )2 ϑ, dV κ2 = ι(θyy ) = D22 + Z1 D1 + (κ2 )2 ϑ,

where ϑ = ι(θ) = ι(du − ux dx − u y dy) is the fundamental invariant contact form. D12 + Z2 D2 + (κ1 )2 . Further, Therefore, the Eulerian operator is A = D22 + Z1 D1 + (κ2 )2 D1 D2 − Z2 D1 ϑ ∧ ̟2 1 1 1 dV ̟ = − κ ϑ ∧ ̟ + , κ1 − κ2 D2 D1 − Z1 D2 ϑ ∧ ̟1 dV ̟ 2 = − κ2 ϑ ∧ ̟2 , κ2 − κ1 which yields the Hamiltonian operator complex

B21 =

κ1

1 − κ2

B11 = − κ1 ,

B22 = − κ2 , 1 D1 D2 − Z2 D1 = 1 D2 D1 − Z1 D2 = − B12 . 2 κ −κ

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Therefore, according to our formula (47), the Euler-Lagrange equation for a Euclideaninvariant variational problem is e 0 = (D1 + Z1 )2 − (D2 + Z2 ) · Z2 + (κ1 )2 E1 (L) e + κ1 H11 (L) e + κ2 H22 (L) e + (D2 + Z2 )2 − (D1 + Z1 ) · Z1 + (κ2 )2 E2 (L) ! e − H2 (L) e H21 (L) 1 + (D2 + Z2 )(D1 + Z1 ) + (D1 + Z1 ) · Z2 · . 1 2 κ −κ e are the invariant Eulerians with respect to the principal curvatures κα , As before, Eα (L) i e are the invariant Hamiltonians. In particular, if L(κ e 1 , κ2 ) does not depend on while Hj (L) any differentiated invariants, the Euler-Lagrange equation reduces to

(D1† )2 + D2† · Z2 + (κ1 )2

e ∂L † 2 e † 2 2 ∂L e = 0. + (D ) + D · Z + (κ ) − (κ1 + κ2 )L 1 2 1 ∂κ1 ∂κ2

e = 1, and For example, the problem of minimizing surface area has invariant Lagrangian L 1 2 so has the well-known Euler-Lagrange equation E(L) = − (κ + κ ) = − 2H = 0, and hence minimal surfaces have vanishing mean curvature. The mean curvature Lagrangian e = H = 1 (κ1 + κ2 ) has Euler-Lagrange equation L 2 1 2 2 2 1 2 2 1 = − κ1 κ2 = −K = 0. 2 (κ ) + (κ ) − (κ + κ ) e = 1 (κ1 )2 + 1 (κ2 )2 , [3, 8], the Euler-Lagrange equation is For the Willmore Lagrangian L 2 2 0 = E(L) = ∆(κ1 + κ2 ) + 12 (κ1 + κ2 )(κ1 − κ2 )2 = 2 ∆H + 4 (H 2 − K)H,

where ∆ = (D1 + Z1 )D1 + (D2 + Z2 )D2 = − D1† · D1 − D2† · D2 is the Laplace–Beltrami operator on our surface.

4.

Invariant Submanifold Flows

In this section, we shift our attention to invariant submanifold flows. Let us single out the m = p + q invariant one-forms ̟1 , . . . , ̟p , ϑ1 , . . . , ϑq

(50)

consisting of the invariant horizontal forms ̟i = ι(dxi ) and the order 0 invariant contact forms ϑα = ι(θα ). Each is a linear combination of the coordinate one-forms dx1 , . . . , dxp , du1 , . . . , duq on M , whose coefficients are certain n-th order differential functions, where n is the order of the underlying moving frame. Let S ⊂ M be a p-dimensional submanifold. Evaluating the coefficients of (50) on the submanifold jet (x, u(n) ) = jn S|z produces a basis for the cotangent space T ∗ M |z of the ambient manifold at z = (x, u) ∈ S, which we continue to denote by (50). By construction, the resulting cotangent space basis is equivariant under the action of G on S ⊂ M . Let t1 , . . . , tp , n1 , . . . , nq , denote the corresponding dual tangent vectors, which form a G–equivariant basis of the bundle T M → S, or frame on S. Since the contact forms

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Peter J. Olver

annihilate the tangent space to S, the vectors t1 , . . . , tp form a basis for the tangent bundle T S, while n1 , . . . , nq form a basis for the complementary G–equivariant normal bundle N S → S induced by the moving frame. In classical geometrical situations, [23], they can be identified with the classical moving frame vectors. Example 8. Let us return to the case of planar Euclidean curves C ⊂ M = R 2 . According to Example 4, the invariant coframe is given by the invariant horizontal form (32) and the order 0 invariant contact form in (33). The corresponding dual frame vectors are the usual (right-handed) Euclidean frame vectors — the unit tangent and unit normal: ∂ 1 ∂ 1 ∂ ∂ t= p + ux , n= p + . − ux (51) ∂u ∂x ∂u 1 + u2x ∂x 1 + u2x In general, let

V = V |S = VT + VN =

p X

j

I tj +

j=1

q X

J α nα

(52)

α=1

be a section of the bundle T M → S, where VT , VN denote, respectively, its tangential and normal components, while I j , J α are differential functions, depending on the submanifold jets. We will, somewhat imprecisely, refer to V as a vector field, even though it is only defined on S. Any such vector field generates a submanifold flow: ∂S = V|S(t) , ∂t

(53)

which forms an n-th order system of partial differential equations, where n refers to the larger of the order of our moving frame and the coefficients I j , J α . Assuming local existence and uniqueness, a solution S(t) to the submanifold flow equations (53) defines a smoothly varying family of p-dimensional submanifolds of M . On the other hand, one typically expects singularities to appear if the flow is continued for a sufficiently long time. The submanifold flow (53) is called G-invariant if G is a symmetry group of the partial differential equation, which requires that its coefficients I j = h V ; ̟j i, J α = h V ; ϑα i, be differential invariants. The tangential components VT do not affect the extrinsic geometry of the submanifold, but only its internal parametrization. Thus, if we are only interested in the images of S(t) under the flow, and not their underlying parametrizations, we can set VT = 0 without loss of generality. Therefore, the normal component VN =

q X

J α nα

(54)

α=1

serves to characterize the same invariant submanifold flow as V, modulo reparametrization. We will say that the vector field VN generates a normal flow, since it only moves the submanifold in its G-equivariant normal direction — as prescribed by the moving frame.

Invariant Variational Problems and Invariant Flows via Moving Frames

225

Example 9. The most well-studied are the Euclidean-invariant curve and surface flows. A plane curve flow is generated by a vector field of the form V=It+Jn

or, equivalently,

VN = J n,

(55)

if we are not concerned about the tangential component’s effect of the parametrization of the curve. In this case, n denotes (one of the two) Euclidean normals to the curve; by convention, we use the inwards normal n when the curve is closed. Particular cases include: i. V = n: this induces the geometric optics or grassfire flow, [7, 61]; ii. V = κ n: this generates the celebrated curve shortening flow, [18, 20], used to great effect in image processing, [55, 61]; iii. V = κ1/3 n: the induced flow is equivalent, modulo reparametrization, to the equiaffine invariant curve shortening flow, also effective in image processing, [4, 55, 61]; iv. V = κs n: this flow induces the modified Korteweg–deVries equation for the curvature evolution, and is the simplest of a large number of soliton equations arising in geometric curve flows, [13, 19, 42]; v. V = κss n: this flow models thermal grooving of metals, [9]. A second important class are the invariant curve flows that preserve arc length. Remarkably, in many classical geometries, certain basic intrinsic curve flows induce integrable, soliton evolutions for the differential invariants. The prototypical example is the Euclidean– invariant vortex filament flow studied by Hasimoto, [25, 35, 36]. The curvature and torsion invariants of the evolving filament satisfy an integrable dynamical system, which can be mapped to the completely integrable nonlinear Schr¨odinger equation, [1]. This led Lamb, [34], to draw attention to the surprisingly common, but still poorly understood connection between invariant curve flows and integrable soliton dynamics; since then, many other examples have been found, [5, 12, 13, 16, 19, 24, 28, 40, 41, 42, 44, 57, 59]. By “integrable”, we shall mean that the evolution equation possesses a recursion operator, [46], inducing an infinite hierarchy of higher order symmetries. However, not all induced differential invariant evolutions are integrable, and, at present, we do not understand the general conditions on the group action and invariant curve flow needed to guarantee integrability. When p = 1, there is only one independent invariant horizontal one-form ̟ = ω + η = ds + η,

(56)

whose horizontal component ω = ds can be identified with the G-invariant arc length element. Invariance requires that the Lie derivative V(ω) vanishes on the submanifold, which (because Lie derivatives preserve the contact ideal) implies the following: Lemma 10. The curve flow induced by V=It+

q X

α=1

J α nα ,

where

I = h V ; ̟ i,

J α = h V ; ϑα i,

preserves arc length if and only if the Lie derivative V(̟) is a contact form.

(57)

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Peter J. Olver

Submanifolds of dimension p ≥ 2 do not have distinguished parametrizations to play the role of the arc length parameter; this is because the invariant horizontal forms are almost never exact on the submanifold. On the other hand, the Lie derivative condition can be straightforwardly mimicked. Definition 11. The invariant submanifold flow induced by V is called intrinsic if V(̟i ) ≡ 0 for all i = 1, . . . , p. Lemma 12. If the vectorfield V defines an intrinsic flow, then it commutes with the invariant differentiations: V, Di = 0 for i = 1, . . . , p. This holds if and only if i

Dj I +

p X

i k Yjk I

+

q X

α=1

j,k=1

i Bjα (J α ) = 0.

(58)

In particular, for curve flows generated by (57), the condition (58) guaranteeing arc length preservation reduces to DI = B(J) =

q X

α=1

Bα (J α ),

(59)

where D is the arc length derivative, while B = (B1 , . . . , Bq ) is the invariant Hamiltonian operator, defined by (30). Example 13. For the Euclidean group action on plane curves, in view of (30), the condition that a curve flow generated by the vector field V = I t + J n be intrinsic is that DI = − κ J.

(60)

Most of the curve flows listed in Example 9 have non-local intrinsic counterparts owing to the non-invertibility of the arc length derivative operator on κ J. An exception is the modified Korteweg-deVries flow, where J = κs , and so I = − 12 κ2 . In general, the normal flow induced by VN = J n has a local intrinsic version if and only if E(κ J) = 0, where E is the invariantized Euler–Lagrange operator, [31]. The next result prescribes the evolution of differential invariants under general intrinsic and normal invariant submanifold flows. See [53] for the proof. Theorem 14. Let K be any differential invariant. If the submanifold flow (53) is intrinsic, then p X ∂K = V(K) = AK (J) + I i Di K. (61) ∂t i=1

If the submanifold flow (53) is normal, then ∂K = V(K) = AK (J). ∂t

(62)

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Example 15. For any of the Euclidean invariant normal plane curve flows Ct = J n listed in Example 9, we have, according to Example 4, ∂κs ∂κ = (D2 + κ2 ) J, = (D3 + κ2 D + 3 κ κs ) J, ∂t ∂t ∂κss = (D4 + κ2 D2 + 5 κ κs D + 4 κ κss + 3 κ2s ) J. ∂t

(63)

For instance, for the grassfire flow J = 1, and so ∂κ = κ2 , ∂t

∂κs = 3 κ κs , ∂t

∂κss = 4 κ κss + 3 κ2s . ∂t

(64)

The first equation immediately implies finite time blow-up at a caustic for a convex initial curve segment, where κ > 0. For the curve shortening flow, J = κ, and ∂κ = κss + κ3 , ∂t

∂κs = κsss + 4 κ2 κs , ∂t

∂κss = κssss + 5 κ2 κss + 8 κ κ2s , (65) ∂t

thereby recovering formulas used in Gage and Hamilton’s analysis, [18]; see also Mikula ˇ coviˇc, [43]. Finally, for the mKdV flow, J = κs , and Sevˇ ∂κ ∂κs = κsss + κ2 κs , = κssss + κ2 κss + 3 κ κ2s , ∂t ∂t ∂κss = κsssss + κ2 κsss + 9 κ κs κss + 3 κ3s . ∂t

(66)

Warning: Normal flows do not preserve arc length, and so the arc length parameter s will vary in time. Or, to phrase it another way, time differentiation ∂t and arc length differentiation D = Ds do not commute — as can easily be seen in the preceding examples. Thus, one must be very careful not to interpret the resulting evolutions (64–66) as partial differential equations in the usual sense. Rather, one should regard the differential invariants κ, κs , κss , . . . as satisfying an infinite dimensional dynamical system of coupled ordinary differential equations. Turning our attention to the intrinsic, arc length preserving curve flow, the complication alluded to in the preceding paragraph does not arise because, by Lemma 12, time differentiation now commutes with arc length differentiation. Substituting (59) in the formula (61): Theorem 16. Under an arc-length preserving flow, κt = Rκ (J)

where

Rκ = Aκ − κs D−1 B

(67)

is the characteristic operator associated with κ. More generally, the time evolution of κn = Dn κ is given by arc length differentiation: ∂κn /∂t = Dn Rκ (J). In this case arc length is preserved, and hence the arc length and time derivatives commute. Thus, unlike (62), the arc-length preserving flow (67) is of a more usual analytical form. However, there is a complication in that the term Z −1 κs D B(J) = κs B(J) ds (68)

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Peter J. Olver

may very well be nonlocal, and so (67) is, in general, an integro-differential equation. Note that any integration constant appearing in (68) just adds in a multiple of κs , which represents the arc length preserving tangential flow κt = κs that just serves to translate the arc length parameter: s 7→ s + c and so can be effectively ignored. Also, on a closed curve, the integral in (68) need not be periodic in s, and so one may not be able to continuously assign a uniquely determined evolution along the entire curve — although, by the preceding remarks, all such evolutions only differ by an overall translation, by an integer multiple of the total length of the curve, of the arc length parameter. In certain situations, (67) turns out to be a well-known local integrable evolution equation, and the characteristic operator R is its recursion operator! Example 17. In the case of Euclidean plane curves, the evolution of the curvature is given by κt = Rκ (J), (69) where Rκ = Aκ − κs D−1 B = D2 + κ2 + κs D−1 · κ = Ds2 + κ2 + κs Ds−1 · κ

(70)

is the modified Korteweg-deVries recursion operator, [48]. In particular, for the mKdV flow, J = κs , and (69) becomes κt = Rκ (κs ) = κsss + 32 κ2 κs , which is the modified Korteweg-deVries equation, and R is its recursion operator, [48]. On the other hand, for the grassfire flow, J = 1, and so κt = Rκ (1) = κ2 + κs Ds−1 κ. For the curve shortening flow, J = κ, and so κt = Rκ (κ) = κss + κ3 + κs Ds−1 κ2 . Finally, for the thermal grooving flow, J = κss and so κt = Rκ (κss ) = κssss + κ2 κss + κs Ds−1 κ κss . As noted above, the induced curvature flow (69) is local if and only if E(κ J) = 0, where E is the invariantized Euler operator or variational derivative, [48]. Clearly not all these local curvature flows will be integrable. Example 18. As another example, consider the action (x, u) 7−→ (α x + β u + a, γ x + δ u + b),

α δ − β γ = 1,

(71)

of the equi-affine group SA(2) = SL(2) ⋉ R2 on plane curves C ⊂ R2 . Applications to computer vision can be found, for instance, in [4, 10, 55, 60]. According to [17, 23, 31], the classical equi-affine moving frame arises from the choice of coordinate cross-section x = u = ux = 0, uxx = 1, uxxx = 0. The fundamental differential invariant is the equi-affine curvature uxx uxxxx − 35 u2xxx . (72) κ = ι(uxxxx ) = u8/3 xx

Invariant Variational Problems and Invariant Flows via Moving Frames

229

All higher order differential invariants are obtained by invariant differentiation with respect to the invariant arc length form ̟ = ι(dx) = ω + η,

where

ω = ds = u1/3 xx dx,

η=

uxxx 3 u5/3 xx

θ,

(73)

Dx being the equi-affine arc length with dual invariant differential operator D = u−1/3 xx derivative. Applying our computational algorithm, but suppressing the details, we obtain dV κ = Aκ (ϑ),

dV ̟ = B(ϑ) ∧ ̟,

where Aκ = D4 + 53 κ D2 + 53 κs D + 13 κss + 49 κ2 ,

B=

1 3

D2 − 29 κ.

The characteristic operator is Rκ = Aκ − κs D−1 B = D4 + 53 κ D2 + 43 κs D + 13 κss + 94 κ2 + 29 κs Ds−1 · κ.

(74)

As in the Euclidean action, both the Eulerian and Hamiltonian operators are invariantly selfadjoint: A = A∗ and B = B ∗ . Therefore, the Euler-Lagrange equation for an equi-affine e κs , . . .) ds takes the invariant form (48), namely, invariant Lagrangian L(κ, e − D4 + 53 κD2 + 53 κs D + 31 κss + 94 κ2 E(L)

The equi-affine arc-length functional hence the Euler-Lagrange equation is

R

1 3

e = 0. D2 − 92 κ H(L)

e = 1 has E(L) e = 0, H(L) e = −1, and ds with L

A∗ (0) − B ∗ (−1) = − 92 κ = 0.

We conclude that the minimal equi-affine curves are those with zero R equi-affine curvature – the conic sections. As another example, the variational problem κ ds has Euler-Lagrange equation A∗ (1) − B ∗ (−κ) = 23 κss + 29 κ2 = 0, the solution to which, [30], gives κ as an elliptic function of s. A general equi-affine invariant curve flow takes the form Ct = I t + J n,

(75)

where t, n are, respectively, the equi-affine tangent and normal directions, [23]. The equiaffine curve shortening flow, [4, 61], is the normal flow with I = 0, J = 1. Under this flow, the equi-affine curvature and its derivative evolves according to ∂κ = Aκ (1) = 13 κss + 49 κ2 , ∂t ∂κs = Aκs (1) = D Aκ (1) − κs B(1) = 13 κsss + ∂t

(76) 10 9

κ κs .

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A second example is the intrinsic (arc-length preserving) flow with J = κs . In this case, the curvature evolution arises from the characteristic operator: κt = R(κs ) = κ5s + 53 κ κsss + 53 κs κss + 59 κ2 κs , which is the integrable Sawada–Kotera equation, [63]. In this case, the characteristic operator R is closely related to, but not the same as the Sawada–Kotera recursion operator, which is given by the following formula, [12]: b = R · (D2 + 1 κ + 1 κs D−1 ). R 3 3

(77)

Example 19. In the case of space curves C ⊂ R 3 , under the Euclidean group G = SE(3) = SO(3) ⋉ R 3 , there are two generating differential invariants, the curvature κ and torsion τ . According to [31], the relevant moving frame formulae are dV κ = Aκ (ϑ),

dV τ = Aτ (ϑ),

dV ̟ = B(ϑ) ∧ ̟,

where ϑ = (ϑ1 , ϑ2 )T is the column vector containing the order 0 invariant contact forms, while the characteristic and Hamiltonian operators are: Aκ = Ds2 + (κ2 − τ 2 ), −2τ Ds − τs , 2τ 2 3κτs − 2κs τ κτss − κs τs + 2κ3 τ Aτ = Ds + D + , s B = κ, 0 . κ κ2 κ2 1 3 κs 2 κ2 − τ 2 κs τ 2 − 2κτ τs Ds − 2 Ds + Ds + , κ κ κ κ2 Thus, under an intrinsic flow with normal component VN = J n1 + K t2 , the curvature and torsion evolve via κt Rκ Aκ κs , where R= = − D−1 B τt Rτ Aτ τs is the recursion operator for the integrable vortex filament flow, with J = κs , K = τs . This flow can be mapped to the nonlinear Schr¨odinger equation via the Hasimoto transformation, [25, 36].

Acknowledgement It is a pleasure to thank Evelyne Hubert, Irina Kogan, Liz Mansfield, Gloria Mar´ı Beffa and Jing Ping Wang for advice and inspiration. Also, thanks to Demeter Krupka and Olga Krupkov´a for their friendship, hospitality and help during my stay in Bratislava and Olomuoc in August, 2007. This research was supported in part by NSF Grant DMS 05–05293.

References [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and the Inverse Scattering Transform (L.M.S. Lecture Notes in Math., vol. 149, Cambridge University Press, Cambridge, 1991).

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[36] J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlin. Sci. 1 (1991) 71–93. [37] J. Langer and D. A. Singer, Lagrangian aspects of the Kirchhoff elastic rod, SIAM Rev. 38 (1996) 605–618. ¨ [38] S. Lie, Uber Integralinvarianten und ihre Verwertung f¨ur die Theorie der Differentialgleichungen, Leipz. Berichte 49 (1897) 369–410; also Gesammelte Abhandlungen 6 B.G. Teubner, Leipzig (1927) 664–701. [39] E. L. Mansfield and P. E. van der Kamp, Evolution of curvature invariants and lifting integrability, J. Geom. Phys. 56 (2006) 1294–1325. [40] G. Mar´ı Beffa, The theory of differential invariants and KdV Hamiltonian evolutions, Bull. Soc. Math. France 127 (1999) 363–391. [41] G. Mar´ı Beffa, Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, Ann. Institut Fourier, to appear. [42] G. Mar´ı Beffa, J. A. Sanders and J. P. Wang, Integrable systems in three-dimensional Riemannian geometry, J. Nonlinear Sci. 12 (2002) 143–167. [43] K. Mikula and D. Sevcovic, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math. 61 (2001) 1473–1501. [44] K. Nakayama, H. Segur and M. Wadati, Integrability and motion of curves, Phys. Rev. Lett. 69 (1992) 2603–2606. [45] M. Niethammer, A. Tannenbaum and S. Angenent, Dynamic active contours for visual tracking, IEEE Trans. Auto. Control 51 (2006) 562–579. [46] P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977) 1212–1215. [47] P. J. Olver, Symmetry groups and group invariant solutions of partial differential equations, J. Diff. Geom. 14 (1979) 497–542. [48] P. J. Olver, Applications of Lie Groups to Differential Equations (Second Edition, Graduate Texts in Mathematics, vol. 107, Springer–Verlag, New York, 1993). [49] P. J. Olver, Equivalence, Invariants, and Symmetry (Cambridge University Press, Cambridge, 1995). [50] P. J. Olver, An introduction to moving frames, In: Geometry, Integrability and Quantization (vol. 5, (I. M. Mladenov and A. C. Hirschfeld, Eds.) Softex, Sofia, Bulgaria, 2004) 67–80. [51] P. J. Olver, A survey of moving frames, In: Computer Algebra and Geometric Algebra with Applications ((H. Li, P. J. Olver and G. Sommer, Eds.) Lecture Notes in Computer Science, vol. 3519, Springer–Verlag, New York, 2005) 105–138.

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[52] P. J. Olver, Generating differential invariants, J. Math. Anal. Appl. 333 (2007) 450– 471. [53] P. J. Olver, Invariant submanifold flows, preprint, 2007. [54] P. J. Olver and J. Pohjanpelto, Moving frames for Lie pseudo–groups, Canadian J. Math., to appear. [55] P. J. Olver, G. Sapiro and A. Tannenbaum, Differential invariant signatures and flows in computer vision: a symmetry group approach, In: Geometry–Driven Diffusion in Computer Vision; ((B. M. Ter Haar Romeny, Ed.) Kluwer Acad. Publ., Dordrecht, the Netherlands, 1994) 255–306. [56] S. J. Osher and J. A. Sethian, Front propagation with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulations, J. Comp. Phys. 79 (1988) 12–49. [57] C. Qu and S. Zhang, Motion of curves and surfaces in affine geometry, Chaos Solitons Fractals 20 (2004) 1013–1019. [58] H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations (D. Van Nostrand Co. Ltd., Princeton, N.J., 1966). [59] J. A. Sanders and J. P. Wang, Integrable systems in n-dimensional Riemannian geometry, Moscow Math. J. 3 (2003) 1369–1393. [60] G. Sapiro, Geometric Partial Differential Equations and Image Analysis (Cambridge University Press, Cambridge, 2001). [61] G. Sapiro and A. Tannenbaum, On affine plane curve evolution, J. Func. Anal. 119 (1994) 79–120. [62] G. Sapiro and A. Tannenbaum, On invariant curve evolution and image analysis, Indiana J. Math. 42 (1993) 985–1009. [63] K. Sawada and T. Kotera, A method for finding N -soliton solutions of the K.d.V. equation and K.d.V.-like equation, Prog. Theor. Physics 51 (1974) 1355–1367. [64] J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science (Cambridge University Press Cambridge, 1996). [65] E. L. Starostin and G. H. M. Van der Heijden, The shape of a M¨obius strip, Nature Materials Lett. 6 (2007) 563–567. [66] T. Tsujishita, On variational bicomplexes associated to differential equations, Osaka J. Math. 19 (1982) 311–363. [67] W. M. Tulczyjew, The Lagrange complex, Bull. Soc. Math. France 105 (1977) 419– 431.

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In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 237-251

Chapter 13

D IFFERENTIAL I NVARIANTS OF THE M OTION G ROUP ACTIONS Boris Kruglikov1∗and Valentin Lychagin2† Institute of Mathematics and Statistics, University of Tromsø Tromsø 90-37, Norway

Abstract Differential invariants of a (pseudo)group action can vary when restricted to invariant submanifolds (differential equations). The algebra is still governed by the LieTresse theorem, but may change a lot. We describe in details the case of the motion group O(n) ⋉ Rn acting on the full (unconstraint) jet-space as well as on some invariant equations.

2000 Mathematics Subject Classification. 35N10, 58A20, 58H10 Key words and phrases. differential invariants, invariant differentiations, Tresse derivatives, PDEs.

Introduction Let G be a pseudogroup acting on a manifold M or a bundle π : E → M . This action can be prolonged to the higher jet-spaces J k (π) (one can also start with an action in some PDE system E ⊂ J k (π) and prolong it). The natural projection πk,k−1 : J k (π) → J k−1 (π) maps the orbits in the former space to the orbits in the latter. If the pseudogroup is of finite type (i.e. a Lie group), this bundle (restricted to orbits) is occasionally a covering outside the singularity set. Otherwise it will become a sequence of bundles for k ≫ 1. Ranks of these bundles varies but it is occasionally given by the Hilbert-Poincar´e polynomial of the pseudogroup action. ∗ †

E-mail address: [email protected] E-mail address: [email protected]

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The orbits can be described via differential invariants, i.e. invariants of the action on some jet level k. Existence and stability of the above mentioned Hilbert-Poincar´e polynomial is a consequence of the Lie-Tresse theorem, which claims that the algebra of differential invariants is finitely generated via the algebraic-functional operations and invariant derivations. This theorem in the ascending degree of generality was proved in different sources [11, 15, 14, 10, 6]. In particular, the latter reference contains the full generality statement, when the pseudogroup acts on a system of differential equations E ⊂ J l (π) (the standard regularity assumption is imposed, which is an open condition in finite jets). In the case the pseudogroup G acts on the jet space, E must be invariant and so consist of the orbits, or equivalently it has an invariant representation E = {J1 = 0, . . . , Jr = 0}, where Jl are (relative) differential invariants. Now the following dichotomy is possible. If the orbits forming E are regular, the structure of the algebra of differential invariants on E can be read off from that one of the pure jet-space. On the other hand if E consists of singular orbits1 (which is often the case when the system is overdetermined, so that differential syzygy should be calculated, which is an invariant count of compatibility conditions), then the structure of the algebra of differential invariants is essentially invisible from the corresponding algebra I of the pure jet-space, because E is the singular locus for differential invariants I ∈ I (if these exist, as was just remarked1 ). In this note we demonstrate this effect on the example of motion group G acting naturally on the Euclidean space Rn . The group is finite dimensional, but even in this case the described effect is visible. For infinite pseudogroups this follow the same route (see, for instance, the pseudogroup of all local diffeomorphisms acting on the bundle of Riemannian metrics in [5]). We lift the action of G to the jets of functions on Rn and describe in details the structure of algebra of scalar differential invariants in the unconstrained (J ∞ Rn ) and constrained (system of PDEs) cases. This motion group was a classical object of investigations (see e.g. the foundational work [12]), but we have never seen the complete description of the differential invariants algebra.

1.

Differential Invariants and Lie-Tresse Theorem

We refer to the basics on pseudogroup actions to [10, 7], but recall the relevant theory about differential invariants (see also [15, 14, 9]). Since we’ll be concerned with a Lie group in this paper, it will be denoted by one symbol G (in infinite case G should be co-filtered as the equations in formal theory). A function I ∈ C ∞ (J ∞ π) (this means that I is a function on a finite jet space J k π for some k > 1) is called a differential invariant if it is constant along the orbits of the lift of the action of G to J k π. For connected groups G we have an equivalent formulation: The Lie derivative vanishes LXˆ (I) = 0 for all vector fields X from the lifted action of the Lie algebra. 1

In this case E can be defined via vanishing of an invariant tensor J, with components Ji , though in general the latter cannot be chosen as scalar differential invariants.

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Note that often functions I are defined only locally near families of orbits. Alternatively we should allow I to have meromorphic behavior over smooth functions (but we’ll be writing though about local functions in what follows, which is a kind of micro-locality, i.e. locality in finite jet-spaces). The space I = {I} forms an algebra with respect to usual algebraic operations of linear combinations over R and multiplication and also the composition I1 , . . . , Is 7→ I = ∞ (Rs , R), s = 1, 2, . . . any finite number. However even F (I1 , . . . , Is ) for any F ∈ Cloc with these operations the algebra I is usually not locally finitely generated. Indeed, the subalgebras Ik ⊂ I of order k differential invariants are finitely generated on non-singular strata with respect to the above operations, but their injective limit I is not. To cure this difficulty S.Lie and later his French student A.Tresse introduced invariant derivatives, i.e. such differentiations ϑ that belong to the centralizer of the Lie algebra g = Lie(G) lifted as the space of vector fields on J ∞ (π). To be more precise we consider the derivations ϑ ∈ C ∞ (J ∞ π) ⊗C ∞ (M ) D(M ) (C -vector fields on π), which commute with the G-action. These operators map differential invariants to differential invariants ϑ : Ik → Ik+1 . We can associate invariant differentiations to a collection of differential invariants I1 , . . . , ˆ 1 ∧ . . . ∧ dI ˆ n 6= 0. Moreover the In (n = dim M ) in general position, meaning dI whole theory discussed above transforms to the action on equations2 E ⊂ J ∞ (π). Namely, given n functionally independent invariants I 1 , . . . , I n we assume their restrictions IE1 , . . . , IEn are functionally independent3 (in fact we can have the latter invariants only without the former), so that they can be considered as local coordinates. ˆ i . Its dual frame Then one can introduce the horizontal basic forms (coframe) ω i = dI E P b )]−1 D . The invariant derivative ˆ ∂I ˆ i = consists of invariant differentiations ∂/ [D (I a j j E E ij of a differential invariant I are just the coefficients of the decomposition of the horizontal differential by the coframe: n X ˆ ∂I ˆ = ωi dI ˆ i ∂I i=1

E

and they are called Tresse derivatives. All invariant tensors and operators can be expressed through the given frame and coframe and this is the base for the solution of the equivalence problem. Lie-Tresse theorem claims that the algebra of differential invariants I is finitely generated with respect to algebraic-functional operations and invariant derivatives.

2.

Motion Group Action

Consider the motion group O(n) ⋉ Rn . It is disconnected and for the purposes of further study of differential invariants we restrict to the component of unity G = SO(n) ⋉ Rn . The two Lie groups have the same Lie algebra g = o(n) ⋉ Rn and the differential invariants of the latter become the differential invariants of the second via squaring. 2

At this point we do not need to require even formal integrability of the system E [6], but this as well as regularity issues will not be discussed here. 3 Here and in what follows one can assume (higher micro-)local treatment.

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Since the latter is inevitable even for the group G, the difference between two algebras of invariants is by an extension via finite group and will be ignored. Below we will make use of the action of G on the space of codimension m affine subspaces of Rn : AGr(m, n) ≡ {Π + c} ≃ {(Π, c) : Π ∈ Gr(n − m, n), c ∈ Π⊥ }. The action of G is x 7→ Ax + b, x ∈ Rn , it is transitive on AGr(m, n) and the stabilizer equals St(Π + c) = {(A, b) ∈ G : AΠ = Π, b ∈ (1 − A)c + Π} ≃ SO(Π) × SO(Π⊥ ) ⋉ Π. We have dim G =

n(n + 1) , dim AGr(m, n) = m(n − m + 1) and 2 AGr(k, n) ≃ G/(SO(m) × SO(n − m) ⋉ Rn−m )

(note that this implies AGr(m, n) 6= AGr(n − m, n) except for n = 2m contrary to the space Gr(m, n)). We can extend the action of G on Rn to the space Rn × Rm by letting g ∈ G act g · (x, u) = (g · x, u). We can prolong the action to the space J k (n, m). For k = 1 the action commutes with the natural Gl(m)-action in fibers of the bundle π10 : J 1 (n, m) → J 0 (n, m) and the action descends on the projectivization, which can be identified with the open subset in Rn × AGr(k, n) by associating the space Ker(dx f ) to a (surjective at x if we assume n > m) function f : Rn → Rm . Thus u is indeed an invariant of the G-action (scalar invariants are its components ui , so that we can assume the fiber Rm being equipped P iwithj coordinates), and the scalar differ4 i j ential invariants of order 1 are h∇u , ∇u i = uxs uxs . These form the generators of scalar differential invariants of order5 ≤ 1. Remark 1. Sophus Lie investigated the vertical actions of G in J 0 (m, n) = Rm × Rn and the invariants of its lift to J ∞ (m, n) [12] (actually in this paper for m = 1, n = 3). This case is easier since the total derivatives D1 , . . . , Dm are obvious invariant derivations.

In what follows we restrict to the case m = 1 and investigate invariants of the G-action in J ∞ (n, 1) = J ∞ (Rn ). Partially the results extend to the case of general m, though the theory of vector-valued symmetric forms S k (Rn )∗ ⊗ Rm is more complicated.

Differential Invariants: Space J ∞ (Rn )

3.

Denote V = T0 Rn . Our affine space Rn (as well as the vector space V ) is equipped with the Euclidean scalar product h, i and G is the symmetry group of it. In what follows we will identify the tangent space Tx Rn with V via translations (using the affine structure on Rn ). Recall that the base space Rn is equipped with the Euclidean metric preserved by G. This claim holds at an open dense subset of J 1 (n, m). However if we restrict to the set of singular orbits with rank(dx u) = r < m, the basic set of invariants will be quite different. 4 5

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The space J ∞ (Rn ), which is the projective limit of the finite-dimensional manifolds J k (Rn ), has coordinates (xi , u, pσ ), where σ = (i1 , . . . , in ) ∈ Zn≥0 is a multiindex with length |σ| = i1 + · · · + in . The only scalar differential invariants6 of order ≤ 1 are I0 = u and I1 = |∇u|2 .

For each x1 ∈ J 1 (Rn ) the group G has a large stabilizer. Provided x1 is non-singular the dimension of the stabilizer St1 is dim G − 2n + 1 = 21 (n − 1)(n − 2). However the stabilizer completely evolves upon the next prolongation: the action of G on an open dense subset of J k (Rn ) for any k ≥ 2 is free. Note that due to the trivial connection in J 0 (Rn ) = Rn × R we can decompose J k (Rn ) = Rn × R × V ∗ × S 2 V ∗ × · · · × S k V ∗ .

(1)

Thus we can represent a point xk ∈ J k (Rn ) as the base projection x ∈ Rn and a sequence of ”pure jets” Qt = dt u ∈ S t V ∗ , t = 0, . . . , k. Covector Q1 can be identified with the vector v = ∇u. Consider the quadric Q2 ∈ S 2 V ∗ . Due to the metric we can identify it with a linear operator A ∈ V ∗ ⊗ V , which has spectrum Sp(A) = {λ1 , . . . , λn } and the normalized eigenbasis e1 , . . . , en (each element defined up to a sign!), provided Q2 is semi-simple. Since Q2 is symmetric, the basis is orthonormal. In what follows we assume to work over the open dense subset U ⊂ J 2 (Rn ), where A is simple, so that the basis is defined (almost) uniquely (this can be relaxed to semi-simplicity, but then the stabilizer is non-trivial and the number of scalar invariants drops a bit). There are precisely (2n − 1) = dim J 2 (Rn ) − dim St1 differential invariants of order ¯ ¯ 2. One choice Pn ¯is2 to take I2,i = λi and I2,(i) = hei , vi, i = 1, . . . , n. There is an obvious relation i=1 I2,(i) = 1, so that we can restrict to the first (n − 1) invariants in this group, but beside this the invariants are functionally independent. Another choice of invariants is provided by the restriction QΠ of Q2 to Π = v⊥ , which ˜1, . . . , λ ˜ n−1 } and has spectrum (again by converting quadric to an operator) Sp(QΠ ) = {λ ˜ i , I˜2,n = ˜ normalized eigenvectors e˜i . So the following invariants can be chosen: I2,i = λ Q2 (v, v) and I˜2,(i) = Q2 (v, e˜i ). Both choices have disadvantages of using transcendental functions (solutions to algebraic equations), but we can overcome this with the following choice: I2,i = Tr(Ai ), I2,(i) = hAi v, vi,

i = 1, . . . , n.

Here the number of invariants is 2n, but they are dependent7 due to Newton-Girard formuP las, which relate the elementary symmetric polynomials Ek (A) = i1 <···
k X (−1)i−1 Si (A)Ek−i (A), i=1

6 7

From now on by this we mean the minimal set of generators. The first (2n − 1) invariants are however independent and algebraic in the jets.

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which together with E0 = 1 gives an infinite chain of formulas E1 = S1 , 2E2 = S12 − S2 , 6E3 = S13 − 3S1 S2 + 2S3 , . . . Now with the help of Cayley-Hamilton formula An = E1 (A)An−1 − E2 (A)An−2 + · · · + (−1)n En−1 (A)A − (−1)n En (A) we can express I2,(n) = E1 (A)I2,(n−1) − E2 (A)I2,(n−2) + · · · − (−1)n det A through our invariants since Ei (A) are functions of I2,i . Remark 2. We could restrict only to invariants I2,(i) , i = 1, . . . , 2n − 1. This is helpful as we shall see. But when we restrict to singular (from the orbits point of view) PDEs these differential invariants may turn to be non-optimal, and this will be precisely the case in the example we investigate. Now there are precisely n+2 = dim S 3 V ∗ differential invariants of order 3, n+3 = 3 4 n+k−1 4 ∗ k ∗ dim S V differential invariants of order 4, . . . , = dim S V differential invarik ants of order k. The third order invariants are the following: I¯3,σ = Q3 (ei , ej , el ), where σ = (ijl) ∈ S 3 {1, . . . , n}. Generating invariants of orders 4 and higher are obtained from the similar formulae, namely as the coefficients qσ of the decomposition X Qk = qσ ω σ , where ω σ = ω i1 · · · ω ik , 1 ≤ i1 ≤ · · · ≤ ik ≤ n. σ=(i1 ,...,ik )

They are again transcendental functions. To get algebraic expressions one can use the third order functions I3,σ = Q3 (Ai v, Aj v, Al v),

σ = (ijk) with 1 ≤ i ≤ j ≤ l ≤ n

and similar expressions for the higher order. Theorem 1. The invariants Ii,σ with i ≤ 3 is the base of differential invariants for the Lie group G action in J ∞ (Rn ) via algebraic-functional operations and Tresse derivatives. This statement is an easy dimensional count8 together with examination of independency condition. To get Tresse derivatives n invariants (for instance of order ≤ 2) should be chosen. However this is not necessary, if one does not care about transcendental functions. Indeed, the vector fields e1 , . . . , en are invariant differentiations (they can be expressed through the total derivatives D1 , . . . , Dn with coefficients of the second order). 8

In fact for n ≤ 4 the same arguments imply that the base can formed only by the invariants Ii,σ with i ≤ 2.

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Remark 3. Notice that the moving frame e1 , . . . , en ∈ C ∞ (U, π2∗ T Rn ) uniquely fixes an element g ∈ G, which transforms it to the standard orthonormal frame at −1 0 ∈ Rn . This leads to the equivariant map defined on the open dense set π∞,2 (U ): J ∞ (Rn ) → J 2 (Rn ) ⊃ U → G. Such map is called the moving frame in the approach of Fells and Olver [2].

4.

Relations in the Algebra I

Since the commutator is an invariant differentiation, decompoP k of invariant 1differentiations sition [ei , ej ] = cij ek yields ≤ 2 n2 (n − 1) (in general precisely this number) 3rd order differential invariants ckij . The number of pure 3rd order invariants obtained via invariant differentiations of the 2nd order invariants is n(2n − 1). So since n2 (n − 1) n(n + 4)(1 − n) n(n + 1)(n + 2) − n(2n − 1) − = ≤0 6 2 3 we can conclude that differential invariants Ii,σ with i ≤ 2 and invariant differentiations ˆ ⊂ J ∞ (Rn ). {ei }ni=1 generate the whole algebra I on an open set U Thus we are lead to the question on relations in this algebra. They can be all deduced from the expressions for pure jets of u ˆ 2 , Q4 = ∇Q ˆ 3 etc Q3 = ∇Q using the structural equations. Here ∗ ˆ : C ∞ (πi∗ S i V ∗ ) → C ∞ (πi+1 ∇ S i+1 V ∗ )

is the symmetric covariant derivative induced by the flat connection ∇ in the trivial bundle J 0 (Rn ) = Rn × R, V = T Rn (the map is the composition of the horizontal differential dˆ and symmetrization). However for the sake of algebraic formulations we change invariant differentiations ei to the following ones: X v1 =ˆ v = v · Dx = u i Di X ˆ = Av · Dx = v2 =Av ui uij Dj X d 2 v = A2 v · D = v3 =A ui uij ujk Dk x ...

...

...

\ n−1 v = An−1 v · D = vn =A x

X

ui1 ui1 i2 . . . uin−1 in Din .

Now we are going to change the basis of differential invariants in Ik to describe the relations in the simplest way.

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Namely for the basis of invariants of order 2 we can take I2,(ij) = Q2 (Ai v, Aj v), 0 ≤ i ≤ j < n. However since Q2 (v, w) = hAv, wi and A is self-adjoint we get I2,(ij) = hAi+1 v, Aj vi = hAi+j+1 v, vi = I2,(i+j+1) , so that the new invariants are precisely the old ones I2,(i) , just with the larger index range i = 1, . . . , 2n − 1 (we can allow arbitrary index i, but the corresponding invariants are expressed via these ones, see Remark 2 and before). Basic higher order invariants are introduced in the same fashion: Is,(i1 ...is ) = Qs (Ai1 v, . . . , Ais v),

0 ≤ i1 ≤ · · · ≤ is < n.

Suppose now that our set of generic (regular) points U ⊂ J 2 (Rn ) is given by not only the constraint that Sp(A) is simple, but also the claim that the n × n matrix kγij k0≤i,j
1

I2,(1) I2,(2) .. .

−1 I2,(n−1) I2,(n)    ..  .

··· ··· .. .

 I2,(1)  [γ ij ] =  .  .. I2,(n−1) I2,(n) · · ·

I2,(2n−2)

be the inverse matrix. Note that all its entries are invariants. Now (Ai0 v · Dx ) Qs (Ai1 v, . . . , Ais v) = Qs+1 (Ai0 v, Ai1 v, . . . , Ais v) s X + Qs (Ai1 v, . . . , Aij −1 v, θi0 ij , Aij +1 v, . . . , Ais v), j=1

where θi0 ij = ∇Ai0 vˆ (Aij v) is the vector which, due to metric duality, is dual to the covector P i0 α β α+β=ij −1 Q3 (A v, A v, A ·). Thus we obtain

Theorem 2. The algebra I is generated by the invariants Is,σ and invariants derivatives v1 , . . . , vn , which are related by the formulae (s ≥ 2): vi0 · Is,(i1 ...is ) = Is+1,(i0 i1 ...is ) +

s n−1 X X

X

Is,(i1 ...ij−1 ,a,ij+1 ...is ) γ ab I3,(i0 ,α,b+β) .

j=1 a,b=0 α+β=ij −1

In this case we can choose Is,σ , s ≤ 3 and vi as the generators. This representation for I via generators and relations is not minimal, as clear from the first part of the section. However the relations are algebraic, explicit and quite simple. To explain how to achieve minimality let us again change the set of generators (basic differential invariants). For the second order we return to I2,i , I2,(i) , 1 ≤ i ≤ n. For the third order we add the invariants I3,[ij]l = Tr(Q3 (Ai ·, Aj ·, Al v)).

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They can indeed be expressed algebraically through the invariants I3,(ijk) together with the lower order invariants. For higher order we have more possibilities of inventing new invariants (which can be described via graphs of the type (k, 1)-tree), but they are again algebraically dependent with already known differential invariants. The relations are as follows (0 ≤ k < n and we show only top of the list): v1 · I0 = I1 , v2 · I0 = I2,(1) , . . . , vn · I0 = I2,(n−1) ,

v1 · I1 = 2I2,(1) , v2 · I1 = 2I2,(2) , . . . , vn · I1 = 2I2,(n) , X X vk+1 · I2,l = I3,[αβ]k , vk+1 · I2,(l) = I3,(αβk) + 2I2,(k+l+1) etc. α+β=l−1

α+β=l−1

Elaborate work with these shows that all P the invariants can be obtained from I0 and k structural constants c¯ij of the frame [vi , vj ] = c¯kij vk .

ˆ ⊂ J ∞ (Rn ) further (but leaving it open dense) we can arrange Corollary 1. By shrinking U that the algebra I of differential invariants is generated only by I0 and the derivations v1 , . . . , vn .

5.

Algebra of Differential Invariants: Equation E

Consider the PDE E = {k∇uk = 1}. By the standard arguments it determines a cofiltered manifold in J ∞ (Rn ) and we identify E with it, so that it consists of the sequence of prolongations Ek ⊂ J k (Rn ) and projections π ¯k,k−1 : Ek → Ek−1 . Since the prolongation of the defining equation for E to the second jets is Q2 (v, ·) = 0 or v ∈ Ker(A) we conclude that most of the invariants, introduced on the previously defined ˆ , vanish: the equation is singular. Indeed, 0 ∈ Sp(A), so that det A = 0, the matrix subset U [γij ] is not invertible etc. In particular, I2,(i) = 0, Is,(i1 ...is ) = 0 if at least one it 6= 0, v2 = · · · = vn = 0. Thus the algebra I description from the previous section does not induce any description of the algebra IE of differential invariants of the group G action on E: the notion of regularity and basic invariants are changed completely! Again the group acts freely on the second jets. So there is 1 invariant of order 0 I0 = u, no invariants of order 1 and (n − 1) invariants of order 2: I2,1 , . . . , I2,n−1

or equivalently

E1 (A), . . . , En−1 (A).

The number of invariants of pure order k > 2 coincides with the ranks of the projections: −1 dim π ¯k,k−1 (∗) =

n+k−2 . k

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The principal axes of Q2 (or normalized eigenbasis of A) are now e1 = v, e2 , . . . , en . These are still the invariant derivations and the invariants of order k > 2 are the coefficients9 of the decomposition by basis in S k Ann(v) ⊂ S k V ∗ : X Qk |E = qσ ω σ , qσ = Qk (vi1 , . . . , vik ). σ=(i1 ...ik ):it >1

Theorem 3. The invariants I0 , I2,i and I3,σ 1 ≤ i < n, σ = (i1 , i2 , i3 ), it 6= 1 form a base of differential invariants of the algebra IE via algebraic-functional operations and Tresse derivatives. Algebra of differential invariants can again be represented in a simpler form via differential invariants and invariant derivatives. If we choose ei for the latter the relations can be read off from the algebra I, though this again involves transcendental functions. ˆ in the basis eα by Γk (these are differential invariDenote the Christoffel symbols of ∇ ij ants of order 3): X X j k j ˆ e ej = ˆ ∇ Γ e ⇐⇒ ∇ ω = − Γik ω k . e k ij i i

Notice that since the connection is torsionless, T∇ = 0, these invariants determine the structure functions ckij = Γkij − Γkji . Let us now substitute the formulas (eigenvalues λi can be expressed through the invariants I2,i , however in a transcendental way; λ1 = 0 corresponds to e1 ) X X Q2 = λi (ω i )2 , Q3 = qijk ω i ω j ω k 1
1
ˆ 2 = Q3 : into the identity ∇Q ˆ ∇

X

λi (ω i )2 =

X

ˆ i )(ω i )2 + 2 (∇λ

We get for 1 < i ≤ j ≤ k ≤ n:

ˆ i λi ω i · ∇ω X X = ∂ek (λi )ω i ω i ω k − 2 λi Γijk ω i ω j ω k .

X

X τ (i) qijk = ∂ek (λi )δij + ∂ei (λk )δjk − ∂ek (λi )δik − 2 λτ (i) Γτ (j)τ (k) τ ∈S3

Since in addition, in general position the invariants λi can be expressed through the invariants ei · I0 (1 < i ≤ n)10 , then by adding decomposition of the covariant derivatives by the frame into the set of operations, we obtain the following ˆ ⊂ E further (but leaving it open dense) we can arrange Corollary 2. By shrinking U that the algebra IE of differential invariants is generated only by I0 and the derivations e1 , . . . , en . 9

Note that these invariants are defined up to ± and so should be squared to become genuine invariants; alternatively certain products/ratios of them define absolute invariants. 10 We have e1 · I0 = 1 on E.

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247

Algebra of Differential Invariants: Equation E˜

Completely new picture for the algebra of differential invariants emerges, when we add one more invariant PDE: the system becomes overdetermined and compatibility conditions (or differential syzygies) come into the play. We will study the following system11 , which comes from application to relativity [1] (when Laplacian ∆ is changed to Dalambertian ): {kuk = 1, ∆u = f (u)} ⊂ E. This equation is a non-empty submanifold in J 2 (Rn ), but when we carry the prolongationprojection scheme, it becomes much smaller. It turns out that for most functions f (u) the resulting submanifold E˜ is just empty. We are going to decompose it into the strata ˜ ∪ · · · ∪ Σn (E), ˜ E˜ = Σ1 (E) ˜ = {x ∈ E˜ : #[Sp(A ˜)] = i} for the operator A ˜ corresponding to the 2-jet where Σi (E) E E Q2 |E˜. It is possible to show that the spectrum of A on E˜ depends on u (and some constants) only. This was done in [3] via the Cayley-Hamilton theorem, though they used the Dalambertian instead of the Laplace operator. In the next section we prove it for the Laplace operator via a different approach. More detailed investigation leads to the following claim: ˜ . . . , Σ3 (E) ˜ are empty, while Σ2 (E), ˜ Σ1 (E) ˜ are Conjecture: The strata Σn (E), not and they are finite-dimensional manifolds. ˜ because on other strata Let us indicate the idea of the proof for the stratum Σn (E) the eigenbasis ei is not defined (but the arguments can be modified). It turns out that the compatibility is related to dramatic collapse of the algebra IE˜ of differential invariants. Indeed, as follows from the discussion above and the next section, there is only one ˜ Since the coefficients of the invariant invariant u of order ≤ 2 for the G-action on E. derivations have the second order, we obtain the following statement: Theorem 4. All differential invariants of the Lie group G-action on the PDE system E˜ can be obtained from the function I0 = u and invariant derivations. Now relations in the algebra IE˜ are differential syzygies for E˜ and they boil down to a system of ODEs on f (u), which completely determines it. The details of this program will be however realized elsewhere. 11

This interesting system was communicated to the first author by Elizabeth Mansfield.

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Boris Kruglikov and Valentin Lychagin

Geometry of the System

In this section we justify the claim from §6. and prove that the spectrum of the operator A = AE˜, obtained from the pure 2-jet Q2 |E˜ via the metric, depends on u only. To do this we reformulate the problem with nonlinear differential equations in the geometric language from contact geometry [13]. P The first equation E we represent as a level surface H = 12 (1 − ni=1 p2i ) = 0 in the jet-space J 1 (Rn ). The second equation from E˜ can be represented as Monge-Ampere type via n-form Ω1 =

n X i=1

dx1 ∧ . . . ∧ dxi−1 ∧ dpi ∧ . . . ∧ dxn − f (u)dx1 ∧ . . . ∧ dxn .

Namely a solution to the system is a Lagrangian submanifold Ln ⊂ {H = 0} such that Ω1 |Ln = 0. Representing Ln = graph[j 1 (u)] we obtain the standard description. The contact Hamiltonian vector field XH preserves the contact structure and being restricted to the surface H =P 0 it coincides with the field of Cauchy characteristic P YH = XH |H=0 = pi Dxi = pi ∂xi + ∂u . Since Cauchy characteristics are always tangent to any solution, the forms Ω1+i = (LXH )i Ω1 also vanish on any solution of the system E. We simplify them modulo the form Ω1 and get: Ω2 = LXH Ω1 + f (u)Ω1 X =2 dx1 ∧ . . . dpi ∧ dxi+1 . . . dpj ∧ dxj+1 . . . ∧ dxn − (f ′ + f 2 ) dx1 ∧ . . . dxn , Ω3 = LXH Ω2 + (f ′ (u) + f 2 (u))Ω1 X = 3! dx1 ∧ . . . dpi ∧ dxi+1 . . . dpj ∧ dxj+1 . . . dpk ∧ dxk+1 . . . ∧ dxn ...

− (D + f )2 (f ) dx1 ∧ . . . ∧ dxn , ...

...

...

Ωn = n! dp1 ∧ . . . dpn − (D + f )n−1 (f ) dx1 ∧ . . . ∧ dxn ,

Ωn+1 = −(D + f )n (f ) dx1 ∧ . . . ∧ dxn ,

where D is the operator of differentiation by u and f is the operator of multiplication by f (u). Thus a necessary condition for solvability is the following non-linear ODE: (D + f )n+1 (1) = 0.

(2)

R This equation can be solved via conjugation D + f = e−g Deg with g(u) = f (u) du [4], which reduces the ODE to the form Dn+1 eg = 0, so that g = Log Pn (u), where Pn (u) is a polynomial of degree n, whence12 f (u) =

n X i=1

12

1 , u − αi

αi = const .

Here we can assume we are working over C, though this turns out to be inessential.

(3)

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249

However there are more compatibility conditions, which produce further constraints on numbers αi . The above relations Ωi = 0 can be used to find Sp(A). Namely let us rewrite them as follows: E1 (A) = E3 (A) =

X

X

λi = f,

E2 (A) =

X

λi λj = 21 (D + f )2 (1),

i<j

λi λj λk =

i<j
1 3! (D

+ f )3 (1), . . . , En (A) = λ1 · · · λn =

1 n! (D

+ 1)n (1).

These, due to Newton-Girard formulas, imply the equivalent identities: I2,1 =

X

X I2,2 = λ2i = −f ′ (u), X X I2,3 = λ3i = 12 f ′′ (u), I2,4 = λ4i = − 3!1 f ′′′ (u),

λi = f (u),

...

In particular we get λi = (u − αi )−1 and so A ∼ Diag

1 1 ,..., u − α1 u − αn

.

The fact that det(A) = 0 on E˜ implies that αn = ∞ and using symmetry ∂u (shift along u) we can arrange α1 = 0 (we use freedom of renumbering the spectral values). The conjecture from the previous section is equivalent to the claim that other αi equal either 0 or ∞. But this will be handled in a separate paper.

8.

Integrating the System Along Characteristics

Let us now consider the quotient of the submanifold {H = 0} ⊂ J 1 (Rn ) by the Cauchy characteristics. We can identify it with the transversal section Σ2n−1 = {H = 0, u = const}. The solutions will be (n − 1)-dimensional manifolds of the induced exterior differential system. P Note that we should augment the system with the contact form ω = du − pi dxi and P i its differential Ω0 = dx ∧ dpi . Note that if we choose f (u) to be the solution of the 1 ODE (2), then n! Ωn = dp1 ∧ . . . ∧ dpn = 0 on solutions. Let us start investigation from the case n = 2. In this case the induced differential system is given by two 1-forms: θ = iXH Ω1 |Σ = p1 dp2 − p2 dp1 −

1 (p1 dx2 − p2 dx1 ) u

and θ0 = iXH Ω0 |Σ = p1 dp1 + p2 dp2 , but it vanishes on Σ. The form θ is contact: θ ∧ dθ 6= 0, so solutions of E are represented by all Legendrian curves on (Σ3 , θ). 1 1 Consider now n = 3. In this case we know that Sp(A) = {0, u−α , u+α } (in fact, α = 0, but let us pretend we do not know it yet). 2 We have: f = u22u , f ′ + f 2 = u2 −α 2. −α2

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Again θ0 = iXH Ω0 vanishes on Σ5 , so the exteriour differential system is generated by two 2-forms: θ1 = iXH Ω1 = (p1 dp2 − p2 dp1 ) ∧ dx3 + (p2 dp3 − p3 dp2 ) ∧ dx1

+(p3 dp1 − p1 dp3 ) ∧ dx2 −

2u (p dx2 u2 −α2 1

∧ dx3 + p2 dx3 ∧ dx1 + p3 dx1 ∧ dx2 );

θ2 = 21 iXH Ω2 = p1 dp2 ∧ dp3 + p2 dp3 ∧ dp1 + p3 dp1 ∧ dp2

1 − u2 −α 2 (p1 dx2 ∧ dx3 + p2 dx3 ∧ dx1 + p3 dx1 ∧ dx2 ).

The integral surfaces of this system integrate to solutions of E. Digression. Let us choose another section for Σ′ ⊂ J 1 (R3 ): since the Cauchy characteristics are given by p the system {x˙ i = pi , u˙ = 1}, we can take in the domain p3 > 0: x3 = const, p3 = 1 − p21 − p22 . Then the forms giving the differential system are given by (being multiplied by p3 ): θ1′ = (1 − p22 ) dp1 + p1 p2 dp2 ∧ dx2 + dx1 ∧ p1 p2 dp1 + (1 − p21 ) dp2 − u22u (1 − p21 − p22 ) dx1 ∧ dx2 ; −α2

θ2′ = dp1 ∧ dp2 −

1 − p21 − p22 dx1 ∧ dx2 . u2 − α 2

If we identify Σ′ ≃ J 1 (R2 ) with the contact form ω ′ = du − p1 dx1 − p2 dx2 , the above 2-forms become represented by the following Monge-Ampere equations: (1 − u2x )uxx + 2ux uy · uxy + (1 − u2x )uyy = uxx uyy − u2xy =

1 (1 u2 −α2

−

2u (1 u2 −α2 2 2 ux − uy ).

− u2x − u2y ),

Compatibility of this pair yields α = 0. Remark 4. The above system is of the kind investigated in [8]: when the surface Σ2 = graph{u : R2 → R1 } ⊂ R3 has prescribed Gaussian and mean curvatures, K and H respectively (this leads to a complicated overdetermined system). In fact the PDEs of the above system can be written in the form H = F1 (u, ∇u), K = F2 (u, ∇u).

References [1] C. B. Collins, Complex potential equations. I. A technique for solution, Math. Proc. Cambridge Philos. Soc. 80 (1) (1976) 165–187. [2] M. Fels and P. Olver, Moving frames and coframes, In: Algebraic methods in physics (Montreal, 1997, CRM Ser. Meth. Phys., Springer, 2001) 47–64. [3] W. I. Fushchich, R. Z. Zhdanov and I. A. Yegorchenko, On the reduction of the nonlinear multi-dimensional wave equations and compatibility of the D’Alembert-Hamilton system, J. Math. Anal. Appl. 161 (2) (1991) 352–360. [4] M. Kontsevich, private communication.

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[5] B. Kruglikov, Invariant characterization of Liouville metrics and polynomial integrals, 2007; arXiv:0709.0423. [6] B. S. Kruglikov and V. V. Lychagin, Invariants of pseudogroup actions: Homological methods and Finiteness theorem, Int. J. Geomet. Meth. Mod. Phys. 3 (5 & 6) (2006) 1131–1165. [7] B. S. Kruglikov and V. V. Lychagin, Geometry of Differential equations, prepr. IHES/M/07/04; In: Handbook of Global Analysis ((D. Krupka and D. Saunders, Eds.) Elsevier, 2008) 725–772. [8] B. S. Kruglikov and V. V. Lychagin, Compatibility, multi-brackets and integrability of systems of PDEs, prepr. Univ. Tromsø 2006-49; ArXive: math.DG/0610930. [9] D. Krupka, J. Janyska, Lectures on Differential Invariants (Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis. Mathematica 1, University J.E. Purkyne, Brno, 1990). [10] A. Kumpera, Invariants differentiels d’un pseudogroupe de Lie. I-II., J. Differential Geometry 10 (2) (1975) 289–345; 10 (3) (1975) 347–416. [11] S. Lie, Ueber Differentialinvarianten, Math. Ann. 24 (4) (1884) 537–578. [12] S. Lie, Zur Invariantenteorie der Gruppe der Bewgungen, Leipzig Ber. 48 (1896) 466– 477; Gesam. Abh. Bd. VI 639–648. [13] V. V. Lychagin, Contact geometry and nonlinear second order differential equations, Uspekhi Mat. Nauk 34 (1) (1979) 137–165 (in Russian); English transl.: Russian Math. Surveys 34 (1979) 149–180. [14] L. V. Ovsiannikov, Group analysis of differential equations (Russian: Nauka, Moscow, 1978; Engl. transl.: Academic Press, New York, 1982). [15] A. Tresse, Sur les invariants differentiels des groupes continus de transformations, Acta Math. 18 (1894) 1–88.

Part III

D IFFERENTIAL E QUATIONS AND G EOMETRICAL S TRUCTURES

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In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 255-259

Chapter 14

R EMARKS ON THE H ISTORY OF THE N OTION OF L IE D IFFERENTIATION Andrzej Trautman∗ Instytut Fizyki Teoretycznej, Uniwersytet Warszawski Ho˙za 69, Warszawa, Poland

1. The derivative X(f ) of a function f , defined on a smooth manifold, in the direction of the vector field X and the bracket of two vector fields, introduced by Sophus Lie himself, are the first examples of what is now called the Lie derivative. Another early example comes from the Killing equation. David Hilbert [1], in his derivation of the Einstein equations, used the expression X ρ ∂ρ g µν − g µρ ∂ρ X ν − g ρν ∂ρ X µ and stated that it is a tensor field for every tensor field g and vector field X. Around 1920, ´ Cartan defined a natural differential operator L(X) acting on fields of exterior forms. Elie He noted that it commutes with the exterior derivative d and gave, in equation (5) on p. 84 in [2], the formula1 L(X) = d ◦ i(X) + i(X) ◦ d, (1) where i(X) is the contraction with X. ´ 2. Władysław Slebodzi´ nski, in his article of 1931 [5], wrote an explicit formula for the Lie derivative (without using that name) in the direction of X of a tensor field A of arbitrary valence. He gave also an equation equivalent to L(X)(A ⊗ B) = (L(X)A) ⊗ B + A ⊗ L(X)B and noted that L(X) commutes with contractions over pairs of tensorial indices. He then applied his results to Hamilton’s canonical equations of motion. For a function H(p, q), ∗

E-mail address: [email protected] In this note, I transcribe all equations from the form given by their authors to the notation in current usage. All manifolds and maps among them are assumed to be smooth. Good references for my notation and terminology are [3] and [4]. 1

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Andrzej Trautman

´ p = (pµ ), q = (q µ ), µ = 1, . . . , n, Slebodzi´ nski defined the vector field XH =

∂H ∂ ∂H ∂ − µ , µ ∂pµ ∂q ∂q ∂pµ

introduced the symplectic form A = dq µ ∧ dpµ , the Poisson bivector B = ∂/∂q µ ∧ ∂/∂pµ and showed that L(XH )A = 0 and L(XH )B = 0. This allowed him to generalize some results of Th´eophile de Donder in the theory of invariants [6]. ´ The priority of Slebodzi´ nski in defining the Lie derivative in the general case was recognized by David van Dantzig who wrote, in footnote on p. 536 of [7], Der Operator [the Lie ´ derivative] wurde zum ersten Mal von W. Slebodzi´ nski eingef¨uhrt. It was van Dantzig who introduced, in the same paper, the name Liesche Ableitung. Also Jan Arnoldus Schouten, ´ in footnote 1 on p. 102 of [8], lists the 1931 paper by Slebodzi´ nski as the first reference for ´ the notion of Lie differentiation. Van Dantzig complemented the approach of Slebodzi´ nski by pointing out that the Lie derivative can be defined as the difference between the value of a geometric object A at a point and the value of that object at the same point obtained by an infinitesimal ‘dragging along’ a vector field. In contemporary notation this is expressed by the formula d L(X)A = ϕ∗t A|t=0 , (2) dt where ϕ∗t A is the pull-back of A by the flow (ϕt , t ∈ R) generated by X. In view of the equation d ∗ ϕ A = ϕ∗t L(X)A, dt t the vanishing of L(X)A is equivalent to the invariance of A with respect to the flow generated by X; see, e.g., §24 in [9]. 3. For quite some time, physicists had been using Lie derivatives, without reference to the work of mathematicians. L´eon Rosenfeld [10] introduced what he called a ‘local variation’ δ ∗ A of a geometric object A induced by an infinitesimal transformation of coordinates generated by X. He noted that δ ∗ commutes with differentiation. It is easily seen that his δ ∗ A is −L(X)A; see, e.g., [11]. Assuming that A is a tensor of type determined by a representation ρ of GL(4, R) in the vector space RN and denoting by ρνµ ∈ End RN the matrices of the corresponding representation of the Lie algebra of GL(4, R), one can deduce from Rosenfeld’s equations the following formula for the Lie derivative L(X)A = X µ ∂µ A − ∂ν X µ ρνµ A. In particular, assuming that L is a Lagrange function depending on the components of A R and on their first derivatives and such that Ld4 x is an invariant, Rosenfeld showed that L(X)L = ∂µ (LX µ )

and used the formula L(X)L =

∂L ∂L L(X)A + ∂µ (L(X)A) ∂A ∂(∂µ A)

Remarks on the History of the Notion of Lie Differentiation

257

to derive a set of identities of the Noether type, and the conservation laws of energymomentum and of angular momentum. One of the main results of that paper was the symmetrization of the canonical energy-momentum tensor t achieved by adding to it an expression linear in the derivatives of the spin tensor s. Incidentally, it is remarkable that this symmetrization, derived independently also by F. J. Belinfante, is a natural consequence of the Einstein–Cartan theory of gravitation. In that theory, based on a metric tensor g and a linear connection ω µ ν = Γµνρ dxρ which is metric, but may have torsion, there are field equations relating curvature and torsion to t and s, respectively; see [12] and the references given there. If these Sciama–Kibble field equations are satisfied and X is a vector field generating a symmetry of space-time so that L(X)g = 0

and L(X)ω = 0

then, denoting by tµ and sµν the 3-forms (densities) of energy-momentum and spin, and the e one has covariant derivative with respect to the transposed connection ω ˜ νµ = Γµρν dxρ by ∇, the conservation law dj = 0, where e ν X µ sµν . j = X µ tµ + 12 ∇

In the limit of special relativity, if X generates a translation, then j reduces to the corresponding component of the density of energy-momentum; for X generating a Lorentz transformation, one obtains a component of the density of total angular momentum. 4. The Lie derivative defines a homomorphism of the Lie algebra V(M ) of all vector fields on an n-dimensional manifold M into the Lie algebra of derivations of the algebra of all tensor fields on M , L([X, Y ]) = [L(X), L(Y )]. The Cartan algebra C(M ) = np=0 C p (M ) of all exterior forms on M is Z-graded by the degree p of the forms. A derivation D of degree q ∈ Z maps linearly C p (M ) to C p+q (M ) and satisfies the graded Leibniz rule, L

D(α ∧ β) = (Dα) ∧ β + (−1)pq α ∧ Dβ

for every α ∈ C p (M ).

Derivations of odd degree are often called antiderivations. The vector space Der C(M ) of all derivations of C(M ) is a super Lie algebra with respect to the bracket ′

[D, D′ ] = D ◦ D′ − (−1)deg D deg D D′ ◦ D.

(3)

The degree of [D, D′ ] is the sum of the degrees of D and D′ and there holds a super Jacobi identity; see [13] for an early review of super Lie algebras, written for physicists. In particular, d is a derivation of degree +1 and, if X ∈ V(M ), then L(X) and i(X) are derivations of degrees 0 and −1, respectively. The Cartan formula (1) represents L(X) as a bracket, as defined in (3), of d and i(X). The contraction i(X) generalizes to fields of vector-valued exterior forms. Let X ∈ V(M ), ξ ∈ C p (M ), p = 0, . . . , n, and Y = X ⊗ ξ, then Y is a vector-valued p-form and i(Y ) is a derivation of the Cartan algebra, of degree p − 1, defined by i(Y )α = ξ ∧ i(X)α,

α ∈ C(M ).

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Andrzej Trautman

By linearity one extends i(Y ) to arbitrary vector-valued p-forms. The bracket [d, i(Y )] is now a derivation of degree p; by the super Jacobi identity its bracket with d is zero and every derivation (super) commuting with d is of this form. If Y and Z are vector-valued forms of degrees p and q, respectively, then the bracket [[d, i(Y )], [d, i(Z)]] super commutes with d and, therefore, there exists a vector-valued (p + q)-form [Y, Z] such that [d, i([Y, Z])] = [[d, i(Y )], [d, i(Z)]].

(4)

The Fr¨ohlicher–Nijenhuis [14] bracket [Y, Z], defined by (4), generalizes the Lie bracket of vector fields; it is super anticommutative, [Z, Y ] = −(−1)pq [Y, Z], and makes the vector space of all vector-valued forms into a super Lie algebra. For example, an almost complex structure J on an even-dimensional manifold is a vector-valued 1-form and [J, J] is its Nijenhuis torsion. 5. A convenient framework to generalize the definition (2) of Lie derivatives is provided by natural bundles. A natural bundle is a functor F from the category of manifolds to that of bundles such that πM : F (M ) → M is a bundle and if ϕ : M → N is a diffeomorphism, then F (ϕ) : F (M ) → F (N ) is an isomorphism of bundles covering ϕ. If A is a section of πN : F (N ) → N , i.e. a field on N of geometric objects of type F , then ϕ∗ A = F (ϕ−1 ) ◦ A ◦ ϕ is its pull-back by ϕ to M . All tensor bundles are natural, but spinor bundles are not. The vertical bundle V F (M ) is the subbundle of the tangent bundle T F (M ) consisting of all vertical vectors, i.e. vectors that are annihilated by T πM . Let (ϕt , t ∈ R) be the flow generated by X ∈ V(M ) and let A be a section of πM . The curve t 7→ (ϕ∗t A)(x) is vertical for every x ∈ M and the Lie derivative L(X)A is now defined as the section of the vector bundle V F (M ) → M such that (L(X)A)(x) is the vector tangent to t 7→ (ϕ∗t A)(x) at t = 0. The monograph by Kol´aˇr, Michor and Slov´ak [15] contains a full account of this approach and, in Ch. XI, an even more general definition of Lie differentiation.

References [1] D. Hilbert, Die Grundlagen der Physik (Erste Mitteilung) (Nachr. G¨ottingen, 1915) 395–407. ´ Cartan, Lec¸ons sur les invariants int´egraux (based on lectures given in 1920-21 in [2] E. Paris, Hermann, Paris 1922, reprinted in 1958). [3] Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick, Analysis, Manifolds and Physics (2nd ed., North-Holland, Amsterdam 1982). [4] I. Agricola and Th. Friedrich, Global analysis: Differential Forms in Analysis, Geometry and Physics (transl. from the 2001 German edition, Graduate Studies in Mathematics, vol. 52, American Mathematical Society, Providence, RI, 2002). ´ [5] W. Slebodzi´ nski, Sur les e´ quations de Hamilton, Bull. Acad. Roy. de Belg. 17 (1931) 864–870.

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[6] Th. de Donder, Th´eorie des invariants int´egraux (Gauthier–Villars, Paris 1927). [7] D. van Dantzig, Zur allgemeinen projektiven Differentialgeometrie, Proc. Roy. Acad. Amsterdam 35 (1932) Part I: 524–534; Part II: 535–542. [8] J. A. Schouten, Ricci-Calculus (2nd ed., Springer-Verlag, Berlin 1954). [9] A. Lichnerowicz, G´eom´etrie des groupes de transformations (Dunod, Paris 1958). [10] L. Rosenfeld, Sur le tenseur d’impulsion-´energie, M´emoires Acad. Roy. Belg., Classe des Sciences 18 Fasc. 6 (1940) 1–30. [11] A. Trautman, Sur les lois de conservation dans les espaces de Riemann, In: Les ´ du CNRS, Paris 1962) Th´eories Relativistes de la Gravitation (Royaumont 1959, Ed. 113–116. [12] A. Trautman, Einstein–Cartan theory, In: Encycl Math. Phys. ((J.-P. Franc¸oise, G. L. Naber and S. T. Tsou, Eds.) Elsevier, Oxford, vol. 2, 2006) 189–195. [13] L. Corwin, S. Sternberg and Y. Neeman, Graded Lie algebras in mathematics and physics, Rev. Mod. Phys. 47 (1975) 573–603. [14] A. Fr¨ohlicher and A. Nijenhuis, Theory of vector-valued differential forms Part I, Indag. Math. 18 (1956) 338–359. [15] I. Kol´aˇr, P. W. Michor and J. Slov´ak, Natural Operations in Differential Geometry (Springer-Verlag, Berlin, 1993).

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 261-275

Chapter 15

S ECOND -O RDER D IFFERENTIAL E QUATION F IELDS WITH S YMMETRY∗ M. Crampin1† and T. Mestdag2‡ Department of Mathematical Physics and Astronomy Ghent University, Krijgslaan 281, B-9000 Ghent, Belgium 2 Department of Mathematics, University of Michigan 530 Church Street, Ann Arbor, MI 48109, USA

1

Abstract We examine the reduction of a system of second-order ordinary differential equations which is invariant under the action of a symmetry group. We describe the reduced system, and show how the integral curves of the original system can be reconstructed from the reduced dynamics. We then specialize to invariant Lagrangian systems. We compare and contrast two approaches to reduction in this case. The first leads to the so-called Lagrange-Poincar´e equations. The second involves an extension of Routh’s reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. Throughout we use a new method of analysis based on adapted frames and associated quasi-velocities.

2000 Mathematics Subject Classification. 34A26, 37J15, 53C05, 70H03 Key words and phrases. Dynamical system, Lagrangian system, Lagrange-Poincar´e equations, symmetry, Routhian, reduction, reconstruction.

1.

Introduction

The concept of symmetry plays an important role in a great number of applications in dynamics. Symmetry properties of dynamical systems have been studied intensively in ∗

For Demo Krupka on his 65th birthday E-mail address: [email protected] ‡ E-mail address: [email protected] †

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recent years: see for example the survey in the recent monograph [6] by Marsden et al., as well as the more long-established reference [7]. Perhaps the most important aspect of symmetry is its use in reduction. When a dynamical system has a Lie group of symmetries, which is to say that considered as a vector field on some manifold it is invariant under the action of the group on the manifold, then the corresponding equations of motion can be reduced to a new set of equations with fewer unknowns. The working assumption is that the reduced system will be simpler to deal with than the original one. The bulk of the literature on symmetry in dynamics concentrates on the Hamiltonian description of dynamical systems with symmetry, in which the theory of Poisson manifolds plays the main role. Less well-known is symmetry reduction for Lagrangian systems. It is the latter that is at the core of the present paper. There are in fact accounts of several different Lagrangian reduction theories to be found in the literature. For example, one distinctive reduction method applies when the configuration space is itself a Lie group; it is called Euler-Poincar´e reduction. The particular issue that we will be concerned with, however, is the following. In rough terminology, the invariance of the Lagrangian leads via the Noether theorem to a set of conserved quantities (the components of momentum). There are two alternative broad types of Lagrangian reduction theory, which differ in whether or not the existence of these conserved quantities is explicitly taken into account in the reduction process. The more direct approach, which effectively ignores conservation laws, is called Lagrange-Poincar´e reduction (and includes Euler-Poincar´e reduction as a special case). Taking account of momentum conservation leads to Routh’s procedure. For more details and some comments on the history of these reduction theories, see e.g. [6] and [8]. One main purpose of our paper is to compare and contrast Lagrange-Poincar´e reduction and Routh’s procedure. Over the last couple of years we have been developing our own techniques for analysing symmetry and reduction of dynamical systems. These techniques, while being well adapted to the discussion of Lagrangian systems, are not restricted to them; in this respect they are different from the techniques usually found in the literature. In fact our techniques are designed to apply to any dynamical system, that is, any vector field, which is invariant under a Lie group. The basic ideas, which we exploit throughout the paper, are most succinctly explained in the simplest context, that of a first-order dynamical system or plain and unadorned vector field. We discuss this case in the following section. The transition to Lagrangian systems, that is, to dynamical systems of Euler-Lagrange type, is made via the consideration of second-order systems. By a second-order system we mean a second-order differential equation field, that is to say, a vector field on a tangent bundle belonging to that special class whose integral curves satisfy a system of second-order differential equations. We describe the general second-order theory in Section 3. Those particular second-order systems defined by Lagrangians are discussed in Section 4, where the two approaches, LagrangePoincar´e and Routh, are explained. We end with an example of a second-order system with symmetry, reduced by all three methods. This paper is in effect a survey and summary of work which has been presented in greater detail in a number of other articles; we draw the reader’s attention in particular to [4], [5] and [9]. Throughout the paper, symmetry groups are supposed to act as follows. We have a

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differentiable manifold M and a connected Lie group G which acts freely and properly to the left on M , so that M is a principal G-bundle. We denote the base by M/G, the projection by π M : M → M/G, and the action by (x, g) 7→ ψgM x. The Lie algebra of G is denoted by g, and for ξ ∈ g, ξ˜ is the fundamental vector field corresponding to ξ (the vector M field whose flow is t 7→ ψexp(tξ) ).

2.

The First-Order Case

Suppose we have a first-order dynamical system, represented by a vector field Z on the manifold M , which is invariant under a symmetry group G acting on M as described above. We can of course express the condition of invariance in terms of the action ψ M ; alternatively, ˜ Z] = 0 for all ξ ∈ g, and conversely when this we can note that if Z is invariant then [ξ, differential condition holds and G is connected (as we assume) then Z is invariant. We will find it convenient always to use the differential version of the condition for invariance. The invariance of Z implies that it ‘passes to the quotient’. That is to say, there is a ˇ which is π M -related to Z: Zˇ = π∗M Z. We call Zˇ well-defined vector field on M/G, say Z, the reduced dynamical system. Two questions arise: reduction: how to describe the reduced system Zˇ explicitly and conveniently; ˇ to obtain an integral curve of Z in a reconstruction: how, given an integral curve of Z, systematic fashion.

2.1.

Reduction

In order to formulate a simple description of the reduced dynamics we introduce and work with a local frame {Ea , Xi }, where the Ea , a = 1, 2, . . . , dim G, are tangent to the fibres of π M : M → M/G, the Xi , i = 1, 2, . . . , dim(M/G) are transverse to the fibres, and all of the members of the basis are G-invariant. We define the Ea as follows. Let {ea } be a basis for g, e˜a the corresponding fundac e c are the structure constants of mental vector fields. Then [˜ ea , e˜b ] = −Cab ˜c , where the Cab g with respect to the given basis. It is clear that in general the e˜a are not invariant. We set Ea = Aba e˜b , and enquire under what conditions on the coefficients Aba the Ea are invariant. We have c [˜ ea , Eb ] = e˜a (Acb ) − Cad Adb e˜c , so the Ea are invariant if and only if

c e˜a (Acb ) = Cad Adb .

The integrability conditions for these equations are satisfied by virtue of the Jacobi identity. There are therefore local solutions, for which the matrix A = (Aba ) is non-singular, and for which A is the identity matrix on some specified local section of π M . Such a local section determines a local trivialization M ≃ G×M/G of M ; identifying the fibres with G, we see that each Ea corresponds to a left-invariant vector field on G. Each e˜a , on the other hand, corresponds to a right-invariant vector field on G (which explains the sign in the expression

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M. Crampin and T. Mestdag

for the bracket); A(g) is the matrix of ad(g) with respect to the basis {ea }. In the literature, {˜ ea } is sometimes referred to as the moving frame, {Ea } as the body-fixed frame (see for example [1]). To define the part of the frame transverse to the fibres, we assume that we have at our disposal a principal connection on the principal G-bundle M ; we take Xi to be the horizontal lift with respect to the connection of a member of some local basis of vector fields on M/G. In particular, we may (and generally will) take this to be a coordinate basis. We may now write Z = Z a Ea + Z i Xi . Since Z, Ea and Xi are all invariant, so also are the coefficients Z a and Z i . We may therefore regard the Z i (in particular) as functions on M/G, and we have ∂ π∗M Z = Zˇ = Z i i , ∂x where the xi are coordinates on M/G. The reduced equations are simply x˙ i = Z i (x).

2.2.

Reconstruction

ˇ say t 7→ zˇ(t) (a curve in M/G). We have to find an Suppose given an integral curve of Z, integral curve of Z, t 7→ z(t) (a curve in M ), over zˇ (so that π M ◦ z = zˇ). We proceed as follows. Take any lift of zˇ to M , t 7→ ζ(t) (so that π M ◦ ζ = zˇ). Then M ζ(t). The next questions therefore are there is a curve t 7→ g(t) ∈ G such that z(t) = ψg(t) how to lift zˇ(t) to ζ(t) in a systematic fashion, and having done so, how to find g(t). Assume as before that we have a principal connection on M , with connection form ω (a g-valued 1-form on M ). Then we can take ζ(t) to be a horizontal lift of zˇ(t). We can now derive a differential equation for g(t). First, differentiate the equation for z(t): z(t) ˙ =

M ψg(t)∗

^ ˙ + ϑ( ζ(t) g(t))| ˙ ζ(t) ,

where ϑ is the Maurer-Cartan form of G (i.e. g −1 g˙ for a matrix group). We want z(t) to be an integral curve of Z, so M z(t) ˙ = Zz(t) = ψg(t)∗ Zζ(t) by invariance. Thus ^ ˙ + ϑ( Zζ(t) = ζ(t) g(t))| ˙ ζ(t) . This formula expresses Zζ(t) in terms of its horizontal and vertical components. We pick out the vertical component, or in other words apply ω: ϑ(g(t)) ˙ = ωζ(t) (Z). The right-hand side is a curve in g, so this is an equation in g, and it has a unique solution for g(t) with g(0) = id. (This is evident for a matrix group, for which the equation is g˙ = gωζ (Z).) M ζ(t) is an integral curve of Z. It is the integral curve through ζ(0): to Then z(t) = ψg(t) find the integral curve over zˇ through some other point in the fibre over zˇ(0), say ψgM ζ(0), we merely have to left translate z(t), that is, take ψgM z(t).

Second-Order Differential Equation Fields with Symmetry

3.

265

The Second-Order Case

A second-order dynamical system determines and is determined by a vector field Γ on a tangent bundle T M , which has the form Γ = uα

∂ ∂ + Γi (x, u) α α ∂x ∂u

when expressed in terms of natural coordinates (xα , uα ); such a vector field is called a second-order differential equation field. In order to consider symmetries of a second-order differential equation field we must extend the group action from M to T M . Suppose G acts on M as before; then the induced M u). (Transformations of T M of action of G on T M is given by ψgT M (x, u) = (ψgM x, ψg∗ this form are sometimes called point transformations.) The fundamental vector fields of the induced action are the complete lifts of the fundamental vector fields of the action on M , which we denote by ξ˜C . Moreover, T M is a principal G-bundle, and we denote by π T M : T M → T M/G the projection (which is not to be confused with the projection T M → M , which we denote by τ .) We assume now that the second-order differential equation field Γ is invariant under the induced action of G: [ξ˜C , Γ] = 0 for all ξ ∈ g.

3.1.

Reduction

We will make extensive use of the complete and vertical lifts of vector fields on M to T M : we denote the vertical lift of a vector field X on M by X V (and its complete lift by X C as above). We recall the following formulae for the brackets of such lifts: [X C , Y C ] = [X, Y ]C ,

[X V , Y C ] = [X, Y ]V ,

[X V , Y V ] = 0.

From these formulae it is clear that the complete and vertical lifts of a G-invariant vector field on M are both invariant under the induced action of G on T M . So if we take an invariant local basis {Ea , Xi } on M as before, then {EaC , XiC , EaV , XiV } is an invariant local basis of vector fields on T M . We now introduce new fibre coordinates with respect to τ , adapted to the invariant basis, which we call quasi-velocities. For any vector field basis {Zα } on M we denote by v α the components of u ∈ Tx M with respect to {Zα |x }: so u = v α Zα |x . Considered as functions on T M the v α are fibre coordinates; these are the quasi-velocities corresponding to the basis {Zα }. Alternatively, let {θα } be the 1-form basis dual to {Zα }; each θα defines a fibre-linear function on T M , θˆα ; then v α = θˆα . We denote by (v a , v i ) the quasi-velocities corresponding to {Ea , Xi }. We need expressions for the derivatives of the quasi-velocities with respect to the members of the invariant basis {EaC , XiC , EaV , XiV }. To find them, the following two fomulae are indispensible: ˆ = Ld ˆ = τ ∗ θ(Z). Z C (θ) Z V (θ) Z θ, i For example, we have e˜Ca (v i ) = L[ e˜a θ = 0 (since the basis dual to an invariant basis is also invariant).

266

M. Crampin and T. Mestdag We also need expressions for the pairwise brackets of {Ea , Xi }: we have [Ea , Eb ] = and we set

c E , Cab c

a [Xi , Xj ] = Kij Ea ,

[Xi , Ea ] = Xi (Aba )˜ eb = Υbia Eb .

It is worth noting that since the vector fields Xi and Ea are G-invariant, so are their brackets, a and Υb . and so are the coefficients Kij ia i Let (x ) be coordinates on M/G, as before. Then we find that e˜Ca (xi ) = 0, EaC (xi ) = 0, EaV (xi ) = 0, XiC (xj ) = δij , XiV (xj ) = 0,

e˜Ca (v i ) = 0, EaC (v i ) = 0, EaV (v i ) = 0, XiC (v j ) = 0, XiV (v j ) = δij ,

e˜Ca (v b ) = 0, b vc, EaC (v b ) = Υbia v i + Cac V b b Ea (v ) = δa , a v j − Υa v b , XiC (v a ) = −Kij ib V a Xi (v ) = 0.

From the first line, (xi , v i , v a ) define coordinates on T M/G. The invariant vector fields of the basis project onto T M/G, and we can read off the coordinate expressions for their projections from the formulae above: π∗T M EaC = π∗T M XiC =

b c Υbia v i + Cac v

∂

∂v b

,

π∗T M EaV =

∂ ∂ a j a b − K v + Υ v , ij ib ∂xi ∂v b

∂ , ∂v a

π∗T M XiV =

∂ . ∂v i

Since Γ is a second-order differential equation field, Γ = v a EaC + v i XiC + Γa EaV + Γi XiV . Each term is invariant, so Γa and Γi define functions on T M/G. We have ∂ ∂ + vi i ∂v b ∂x ∂ ∂ ∂ a j − v i Kij v + Υaib v b + Γ a a + Γi i b ∂v ∂v ∂v i ∂ i ∂ a ∂ = v +Γ +Γ . ∂xi ∂v i ∂v a

b c ˇ = v a (Υbia v i + Cac π∗T M Γ = Γ v )

The reduced equations are x˙ i = v i , v˙ i = Γi (xj , v j , v b ), v˙ a = Γa (xj , v j , v b ), or x ¨i = Γi (xj , x˙ j , v b ),

v˙ a = Γa (xj , x˙ j , v b );

they are of mixed first- and second-order type. So far as we are aware, the study of the reduction of general second-order dynamical systems with symmetry by methods similar to ours has been attempted by other authors only for single symmetries (that is, 1-dimensional symmetry groups), in [2]. For a more detailed account of our approach, see [4].

Second-Order Differential Equation Fields with Symmetry

3.2.

267

Reconstruction

In order to carry out reconstruction using the method described in Section 2 we need a principal connection on the bundle π T M : T M → T M/G. We have already assumed that we have at our disposal a principal connection on π M : M → M/G. There is in fact a simple method of lifting such a connection to one on π T M . The initial connection is specified by its connection form ω. We show that the pull-back τ ∗ ω of ω to T M by the tangent bundle projectionτ is the connection form of a principal connection on the principal G-bundle π T M . Clearly, τ ∗ ω is a g-valued 1-form on T M . The action of G on T M is τ -related to the action on M . Likewise, for any ξ ∈ g the fundamental vector ˜ the fundamental vector field field ξ˜C corresponding to the action on T M is τ -related to ξ, corresponding to the action on M . Thus ˜ = ξ, τ ∗ ω(ξ˜C ) = ω(τ∗ ξ˜C ) = ω(ξ) while ψgT M ∗ τ ∗ ω = τ ∗ ψgM ∗ ω = ad(g −1 )τ ∗ ω, as required. The connection defined by τ ∗ ω is called the vertical lift of the original connection, and its connection 1-form is denoted by ω V . When we use ω V in the reconstruction process, the right-hand side of the reconstruction equation is ω V (Γ). The special natures of Γ (that it is a second-order differential equation field) and ω V (that it is a vertical lift connection) now come into play. For at any point u ∈ T M , ωuV (Γ) = ωτ (u) (τ∗ Γu ) = ωτ (u) (u); that is to say, ωuV (Γ) is just the vertical part of u (considered as an element of g), and in particular is the same for all G-invariant second-order differential equation fields on T M .

4.

Lagrangian Systems

We now suppose that we are dealing with a second-order dynamical system Γ defined by a regular Lagrangian L on T M . Thus Γ is the Euler-Lagrange field of L, and satisfies the Euler-Lagrange equations, which in terms of coordinates (xα , uα ) can be written Γ

∂L ∂uα

−

∂L = 0. ∂xα

We assume that L is regular, which is to say that its Hessian with respect to the fibre coordinates, the symmetric matrix with entries ∂2L , ∂uα ∂uβ is non-singular. Then Γ is uniquely determined by the Euler-Lagrange equations (and the fact that it is a second-order differential equation field). In order to use the methods described in the previous sections we have to express the Euler-Lagrange equations in terms of a vector field basis on M which is not of coordinate type. With respect to the basis {Zα } they take the form Γ(ZαV (L)) − ZαC (L) = 0.

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Assume that the regular Lagrangian L is G-invariant: ξ˜C (L) = 0. Then the EulerLagrange field Γ is also G-invariant, as one would expect. We wish to carry out a reduction, and to express the reduced equations in terms of an appropriate reduced version of the Lagrangian. As we mentioned in the Introduction, there are in fact two different ways of proceeding. In the first, which is called Lagrange-Poincar´e reduction, we work with the invariant basis {Ea , Xi }, as before. The Euler-Lagrange equations become Γ(XiV (L)) − XiC (L) = 0,

Γ(EaV (L)) − EaC (L) = 0,

ˇ on T M/G. and the reduced equations determine a vector field Γ The second approach could be characterized as making more direct use of the particular properties of the Euler-Lagrange formalism. This time we use a mixed basis {˜ ea , Xi } (mixed in the sense that only part of it is invariant); since e˜Ca (L) = 0, the Euler-Lagrange equations are Γ(XiV (L)) − XiC (L) = 0, Γ(˜ eVa (L)) = 0. Thus the momentum, whose components are e˜Va (L), is conserved. The first step in the reduction process in this case (if ‘reduction’ is the right word) just consists in restriction to a level set of momentum. The process as a whole is called Routh’s procedure; it generalizes the elimination of the momentum conjugate to a cyclic coordinate which was Routh’s original version of the procedure [11].

4.1.

Lagrange-Poincar´e Reduction

ˇ on T M/G. The Euler-Lagrange equations Since it is invariant, L defines a function L reduce directly to ˇ X ˇ iV (L)) ˇ −X ˇ iC (L) ˇ = 0, Γ(

ˇ E ˇaV (L)) ˇ −E ˇaC (L) ˇ =0 Γ(

ˇ C = π∗T M X C etc. Using the formulae from the previous section we obtain where X i i ˇ ˇ ˇ a j a b ∂L ˇ ∂ L − ∂ L = −(Kij v + Υ v ) Γ ib ∂v i ∂xi ∂v a ! ˇ ˇ b c ∂L ˇ ∂ L = (Υbia v i + Cac Γ v ) b; a ∂v ∂v !

and as before,

ˇ = v i ∂ + Γi ∂ + Γa ∂ . Γ ∂xi ∂v i ∂v a These are the Lagrange-Poincar´e equations [3], though they are usually written with d/dt ˇ see also [9]. in place of Γ;

4.2.

Routh’s Procedure

We set pa = e˜Ca (L). Considered as a vector, (pa ) takes its values in g∗ , the dual of the Lie algebra: it is the (generalized) momentum. Since Γ(pa ) = 0, the vector field Γ is tangent

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269

to the level sets of momentum; we will concentrate on its restriction to one level set, say Nµ : pa = µa . We work now with the mixed basis {˜ ea , Xi }. The quasi-velocities are (˜ v a , v i ), where v˜a = Aab v b ; the v˜a are not invariant. The pairwise brackets of elements of the basis are [˜ ea , Xi ] = 0,

a [Xi , Xj ] = Rij e˜a ,

a b Rij = Aba Kij .

a as the components of curvature of the con(The expression for [Xi , Xj ] identifies the Rij nection on π M , regarded as a g-valued 2-form on M/G.) The derivatives of the quasi-velocities are

XiC (v j ) = 0, XiV (v j ) = δij , e˜Ca (v i ) = 0, e˜Va (v i ) = 0,

a vj , XiC (˜ v a ) = −Rij XiV (˜ v a ) = 0, C b v e˜a (˜ v b ) = Cac ˜c , V b b e˜a (˜ v ) = δa .

Set gab = e˜Va (pb ) = e˜Va (˜ eVb (L)). Since vertical lifts commute, gba = gab . We assume that the symmetric matrix (gab ) is everywhere non-singular. Then e˜Va is transverse to Nµ , and in principle we can solve the equations pa = µa for v˜a . Thus restricting to a level set of momentum is a form of reduction, in the sense that by doing so we reduce the number of variables, and presumably thereby the difficulty of the problem. It is however a somewhat different form of reduction from those discussed so far: reduction by restriction rather than projection. The gab are in fact components of the Hessian of L. The Hessian of L can be defined in a coordinate-independent way as the symmetric covariant 2-tensor g along τ given by g(u, v) = uV (v V (L)), for any vectors u, v at the same point of M . We have gab = g(˜ ea , e˜b ). We may use gab to define vector fields tangent to Nµ . Denote by (g ab ) the matrix inverse to (gab ). For any vector field Y on T M , the vector field Y − g ab Y (pa )˜ eVb annihilates pa and is therefore tangent to each level set of momentum. In particular, set ¯ iC = XiC − g ab XiC (pb )˜ X eVa = XiC − Pia e˜Va ¯ iV = XiV − g ab XiV (pb )˜ X eVa = XiV − Qai e˜Va . ¯ C, X ¯ V are tangent to Nµ . We have Then X i i a j ¯ iC (˜ X v a ) = −Rij v − Pia ,

¯ iV (˜ X v a ) = −Qai .

We now derive some Euler-Lagrange-like equations which determine the restriction of Γ to the level set of momentum Nµ . These equations involve, not the Lagrangian itself, but a modification of it called the Routhian, which is given by R = L − pa v˜a . Now a j ¯ iC (˜ ¯ iC (L) − pa (X ¯ iC (L) + Pia pa = X v ) v a ) + Rij XiC (L) = X C a j ¯ i (R) − pa Rij v ; = X

¯ iV (L) + Qai pa = X ¯ iV (L) − pa X ¯ iV (˜ XiV (L) = X va ) ¯ iV (R). = X

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But Γ(XiV (L)) − XiC (L) = 0, and Γ is tangent to Nµ . So if Rµ is the restriction of the Routhian to Nµ we have a j ¯ iV (Rµ )) − X ¯ iC (Rµ ) = −µa Rij Γ(X v .

These are the required equations; we call them the generalized Routh equations. The generalized Routh equations may appear to be straightforwardly second-order differential equations, unlike the other reduced equations for second-order differential equation fields, which are mixed first- and second-order equations. This appearance is deceptive. In the first place, the generalized Routh equations (when expressed explicitly as differential equations) are equations on Nµ , not T M/G as is the case for the other reduced equations. Now Nµ can be locally identified with M ×M/G T (M/G). For local coordinates on Nµ we may take (xi , θa , v i ), where (θa ) are fibre coordinates on M , so that (xi , θa ) are coordinates on M and (xi , v i ) coordinates on T (M/G). The quasi-coordinates (v i , v a ) on T M are linear combinations of x˙ i and θ˙a , and in fact v i = x˙ i . So we can express v a in terms of x˙ i and θ˙a . On T M the resulting expression is an identity; but on restricting to Nµ , the equations pa = µa , when expressed in this way in terms of x˙ i and θ˙a , become additional implicit first-order differential equations, which we may regard as equations for the θ˙a (since the equations v i = x˙ i are already subsumed in the representation of the generalized Routh equations as second-order equations). The level set Nµ is not in general G-invariant: ξ˜C is not in general tangent to Nµ . In fact c ξ˜C (pa ) = ξ b e˜Cb (pa ) = ξ b [˜ eCb , e˜Va ](L) = ξ b Cab pc . c µ = 0. The set of ξ ∈ g which satisfy Thus ξ˜C will be tangent to Nµ if and only if ξ b Cab c this condition forms a subalgebra gµ of g. It is in fact the algebra of Gµ , the isotropy group of µ ∈ g∗ under the coadjoint action of G in g∗ . Now Γ|Nµ , Rµ , and the generalized Routh equations, are all invariant by Gµ . We can therefore carry out a further reduction, by Gµ , in the manner described earlier, to obtain a reduced system on Nµ /Gµ . The resulting reduced equations have been called the Lagrange-Routh equations [8]. We do not give the derivation here, but refer the reader to [5], as well as [8], for the details. In fact [8] contains an extensive discussion of the background to Routh’s procedure and its modern generalization. The methods used in this paper are quite different from ours, however, and it deals only with so-called simple mechanical systems. For a more detailed account of all aspects of our approach see [5].

4.3.

Reconstruction

The same method of reconstruction as was described for second-order differential equations in the previous section, namely using the vertical lift connection, can be used for LagrangePoincar´e reduction. For Routh’s procedure it is necessary to carry out reconstruction only for the final stage of reduction by Gµ : an integral curve of the restriction of Γ to Nµ is, after all, an integral curve of Γ. It is not so obvious how to adapt the vertical lift connection to this situation, though it can be done. We will now describe an alternative way of constructing a connection, which is based more closely on the fact that we are dealing with a Lagrangian system, and applies more-or-less directly to both reconstruction problems.

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We consider first the case of a simple mechanical system, which is one in which the Lagrangian takes the simple form L = T − V where T is a kinetic energy function derived from a Riemannian metric g, and V a function on M defining the potential energy. The symmetry group G consists of all isometries of the metric g leaving V invariant. Then the distribution on M consisting of all vectors orthogonal (with respect to g) to the fibres of π M : M → M/G is G-invariant, and defines a principal connection (of which it is the horizontal distribution). This is the so-called mechanical connection on π M . It can be lifted to a principal connection on π T M : T M → T M/G, as before. For the vertical lift of the ˜ = 0 for all ξ ∈ g. mechanical connection, v ∈ Tu T M is horizontal just when gτ (u) (τ∗ v, ξ) This connection can be used for reconstruction in the Lagrange-Poincar´e case. For Routh’s procedure we define the required connection by saying that v ∈ Tu Nµ is horizontal just ˜ = 0 for all ξ ∈ gµ . when gτ (u) (τ∗ v, ξ) In general, we can use the Hessian of L in place of the Riemannian metric. This doesn’t give a connection on M , but does give connections on T M → T M/G and Nµ → Nµ /Gµ . Indeed, since we have (wittingly) used the same symbol, g, for both the metric in the case of a simple mechanical system and the Hessian in general, the definitions are almost identical: the only difference is that in general g is not projectable. For the Lagrange-Poincar´e case, ˜ = 0 for all ξ ∈ g. For the we say that v ∈ Tu T M is horizontal just when gu (τ∗ v, ξ) ˜ = 0 for all ξ ∈ gµ . Routhian case, we say that v ∈ Tu Nµ is horizontal just when gu (τ∗ v, ξ) Both of these specifications define principal connections, which we call collectively the generalized mechanical connection. A fuller account of this construction can be found in [9].

5.

An Example: Wong’s Equations

In this final section we determine the reduced equations for an interesting second-order differential equation field, namely the geodesic field for a Riemannian manifold on which a group G acts freely and properly to the left as isometries. We make the further stipulation that the vertical part of the metric (that is, its restriction to the fibres of π M : M → M/G) comes from a bi-invariant metric on G. The reduced equations in such a case are known as Wong’s equations [3, 10]. We will derive the reduced equations by each of the three methods discussed above. We will denote the metric by g. The fact that the symmetry group acts as isometries means that the fundamental vector fields ξ˜ are Killing fields: Lξ˜g = 0. It follows that the components of g with respect to the members of an invariant basis {Ea , Xi } are themselves invariant. We have a small notational problem to deal with here: we will need to distinguish between the components of g with respect to the fundamental vector fields e˜a and those with respect to the Ea . We will set g(˜ ea , e˜b ) = gab , as before. For g(Ea , Eb ) we will write hab . We set g(Xi , Xj ) = gij . We will use the mechanical connection, which means that g(˜ ea , Xi ) = 0. Since both hab and gij are G-invariant functions, they pass to the quotient; in particular, the gij are the components with respect to the coordinate fields of a metric on M/G, the reduced metric. The further assumption about the vertical part of the metric has the following implications. It means in the first place that LEc g(Ea , Eb ) = 0 (as well as Le˜c g(Ea , Eb ) = 0), and secondly that the hab must be independent of the coordinates xi on M/G, which is to

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say that they must be constants. From the first condition, taking into account the bracket c E , we easily find that the h must satisfy h C d + h C d = 0. relations [Ea , Eb ] = Cab c ab ad bc bd ac It is implicit in our choice of an invariant basis that we are working in a local trivialization of M → M/G. Then e˜a , Ea and Aba are all objects defined on the G factor, and so are independent of the xi . We may write Xi =

∂ − γia Ea ∂xi

for some coefficients γia which are clearly G-invariant; moreover b [Xi , Ea ] = γic Cac Eb = Υbia Eb . b , and therefore h Υc + h Υc = 0. Thus Υbia = γic Cac ac ib bc ib The second-order differential equation field Γ of interest is the geodesic field of the Riemannian metric g. To find the reduced equations by the direct method we have to express Γ in terms of the invariant basis, and for this purpose we need the connection coefficients of the Levi-Civita connection in terms of this basis. Using the data above in the standard Koszul formulae for the Levi-Civita connection coefficients of g with respect to the basis i , which are just the connection {Ea , Xi } we find that the only non-zero coefficients are Γjk coefficients of the Levi-Civita connection of the reduced metric gij , and a a Γbc = 21 Cbc ,

a Γjb = Υajb ,

a a Γjk = 12 Kjk ,

c Γjbi = − 21 g ik hbc Kjk = Γbji .

It follows that Γ = v i XiC + v a EaC

i j k − Γjk v v + (Γjbi + Γbji )v j v b + Γbci v b v c XiV

a j k a a j b a b c − Γjk v v + (Γjb + Γbj )v v + Γbc v v EaV

i j k c j b = v i XiC + v a EaC − Γjk v v − g ik hbc Kjk v v XiV − Υajb v j v b EaV .

The reduced vector field on T M/G is therefore

∂ ∂ i j k ik c j b ˇ = v i ∂ − Γjk Γ v v − g h K v v − Υajb v j v b a bc jk i i ∂x ∂v ∂v

and the reduced equations are

i j k c j b x ¨i + Γjk x˙ x˙ = g ik hbc Kjk x˙ v

v˙a + Υajb x˙ j v b = 0. These are Wong’s equations. (The form of the second of these equations suggests that the Υajb should be regarded as connection coefficients. It is indeed the case that they are: the connection in question is that induced by ω on the adjoint bundle, that is, the vector bundle associated with the principal G-bundle π M by the adjoint action of G on g.) The geodesic equations may also be derived from the Lagrangian L = 21 gαβ uα uβ = 12 gij v i v j + 12 hab v a v b .

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It is of course G-invariant. We may therefore apply Lagrange-Poincar´e reduction, which gives the reduced equations ∂gjk j k d a j (gij v j ) − 12 v v = −(Kij v + Υaib v b )hac v c dt ∂xi d b c (hab v b ) = (Υbia v i + Cac v )hbd v d . dt b h is skew-symmetric in c and d, so the Now Υaib hac is skew-symmetric in b and c, and Cac bd final terms in each equation vanish identically, and we may write the equations in the form

j k l gij x ¨j + Γkl x˙ x˙

hab v˙ b + Υbic x˙ i v c

c j b = −hbc Kij x˙ v

= 0,

using the skew-symmetry of Υcib hac again in the second equation. These equations are c is of course skew-symmetric in equivalent to the ones obtained by the direct method (Kij its lower indices). In order to use Routh’s procedure we must rewrite the Lagrangian in terms of the quasivelocities associated with the mixed basis: it is given by L = 21 gij v i v j + 12 gab v˜a v˜b . The momentum is given by pa = gab v˜b , and the Routhian by R = L − pa v˜a = 12 gij v i v j − 12 g ab pa pb . ¯ V (R) and X ¯ C (R). In fact, it is easy to see that X ¯ V (R) = The next problem is to calculate X i i i j C ¯ gij v . The calculation of Xi (R) reduces to the calculation of Xi (gij ) and Xi (g ab ). The first is straightforward. For the second, we note that gab = A¯ca A¯db hcd , where (A¯ba ) is the matrix inverse to (Aba ); since the right-hand side is independent of the xi , so is gab , and so equally is g ab . It follows that ¯ iC (R) = 1 ∂gjk v j v k − 1 γic Ec (g ab )pa pb . X 2 ∂xi 2 b + g be C a ), from Killing’s equations. Using the relation Now Ec (g ab ) = −Adc (g ae Cde de between gab and hab , and the fact that ad is a Lie algebra homomorphism, we find that e d Ec (g ab ) = −Aad Abe (hdf Ccf + hef Ccf ).

The expression in the brackets vanishes, as follows easily from the properties of hab . Thus the generalized Routh equation is ∂gjk j k d j k l j a j (gij v j ) − 21 v v = g v ˙ + Γ v v = −µa Rij v . ij kl dt ∂xi

a = Aa K b , and µ = g v b b a c b ¯c But Rij a ab ˜ = Aa hbc v , so µa Rij = hbc Kij v . The generalized b ij Routh equation is therefore equivalent to

j k l c j b gij x ¨j + Γkl x˙ x˙ = −hbc Kij x˙ v

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again. On the other hand, the constancy of µa gives hbc

d ¯c b (A v ) = 0. dt a

If we are to understand this equation in the present context, we evidently need to calculate A˙ ba . Now b A˙ ba = v i Xi (Aba ) + v˜c e˜c (Aba ) = v i Υcia Abc + v˜c Ccd Ada . It follows that hbc

d ¯c (A ) = −hbc A¯da A¯ce A˙ ed = −hbc A¯da A¯ce (v i Υfid Aef + v˜f Cfeg Agd ) dt a c = −hbc A¯da (v i Υcid + v e Ced ),

where in the last step we have again used the fact that ad is a Lie algebra homomorphism. Now from the skew-symmetry properties of hab we obtain hbc

d ¯c c (A ) = hcd A¯da (v i Υcib + v e Ceb ), dt a

and therefore

d ¯c b (A v ) = hcd A¯da (v˙ c + Υcib v i v b ). dt a The first-order part of Wong’s equations is thus equivalent to the constancy of momentum. hbc

Acknowledgement The first author is a Guest Professor at Ghent University: he is grateful to the Department of Mathematical Physics and Astronomy at Ghent for its hospitality. The second author is currently a Research Fellow at The University of Michigan through a Marie Curie Fellowship. He is grateful to the Department of Mathematics for its hospitality. He also acknowledges a research grant (Krediet aan Navorsers) from the Fund for Scientific Research - Flanders (FWO-Vlaanderen), where he is an Honorary Postdoctoral Fellow.

References [1] A. M. Bloch, Nonholonomic Mechanics and Control (Interdisciplinary Applied Mathematics 24, Springer 2003). [2] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems (Texts in Applied Mathematics 49, Springer 2004). [3] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian Reduction by Stages (Memoirs of the American Mathematical Society 152, AMS 2001). [4] M. Crampin and T. Mestdag, Reduction and reconstruction aspects of second-order dynamical systems with symmetry, preprint (2006); available at maphyast.ugent.be.

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[5] M. Crampin and T. Mestdag, Routh’s procedure for non-Abelian gymmetry groups, to appear in J. Math. Phys.; arXiv:0802.0528. [6] J. E. Marsden, G. Misiolek, J-P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages (Lecture Notes in Mathematics 1913, Springer 2007). [7] J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry (Texts in Applied Mathematics 17, Springer 1999). [8] J. E. Marsden, T. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys. 41 (2000) 3379–3429. [9] T. Mestdag and M. Crampin, Invariant Lagrangians, mechanical connections and the Lagrange-Poincar´e equations, to appear in J Phys. A: Math. Theor.; arXiv:0802.0146. [10] R. Montgomery, Canonical formulations of a classical particle in a Yang-Mills field and Wong’s equations, Lett. Math. Phys. 8 (1984) 59–67. [11] E. J. Routh, A Treatise on the Stability of a Given State of Motion (MacMillan 1877, available on google.books.com).

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 277-291

Chapter 16

D IMENSIONAL R EDUCTION OF C URVATURE -D EPENDENT C ENTRAL P OTENTIALS ON S PACES OF C ONSTANT C URVATURE 1∗ 1† Jos´e F. Carinena , Manuel F. Ranada and Mariano Santander2‡ ˜ ˜ 1 Departamento de F´ısica Te´orica, Facultad de Ciencias Universidad de Zaragoza, 50009 Zaragoza, Spain 2 Departamento de F´ısica Te´orica, Facultad de Ciencias Universidad de Valladolid, 47011 Valladolid, Spain

Abstract The motion of a particle on a curvature-dependent central potential in a configuration spaces of constant curvature will be described, as well as the corresponding reduction process.

2000 Mathematics Subject Classification. 37J35, 70H06, 37J15, 70G45 Key words and phrases. Spaces of constant curvature, central potentials, spheres and hyperbolic planes.

1.

Introduction and Motivation

A very well-known result of classical mechanics in three-dimensional Euclidean space is that the motion of a particle in a central force problem takes place in a plane through the centre of force. One can also ask what happens in spaces of constant curvature. Actually, in the usual Euclidean case the reasoning turns out to be very simple because both the position vector and the linear momentum vector of the particle lie always in a plane that is orthogonal to the angular momentum vector, which is a fixed vector in space [1, 2, 3]. This fact however does not remain true anymore in the general case of a nonzero constant ∗

E-mail address: [email protected] E-mail address: [email protected] ‡ E-mail address: [email protected] †

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Jos´e F. Cari˜nena, Manuel F. Ra˜nada and Mariano Santande

curvature Riemannian space. It is therefore necessary to use a more geometric approach to mechanics [4, 5] and to handle geometric tools for dealing with such a problem. The theory of central potentials in constant curvature spaces has been developed by many authors using different approaches (see Refs. [6]-[22]) but most cases are focused on the Kepler problem, the hydrogen atom or the harmonic oscillator, with some extensions to other particular Hamilton–Jacobi separable (or super-separable) potentials. We have recently studied in Ref. [23] the theory of central potentials in two-dimensional spaces with a constant curvature, the sphere S 2 and the hyperbolic plane H 2 , using a formalism introduced for the theory of super-integrable systems [19], and applied to the harmonic oscillator in previous articles [24, 25]. The corresponding quantum case has also been studied in [26]. The surprising fact is that even if apparently the three manifolds look different from a geometric perspective, they share some dynamical properties. For instance, it was proved in [23] that Binet equation for a central potential is not only valid in the Euclidean space but in all the spaces with a nonzero constant curvature κ too. Once the well-known reduction from motion in a three-dimensional Euclidean space to a motion in a two-dimensional Euclidean plane has been recalled, we set as our aim here to study the curvature dependent generalization of this reduction, which should lead from the motion in a three dimensional constant curvature space to the motion in a two-dimensional constant curvature submanifold of the former. We present all the mathematical expressions using the curvature κ as a parameter. They reduce to the appropriate property for the system on the sphere S 3 , or on the hyperbolic plane H 3 , when particularised for κ > 0, or κ < 0, respectively. The Euclidean case arises as the particular case κ = 0. The use of not only Riemannian but also pseudo-Riemannian metrics is also possible and we shortly describe how to develop a theory in 2d constant curvature spaces and in particular we summarize some results on the ‘curved’ Kepler problem recently published in [27].

2.

Tagged Trigonometric Functions

In order to deal simultaneously with all possible values of κ, it is convenient to use the following ‘tagged’ trigonometric functions   √ √ √1 sin κ x if κ > 0,   cos κ x if κ > 0,   κ x if κ = 0, 1 if κ = 0, Sκ (x) =  Cκ (x) =   √1 sinh√−κ x if κ < 0.  cosh√−κ x if κ < 0, −κ Note that for κ = 1 we have the usual trigonometrical functions, S1 (x) = sin x,

C1 (x) = cos x ,

while for κ = 0 one gets the ‘parabolic’ sine and cosine, S0 (x) = x,

C0 (x) = 1 ,

and finally, for κ = −1, they are the hyperbolic functions: S−1 (x) = sinh x,

C−1 (x) = cosh x .

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These tagged trigonometric functions have slightly modified κ-dependent relations but similar to those of the classical trigonometric ones, namely, 2

2

Cκ (x) + κ Sκ (x) = 1 ,

d Sκ (x) = Cκ (x) , dx

d Cκ (x) = −κ Sκ (x) , dx

and 2

2

Cκ (2x) = Cκ (x) − κ Sκ (x) ,

Sκ (2x) = 2 Sκ (x) Cκ (x) .

Note that in the flat case κ = 0 all Cκ (x) are replaced by 1, while all Sκ (x) are replaced by its variable x. In this sense Cκ (x) may be looked at as a kind of ‘curved’ deformation of the function 1, while Sκ (x) is a deformation of the linear function x.

3.

Central Potentials in 3d Constant Curvature Manifolds

The three-dimensional sphere S 3 of radius one can be seen as a submanifold embedded in the ambient space R4 = (x0 , x1 , x2 , x3 ) through the constraint equation x20 +x21 +x22 +x32 = 1. We can consider a local coordinate system with three orthogonal polar coordinates (r, θ, φ), 0 < r < π, 0 < θ < π, 0 < φ < 2π, given by x0 = cos r ,

x1 = sin r sin θ cos φ ,

x2 = sin r sin θ sin φ ,

x3 = sin r cos θ ,

(1)

i.e. r is defined by x0 = cos r and θ and φ are the usual coordinates in a sphere of radius sin r, x21 + x22 + x32 = sin2 r. The North pole (1, 0, 0, 0) is outside the domain of this chart but it is obtained as the limit when r goes to zero. It can be considered as the origin of these new coordinates. The expression of the induced Riemannian metric in S 3 is given by ds2 = dr2 + sin2 r dθ2 + sin2 r sin2 θ dφ2 ,

(2)

from which the following expression for the kinetic energy T in S 3 , the Lagrangian for the free system, is obtained T =

1 2 vr + sin2 r vθ2 + sin2 r sin2 θ vφ2 . 2

(3)

The dynamical vector field is given by X T = vr

∂ ∂ ∂ ∂ ∂ ∂ + vθ + vφ + fr + fφ , + fθ ∂r ∂θ ∂φ ∂vr ∂vθ ∂vφ

(4)

where the functions fr , fθ and fφ are given by  fr        

= sin r cos r (vθ2 + sin2 θ vφ2 ) ,

fθ = −2

        fφ

= −2

cos r vθ vr + sin θ cos θ vφ2 , sin r cos r vφ vr . sin r

(5)

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Jos´e F. Cari˜nena, Manuel F. Ra˜nada and Mariano Santande

The trajectories of this ‘free system’ are then given by the solutions of the system of secondorder differential equations  r¨ = sin r cos r (θ˙2 + sin2 θ φ˙ 2 ) ,         cos r ˙ ¨ ˙2

θ = −2

θ r˙ + sin θ cos θ φ ,

(6)

sin r        ¨ = −2 cos r φ˙ r˙ ,  φ sin r

which coincide with the geodesics of the metric. Note that the curves θ = θ0 , φ = φ0 , r˙ = 1 are geodesics through the North pole and the distance from this point to (r0 , θ0 , φ0 ) is r0 . A natural Lagrangian (kinetic term minus a potential) is then of the following form: L=

1 2 vr + sin2 r vθ2 + sin2 r sin2 θ vφ2 − k U (r, θ, φ) , 2

(7)

and the corresponding dynamics is given in the velocity phase space by the associated Euler-Lagrange vector field X L = vr

∂ ∂ ∂ ∂ ∂ ∂ + Fθ + vθ + vφ + Fr + Fφ , ∂r ∂θ ∂φ ∂vr ∂vθ ∂vφ

(8)

where (Fr , Fθ , Fφ ) denote the forces   Fr    

Fθ

     Fφ

= fr − k Ur′ , k = fθ − U′ , sin2 r θ k U′ . = fφ − 2 sin r sin2 θ φ

(9)

The Euler-Lagrange equations of motion are given by  ˙ φ) ˙ ,  ˙ θ,  r¨ = Fr (r, θ, φ, r,

˙ φ) ˙ , θ¨ = F (r, θ, φ, r, ˙ θ,

θ   ¨ ˙ φ) ˙ . φ = Fφ (r, θ, φ, r, ˙ θ,

(10)

Note that XL is a second-order differential equation vector field, because the Lagrangian is regular. Suppose now that instead of considering R3 we consider a family of three-dimensional submanifolds Q3κ of R4 defined by x20 + κ (x21 + x22 + x23 ) = 1, where κ ∈ R. We can use the above mentioned tagged functions to parametrise such κ-dependent submanifold Q3κ with local coordinates (r, θ, φ), 0 < r < π, 0 < θ < π, 0 < φ < 2π, as follows x1 = Sκ (r) sin θ cos φ ,

x2 = Sκ (r) sin θ sin φ ,

x3 = Sκ (r) cos θ ,

x0 = Cκ (r) , (11)

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i.e. we are using the usual spherical coordinates (θ, φ) on the sphere of radius Sκ (r) with the ambient metric. Note however that in the induced metric the curvature is κ. In this way, if we use the notation of Q3κ for representing the three three-dimensional spaces with constant curvature κ, then the following κ-dependent expression for the induced Riemannian metric on the manifold Q3κ , that is, in Sκ3 , E3 and Hκ3 , respectively, is obtained: ds2 = dr2 + S2κ (r) dθ2 + S2κ (r) sin2 θ dφ2 ,

(12)

which reduces respectively to ds21 = dr2 + (sin2 r) dθ2 + (sin2 r sin2 θ) dφ2 , ds20 = dr2 + r2 dθ2 + (r2 sin2 θ) dφ2 , ds2−1 = dr2 + (sinh2 r) + (sinh2 r sin2 θ) dφ2 , and we can study simultaneously the three-dimensional sphere Sκ3 , the Euclidean space E3 , and the hyperbolic space Hκ3 , just particularising the parameter κ for the values κ > 0, κ = 0 or κ < 0, respectively. In the mechanical setting, the free motion on such manifolds is described by the Lagrangian L0 corresponding to the metric (12) which is given by L0 (κ) =

1 2 vr + S2κ (r) vθ2 + S2κ (r) sin2 θ vφ2 . 2

(13)

The dynamical vector field is XL0 = vr

∂ ∂ ∂ ∂ ∂ ∂ + vθ + vφ + fr + fθ + fφ , ∂r ∂θ ∂φ ∂vr ∂vθ ∂vφ

(14)

where (fr , fθ , fφ ) represent the ‘κ-forces’ characterising the free geodesic motion on the configuration space Q3κ , explicitly given by   fr     

fθ

      fφ

= Sκ (r) Cκ (r) (vθ2 + sin2 θ vφ2 ) , Cκ (r) = −2 vθ vr + sin θ cos θ vφ2 , Sκ (r) cos θ Cκ (r) = −2 vφ vr − 2 vθ vφ , sin θ Sκ (r)

(15)

and the trajectories of the free motion are the solutions of the system of second-order differential equations   r¨ = Sκ (r) Cκ (r) (θ˙2 + sin2 θ φ˙ 2 ) ,     Cκ (r) ˙  ¨ θ r˙ + sin θ cos θ φ˙ 2 , θ = −2 (16) (r) S κ    (r) cos θ  ¨ = −2 Cκ  φ˙ r˙ − 2 θ˙ φ˙ ,  φ sin θ Sκ (r)

which coincide with the equations for the geodesics of the metric. Note that the curves starting from the North pole θ = θ0 , φ = φ0 and r˙ = 1, are geodesics and the distance from this point to (r0 , θ0 , φ0 ) is r0 .

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Jos´e F. Cari˜nena, Manuel F. Ra˜nada and Mariano Santande

A more general natural Lagrangian (κ-dependent kinetic term minus a potential) for a system in the configuration space Q3κ , has the following form L(κ) =

1 2 vr + S2κ (r) vθ2 + S2κ (r) sin2 θ vφ2 − k U (r, θ, φ; κ) , 2

(17)

in such a way that for κ = 0 we recover a standard Euclidean system lim L(κ) =

κ→0

1 2 (v + r2 vθ2 + r2 sin2 θ vφ2 ) − k V (r, θ, φ) , 2 r

V (r, θ, φ) = U (r, θ, φ; 0) .

The dynamics is then represented by the κ-dependent second-order differential equation vector field X L = vr

∂ ∂ ∂ ∂ ∂ ∂ + vθ + vφ + Fr + Fθ + Fφ , ∂r ∂θ ∂φ ∂vr ∂vθ ∂vφ

(18)

where the three forces (Fr , Fθ , Fφ ) are respectively given by   Fr     

Fθ

      Fφ

= Sκ (r) Cκ (r) (vθ2 + sin2 θ vφ2 ) − k Ur′ , k Cκ (r) = −2 vθ vr + sin θ cos θ vφ2 − 2 U′ , Sκ (r) Sκ (r) θ Cκ (r) k θ ′ = −2 vφ vr − 2 cos sin θ vθ vφ − S2κ (r) sin2 θ Uφ , Sκ (r)

(19)

and therefore the trajectories of the motion under the action of a potential k U (r, θ, φ; κ) are the solutions of the system of second-order differential equations   r¨ = Sκ (r) Cκ (r) (θ˙2 + sin2 θ φ˙ 2 ) − k Ur′ ,     k Cκ (r) ˙  ¨ ˙2

Uθ′ , θ r˙ + sin θ cos θ φ − 2 (r) (r) S S κ κ    k cos θ ˙ ˙ k Cκ (r) ˙  ¨  Uφ′ − 2 Uφ′ . φ r˙ − 2 θφ− 2  φ = −2 2 sin θ Sκ (r) Sκ (r) sin θ Sκ (r) sin2 θ θ = −2

(20)

Now, having in mind the meaning of the coordinate r as distance from the point to the North pole, we call central a potential which as a function on the configuration space depends only on the distance r. Our aim is to prove that the dynamical vector field XL arising from a central potential U (r; κ), i.e. described by a Lagrangian L(κ) = with forces

1 2 vr + S2κ (r) vθ2 + S2κ (r) sin2 θ vφ2 − k U (r; κ) , 2    Fr

F

θ   F φ

= fr − k Ur′ , = fθ , = fφ ,

(21)

(22)

is tangent to a κ-dependent foliation M ′ (κ) of the phase space T Q3κ whose leaves are tangent bundles of a family M (κ) of submanifolds of the configuration space Q3κ and that this property is true no matter the possible values of κ.

Dimensional Reduction of Curvature-Dependent Central Potentials

283

We recall that a vector field X ∈ X(P ) in a manifold P is tangent to a submanifold N of P when X|N takes values in T N . If N is determined by a rank one function Ψ : P → R as N = Ψ−1 (0), this is characterized by the weak (in Dirac’s sense) equality XΨ ≈ 0 ⇐⇒ (XΨ)(n) = 0 , ∀n ∈ N .

(23)

e µ,~λ : Q3 → R The family M (κ) is defined as follows: Consider first the function Φ κ κ given by ~

e µ,λ (r, θ, φ) ≡ µ Cκ (r) + Sκ (r) (λ1 sin θ cos φ + λ2 sin θ sin φ + λ3 cos θ) , Φ κ

(24)

and its zero value level set, which is the intersection of Q3κ with the hyperplane µ x0 + fκ λ1 x1 + λ2 x2 + λ3 x3 = 0. Varying the parameters (µ, λ1 , λ2 , λ3 ), we obtain a family M 2 3 2 of either two-dimensional spheres Sκ inside Sκ when κ > 0, or Euclidean planes E inside E3 when κ = 0, or hyperbolic planes Hκ2 inside Hκ3 when κ < 0. Furthermore, in the three cases the centre of the potential is placed at the particular point characterised by r = 0, that in the spherical κ > 0 case, corresponds to the North pole N of Sκ3 . f(κ) given by all Consequently, we can restrict ourselves to the subfamily M (κ) of M f those submanifolds in M (κ) passing through this particular point, i.e. we put µ = 0, and consider the intersection with the zero level set of the function. ~

Φλκ ≡ Sκ (r) (λ1 sin θ cos φ + λ2 sin θ sin φ + λ3 cos θ) .

(25)

So, in the following we restrict our study to this two-parameter family M (κ). The tangency conditions for the second-order vector field XL is ~

~

XL (Φλκ ) ≡ Ψλκ = 0 ,

~

~

XL (Ψλκ ) ≡ Ωλκ ≈ 0 .

(26)

The first tangency condition has a geometric or kinematic meaning, independent of the potential function U (r; k), vr Φ′r + vθ Φ′θ + vφ Φ′φ = 0, (27) while the second condition for the vector field XL leads to vr2 Φ′′rr + vθ2 Φ′′θθ + vφ2 Φ′′φφ + 2vr vθ Φ′′rθ + 2vr vφ Φ′′rφ + 2vθ vφ Φ′′θφ + +(fr − k Ur′ )Φ′r + fθ Φ′θ + fφ Φ′φ ≈ 0 , which splits into a system of seven κ-dependent equations. Three of these equations, Φ′′rθ −

Cκ (r) ′ Φθ = 0 , Sκ (r)

Φ′′rφ −

Cκ (r) ′ Φφ = 0 , Sκ (r)

Φ′′θφ −

cos θ ′ Φ = 0, sin θ φ

(28)

are identically satisfied by the own definition of the function Φ, and the other four equations turn out to be Φ′′rr = −κ Φ ≈ 0 , Φ′′θθ + Sκ (r) Cκ (r) Φ′r = −κ S2κ (r) Φ ≈ 0 , Φ′′φφ + Sκ (r) Cκ (r) sin2 θ Φ′r + sin θ cos θ Φ′θ = −κ S2κ (r) sin2 θ Φ ≈ 0 ,

284

Jos´e F. Cari˜nena, Manuel F. Ra˜nada and Mariano Santande Φ′r Ur′ =

Cκ (r) Φ Ur′ ≈ 0 . (r) Sκ

We see from Φ ≈ 0 that the conditions are satisfied in the weak sense. Therefore, the trajectories of a particle on Q3κ under the action of a central potential U (r; κ) are always on a two-dimensional submanifold Q2 inside the initial configuration space Q3κ in such a way that: (1) If the configuration space is the sphere Sκ3 then the trajectories of the particle are curves on a two-dimensional sphere S 2 passing through the center of forces. (2) If the configuration space is the Euclidean space E3 then the trajectories are curves on a Euclidean plane E2 passing through the center of forces. (3) If the configuration space is the hyperbolic space Hκ3 then the trajectories of the particle are curves on a hyperbolic plane H 2 passing through the center of forces. Hence this property is not a specific or special characteristic of the Euclidean world, but it also holds in all the three spaces of constant curvature; therefore, the three mentioned situations are but three particular instances of a more general property. Also we note that, in all the three cases, the value of the curvature of the two-dimensional submanifold, S 2 , E2 or H 2 , is the same κ as in the original three-dimensional space, i.e. there is a reduction of the dimension but not a change of the curvature.

4.

Motions on 2d Constant Curvature Submanifolds

Instead of considering 2d constant curvature surfaces in a 3d Euclidean space we can study the more general case of the Cayley–Klein (hereafter shortened as CK) 2d spaces Sκ21[κ2 ] [28] with constant curvature κ1 and metric of a signature type (1, κ2 ). Each one of the two parameters κ1 and κ2 can be brought to a standard value 1, 0 or −1, and the nine combinations of these values correspond to the so called ‘standard’ CK spaces. We display these nine spaces in a Table. The three rows accommodate spaces with either a Riemannian, or a degenerate, or a pseudo-Riemannian (Lorentzian) metric, respectively, according to the sign of κ2 . The three instances along each row are the spaces with positive, zero or negative constant curvature (for more details see [29]-[31]. The symmetric homogeneous space Sκ21[κ2 ] admits a maximal three-dimensional isometry group denoted SOκ1 ,κ2 (3), whose three dimensional Lie algebra is soκ1 ,κ2 (3) with generators P1 , P2 and J given in the matrix realization by: 



0 −κ1 0   0 0  P1 =  1 0 0 0





0 0 −κ1 κ2   0 P2 =  0 0  1 0 0



with the following commutation relations among the generators J, P1 and P2 : [J, P1 ] = P2 ,

[J, P2 ] = −κ2 P1 ,



0 0 0   J =  0 0 −κ2  , 0 1 0

[P1 , P2 ] = κ1 J .

Dimensional Reduction of Curvature-Dependent Central Potentials

285

Table 1. The nine standard two-dimensional CK spaces Sκ21[κ2 ] . Measure of distance & Sign of κ1 Measure of angle & Sign of κ2

Elliptic κ1 = 1

Parabolic κ1 = 0

Hyperbolic κ1 = −1

Elliptic κ2 = 1

Elliptic S2

Euclidean E2

Hyperbolic H2

Galilean

Parabolic κ2 = 0

Co-Euclidean Oscillating NH ANH1+1

Co-Minkowskian Expanding NH NH1+1

Minkowskian

Hyperbolic κ2 = −1

Co-Hyperbolic Anti-de Sitter AdS1+1

G1+1

M1+1

Doubly Hyperbolic De Sitter dS1+1

The group SOκ1 ,κ2 (3) acts by matrix multiplication on a R3 ambient space by isometries of the ‘ambient space’ metric: dl2 = (ds0 )2 + κ1 (ds1 )2 + κ1 κ2 (ds2 )2 , and the space Sκ21[κ2 ] is the homogeneous space Sκ21[κ2 ] ≈ SOκ1 ,κ2 (3)/SOκ2 (2), where SOκ2 (2) is the subgroup generated by J. The relation among ambient coordinates and polar coordinates is given by: 







s0 Cκ1 (r)   1    s  =  Sκ1 (r) Cκ2 (φ)  , s2 Sκ1 (r) Cκ2 (φ) and therefore the induced metric is given by: 2 dl|S 2

κ1[κ2 ]

= dr2 + κ2 S2κ1 (r) dφ2 .

The Killing vector fields are the first-order differential operators in polar coordinates: X P1 X P2 XJ

∂ Sκ (φ) ∂ − 2 , ∂r Tκ1 (r) ∂φ ∂ Cκ2 (φ) ∂ = κ2 Sκ2 (φ) + , ∂r Tκ1 (r) ∂φ ∂ = . ∂φ

= Cκ2 (φ)

and close on a soκ1 ,κ2 (3) Lie algebra. The associated momenta are: 







P1 Cκ2 (φ) vr − κ2 Cκ1 (r) Sκ1 (r) Sκ2 (φ)vφ     P κ =  2   2 Sκ2 (φ)vr + κ2 Cκ1 (r) Sκ1 (r) Cκ2 (φ)vφ  , J κ2 S2κ1 (r)vφ

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Jos´e F. Cari˜nena, Manuel F. Ra˜nada and Mariano Santande

but it is more convenient to introduce new variables, to be called CK Noether momenta: P1 := P1 ,

P2 :=

P2 , κ2

J :=

J , κ2

which are essentially equivalent to the previous ones when κ2 6= 0 but admit a limit when κ2 → 0. The three CK momenta are linked, for all values of κ1 and κ2 by a fundamental relation: s2 P1 − s1 P2 + s0 J = 0, which reduces in the standard Euclidean case to the well known (Euclidean) relation between angular and linear momentum J = xP2 − yP1 . The Kepler potential in a Sκ21[κ2 ] , defined to be [6, 9]: VK = −

k . Tκ1 (r)

(29)

has recently been studied in [27] and it has been shown to be super-integrable. A first integral is linked to the invariance of VK under rotations around the potential center, and leads to the constancy of angular momentum J , and to the quadratic constant IJ 2 = J 2 . In any Sκ21[κ2 ] , the potential (29) allows for two additional constants of motion of types IJ P1 , IJ P2 , which are associated to the separability of the Kepler potential in two equiparabolic ‘01’ and ‘20’ coordinate systems (see [32]). These constants of motion are: IJ P1 = J P1 + W01 , IJ P2 = J P2 + W02 ,

W01 = k Sκ2 (φ) W02 = k Vκ2 (φ) ,

(30)

where Vκ2 (φ) is the CK version of the ‘versed sine’, Vκ2 (φ) = (1−Cκ2 (φ))/κ2 ; for κ2 = 0 this reduces to the function φ2 /2. There is a relation among the functions W01 , W02 as a consequence of C2κ2 (φ) + κ2 S2κ2 (φ) = 1: (W01 )2 + W02 (κ2 W02 − 2) = 0 .

(31)

The two constants IJ P1 , IJ P2 , together with the energy and the angular momentum: 1 2 k (P1 + κ2 P22 + κ1 κ2 J 2 ) − , 2 Tκ1 (r) = J 2,

IE = IJ 2

(32)

provide a set of four constants of motion. As the maximal number of functionally independent constants of motion for this system is three, a single relation among the four I above should exist. This relation, which is quadratic in the I’s can be checked using (31): (IJ P1 )2 + IJ P2 (κ2 IJ P2 − 2k) = (2IE − κ1 κ2 IJ 2 ) IJ 2 .

(33)

Dimensional Reduction of Curvature-Dependent Central Potentials

287

The two constants IJ P1 , IJ P2 can be seen as the components of a single Keplerian (conserved) vector under the (sub)group SOκ2(2) of rotations around the origin in the space Sκ21[κ2 ] . This vector will be called here the CK eccentricity vector E ; along any evolution under the Kepler potential, the (constant) values of the components of E will be denoted E01 , E02 : ! ! ! E01 IJ P1 J P1 + W01 = = , (34) E02 IJ P2 J P2 + W02 and in terms of E01 , E02 and of the values of energy and angular momentum, (33) is written

2 E01 + E02 (κ2 E02 − 2k) = 2E − κ1 κ2 J 2 J 2 ,

(35)

which is well defined in all CK spaces Sκ21[κ2 ] . The vector E is related to the ordinary LRL vector in the Euclidean case; notice also the appearance in the r.h.s. of the specific combination 2E − κ1 κ2 J 2 , which for the Euclidean space reduces to 2E. Hence, the existence of a Keplerian additional conserved vector is not a specifically Euclidean property, but still holds even if the configuration space is the more general space Sκ21[κ2 ] , with any constant curvature and any signature. The ‘Riemannian’ part of this statement has been known since a long time for the Kepler problem in S2κ1 and in H2κ1 ; these cases appear in our approach when κ2 = 1. This CK formalism also covers the cases where κ2 < 0, i.e. the Kepler problem in a locally Minkowskian constant curvature configuration space. For the motion of a particle under the action of the ‘curved’ Kepler potential in a ‘curved’ 3-d configuration space of Cayley–Klein type, there also exists a ‘curved’ form for the Laplace–Runge–Lenz vector, which has been obtained by Herranz and Ballesteros [29]. We also point out that the super-integrability of the problem can be used to find the orbits in this curved Kepler potential, as it has been shown in [27], by introducing the CK Laplace–Runge–Lenz vector a vector A : A1 A2

!

:=

κ2 E02 − k −E01

!

=

κ2 (J P2 + W02 ) − k −J P1 − W01

!

.

(36)

In the Euclidean plane this vector reduces precisely to the Laplace–Runge–Lenz vector. In intrinsic terms, the CK Laplace–Runge–Lenz vector is the Hodge κ2 -* of the eccentricity vector shifted by the constant vector (−k, 0). Moreover, it has also been proved that the momentum hodographs for this curved Kepler problem are also ‘cycles’. The details can be found in the recent paper [27].

5.

Final Comments

The Euclidean space E3 can be considered as a very particular or ‘limiting case’ of the constant curvature spaces. The curvature κ modifies some details but preserves the fundamental structures. There is a sound geometrical reason for this situation. If we consider a particular potential in the flat Euclidean space that is invariant under reflection in a given 2-plane, then

288

Jos´e F. Cari˜nena, Manuel F. Ra˜nada and Mariano Santande

the reflected of any possible motion is a possible motion too, the one determined by the reflected initial conditions. If we choose the initial conditions to be a position in the plane and a velocity tangent to the plane, such conditions are invariant under reflection, and therefore the reflected motion has the same initial conditions as the original one, hence both motions coincide, i.e. for these special initial conditions the whole trajectory is contained in that plane. A central potential is invariant under any reflection in a 2-plane through the origin. Thus any generic set of initial conditions (position not on the centre, nonzero velocity) determines a particular plane passing through the centre of forces and the initial point, and containing the initial velocity. The previous reasoning when applied to this plane implies that the motion is contained in such a plane. Assume now that the 3d configuration space is not R3 , but it has a constant curvature κ. Then a similar reasoning can be made, with totally geodesic 2-dimensional submanifolds playing the rˆole of Euclidean 2-planes. This follows directly from the Weierstrass ambient space model, where the totally geodesic 2-dimensional submanifolds are the intersection of the ‘sphere’ x20 +κ (x21 +x22 +x23 ) = 1 with a 3-plane (just in the same way as geodesics S 1 ⊂ S 2 are the intersection of S 2 ⊂ R3 with 2-planes through the origin in R3 ). Reflections in that 3-plane in the ambient space represent reflections on the totally geodesic S 2 ⊂ S 3 or H 2 ⊂ H 3 . Hence motion on a space S 3 or H 3 , in a potential invariant under geodesic reflection on a fixed totally geodesic S 2 or H 2 with initial position on that submanifold and initial velocity tangent to it, is always fully contained in that submanifold. As an arbitrary central potential is invariant under reflection in any totally geodesic 2d submanifold, either S 2 or H 2 respectively, through the centre of forces, the previous reasoning, when applied to these submanifolds, implies that the motion stays in such submanifold. In one of our previous articles we quote a statement by Fronsdal [33]: “A physical theory that treats space-time as Minkowskian flat must be obtainable as a well-defined limit of a more general physical theory, for which the assumption of flatness is not essential”. Our study is not a relativistic one, but, in a sense, this statement looks rather similar (Minkowskian must be changed to Euclidean) to our idea of using the concept of deformation as an approach with κ as the parameter of the deformation. The flatness assumption κ = 0 is not necessary for proving that the motion of a particle in a central force problem is always a motion in an analogous two-dimensional submanifold with the same curvature as in the total space. A similar situation was found in [23] for the Kepler problem where it was proved that even if κ 6= 0 the particle moves in conics. Also in this case the property of κ = 0 flatness was not a necessary assumption for arriving to such an important characteristic. We end this paper with two comments. In the first place let us recall that gravitation introduces curvature not only in the four-dimensional space-time but also in the threedimensional space-like surfaces. So, although this study has been a non-relativistic one, we think that the results obtained in this article could also be of use in the study of relativistic gravitation models. Secondly, we have presented an approach based on tangency conditions that has proved to be successful; nevertheless it is natural to look also at this geometric problem from the more traditional viewpoint of Noether’s theorem and the theory of symmetries (in this case κ-dependent symmetries). This alternative approach remains as an interesting matter to be studied.

Dimensional Reduction of Curvature-Dependent Central Potentials

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Acknowledgement Support of projects E24/1 (DGA), MTM-2005-09183, MTM-2006-10531, and VA-013C05 is acknowledged.

References [1] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (4th ed., Cambridge University Press, 1965). [2] H. Goldstein, Classical Mechanics (2nd ed., Addison-Wesley, Reading, Mass., 1980). [3] J. V. Jos´e and E. J. Saletan, Classical Dynamics. A contemporary approach (Cambridge University Press, Cambridge, 1998). [4] R. Abraham and J. E. Marsden, Foundations of Mechanics (2nd ed., Benjamin/Cummings, Reading, Mass., 1978). [5] G. Marmo, E. Saletan, A. Simoni and B. Vitale, Dynamical Systems. A Differential Geometric Approach to Symmetry and Reduction (John Wiley, Chichester, 1985). [6] E. Schr¨odinger, A method of determining quantum mechanical eigenvalues and eigenfunctions, Proc. R. Ir. Acad. A 46 (1940) 9–16. [7] L. Infeld, On a new treatment of some eigenvalue problems, Phys. Rev. 59 (1941) 737–747. [8] A. F. Stevenson, Note on the ‘Kepler Problem’ in a spherical space, and the factorization method of solving eigenvalue problems, Phys. Rev. 59 (1941) 842–843. [9] L. Infeld and A. Schild, A note on the Kepler problem in a space of constant negative curvature, Phys. Rev. 67 (1945) 121–122. [10] P. W. Higgs, Dynamical symmetries in a spherical geometry I, J. Phys. A: Math. Gen. 12 (1979) 309–323. [11] H. I. Leemon, Dynamical symmetries in a spherical geometry II, J. Phys. A: Math. Gen. 12 (1979) 489–501. [12] E. G. Kalnins, W. Miller and P. Winternitz, The group O(4), separation of variables and the hydrogen atom, SIAM J. Appl. Math. 30 (1976) 630–664. [13] A. O. Barut and R. Wilson, On the dynamical group of the Kepler problem in a curved space of constant curvature, Phys. Lett. A 110 (1985) 351–354. [14] A. O. Barut, A. Inomata and G. Junker, Path integral treatment of the hydrogen atom in a curved space of constant curvature, J. Phys. A: Math. Gen. 20 (1985) 6271–6280. [15] A. O. Barut, A. Inomata and G. Junker, Path integral treatment of the hydrogen atom in a curved space of constant curvature: II, J. Phys. A: Math. Gen. 23 (1990) 1179–1190.

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[16] C. Grosche, The path integral for the Kepler problem on the pseudosphere, Ann. Phys. 204 (1990) 208–222. [17] C. Grosche, On the path integral in imaginary Lobachevsky space, J. Phys. A: Math. Gen. 27 (1994) 3475–3489. [18] C. Grosche, G. S. Pogosyan and A. N. Sissakian, Path integral discussion for Smorodinsky-Winternitz potentials II, Fortschr. Phys. 43 (1995) 523–563. [19] M. F. Ra˜nada and M. Santander, Superintegrable systems on the two-dimensional sphere S 2 and the hyperbolic plane H 2 , J. Math. Phys. 40 (1999) 5026–5057. [20] J. J. Slawianowski, Bertrand systems on spaces of constant sectional curvature, Rep. Math. Phys. 46 (2000) 429–460. [21] E. G. Kalnins, J. M. Kress, G. S. Pogosyan and W. Miller, Completeness of superintegrability in two-dimensional constant-curvature spaces, J. Phys. A: Math. Gen. 34 (2001) 4705–4720. [22] E. G. Kalnins, W. Miller and G. S. Pogosyan, The Coulomb-oscillator relation on ndimensional spheres and hyperboloids, Phys. of Atomic Nuclei 65 (2002) 1119–1127. [23] J. F. Cari˜nena, M. F. Ra˜nada and M. Santander, Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S 2 and the hyperbolic plane H 2 , J. Math. Phys. 46 (2005) 052702, 1–25. [24] M. F. Ra˜nada and M. Santander, On the Harmonic Oscillator on the two-dimensional sphere S 2 and the hyperbolic plane H 2 , J. Math. Phys. 43 (2002) 431–451. [25] M. F. Ra˜nada and M. Santander, On the Harmonic Oscillator on the two-dimensional sphere S 2 and the hyperbolic plane H 2 II, J. Math. Phys. 44 (2003) 2149–2167. [26] J. F. Cari˜nena, M. F. Ra˜nada and M. Santander, The quantum harmonic oscillator on the sphere and the hyperbolic plane: κ-dependent formalism, polar coordinates, and hypergeometric functions, J. Math. Phys. 48 (2007) 102106. [27] J. F. Cari˜nena, M. F. Ra˜nada and M. Santander, Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton, J. Phys. A: Math. Theor. 40 (2007) 13645–13666. [28] J. F. Cari˜nena, M. A. del Olmo and M. Santander, A new look at Dimensional Analysis from a group-theoretical viewpoint, J. Phys. A: Math. Gen. 18 (1985) 1855–1872. [29] F. J. Herranz and A. Ballesteros, Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature, SIGMA 2 (2006) 010 pp. 22. [30] F. J. Herranz, R. Ortega and M. Santander, Trigonometry of space-times: a new selfdual approach to a curvature/signature (in)dependent trigonometry, J. Phys. A: Math. Gen. 33 (2000) 4525–4551.

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[31] F. J. Herranz, M. Santander, Casimir Invariants for the complete family of quasisimple orthogonal algebras, J. Phys. A: Math. Gen. 30 (1997) 5411–5426. [32] J. F. Cari˜nena, M. F. Ra˜nada, M. Santander and T. Sanz-Gil, Separable potentials and triality in 2d-spaces of constant curvature, J. Nonlinear Math. Phys. 12 (2005) 230– 252. [33] C. Fronsdal, Elementary particles in a curved space, Rev. Mod. Phys. 37 (1965) 221– 224.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 293-314

Chapter 17

D IRECT G EOMETRICAL M ETHOD IN F INSLER G EOMETRY L. Tam´assy ∗ Debrecen University

I.

Minkowski Angle and Related Problems

Introduction Finsler geometry is a very natural generalization of Riemannian geometry. Both are built on the arc length of curves. A Finsler space F n = (M, F) is given by an n-dimensional manifold M and a fundamental (metric) function F : T M → R+ ,

(p, y) 7→ F(p, y),

p ∈ M, y ∈ Tp M,

where R+ means the non-negative reals. F must satisfy the following requirements [BSC]: (i) regularity: F ∈ C ∞ on the slit tangent bundle T M ◦ : T M without the null section {(p, 0)} ⊂ T M , and F ∈ C ◦ on the null section (ii) positive homogeneity: F(p, λy) = λF(p, y), λ ∈ R+ (iii) strong convexity:

∂2F 2 (p, y)v i v j ∂y i ∂y j

≡ 12 gij (p, y)v i v j ≥ 0, ∀ v ∈ Tp M .

An important, more restrictive version of homogeneity is the (iv) absolute homogeneity: F(p, λy) = |λ|F(p, y), λ ∈ R. The Finsler norm of a vector y ∈ Tp M is kyk := F(p, y). ∗

E-mail address: [email protected]

(1)

294

L. Tam´assy

The Finsler arc length of a curve c : [a, b] → M is defined by the integral of the norm kc(t)k ˙ of the tangent vector of the curve c(t): s :=

Z

b

a

F(c(t), c(t))dt. ˙

(2)

(1) means that a Finsler space endows the vectors y ∈ Tp M with a norm with the properties: a) kyk ≤ 0, and kyk = 0 ⇐⇒ y = 0; b) kλyk = λkyk, λ ∈ R+ ; c) ky1 + y2 k ≤ ky1 k + ky2 k, y1 , y2 ∈ Tp M ; and d) a differentiability (continuity) property in p and y following from (i) and (1). Conversely, if Tp M are endowed with such a norm, then M and F given by (1) is a Finsler space. A Riemann space V n = (M, g) is that special case where R ˙ ci ˙ g )1/2 dt. F 2 is quadratic in c. ˙ Then the Riemann arc length has the form s = ab (hc, An important notion is the indicatrix hypersurface I(p0 ) := {y ∈ Tp0 M | F(p0 , y) = 1}. It plays the role of the Euclidean unit sphere S n−1 ⊂ E n . In consequence of (iii) I(p) is strictly convex. (iv) is equivalent to the symmetry of the indicatrix, or to F(p, y) = F(p, −y), ∀ p, y. (ii) is equivalent to the invariance of the arc length s with respect to the orientation preserving parameter transformations. (iv) is equivalent to the invariance of the arc length s with respect to any parameter transformation. In case of (iv) the norm induced by F on Tp M has the property: eb) kλyk = |λ| kyk, λ ∈ R, and thus in this case Tp M becomes a Banach space. In the case of a Riemann space the indicatrices are ellipsoids varying with the point p, and the tangent spaces become Euclidean spaces. So we can say with S. S. Chern [7] that Finsler geometry is nothing but Riemannian geometry without the quadratic restriction. Finsler geometry originsted from a variational problem. The first steps were done by P . Finsler in his thesis written under the guidance of C. Charatheodory in G¨ottingen in 1918, and thus Finsler geometry has close relation to the field of Demeter Krupka to whom this volume is dedicated. Later Finsler geometry was developed on the analogy of Riemann geometry. Thus the existence of a linear metrical connection between the tangent vectors of the base manifold is of basic importance in this theory. Such a connection is metrical if the length (the norm) of the parallel translated tangent vectors remains unaltered. It means that by a metrical connection indicatrices (the ensemble of the unit Finsler vectors) are taken into indicatrices. However in a Finsler space in the general case this can not be satisfied by a linear connection, for indicatrices may be arbitrary, and thus they cannot be taken into each other by linear transformations. Indeed, it is easy to see that by the homogeneity condition (ii) indicatrix bundle and fundamental function determine each other. Thus, if we give a Finsler space by such indicatrices which are not in affine (linear) relation to each other, then in this Finsler space there exists no linear metrical connection between the tangent vectors. This problem was eliminated by E. Cartan by introducing the line elements (p, y). (p, y) is equivalent to (p, λy), λ ∈ R, λ 6= 0. Then the n-dimensional vectors sitting on the line elements (according to R. Ingarden the “Finsler vectors”) form a vector bundle Ξ = (E, π, T M, V n ) with the 2n-dimensional base space T M , and with n-dimensional vector spaces as fibers: π −1 (p, y) = V(p,y) T M ⊂ T T M . The square of the Finsler norm of such a

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vector is kξ(p, y)k2 = gij (p, y)ξ i (p, y)ξ j (p, y), gij (p, y) taken from to (ii). With this norm one can construct metrical linear connections in VT M = {V(p,y) T M | (p, y) ∈ T M }. On the other hand, introduction of line elements makes the apparatus and the theory of Finsler geometry a little more complicated. This inconvenience does not arise if we consider purely metrical questions, only, as arc length, area, angle, geodesic, isometry, etc., or if we investigate Finsler spaces which admit metrical linear connections between the tangent vectors ξ(p) of the base manifold M . There are a number of such Finsler spaces, and just they are the ones most near to Riemannian spaces, and that gives them a certain significance. In this paper we investigate questions which do not involve line elements (or involve them very rarely), i.e. we consider purely metrical questions, and such Finsler spaces, which admit metrical linear connections in the tangent bundle T M . We often use direct geometrical considerations in place of analytical calculations. In spite of the fact that the analytical method is in general more powerful than the geometrical method, the last one turns out in several cases to be surprisingly effective.

1.

Angle in Minkowski and Finsler Spaces

In a Finsler space F n = (M, F) angle ∡α usually means the angle of two vectors ξ(p, y), η(p, y) of V(p,y) T M ⊂ T T M . The cosine of ∡α = ∡(ξ, η) is given by cos α :=

gij (p, y)ξ i (p, y)η j (p, y) , kξk.kηk

(∗)

where gij (p, y) is given by (iii). This gij is a Riemannian metric in V(p,y) T M . In this paragraph we consider the angle of two vectors ξ(p), η(p) of a tangent space Tp M of the base manifold of a Finsler space. This Minkowski angle has attracted less interest. Since the Finsler space makes its tangent space into a Minkowski space, measuring of such angles in a Finsler space reduces to that in a Minkowski space. In Section 2 we prove that a diffeomorphism between two Finsler spaces is an isometry iff it keeps angle (in the above sense) and area, similary to the well-known result of Riemannian geometry. We also show (Section 3) that this angle is applicable in measuring the deviation of a Finsler space from being Riemannian. Given a Finsler space F n = (M, F) we consider an angle α = ∡(a, b) between two rays a, b ∈ Tp0 M emanating from the origin 0 = p0 of Tp0 M . Tp0 M is an n-dimensional vector space V n . a and b span a two-dimensional linear subspace Σ of Tp0 M , provided a is not parallel to b : a ∦ b. The convex domain of Σ bounded by a and b will be denoted by A. This is unambigous if a ∦ b. If a = b, then A = ∅. If a, b ⊂ g, a 6= b, then the straight line g cuts Σ into Σ+ and Σ− . Then A = Σ+ or A = Σ− . Minkowski space is a special Finsler space. If the fundamental function F of the Finsler space F n = (Rn , F) (where Rn is the n-dimensional number space with the cannonical topology) has the property that in a coordinate system (x) F(x, y) is independent of the points x, then it is a Minkowski space Mn = (Rn , F), and the coordinate system (x) is called adapted. F n makes each Tx0 M into a Minkowski space Mnx0 with indicatrix body Bxn0 (1) := {y | F(x0 , y) ≤ 1} ⊂ Tx0 M and with the Minkowski functional F(y) =

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F(x0 , y) : Tx0 M → R+ . Bxn (1) is then a Minkowski ball of radius 1, and ∂Bxn (1) = I is the indicatrix (hyper) surface. By Bx2 (1) = Bxn (1) ∩ Σ, Mnx (or F n ) induces on Σ ⊂ Tx M a two-dimensional Minkowski metric and thus an M2x . Bx2 (1) ∩ A = D is a segment of the indicatrix body of M2x belonging to ∡α(a, b). We denote the Minkowski area in M2x by k · kM , and the Euclidean area in Σ equipped with a Euclidean metric by k · kE . Then the 2-dimensional Minkowski area of D in Mnx is the Minkowski area of D in M2 ≡ M: kDkM =

Z

σdy 1 dy 2 ,

σ=

D

π , kB 2 kE

(3)

(Z. Shen [15, §1.3], or H. Busemann [6], H. Rund [14, Chap. I, §8], D. Bao – R S. S. Chern – Z. Shen [2, §1.4], and many other places.) Since D dy 1 dy 2 is the Euclidean area of D, (3) is equivalent to πkDkE kDkM = . (3′ ) kB 2 kE (3) and (3′ ) are true for any domain G ⊂ Σ in place of D. The Minkowski measure of the angle ∡M α(a, b) can be defined as follows: Definition. ∡M α(a, b) := ǫ2kDkM . ǫ = 1 or − 1.

(4)

The sign ǫ depends on the orientation of the angle. (4) is basicaly the Landsberg angle [14, p. 33]. ∡M α can be expressed by the Minkowski functional F and the data of the two legs a and b [5, (3.a) and (3.b)]. If Mnx is a Euclidean space E n , then (4) reduces to the Euclidean measure ∡E α of the angle α. Indeed, if Mnx = E n , then B 2 is the Euclidean unit ball. Thus ∡M α = 2ǫkDkM is a generalization of the Euclidean measure of α. kDkM has a sign, and because of the additivity of the area, ∡M α is additive: ∡M α1 + ∡M α2 = ∡M (α1 + α2 ). Also ∡M α is symmetric in the sense that |∡M (a, b)| = |∡M (b, a)|. The case of the straight angle: In this case a ∪ b = g is a line through 0 ∈ Tx M . Let + − ∡(a, b) = α+ be the straight angle with the domain Σ+ g = A , and ∡(b, a) = α the − − straight angle with the domain Σg = A . Because of the additivity ∡M α+ + ∡M α− = 2π

∀ g.

Therefore the equality ∡M α+ = ∡M α− of the Minkowski measure of the two straight angles α+ and α− implies kB 2 ∩ A+ kE = k+ D2 kE = k− D2 kE = kB 2 ∩ A− kE , and conversely. In other words: ∡M α+ = ∡M α− iff g bisects B 2 . If B 2 is symmetric, then every line g through O bisects B 2 . We show that also conversely, if every g through O bisects B 2 , then B 2 is symmetric. Suppose that B 2 is non-symmetric. Then there exists a ϕ0 , such that in a Euclidean polar coordinate system r(ϕ0 ) > r(ϕ0 + π), where (r(ϕ), ϕ) ∈ ∂B 2 , ∀ ϕ. A g is fixed by its direction ϕ. Then for every g(ϕ) Z 1 ϕ+π 2 1 r (ϕ)dϕ = kB 2 kE , ∀0 ≤ ϕ < π. 2 ϕ 2

Direct Geometrical Method in Finsler Geometry Especially Z

ϕ0 −ǫ+π

Z

r2 (ϕ)dϕ =

ϕ0 −ǫ

Hence

Z

ϕ0 +ǫ

ϕ0 +ǫ+π

297

r2 (ϕ)dϕ.

ϕ0 +ǫ

r2 (ϕ)dϕ =

ϕ0 −ǫ

Z

ϕ0 +ǫ+π

r2 (ϕ)dϕ.

ϕ0 −ǫ+π

By the integral mean value theorem 2ǫr2 (ϕ1 ) = 2ǫr2 (ϕ2 ),

ϕ0 − ǫ ≤ ϕ1 ≤ ϕ0 + ǫ

ϕ0 − ǫ + π ≤ ϕ2 ≤ ϕ0 + ǫ + π,

and because of the continuity of r(ϕ), ǫ → 0 yields r(ϕ0 ) = r(ϕ0 + π) in contradiction to our assumption. Therefore B 2 is symmetric. This is equivalent to the absolute homogeneity of F. These statements are summed up in Theorem 1. ∡M α = ǫ2kDkM is an additive, symmetric measure of the angles in Minkowski or Finsler spaces. In a Euclidean space this reduces to the Euclidean measure e = ±π for every straight angle α e if and only if the Finsler metric is absolute of α. ∡M α homogeneous.

2.

Diffeomorphisms which Keep Angle and Area

Using the notations of the previous paragraph, let F n = (M, F) and F n = (M , F) be two connected paracompact Finsler spaces, ϕ : M → M a diffeomorphism, I(p0 ) := {y ∈ Tp0 M | F(p0 , y) = 1} and I(p0 ) := {y ∈ Tp0 M | F(p0 , y) = 1}, p0 = ϕ(p0 ) are indicatrix hypersurfaces (indicatrices) of F n and F n resp. I(p) ∩ Σ = I 2 (p) is the 2 2 indicatrix of M2p , and I(p) ∩ Σ = I (p), p = ϕ(p), Σ = ϕ(Σ) is the indicatrix of Mp . ϕ is an isometry iff (dϕ)I(p) = I(p), ∀ p ∈ M. (5) Theorem 2. The diffeomorphism ϕ : M → M is an isometry between the Finsler spaces F n and F n iff ϕ keeps angle (in the sense of paragraph 1) and (2-dimensional) area. A) Suppose that ϕ is an isometry. By (5) 2

(dϕ)I 2 (p) = (dϕ)I(p) ∩ (dϕ)Σ = I(p) ∩ Σ = I (p). 2

Σ and Σ equipped with Euclidean metrics are E 2 and E resp. Then by (3′ ) kDkM = kDkM2 = π

kDkE , kB 2 kE

and, since dϕ is a linear mapping which keeps the ratio of areas, kDkM = π

k(dϕ)DkE . k(dϕ)B 2 kE

(6)

298

L. Tam´assy Finally, in consequence of (6), we obtain kDkM = π

kDkE 2

kB kE

= kDkM ,

D = (dϕ)D.

This means that ϕ keeps (2-dimensional) area. (It is easy to see that an isometry keeps also the k-dimensional (1 ≤ k ≤ n) area.) According to (4) ∡M α is defined by area. Thus, if ϕ keeps area, then ϕ keeps angle too. Indeed, we know that ∡M α = 2ǫkDkM and ∡M α = 2ǫkDkM , where α = (dϕ)α. Then, from kDkM = k(dϕ)DkM = kDkM (ϕ keeps area) we obtain ∡M α = ∡M α, that is ϕ keeps angle too. B) Suppose that ϕ keeps area (r) and angle (n). Let us denote (dϕ)I 2 (p) =: Ie2 (p). 2 2 F n determines the indicatrix I (p) = I(p) ∩ Σ. We denote (dϕ)−1 I (p) =: Ib2 (p) and b 2 ∩ A =: D, b where B b 2 means I b2 and the points in its inside, and A is the domain of the B angle ∡α(a, b). e Furthermore D e := A e∩B e 2 and e, e dϕ maps a, b into a b, and the domain A into A. 2 2 2 e 2 (resp. B ) means I e2 (resp. I ) and the points in its inside. D := Ae ∩ B , where B −1 2 e e e, b into a, b, and I (p) into I 2 (p). (Figure 1) Moreover (dϕ) takes a b

I2

c

A

D C p

Ib2

b C

b D

a eb

dϕ

e D

Ie2

e A

D

e a

ϕ(p) = p

I2

Figure 1. By our assumption (r) and (n) we obtain (r)

Thus

e kDkM = kDk M

(n)

(a)

b M. and kDkM = kDkM = kDk

b kDkE b M = kDkE =⇒ kDE k = kDk b E = kDk = k Dk M kB 2 (p)kE kB 2 (p)kE

(7)

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299

for any ∡α(a, b). b Suppose that there exist p and Let c be a ray in Σ, c ∩ I 2 (p) = C, and c ∩ Ib = C. 2 b and let us say that C b is outside B . Then, because of the continuity, c such that C 6= C, p b (H b := h ∩ I) b is outside B 2 . Now there exists a ray h(6= c), such that the whole arc Cb H p b the segment D(c, h) of Bp2 is a proper part of the segment D(c, h) bounded by c, h and b Then kD(c, h)kE < kD(c, b b ∀ c, p. Thus I. h)kE , what contradicts (7). Therefore C = C, 2 ∂Bp2 = I 2 (p) = Ib2 . Consequently (dϕ)I 2 (p) = Ie = (dϕ)Ib = I (p), ∀ p ∈ M . This yields (5), and so ϕ is an isometry.

In Theorem 2 we used the Minkowski angle ∡M α(µ, ν) of the tangent vectors µ(p), ν(p) ∈ Tp M . Nevertheless in the case of a Finsler space the Finsler angle ∡F α(ξ, η) of ξ(p, y), η(p, y) ∈ V(p,y) T M is used in general. Its cos is given in (∗) (at the beginning of Section 1). It is easy to see that if a diffeomorphism ϕ keeps area and dϕ keeps Minkowski angle, then dϕ keeps Finsler angle too. Namely in this case ϕ is an isometry (Theorem 2). Then the metric tensors g(p, y) of F n and the metric tensor g(p, y), p = ϕ(p), y = (dϕ)y of F n are related by (dϕ)g(p, y) = g(p, y), and this implies that dϕ keeps the Finsler angle ∡F α(ξ(p, y), η(p, y)). Also conversely, if dϕ keeps area and Finsler angle, then it keeps Minkowski angle too. We know that F n induces by the metric tensor g(p0 , y) on V(p0 ,y) T M a Riemannian space V n (p0 ), and F n induces by g(p0 , y), p0 = ϕ(p0 ), y = (dϕ)y on V(p0 ,y) T M a V n (p0 ). If dϕ keeps Finsler angle, then dϕ : V n (p0 ) → V n (p0 ) is a conformal mapping, and thus g(p0 , y) = c(dϕ)g(p0 , y), ∀ y ∈ Tp0 M . But dϕ keeps area. Thus the ellipses Q(p0 , y) and Q(p0 , y) corresponding to g(p0 , y) and g(p0 , y) have the same Minkowski area. Hence c = 1. This implies (dϕ)I(p) = I(p), that is ϕ is an isometry, and then dϕ keeps Minkowski angle.

3.

Deviation of Finsler Spaces from Riemannian Spaces

There are known several conditions which imply the reduction of a Finsler space F n to a Riemannian space V n . Such a condition is the vanishing of the Cartan tensor Cijk or the constantness of the distortion τ (x, y) [17]. Many other quantities, such as the Scurvature [16], Landsberg curvature, Cartan torsion, etc. can be coupled with this problem. Also, a Finsler space is a Riemann space iff the indicatrices are ellipsoids. We want to present conditions expressed by the Minkowskian angle which imply the reduction of the indicatrices to ellipsoids and thus the F n to a V n . We consider a Finsler space F n = (M, F) and its tangent space as a Minkowski space n M = (Tp0 M, F(p0 , y)) and a 2-dimensional linear subspace Σ of Tp0 M . Tp0 M can be indentified with a vector space V n or the coordinate space Rn (x) which can be equipped with a Euclidean metric, yielding E n (x). B n is the indicatrix body of Mn , ∂B n = I the indicatrix surface, and I ∩ Σ = I 2 is the indicatrix of the M2 induced by Mn on Σ. If F n is a Riemannian space V n , then M2 is a Euclidean space, and I reduces to an ellipse. In this case the Minkowskian and the Euclidean angle are the same, ∡M α(a, b) = ∡E α(a, b), and it equals π iff α is a straight angle: its two legs a, b are two half lines of a straight line

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g trough the origin: a ∪ b = g. As we have shown in Section 1 ∡M α(a, b) = π

if

a ∪ b = g,

∀ g ⊂ Σ ⊂ Tp0 M,

∀p ∈ M

is necessary for an F n to be a V n . Given an arbitrary ray a ⊂ Σ, let a be the other ray, such that a ∪ a is a line g, and let b ⊂ Σ be such a ray in Σ that ∡M α(a, b) = π. b depends on a, and |∡M (b, a)| =: f (a) ≥ 0 is a function of a ⊂ Σ. f (a) = 0, ∀ a ⊂ Σ is necessary for F n = V n . Let in Σ d0 be an initial ray, y a point, 0y the ray trough y, r = F(p0 , y), and ν = ∡M (0y, d0 ). Then (r, ν) is a Minkowskian polar coordinate system in Σ. The arbitrary a is a function of ν. Denoting f (a(ν)) by f (ν), we obtain that G(p, Σ) :=

Z

2π

ν=0

f (ν)dν = 0,

f (ν) ≡ f (a(ν)),

∀Σ ⊂ Tp0 M,

∀p ∈ M

(8)

is necessary for F n = V n . This and Section 1 of this Chapter yield Proposition 1. (8) is equivalent to: 1) b = a, ∀ a, 2) ∡M (a, a) = π, ∀ a, 3) ∀ g bysects I 2 , 4) I(p) is symmetric, 5) F n is absolutely homogeneous. All these are necessary for a Finsler space to be Riemannian. G(p, Σ) ≥ 0 measures the deviation of an F n from being absolutely homogeneous in Σ at p. We want to obtain sufficient conditions for F n = V n . Our tool for this will be the difference between Minkowski orthogonality and transversality. Since the properties listed in the Proposition 1 are necessary, we suppose that the indicatrices of F n = (M, F) are symmetric. Let g = a∪a, h = b∪b be lines and rays in Σ ⊂ Tp M , M2p = (Tp M, F(p, y)). Our considerations will be restricted to Σ. Because of the symmetry of I 2 (p) the Minkowskian perpendicularities a⊥M b, i.e. ∡M α(a, b) = π2 , a⊥M b, a⊥M b, a⊥M b are equivalent. They mean g⊥M h. So, in the case of the symmetry of I 2 (p) we can speak of the perpendicularity of lines in place of rays. Denoting by g ⊥ a line perpendicular to g, we obtain (g ⊥ )⊥ k g. – Another notion is transversality. Let g ∩ I 2 (p) = G, G ′ . Then the tangent TG I 2 (p) =: g ∗ is called transversal to g. Because of the symmetry of I 2 (p), TG ′ I 2 (p) =: (g ′ )∗ is parallel to g ∗ . Also, any line parallel to g ∗ is said to be transversal to g. So we can speak of transversality of a direction to another direction. Nevertheless, this relation is not symmetrical, that is the direction transversal to g ∗ is in general not g : (g ∗ )∗ ∦ g. The relation (g ∗ )∗ k g, ∀ g ⊂ Σ (9)

means that in M2p the transversality operation ∗ is involutive. A strictly convex, closed, differentiable curve with O in its interior and with the property (9) is called a Radon curve. Every ellipse is a Radon curve, but not conversely. This shows that if the indicatrices of an F 2 = (M, F) satisfy (9) at every point p ∈ M , then these indicatrices need not to be ellipses, and thus F 2 needs not to be a Riemannian space V 2 = (M, g). We claim that if n > 2 and (9) is satisfied in every Σ ⊂ I 2 (p), then F n is a V n . Indeed, under these conditions every I 2 (p) = I(p) ∩ Σ, ∀ p, Σ is a Radon curve. Then in Tp M every cylinder osculating to Bpn osculates along a planar curve [19]. In this case, according to W. Blaschke [4, pp. 157–159], every I(p) is an ellipsoid, and thus F n = V n .

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301

If F n = V n , then ∀ I 2 (p) is an ellipse, and (9) is satisfied. But (g ⊥ )⊥ k g is always true if I 2 (p) is symmetric. Hence in this case g ⊥ k g ∗ .∀ g. If ∡M α(g ∗ , g ⊥ ) = 0, i.e. if g ∗ = g ⊥ , then (9) is realized, for (g ⊥ )⊥ k g is true. Hence K(p, Σ) :=

Z

π

0

g ⊂ Σ,

|∡M α(g ∗ (ν), g ⊥ (ν))|dν = 0,

(10)

∀ Σ ⊂ Tp0 M, ∀ p ∈ M

is sufficient for F n = V n . Conversely, (10) is always satisfied in a V n . Thus we obtain Theorem 3. An absolutely homogeneous Finsler space F n , n > 2 reduces to a Riemann space if and only if K(p, Σ) = 0, ∀Σ ⊂ Tp0 M , ∀ p ∈ M . The deviation of an absolutely homogeneous F n from being a Riemannian space on Σ ⊂ Tp0 M can be measured by K(Σ). Thus K(Σ) can be considered as a kind of sectional curvature. The deviation at a point p0 ∈ M can be measured by the integral G(p0 ) = R

1

Gn,2

dσ

Z

Gn,2

K(Σ)dσ ≥ 0,

where Gn,2 is the Grassmann manifold of theR2-dimensional linear subspaces of Tp0 M , and dσ is a positive measure on Gn,2 , such that Gn,2 dσ is finite and invariant with respect to linear transformations in Tp0 M . The deviation of F n from being Riemannian on M (the global case) can be measured by the integral H(M ) = R

1 M dµ

where dµ is the Finsler volume element, and

II.

Z

M

R

M

G(x)dµ ≥ 0,

dµ is supposed to be finite.

Isometry between Finsler Spaces

Introduction We consider two Finsler spaces F n = (M, F) and F n = (M , F) with base manifolds M and M , and with fundamental functions (Finsler metrics) F and F resp. Since fundamental function F and indicatrices I(p), p ∈ M determine each other, we write F n = (M, I) and F n = (M , I). Let ϕ : M → M be a diffeomorphism. In this article we want to investigate the case when ϕ is a (global) isometry. Since global isometry induces local isometries, and in our case (ϕ is a diffeomorphism) local isometry implies global isometry, we will consider often the local case only. ϕ takes the point P (x) ∈ M into the point P (x) ∈ M . This is a point transformation. However, ϕ can be considered also as a coordinate transformation (x) → (x). After this coordinate transformation F n = (M (x), I(x)) appears in the form (M (x), I(x)), where ∂x I(x) = ∂x I(x). But this coordinate transformation does not change the metric of the

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Finsler space. Now, if we consider x = ϕ(x) as the coordinates of a point of another space M (M ∋ P (x) 7→ P (x) ∈ M ), then we get F n = (M , I), which is clearly isometric to F n by ϕ. This clearly can be reversed. If ϕ : M → M is an isometry between F n and F n , then F n can be considered as F n (on ϕ−1 (M ) = M ) in another coordinate system (x). This gives the following: Any pair of Finsler spaces F n = (M, I) and F n = (M , I) isometric by ϕ : M → M can be represented by a single Finsler space F n in two different coordinate systems: F n = (M (x), I(x)) and F n = (M (x), I(x)). Conversely, if a Finsler space is represented in two different coordinate systems, then these representations are isometric. Nevertheless with this the problem is not settled. Isometries of Finsler spaces have been investigated from different points of view by a number of geometers. Here we mention only a few of them, as Shaoquiang Deng and Zixin Hou [8] proved that diffeomorphisms of a Finsler space F n = (M, F) onto itself which preserve F coincide with the diffeomorphisms which preserve distance ̺(p, q), p, q ∈ M . Other interesting problems of Finsler isometries are investigated among others by M. Matsumoto [13], L. Kozma and P. I. Radu [11], Z. I. Szab´o [18], Lovas [12], etc. In this chapter we investigate diffeomorphisms ϕ : M → M , which yield isometries between Finsler spaces F and F . We use indicatrices I in place of fundamental functions F, and prefer direct geometric considerations rather than analytic calculations. First we point out that if ϕ : M → M yields an isometry between F n = (M, I) and n F = (M , I), then the corresponding indicatrices I(x) and I(ϕ(x)) are affine equivalent: I(x) ae I(ϕ(x)) (i.e. there exists an affine transformation taking I(x) into I(ϕ(x)). This is a necessary condition in order that ϕ generates an isometric mapping. Fulfilment of this condition will nearly always be presupposed. In Section 4 we show that the affine equivalence yields a decomposition of M into disjunct subsets Mλ . In Section 5 we show that any Finsler space over each Mλ is the affine deformation of a locally Minkowski space. (This is the decomposition of F ). Also we show that at an isometry each Mλ is mapped into a subset of M with similar property. If M = M , then we obtain motions. Section 6 is devoted to the study of several motions of Finsler spaces. We consider a connected geodesically complete F 2 = (M, I), which admits a non-trivial 1-parameter group of continuous motions ϕt , and in the decomposition of M appear two subsets M1 and M2 consisting of a single point only. We show that this F 2 is diffeomorphic to the sphere S 2 (and ϕt are “rotations”). In Section 7 we construct to F n and F n two Riemannian spaces V n and V n (V n is arbitrary), and we show that F n is isometric to F n iff V n is isometric to V n . This reduces the isometry of Finsler spaces to that of Riemannian spaces. In the final Section 8 we obtain analytical conditions for the considered isometries, and in case if F n is an affine deformation of a locally Minkowski space (in this case the assumption I(x) ae I(ϕ(x)) is satisfied automatically) we investigate the existence of a 1-parameter group ϕt of continuous motions, and we obtain Killing type equations.

4.

Decomposition of the Base Manifold

Let F n = (M F) and F n = (M , F) be two n-dimensional Finsler spaces with fundamental functions F and F, and base manifold M and M resp. In this chapter Finsler spaces will

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be supposed to be regular and positively homogeneous. One can suppose strict convexity too, but this will not be exploited. We suppose that ϕ : M → M is a diffeomorphism. First we give a necessary condition in order that ϕ be an isometry. Locally let ϕ be given by xi → xi (x), i = 1, 2, . . . , n, where xi and xi are local coordinates on U ⊂ M and ϕ(U ) = U ⊂ M resp. Then F(x, y) = F(ϕ(x), (dϕ)y) ≡ F(x, y),

y ∈ Tx M, y ∈ Tx M .

(11)

The indicatrix I(x) of F n at x consists of those y ∈ Tx M for which F(x, y) = 1. Then from (11) we obtain (dϕ)I(x) = I(ϕ(x)) = I(x). (12) dϕ is a (centro) affine transformation a(x, x) taking Tx M into Tx M , i.e. dϕ is a regular linear transformation, which takes the origo O ≈ x of Tx M into the origo O ≈ x of Tx M . The attribute “centro” relates to this property. In the following affine transformations a always mean centro-affine transformations. Thus at any isometry I(ϕ(x)) = a(x, x)I(x).

(13)

If for two configurations C ⊂ Tx M and C ⊂ Tx M the relation a(x, x)y = y ∈ C

and a−1 (x, x)y = y ∈ C

holds, for ∀ y ∈ C and y ∈ C, then we say that C and C are affine equivalent: C ae C. In a Finsler space F n = (M, F) fundamental function F and indicatrices I(p), p ∈ M reciprocally determine each other. So we may write (M, I) in place of (M, F). So we obtain Theorem 4. If the diffeormorphism ϕ : M → M is an isometry between the Finsler spaces F n = (M, I) and F n = (M , I), then I(p) and I(ϕ(p)) are affine equivalent: I(p) ae I(ϕ(p)).

(14)

(14) is a simple, but essential necessary condition of the isometry between F n and F n . (It means the linearization of the problem.) Nevertheless (14) is far from sufficient for the isometry. Indicatrices of a Riemannian space V n = (M, g) are ellipsoids (i.e. special quadrics) Q(x) ⊂ Tx M given by gik (x)v i v k = 1, v ∈ Tx M . So, given two Riemannian spaces V n = (M, Q) and V n = (M , Q), and a diffeomorphism ϕ : M → M ; Q(x) ae Q(ϕ(x)) is always satisfied, nevertheless two diffeomorphic Riemannian spaces (over the same base manifold) are not isometric in general. On the base manifold M of a Finsler space F n = (M, I) (14) is an equivalence relation. (14) splits M into equivalence classes. Let p1 be an arbitrary point of M . Then we define M1 := {p ∈ M | I(p) ae I(p1 )}, M2 := {p ∈ M | I(p) ae I(p2 )},

p2 ∈ M, p2 ∈ / M1

M3 := {p ∈ M | I(p) ae I(p3 )}, .. .

p3 ∈ M, p3 ∈ / (M1 ∪ M2 )

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Theorem 5. M=

[

Mλ

(15)

λ∈L

is a decomposition of M . This decomposition is unique, and Mλ λ ∈ L are equivalence classes.

5.

Affine Deformation of a Finsler Space

Tx M can be identified with a real vector space V n (y) consisting of the n-dimensional vectors y. Let φ(x) = {yφ } ⊂ Tx M be a subset (a configuration) of Tx M ≈ V n (y), and a(x) : Tx M → Tx M an affine transformation. Then a(x) : φ(x) → φ(x) = {y φ = a(x)yφ }. Let A be a section of the bundle M fibered by GL(n, R). The elements of A can be considered as affine transformations: a(p) : Tx M → Tx M . The components aij (p) of a(p) are continuous. If the (previous) configurations are the indicatrices I(p) of a Finsler space F n = (M, I), then (16) a(p)I(p) =: I(p) yields new indicatrices, and the Finsler space F n = (M, I) is called the affine deformation of F n and denoted by AF n . ∗

A Finsler space F n = (M, I) is a locally Minkowski space ℓMn = (M, I) if for each point p ∈ M there exists a chart U ∋ p with a coordinate system (x), such that F(x, y) is independent of x, or equivalently, the indicatrices I(x) are parallel translates of each other, (x) being considered as an affine coordinate system. Such a local coordinate system is called adapted. F n is a Minkowski space, if M = U is Rn or homeomorphic to it. Lemma. If Mλ admits a locally Minkowski structure, then F n = (M, I) restricted to Mλ ∗

is the affine deformation of a locally Minkowski space ℓ Mn (Mλ , I) on Mλ : ∗

F n ↾ Mλ = A (ℓMn (Mλ , I)).

(17)

Proof. Given F n = (M, I) and Mλ ⊂ M , first we construct a locally Minkowski space ∗

ℓ Mn (Mλ , I), and then an affine deformation A. We supposed that Mλ admits a locally Minkowski structure. In this case Mλ is an affine manifold [20, Chap. III. Section 1]. An affine manifold has a system of local charts Uα (xα ), α ∈ A covering Mλ , such that the transition xα1 ↔ xα2 between any two coordinate systems (on Uα1 ∩ Uα2 ) is linear. Let us consider such a system of local charts {Uα (xα )}, and let U (x) be one of them. We ∗

endow Tx0 M , x0 ∈ U with an indicatrix I(x0 ), which should be the indicatrix I(x0 ) of F n at x0 . Considering the coordinate system (x) on U as an affine coordinate system, we ∗

∗

define I(x), x ∈ U by parallel translation of I(x0 ). Thus we obtain on U a Minkowski ∗

space Mn (U, I), for which (x) is an adapted coordinate system. The fact that the transition ∗

between the different coordinate systems (xα ) is linear, makes it possible to extend I from

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∗

U to the whole Mλ in such a way that I is independent of xα on Uα , ∀α ∈ A (see [20, ∗

Chap. III., Theorem 8]). Thus we obtain a locally Minkowski space ℓ Mn (Mλ , I), where every (xα ) is an adapted coordinate system. The affine deformation A is easier to obtain. According to the construction of Mλ the indicatrices of F n at p0 and at an arbitrary point p ∈ Mλ are affine equivalent: I(p) = m1 (p0 , p)I(p0 ), where m1 (p0 , p) : Tp0 M → Tp M is an affine transformation. Furthermore, since I(p) ∈ C ◦ on M , also m1 (p0 , p) must be continuous: m1 ∈ C ◦ . ∗

∗

I(p) arises from I(p0 ) by (finitely many) successive linear transitions from an adapted ∗

∗

coordinate system to the next one. Thus I(p) = m2 (p0 , p)I(p0 ), and thus I(p) = ∗

∗

m1 (p0 , p) ◦ m−1 2 (p, p0 )I(p) ≡ a(p)I(p).

a(p) := m1 (p0 , p) ◦ m−1 2 (p, p0 ) yields A, which satisfies (17). A relation similar to (17) is true even if Mλ admits no locally Minkowski structure (e.g. if Mλ is no manifold), but in this case the consideration is a little more intricate. If fλ which admits a locally Minkowski structure, then Mλ can be covered by a manifold M ∗

∗

fλ , I) starting with I(p0 ) = I(p0 ), p0 ∈ we construct a locally Minkowski space ℓ Mn (M f Mλ ⊂ Mλ , and we obtain ∗

fλ , I)] ↾ Mλ . F n ↾ Mλ = [A ℓMn (M

fλ admitting a locally MinIt may happen that Mλ cannot be covered by a manifold M kowski structure (e.g. a sphere admits no locally Minkowski structure (see [1, p. 250]). Then Mλ can be split into (possibly few) parts: Mλκ : ∪κ∈K Mλκ = Mλ , such that Mλκ1 ∩ fλκ admitting a Mλκ2 = ∅, ∀ κ1 , κ2 ∈ K, and each Mλκ can be covered by a manifold M locally Minkowski structure. This is certainly so, for Mλ is covered by the union of local fλκ = Uλκ ⊃ Mλκ . charts Uλκ , and any chart admits a locally Minkowski structure. Then M Denoting λκ by α ∈ A, we can write [[ λ

κ

and by the Lemma we obtain:

fλκ ≡ M

[ α

fα ⊃ M

[

Mλ = M,

λ

Theorem 6. Any Finsler space is the union of affine deformations of locally Minkowski spaces Fn =

[

α∈A

∗

fα , I) ↾ Mα . Aα (ℓMn (M

(18)

Proposition 2. Under the condition (C) given below, Mλ of the decomposition (15) is closed. Proof. Let us consider a sequence pn of points in Mλ convergent in the topology of M to a point q ∈ M . According to the construction of the Mλ the indicatrices at p0 ∈ Mλ and

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at an arbitrary point p ∈ Mλ are affine equivalent: I(p) = m1 (p0 , p)I(p0 ). Furthermore, since I(p) ∈ C ∞ on M , m1 (p0 , p) must be continuous: m1 ∈ C ◦ on Mλ . Now we suppose a little more: (C)

m1 (p0 , p) can be extended continuously to the closure of Mλ .

Hence there exists lim m1 (p0 , pn ) = m1 (p0 , q).

n→∞

Then from lim I(pn ) =

n→∞

we obtain

h

i

lim m1 (p0 , pn ) I(p0 )

n→∞

I(q) = m1 (p0 , q)I(p0 ), which means that q ∈ Mλ . Then Mλ is closed. – This means a certain restriction on the possible Mλ in the decomposition (15). According to (15) for the base manifolds of two Finsler spaces F n = (M, I) and F n = (N, I) we have the decompositions a) M =

[

Mλ

and

b) N =

[

N̺ .

(19)

̺∈R

λ∈L

Mλ is determined by a point pλ ∈ M , and N̺ is determined by a point p̺ ∈ N . Theorem 7. If ϕ : M → N is an isometry and hence a diffeomorphism between F n and F n , then ϕ(Mλ ) = Nλ is a bijective relation between {Mλ } and {N̺ }, and consequently L = R. Also the Finsler areas kMλ kF and kNλ kF equal. Proof. We show that ϕ(Mλ ) = {ϕ(p) | p ∈ Mλ } ⊂ Nλ

(20.a)

ϕ−1 (Nλ ) = {ϕ−1 (p) | p ∈ Nλ } ⊂ Mλ .

(20.b)

and

First we prove (20.a). Let ϕ be an isometry, p an arbitrary and pλ a fixed point of Mλ . Then (dϕ)I(p) = I(ϕ(p)) = I(p) and

(dϕ)I(pλ ) = I(ϕ(pλ )) = I(pλ ).

Thus a)

I(p) ae I(p)

and

b)

I(pλ ) ae I(pλ ).

(21)

According to the construction of Mλ I(p) ae I(pλ ). Then

(21.a)

(21.c)

(21.b)

I(ϕ(p)) = I(p) ae I(p) ae I(pλ ) ae I(pλ ) = I(ϕ(pλ )).

(21.c)

(21.d)

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This yields (20.a). Conversely, if p, pλ ∈ Nλ , then according to the construction of Nλ I(p) ae I(pλ ). With respect to this and (21.a,b), from (21.d) we obtain (21.c), which yields (20.b). Since Mλ in (20.a), and later Nλ in (20.b) were arbitrarily chosen sets of the decomposition (19), the above considerations prove the 1 : 1 relation between {Mλ } and {Nλ }. So ϕ(Mλ ) = Nλ is a necessary condition for ϕ : M → N to be an isometry between F n = (M, I) and F n = (N, I), but it is not sufficient. kMλ kF = kNλ kF is obvious from ϕ(Mλ ) = Mλ , since ϕ is an isometry. Corollary 1. If ϕ : M → M is a motion on F n = (M, I), then any Mλ from (15) must be mapped on itself: ϕ(Mλ ) = Mλ , ∀ λ ∈ L. (22) So (22) is a necessary condition for ϕ to be a motion in F n . Corollary 2. If an Mλ consists of a single point pλ only, then pλ must be a fix point of any motion of F n . The role of the fix points of a motion of a Finsler space is investigated also by L. Kozma and P. I. Radu [11], and in a most recent work of Shaoquiang Deng.

6.

1-Parameter Continuous Group of Motions

In this section we present a result concerning the 1-parameter continuous group of motions in Finsler spaces. Theorem 7 has a consequence also in this case Corollary 3. If ϕt is a 1-parameter continuous group of motions of F n , then any orbit lies in an Mλ which contains a subset diffeomorphic to the line R1 or to the circle S 1 . Theorem 8. If ϕt 6= id is a 1-parameter continuous group of motions on a geodesically complete connected 2-dimensional Finsler space F 2 = (M, I), and in the decomposition of M according to (15) M1 and M2 consist of p1 resp. p2 only, and the sum r1+ + r2− , resp. r2+ + r1− of the injectively radii at p1 and p2 is greater than the distance ̺(p1 , p2 ) resp. ̺(p2 , p1 ) then M is diffeomorphic to the sphere S 2 , and ϕt consists of “rotations” around p1 and p2 . Proof. Let M1 = p1 , M2 = p2 and g(p1 , p2 ) = g(s) be a short geodesic of F 2 in arc length parameter connecting p1 and p2 . Let g(0) = p1 , g(D) = p2 , and q0 = g(a), such that a < r1+ and D − a < r2− . Then according to our assumption on the injectivity radii there exist the following two geodesic circles through q0 . The first one is Sp+1 (a) := {q ∈ M | ̺(p1 , q) = a} the geodesic (forward) circle centered at p1 with radius a, where ̺(p1 , q) designates the Finsler distance from p1 to q. Sp+1 (a) and its interior is a geodesic ball Bp+1 (a). The second one is Sp−2 (b) := {q ∈ M | ̺(q, p2 ) = b}, b = D − a the geodesic (backward) circle centered at p2 with radius b, and Bp−2 (b) is the corresponding geodesic ball. Then q0 is a common point of the two geodesic balls, and these balls have no common inner point.

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We claim that q0 is no fix point of ϕt . We show that the assumption that q0 is a fix point of ϕt leads to a contradiction. We know that ϕt (Sp+1 (a)) = Sp+1 (a) globally, for if q1 ∈ Sp+1 (a), then ̺(p1 , q1 ) = ̺(p1 , ϕt (q1 )) implies ϕt (q1 ) ∈ Sp+1 (a). If we suppose that there exist q1 ∈ Sp+1 (a) and ϕt such that ϕt (q1 ) = q2 6= q1 , then the arcs q0 q1 and q0 q2 = ϕt (q0 ) ϕt (q1 ) have different arc length. But this is impossible for a motion ϕt 6= id. Therefore if q0 is fix, then Sp+1 (a) is pointwise fix at any ϕt . But then ϕt = id on the whole M too. Let namely p be an arbitrary point of M . Then the (prolonged) geodesic g(p1 , p) joining p1 and p intersects Sp+1 (a) in a point q3 . p1 and q3 are fix at ϕt . Consequently also p is fix. Nevertheless we supposed that ϕt 6= id. Thus q0 cannot be a fix point. Now ϕt (q0 ) = q ∗ ∈ Sp+1 (a), q ∗ 6= q0 and for an appropriate t ϕt (q0 ) may be any point of the arc q0 q ∗ . With an appropriate choice of t and by several repetitions of ϕt any point of Sp+1 (a) can be reached. But any ϕt keeps the image of q0 also on Sp−2 (b). Therefore Sp+1 (a) = Sp−2 (b), which are diffeomorphic to S 1 . Also the geodesic circles Sp+1 (a), a < a, and Sp−2 (b), b < b are diffeomorphic to S 1 , and they fill Bp+1 (a) and Bp−2 (b). These geodesic balls are diffeomorphic to a hemisphere. Their union is diffeomorphic to S 2 , and the orbits of ϕt on Bp+1 (a) ∪ Bp−2 (b) ≡ B are geodesic circles, that is ϕt is a “rotation” on B. Finally we show that Bp+1 (a) ∪ Bp−2 (b) = M. (23) Let again p be an arbitrary point of M , and g(s) in arc length parameter s the short geodesic connecting p1 = g(0) with p = g(s0 ). If s0 ≤ a, then p ∈ Bp+1 (a). If s0 > a, then g(s) crosses Sp+1 (a) = Sp−2 (b) at g(a), and goes over to Bp−2 (b). If s0 is big enough, then g(s) crosses again Sp+1 (a) = Sp−2 (b), and goes over to Bp+1 (a), and so on. This gives that any p ∈ M is a point of Bp+1 (a) or of Bp−2 (b), and this yields (23). Corollary 4. If in an F 2 = (M, I) in the decomposition of M according to (15) there are three sets consisting of one point: M1 = p1 , M2 = p2 , M3 = p3 , then F 2 admits no 1-parameter continuous group of motions ϕt (except the trivial case ϕt = id). It is rather easy to see this. The short proof is omitted. One can see also that the cut locus of p1 is p2 , and conversely. rem In case of a Riemannian space V 2 the conditions M1 = p1 and M2 = p2 cannot be satisfied, for in a V n any two indicatrices Q(p1 ), Q(p2 ) are affine equivalent. However, Theorem 8 and Corollary 4 remain valid if M1 = p1 , M2 = p2 are replaced by the condition that ϕt has two fix points p1 and p2 . Also the proof remains unaltered. In case of n = 3 the situation is different. In an E 3 which is a special F 3 , a rotation around a line L is a 1-parameter continuous group of motions, which leaves fix the line L, and thus three of its points p1 , p2 , p3 . rem In the case of an absolutely homogeneous F 2 Theorem 8 can be related to some other results. Any geodesic g(s) emanating from p1 is a forward geodesic with respect to p1 until g(a) = q ∈ Sp+1 (a) = Sp−2 (b), and it is a backward geodesic with respect to p2 from a until g(D) = p2 . It is easy to see that the cut locus of p1 is p2 . If we assume additionally that also the sum of the injectivity radii r2+ + r1− is greater than ̺(p2 , p1 ), then after a consideration analoguos to the previous one we find that the cut locus of p2 is p1 , that is the cut loci both of p1 and p2 are a unique point. Manifolds with this property were investigated by several authors. L. Green [9] showed that a compact 2-dimensional Riemannian manifold

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V 2 for which the cut locus of any point consists of a unique point only (called wiedersehen manifold) is isometric to the sphere. In Theorem 8 (beside of the existence of a 1-parameter continuous group of motions) the one point cut locus property holds for the points p1 , p2 only, and also the consequence is weaker: F 2 is only diffeomorphic to the sphere. Green’s theorem was later generalized to compact V n of even dimension by M. Berger and J. Kazdan [3] and to similar manifolds of odd dimension by C. T. Yang [21]. A survey on these problems can be found in J. Kazdan’s paper [10].

7.

Isometries between Finsler Spaces

In Theorem 4 we have shown that if a diffeomorphism ϕ : M → M is an isometry between two Finsler spaces F n = (M, I) and F n = (M , I), then there exists an A, such that I(ϕ(x)) = a(x, ϕ(x))I(x),

a ∈ A or

I(ϕ(x)) ae I(x),

(13)

where A : M → GL(n, R), M ∋ x 7→ a(x), and a(x) is a regular n × n matrix, which can be considered as an affine transformation a(x, ϕ(x)) : Tx M → Tϕ(x) M . As we pointed out in Section 4, the relation (13) or (14) is a necessary, but far from sufficient condition in order that ϕ be an isometry between F n and F n . We remark that a may not be unique in (13). Let k(x) : Tx M → Tx M be an affine transformation taking I ⊂ Tx M globally into itself: kI = I (e.g. if I is the unit sphere S of the Euclidean space E n , and k is a rotation, then kS = S). Such a k is called affine ∗ automorphism of I. Then for a ◦ k =: a ∗

aI = I

and {k} =: K ⊂ GL(n, R)

is a subgroup of the group of all affine transformations. So in place of (13) the more comprehensive and more appropriate relation is ∗

I(ϕ(x)) = a(x, ϕ(x))I(x).

(24)

Now we want to complete (24) to a sufficient condition. Clearly, ∗

a(x, ϕ(x)) = (dϕ)(x)

(25)

together with (24) is sufficient for ϕ to be an isometry between F n and F n . – This must be ∗ understood in such a way that there exists a k(x), such that a(x, ϕ(x)) = a(x, ϕ(x)) ◦ k(x) satisfies (25). Let V n = (M, g) be an arbitrary Riemannian space on M . Indicatrices of V n are the ellipsoids gij (x)v i v j = 1, v ∈ Tx M . They will be denoted by Q(x). So V n = (M, Q). We define V n = (M, Q) by ∗

Q(ϕ(x)) := a(x, ϕ(x))Q(x).

(26)

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∗

∗

dϕ takes Q(x) into Q(ϕ(x)) = (dϕ)x Q(x). Thus V n = (M , Q) is isometric to V n by ϕ. If ϕ is an isometry between V n and V n too, then also Q(ϕ(x)) = (dϕ)x Q(x)

(27)

must hold. Then from (26) and (27) we obtain (25). In this case (24) yields the isometry of F n and F n . Thus the isometry between V n and V n implies the isometry between F n and n F . If M admits a locally Minkowski structure [1, 19] then V n can be chosen as a locally Euclidean space ℓE n (a space which can be covered by local charts Uα (x), Uβ (y), Uγ (z), . . . , such that any transition function y i = y i (x) on Uα ∩ Uβ etc., is linear and the metric on each chart is Euclidean). Then V n = ℓE n has vanishing curvature R = 0, and if ϕ is an isometry between V n and V n , then also R = 0 on M . Thus in this case the condition for the isometry between F n and F n (beside of (24)) is R = 0. The converse is rather simple. We show that if ϕ is a Finsler isometry between F n and F n , then ϕ is an isometry between V n and the constructed V n , respectively if V n = ℓE n , then R = 0. – If ϕ is a Finsler isometry between F n and F n , then I(x) = (dϕ)I(x). ∗ ∗ ∗ This dϕ is an affine transformation a. Thus I = aI. From these we obtain (25): a = dϕ. ∗ V n was arbitrary, and V n was constructed by Q = aQ. From this and (25) we obtain Q = (dϕ)Q, and this means that V n and V n are isometric by ϕ. In the special case of ∗ V n = ℓE n we obtain R = R = 0. We remark that in the reversed considerations I = aI was no condition. This follows from the assumption that ϕ is a Finsler isometry between F n and F n . Now we summarize our result. We know that for the isometry of the Finsler spaces n F = (M, I) and F n = (M , I) by the diffeomorphism ϕ : M → M (24) is a necessary condition. Let V n = (M, Q) be an arbitrary Riemannian space, and V n = (M , Q) given by (26). Then we obtain Theorem 9. Under the above conditions F n is isometric to F n by ϕ if and only if there ∗ exists an a such that V n is isometric to V n . This theorem reduces the isometry of Finsler spaces to isometry of Riemannian spaces. ∗ In the “only if” case (24) is automatially satisfied with a = a (k = id). If V n is chosen as a locally Euclidean space V n = ℓE n , then the isometry between V n and V n is equivalent to R = 0, the vanishing of the curvature tensor of V n . We remark that V n can be replaced by any other Finsler space (M, J). Then V n is ∗ replaced by the Finsler space (M , J), where J := aJ. Such a choice is favorable if the isometry of (M, J) and (M , J) can easily be decided.

8.

Analytical Condition

As we have seen in the previous Section, ϕ : M → M is an isometry between the Finsler spaces F n = (M, I) and F n = (M , I) iff (24) and (25) hold. (24) assures the linear relation between I and I, and (25) has the form ∗

aij (x) =

∂xi (x). ∂xj

(28)

Direct Geometrical Method in Finsler Geometry ∗

311 ∗

Here ϕ is written in the form xi = xi (x), and aij (x) are the components of a(x, ϕ(x)). (28) ∗

has a solution for xi (x) iff aij (x) is a gradient vector of a function f i (x) for any fix i. Thus we obtain Theorem 10. ϕ : M → M is an isometry between F n = (M, I) and F n = (M , I) if and ∗ ∗ only if there exists an aij (x) satisfying (24) and (25). To this aij must be a gradient vector for each i. In the case if F n is the affine deformation of a locally Minkowski spaces we can say a little more: Proposition 3. A) If F n = (M, I) is the affine deformation of a locally Minkowski space: F n = A[ℓMn (M, J)], then it can be mapped isometrically only on another Finsler space F n = (M , I), which is also affine deformation of a locally Minkowski space. n B) In order that two Finsler spaces A[ℓ Mn (M, J)] and A [ℓ Mn (M , J)] admit an isometric mapping ϕ : M → M it is necessary that any pair J(p) and J(q) of their indicatrices be affine equivalent. Proof. A) According to the Introduction of this Chapter to F n = (M, I) = A[ℓMn (M, J)] are those spaces isometric which can be represented (locally) as F n = (M (x), I(x)) in other coordinate systems (x). ℓMn (M, J) is a locally Minkowski space in any other coordinate system. Then the isometric images of our F n are again affine deformations of a locally Minkowski space in the form A(x) ℓMn (M, (x), J(x)) x e x e) = ∂∂xe A( A(x). B) Suppose that ϕ : M → M is an isometry between F n = A[ℓMn (M, J)] and F n = A[ℓMn [M , J)]. First we remark that in F n the indicatrices J(p), J(q), are affine equivalent: J(p) ae J(q), ∀ p, q ∈ M . Namely let us connect p and q by an arc. This can be covered by finitely many adapted coordinate systems Uα (α), α ∈ A. Within one adapted coordinate system, considered as an affine coordinate system, the equation of the indicatrices is the same. On transition from one adapted coordinate system to another the transition is linear. The number of the transitions (linear transformations), is finite. Denoting the product of these linear transformations by L, we obtain J(q) = LJ(p), that is I(p) ae I(q). The affine deformation A does not hurt this relation. The same is true for F n = A[ℓMn (M , J)] too, that is J(p)aeJ(q), ∀ p, q ∈ M . – Since ϕ : M → M is an isometry between F n and F n , J(p) ae J(ϕ(p)). Combination of this relation with the previous two ones yields that J(p) ae J(q) with arbitrary p ∈ M and q ∈ M . From these it follows that if two indicatrices I(p) and J(q) of the above F n and F n are not affine equivalent, then these affine deformations of locally Minkowski spaces cannot be isometric. Now we want to obtain a sufficient analytical condition for the isometry between F n = A[ℓMn (M, J)] and F n = A[ℓMn (M , J)]. Let U (x) and U (y) be local charts on M and M resp, and let the necessary condition (13) of the isometry between F n = A[ℓMn (U, J)] and F n = A[ℓ Mn (U , J)] be satisfied, that is let there exist points x0 ∈ U and y0 ∈ U such that J(x0 ) ae J(y0 ) or J(y0 ) = a0 J(x0 ). We give a sufficient condition for the isometry of F n ↾ U and F n ↾ U . Since

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in a locally Minkowski space indicatrices of any two points are affine equivalent, we have: J(x0 ) ae J(x) with arbitrary x ∈ U , and also J(y) ae J(y0 ) with arbitrary y ∈ U . Then there exist matrices a(x) and a(y) such that J(x) = a(x)J(x0 ) and J(y) = a(y)J(y0 ). Then a(y) ◦ a0 ◦ a−1 (x)J(x) = J(y) or m(x, y)J(x) = J(y) (29) m(x, y) = a(y) ◦ a0 ◦ a−1 (x).

∂y If y = y(x) is an isometry between A[ℓMn (U, J)] and A[ℓMn (U , J)], then m = ∂x , ∂y and conversely, from m = ∂x follows the isometry. Hence the sufficient condition for the isometry (beside the necessary condition (13)) is the solvability of the ODE system

mij (x, y(x)) =

∂y 0 (x) ∂xj

(30)

for y i = y i (x), where mij is given by (29). Using this result and Theorems 6 and 7 we can say that F n = (M, I) is isometric to F n = (N, I) iff in the decompositions (19) to each Mλ there exists an Nλ , and for a pair of points x ∈ Mλ , y ∈ Nλ I(x)aeI(y) holds, and the PDE system (30) is solvable for y = y(x), and the solutions match differentiably. In this result the condition of the isometry is explicitely expressed by PDE systems. Thus we obtain Theorem 11. Two Finsler spaces F n = (M, I) and F n = (N, I) are isometric iff for the corresponding pairs Mλ and Nλ of the decompositions M = ∪Mλ and N = ∪Nλ (see (19)) the PDE systems (30) are solvable for y = y(x), and the solutions match differentiably. This result has some application to the group ϕt of 1-parameter continuous motions on a Finsler space F n = (M, I). Let ϕt : M ×R → M be a 1-parameter group of transformations on M . ϕt takes a point x ∈ M into another point y ∈ M which is determined by x and t : ϕt : x 7→ y = y(t, x). dϕt takes Tx M into Ty(t,x) M . If F n is an affine deformation of a locally Minkowski space, which is a special Finsler space: F n = A[ℓMn (M, I)], then the indicatrices of F n are affine equivalent to each other. Then I(y(t, x)) = a(x, y(t, x))I(x). If ϕt is a motion, then dϕt takes indicatrix into indicatrix. In this case ∂y ∂y i (t, x) or aik (x, y(t, x)) = (t, x) (31) ∂x ∂xk (c.f. (13)). Also conversely, if (31) holds, then ϕt is a 1-parameter continuous group of motions. (31) can be considered as a PDE system for y(t, x). a(x, y(t, x)) =

d i y (t, x) =: v i (t, x) dt are the components of the velocity vector v of the motion at y(t, x). Thus i

y (t, x) =

Z

0

where y i (t, x) is the solution of (31).

t

v i (τ, x) dτ,

(32)

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Proposition 4. (31) and (32) are a kind of Killing equations for F n = A ℓMn . For an arbitrary F n = (M, I) we have the decomposition (18). In order that ϕt be a 1parameter continuous group of motions on F n (31) must be satisfied on each subset Mλ of the decomposition (15). The different solutions on Mλ (resp on Mα ) must fit differentiably.

References [1] D. Bao and S. S. Chern, A note on the Gauss-Bonnet therorem for Finsler spaces, Ann. Math. 143 (1946) 233–252. [2] D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry (Springer, New York, 2000). [3] M. Berger and J. L. Kazdan, A Strum-Liouville inequality with application to an isoperimetric inequality for volume in terms of injectivity radius, and to wiedersehen manifolds, In: General Inequalities 2 (Birkh¨auser, 1980) 367–377. [4] W. Blaschke, Kreis und Kugel (Leipzig, 1916). [5] T. Q. Binh and L. Tam´assy, Deviation of a Finsler space from a Riemannian one, Tensor N. S. 68 (2007) 44–50. [6] H. Busemann, Intrinsic area, Ann. Math. 48 (1947) 234–267. [7] S. S. Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices Amer. Math. Soc. 43 (1996) 953–959. [8] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific J. Math. 207 (2002) 149–155. [9] L. W. Green, Auf Wiedersehensfl¨achen, Ann. of Math. 78 (1963) 289–299. [10] J. L. Kazdan, An isoperimetric inequality and wiedersehen manifolds, In: Seminar on Diff. Geom. ((S. T. Yau, Ed.) Annals of Math. Studies 102, 1982) 143–157. [11] L. Kozma and P. I. Radu, Weinstein’s theorem for Finsler manifolds, J. Math. Kyoto Univ. 42 (2006) 377–382. [12] R. Lovas, On the Killing vector fields of generalized metrics, SUT J. Math. 40 (2004) 133–156. [13] M. Matsumoto, V -transformations of Finsler spaces I., Definitions, infinitesimal transformations and isometries, J. Math. Kyoto Univ. 12 (1978) 479–512. [14] H. Rund, The Differential Geometry of Finsler Spaces (Springer, 1959). [15] Z. Shen, Lecture notes on Finsler geometry (Indiana Univ., 1998). [16] Z. Shen, Differential Geometry of Spray and Finsler Spaces (Kluwer, 2001).

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[17] Z. Shen, Lectures on Finsler Geometry (Singapore, 2001). [18] Z. I. Szab´o, Generalized spaces with many isometries, Geom. Dedicata 11 (1981) 369–383. [19] L. Tam´assy, Ein Problem der zweidimensionalen Minkowskischen Geometrie, Ann. Polon. Math. 9 (1960) 39–48. [20] L. Tam´assy, Finsler geometry in the tangent bundle, In: Finsler Geometry (Advanced Studies in Pure Math., Japan, 48, Sapporo 2005) 168–194. [21] C. T. Yang, Odd-dimensional wiedersehen manifolds are spheres, J. Diff. Geom. 15 (1980) 91–96.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 315-340

Chapter 18

L INEAR C ONNECTIONS A LONG THE TANGENT B UNDLE P ROJECTION W. Sarlet∗ Department of Mathematical Physics and Astronomy Ghent University, Krijgslaan 281 B-9000 Ghent, Belgium

Abstract An intrinsic characterization is given of the concept of linear connection along the tangent bundle projection τ : T M → M . The main observation thereby is that every such connection D gives rise to a horizontal lift, which is needed to extend the action of the associated covariant derivative operator to tensor fields along τ in a meaningful way. The interplay is discussed between the given D and various related connections, such as the canonical non-linear connection of the geodesic equations and certain linear connections on the pullback bundle τ ∗ τ . This is particularly relevant to understand similarities and differences between various notions of torsion and curvature. I further discuss aspects of variationality and metrizability of a linear D along τ and let me guide for the selected topics by a very short, old paper of Krupka and Sattarov.

1.

Introduction

On the occasion of celebrating a scientist’s 65th birthday, it is respectable to look back at the history of the person’s involvement in science and it is definitely a good sign if one can easily detect older work which still raises interesting questions or challenges. I recently laid my hands on what is in fact a rather minor contribution of Demeter Krupka [10], also one of the first reprints he gave me personally, and I was astonished to see that, looking at it now, it confronts me with questions I had not thought of before, even though they are directly related to my own research of the past 10 to 15 years. Section 2 in [10] carries “connections on the tangent bundle” in the title, but is about maps from a tangent bundle T M into the fibre bundle ΓM → M of linear connections on M , which make the following diagram commutative: ∗

E-mail address: [email protected]

316

W. Sarlet ΓM >

D

TM

?

τ

-

M

In my opinion, the concept of a ‘connection on a manifold’ has an unambiguous meaning in the literature, and that is not what the above diagram is about. Instead, a much better name for the D under consideration here is linear connection along the tangent bundle projection. The surprising observation for me, however, is that, having been involved in the development of a comprehensive theory of derivations of forms along the tangent bundle projection in [15, 16], and having made use of the calculus along τ in many applications since then, the idea of such a linear connection along τ never came up. Needless to say, other types of connections frequently play a role in my use of the calculus along τ , such as what are called linear connections on the pullback bundle τ ∗ τ : τ ∗ T M → T M , so it becomes intriguing to understand the difference or interplay between all such related, but different concepts. What is more, it turns out that (to the best of my knowledge) not much can be found in the literature about linear connections along τ and what is available all seems based on (sometimes rather untidy) ad hoc coordinate constructions, i.e. seems to lack a proper coordinate-free foundation. For example, going back to section 2 of [10] i (q, v) are connection coagain, if (q, v) is taken as notation for coordinates on T M and γjk efficients of D, the authors state, as though it should be common knowledge, that geodesics are curves in M , satisfying the equations k q¨k + γij (q, q) ˙ q˙i q˙j = 0,

(1)

and that there is a covariant derivative operator for tensor fields along τ , “defined in a standard manner”, which in the case of a metric tensor g along τ is given by gij;k =

∂gij ∂gij s r m m − γ q˙ − gim γjk − gjm γik . k ∂q ∂ q˙s rk

(2)

But is that common knowledge? It seems to me that the term ‘geodesic’ should be used only if there is a clear notion of parallel transport first, leading subsequently to geodesics as auto-parallel curves. This, plus a coordinate-free backing for the “standard manner” in which gij;k should be defined, I was unable to find in the literature on which the statements in [10] must have been based. Linear connections along τ were probably introduced for the first time by Hanno Rund, who called them “direction-dependent connections” [21]. Unfortunately, what is probably a comprehensive account of Rund’s involvement in this theory, seems to have appeared only in extra chapters of the Russian translation (by Asanov) of his book on Finsler spaces (see the Math. Review MR0641695 (83i:53097)). So probably, the best source now (for illiterates in Russian) is Appendix A of Asanov’s own book [4]. There, one will find corresponding concepts of torsion, curvature, Bianchi identities, etcetera explained. The torsion k − γ k , as one might expect, and tensor, for example, is defined as having components γij ji

Linear Connections Along the Tangent Bundle Projection

317

the components of the curvature tensor are given by k k ∂γij ∂γij − γ mvr ∂q l ∂v m rl

!

−

∂γilk ∂γilk m r − m γ v j ∂q ∂v rj

!

k k m γilm . γij − γmj + γml

(3)

But although an attempt is made to construct such tensors in an intrinsic way, the result is rather unsatisfactory for several reasons: to begin with, objects which should be regarded as living along the tangent bundle projection are often subjected to operations (such as an exterior derivative) which act on the full tangent bundle; the result then is usually not a tensor along τ ; this in turn prompts the author to add corrective terms in a rather ad hoc manner, in order to arrive at a quantity with a proper tensorial meaning. Observe for later that such corrections always involve derivatives with respect to the fibre coordinates v i on T M . My aim is to shed a refreshing light on all such concepts by making use of the calculus along τ in a systematic way. This will lead to new questions which cannot all be exhaustively discussed in the course of the present paper. In selecting topics for discussion, therefore, I will let the further aspects treated in Krupka’s paper [10] be my guidance. It is unfortunate that the literature is full of rather strange terminology for things which are (closely or not) related to the topics under consideration here. The point is that in all such cases, the issue is about tensor fields along τ and operations on them, which have not been properly identified or recognized as such. In [2], for example, it is mentioned that what I would call objects along τ are sometimes called d-objects, or M -objects, or even Finsler objects (though they have nothing whatsoever to do with Finsler spaces). I dare hope that the present paper can inspire to more unification in this terminology as well.

2.

Elements of the Calculus of Forms Along τ

Let X (τ ) denote the C ∞ (T M )-module of vector fields along τ , which are maps fitting in a similar commutative diagram as above, with T M replacing ΓM , or equivalently are V sections of the pullback bundle τ ∗ τ : τ ∗ T M → T M . Likewise, k (τ ) and V k (τ ) will refer to scalar and vector-valued k-forms along τ respectively. V In coordinates, elements X ∈ X (τ ) and α ∈ 1 (τ ) are of the form, X = X i (q, v)

∂ , ∂q i

α = αi (q, v) dq i ,

(4)

while, more generally, an element L ∈ V ℓ (τ ) is of the form L = λi ⊗

∂ ∂q i

with λi = λij1 ···jℓ dq j1 ∧ · · · ∧ dq jℓ ∈

where the λij1 ···jℓ again are functions on T M . Definition. D : Vp

1. D(

V

(τ ) →

(τ )) ⊂

Vp+r

V

(τ ) is a derivation of degree r if

(τ )

2. D(α + λ β) = Dα + λ Dβ,

λ ∈ IR

Vℓ

(τ ),

(5)

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3. D(α ∧ γ) = Dα ∧ γ + (−1)pr α ∧ Dγ,

α∈

Vp

(τ ). V

A derivation D of degree r of V (τ ) has an associated derivation of (τ ), also denoted by V D, such that in addition to the above rules: for L ∈ V ℓ (τ ) and ω ∈ p (τ ), D(ω ∧ L) = Dω ∧ L + (−1)pr ω ∧ DL.

V

For practical purposes, it is of interest to know that every D of (τ ) is completely determined by its action on functions on T M and on basic 1-forms, i.e. 1-forms on M regarded as 1-forms along τ by composition with τ . For an extension to V (τ ), it suffices to specify a consistent action on basic vector fields (vector fields on M ). The commutator of D1 and D2 (of degree r1 and r2 respectively) is the degree r1 + r2 derivation, defined by [D1 , D2 ] = D1 ◦ D2 − (−1)r1 r2 D2 ◦ D1 ,

(6)

and satisfies a graded Jacobi identity. There is a canonically defined vertical exterior derivative dV on V (τ ). In the light of what was said above, it suffices to know that dVF = Vi (F ) dq i , dV α = 0

for α ∈

with Vi = V1

(M ),

∂ , ∂v i dV

∀F ∈ C ∞ (T M ), ∂ ∂q i

= 0.

(7) (8)

The classification of derivations of forms along τ and many related issues were discussed in great detail in [15, 16]. I shall limit myself here to recalling the essentials of this theory which will be needed further on. The first point to observe is that a classification requires the availability of a (non-linear or Ehresmann) connection on τ : T M → M . As a matter of fact, any choice of a connection, giving rise to a local basis of horizontal vector fields Hi =

∂ ∂ − Γji (q, v) j , i ∂q ∂v

(9)

allows to construct a horizontal exterior derivative dH as follows: dHF = Hi (F ) dq i ,

(10)

dHα = dα

(11)

F ∈ C ∞ (T M ), V1 ∂ ∂ H for α ∈ (M ), d = Vi (Γkj ) dq j ⊗ k . i ∂q ∂q

Inspired by the standard Fr¨olicher and Nijenhuis theory of derivations of (scalar) forms [8], one is then led to distinguish four types of derivations. • Type i⋆ derivations are those which vanish on functions; they are determined by some L ∈ V (τ ), written as iL , and defined exactly as in the standard theory. That is to say, V for L ∈ V r (τ ), α ∈ 1 (τ ), iL α(X1 , . . . , Xr ) = α(L(X1 , . . . , Xr )),

which extends to a derivation of degree r − 1 on scalar forms, and is taken to be zero also on basic vector fields.

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319

• Type dV⋆ derivations are those of the form dVL = [iL , dV] for some L. H H • Likewise, type dH ⋆ derivations are those of the form dL = [iL , d ].

• Finally, the extension to vector-valued forms requires an extra class of derivations, V said to be of type a⋆ . By definition, these vanish on (τ ), they are denoted as aQ V for some Q ∈ r (τ ) ⊗ V 1 (τ ) and (in view of what has been said before) are further completely determined by the following action on X ∈ X (τ ): aQ X(X1 , . . . , Xr ) = Q(X1 , . . . , Xr )(X).

(12)

The classification theorem proved in [15] states that every derivation of V (τ ) has a unique representation as the sum of one of each of the above four types of derivations. The torsion T and curvature R of the non-linear connection we started from, make their appearance within this theory as vector-valued 2-forms along τ (as opposed to verticalvector-valued semi-basic forms on T M in other approaches, for example). In fact, T and V R are uniquely determined by the following commutators on (τ ) (extra terms of type a⋆ come in when the same commutators are regarded as derivations on V (τ )): [dH, dV] = dVT ,

1 H H 2 [d , d ]

= −idV R + dVR .

(13)

In coordinates, T

=

R =

∂ , ∂q k ∂ i j 1 k 2 Rij dq ∧ dq ⊗ ∂q k , 1 k 2 Tij

dq i ∧ dq j ⊗

Tijk = Vj (Γki ) − Vi (Γkj ),

(14)

k Rij = Hj (Γki ) − Hi (Γkj ).

(15)

A concise formulation of Bianchi identities then is obtained as follows: dHR = 0,

dHT + dVR = 0.

(16)

The connection we are using to develop these ideas of course also provides ways to pass from objects along τ to objects on the full tangent bundle and vice versa. This really works both ways: for example, if X = X i (q, v)∂/∂q i is any vector field along τ , we have vertical and horizontal lifts to vector fields on T M , which in coordinates are given by X V = X i Vi ,

X H = X i Hi ,

(17)

but conversely, every vector field on T M has a unique decomposition into a horizontal and vertical part and this may reveal new interesting objects along τ . To see this interplay at work, consider the brackets of horizontal and vertical lifts on T M . We have [X V , Y V ] = ([X, Y ]V )V , H

V

H

H

H

V

(18) V

H

[X , Y ] = (DX Y ) − (DY X) , H

(19) V

[X , Y ] = ([X, Y ]H ) + R(X, Y ) .

(20)

We see, for example, that the decomposition of the bracket of a horizontal and a vertical lift inevitably leads to the identification of two important derivations of degree zero, the

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horizontal and vertical covariant derivative : DVX and DH X depend linearly on their vector field argument, are determined by the following action on functions F ∈ C ∞ (T M ) and coordinate vector fields DVX F = X V (F ), H DH X F = X (F ),

∂ = 0, ∂q i ∂ ∂ DH = X j Vi (Γkj ) k , X i ∂q ∂q DVX

(21) (22)

extend to 1-forms along τ by the duality rule DhX, αi = hDX, αi + hX, Dαi,

(23)

and then further to arbitrary tensor fields along τ in the usual way. The other two brackets above identify the curvature tensor R again, plus horizontal and vertical brackets of vector fields along τ , which are given by [X, Y ]V = (X V (Y i ) − Y V (X i ))

∂ , ∂q i

[X, Y ]H = (X H (Y i ) − Y H (X i ))

∂ . (24) ∂q i

Note that the vertical bracket satisfies a Jacobi identity, but the horizontal one doesn’t, unless R is zero. The covariant derivative operators DV and DH in turn define a linear connection on the pull-back bundle τ ∗ T M → T M , as follows: every ξ ∈ X (T M ) has its unique decomposition in the form ξ = X H + Y V , with X, Y ∈ X (τ ), define ∇ξ : X (τ ) → X (τ ) by (see [14]) V ∇ξ = DH (25) X + DY . ∇ξ is said to be a connection of Berwald type. For more insight in the geometric features of such connections, see [7]. Since ∇ξ acts on vector fields along τ , one may raise the question: should this be called a linear connection along τ ? As explained in the introduction, however, it is only after looking back at Krupka’s old paper [10], that I realized that there is something else which corresponds better to this terminology, although there should be links with what has just been recalled. Before entering into the subtleties of this discussion, I need to say a few words about the special case that the non-linear connection which has been used so far, is the canonical one associated to a second-order equation field (S ODE). To keep it well distinguished from the general case, I will do this in a separate section.

3.

The Case of a Bsode Connection

As is well known, the tangent bundle T M of a manifold comes equipped with an intrinsically defined type (1,1) tensor field S, usually called the vertical endomorphism, which has the coordinate expression S = dq i ⊗ ∂/∂v i . A S ODE Γ is characterized by the fact that S(Γ) is the dilation vector field ∆ = v i ∂/∂v i . It has the property (LΓ S)2 = I (the identity tensor), from which it follows that 1 PH = (I − LΓ S), 2

1 PV = (I + LΓ S) 2

Linear Connections Along the Tangent Bundle Projection

321

are complementary projection operators and thus define a non-linear connection. If Γ has the coordinate representation Γ = vi

∂ ∂ + f i (q, v) i , ∂q i ∂v

(26)

the connection coefficients of the S ODE connection are given by Γji = −

1 ∂f j . 2 ∂v i

(27)

A S ODE connection is characterized by the fact that it has zero torsion T . Note that there exists a canonical vector field along τ , namely the identity map on T M , which will be denoted by ∂ T = vi i . (28) ∂q It is of some interest to point out that for any non-linear connection, TH is a S ODE, let us call it the associated S ODE, but if the connection we start from is a S ODE connection, its associated TH will in general not coincide with the original Γ, as is obvious from the coordinate expressions. In addition to the machinery developed for arbitrary connections, the case of a S ODE connection has two very important extra tools to offer: one is the dynamical covariant derivative ∇, which is a degree 0 derivation, the other is a (1,1) tensor Φ ∈ V 1 (τ ), called the Jacobi endomorphism. They are forced upon us, for example, via the same sort of interplay between the calculus along τ and standard calculus on T M , by looking at the decomposition of LΓ X H . Indeed, it turns out that the vertical part in this decomposition depends tensorially on X, while the horizontal part identifies a derivation. So, we can write, LΓ X H = (∇X)H + Φ(X)V ,

(29)

which defines Φ and ∇ on X (τ ). ∇ is further determined by the duality rule (23) and the fact that ∇F = Γ(F ) on functions F ∈ C ∞ (T M ). For computational purposes: ∂ ∇ ∂q i

= Γki

∂ , ∂q k

∇(dq i ) = −Γik dq k ,

(30)

and Φ = Φij dq j ⊗

∂ , ∂q i

with Φij = −

∂f i − Γkj Γik − Γ(Γij ). ∂q j

(31)

The relevance of Φ and ∇ is already obvious from the properties: dVΦ = 3R,

dHΦ = ∇R.

(32)

Since variationality will be one of the topics under discussion later on, I conclude this section with the very concise formulation of the so-called Helmholtz conditions within this geometric approach: the necessary and sufficient conditions for the existence of a (regular)

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Lagrangian formulation of a given S ODE Γ are the existence of a non-degenerate, symmetric type (0,2) tensor field g along τ , satisfying the requirements [16]:

V

∇g = 0,

(33)

V

DX g(Y, Z) = DZ g(Y, X),

(34)

g(ΦX, Y ) = g(ΦY, X).

(35)

To close the sections about the main ingredients of the calculus along τ and its relevance in the study of second-order dynamics, I should say that the calculus along τ is being used systematically also in Szilasi’s magnum opus [23].

4.

Linear Connections Along τ

Let us go back now to Rund’s direction-dependent connections, i.e. maps D : T M → ΓM fitting in the commutative diagram of the introduction. My first aim is to find a coordinatefree justification for the equations (1), as geodesic equations, and for (2) as defining relation of the covariant derivative of a metric along τ . The fibre of ΓM at m ∈ M consist of maps D : Tm M × Xm → Tm M, where Xm denotes the module of vector fields on M defined in a neighbourhood of m, which have the properties Dλvm Y

= λDvm Y,

λ ∈ IR

Dvm (f Y ) = f (m)Dvm Y + vm (f )Y (m),

f ∈ C ∞ (M )

plus linearity with respect to the sum in both arguments. So for each wm ∈ Tm M , D(wm ) k on T M , such that is such a map, and is locally defined by functions γij ∂ ∂ k D(wm ) ∂ = γij (wm ) k . ∂q m ∂q j ∂q i

(36)

m

For an alternative view, given a D in the above sense, define for all X ∈ X (τ ), and Y ∈ X (M ), a map D : X (τ ) × X (M ) → X (τ ), by (DX Y )(wm ) = D(wm )Xwm Y.

(37)

By construction, DX Y will be IR-linear in both arguments and further satisfies DF X Y

= F DX Y,

∀F ∈ C ∞ (T M )

DX (f Y ) = f DX Y + X(f ) Y,

(38) ∞

∀f ∈ C (M ).

(39)

Proposition 1. A linear connection D along τ is a map D : X (τ ) × X (M ) → X (τ ) which is IR-linear in both arguments and has the properties (38, 39).

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Proof. Indeed, conversely, using the standard trick with a bump function, the above properties imply that the value of DX Y at a point wm ∈ Tm M only depends on the value of X at wm . As a result, it makes sense to define a map D : T M → ΓM by D(wm )vm Y = (DX Y )(wm ), where X is any vector field along τ such that Xwm = vm , and this provides a linear connection along τ , in the sense of the commutative diagram we started from. To extend D further as covariant derivative operator, we need to extend the second argument to X (τ ), which in turn requires some notion of horizontal lift for the action on functions on T M . Given a curve σ : t 7→ q i (t) in M , take a vector field along σ, i.e. a curve η : t 7→ i (q (t), η i (t)) in T M which projects onto σ, and define another vector field Dσ˙ η along σ by (Dσ˙ η)(t) = D(η(t))σ(t) ˙ Y.

(40)

Here, Y is any vector field, defined in a neighbourhood of σ(t), such that Y (σ(t)) = η(t). In coordinates:

k

Dσ˙ η(t) = η˙ (t) +

∂ ∂q k

k γij (η(t))q˙i (t)η j (t)

where = (q(t), q(t)) ˙ and η˙ k (t) = σ(t) ˙

,

(41)

σ(t)

∂Y k (q(t))q˙i (t). ∂q i

We can now come to a notion of parallel transport in the usual way: η is said to be parallel along σ if Dσ˙ η(t) = 0 for all t. As in the standard theory, for a given curve σ in M and an arbitrary point v in the fibre of σ(t0 ) say, there is a unique η along σ, which passes through v and is parallel. This η is called the horizontal lift of σ (through v): η = σ h . Note, however, that the differential equations to be solved for the η i are non-linear here. Definition. A curve γ in M is said to be a geodesic of the linear connection D along τ if γ h = γ, ˙ i.e. Dγ˙ γ˙ = 0. (42) From (41), it is clear that in coordinates, t 7→ q i (t) is a geodesic if it satisfies the S ODE equations k q¨k + γij (q, q) ˙ q˙i q˙j = 0, which are indeed the equations (1). This resolves our first query. In order to obtain a covariant derivative operator, acting on tensorial objects along τ , it suffices now to extend the horizontal lift construction to vector fields. Definition. The horizontal lift of X ∈ X (M ) is the vector field X h on T M , which projects onto X and is further determined by the requirement that its integral curves are horizontal lifts of integral curves of X.

324

W. Sarlet It follows that, in coordinates: X h = X i (q)

∂ ∂ k − γij (q, v)v j k i ∂q ∂v

put

= X i hi ,

(43)

and this horizontal lift naturally extends to X (τ ). We can then, in the first place, extend the action of D to an operator D : X (τ ) × X (τ ) → X (τ ),

(44)

by putting DX F = X h (F )

X ∈ X (τ ), F ∈ C ∞ (T M ),

(45)

and DX (F Y ) = F DX Y + X h (F )Y

Y ∈ X (M ), F ∈ C ∞ (T M ).

(46)

Finally, DX further extends to 1-forms along τ by duality, and subsequently to arbitrary tensor fields along τ . In particular, for g ∈ T20 (τ ), an intrinsic definition of DX g becomes: (DX g)(Y, Z) = X h (g(Y, Z)) − g(DX Y, Z) − g(Y, DX Z).

(47)

This justifies the covariant derivative formula (2) of Rund (as found in [4] and [10], for example), except for a difference in convention! Indeed, the coordinate expression of (47) reads ∂gij s r ∂gij m m − γ v − γki gjm − γkj gim , (48) gij|k = (D∂/∂qk g)ij = k ∂q ∂v s kr and has a different order for the bottom indices of the connection coefficients, an issue of course which depends on the convention adopted at the very start, namely with the defining relations (36).

5.

A Variety of Related Connections

We have seen in the previous section that a linear connection D along τ comes with an associated notion of horizontal lift, as it should, and defines a S ODE, namely the equations for geodesics. But that inevitably means that there are a number of other connections around. Studying the relationship between those connections is a somewhat slippery domain, because it is easy to get dragged away into identifying all sorts of tensors, which are perhaps only marginally interesting. I shall try to limit myself here to what seem to me the bare essentials of such a discussion. The horizontal lift (43) was indispensable to arrive at the covariant derivative operator DX . It defines at the same time a non-linear connection, however, with connection coefficients h k k j Γi = γij v , (49) and that in turn, in agreement with the general formulas (22), defines a horizontal covariant derivative DhX . Since DX and DhX agree on functions, their difference, when acting on some Y ∈ X (τ ), depends tensorially on Y . Hence, every linear connection D along τ has an associated type (1,2) tensor field along τ , which I shall denote by K, and seems not to have been noticed or highlighted before in the literature.

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325

Definition. The fundamental tensor field K of a linear connection D along τ is the type (1, 2) tensor along τ determined by K(X, Y ) = DX Y − DhX Y.

(50)

In coordinates K = Kkij dq i ⊗ dq j ⊗

∂ , ∂q k

Kkij = −

∂γilk l ∂ h k k Γi + γij . v = − ∂v j ∂v j

(51)

In terms of the different types of derivations, discussed in section 2, we can write that DX = DhX + aX K for the action on X (τ ) (with X K(Y ) = K(X, Y )). By the duality rule (23), V this implies that for the action on 1-forms α ∈ 1 (τ ), we will have DX α = DhX α − iX K α. It follows that for the action on arbitrary tensor fields, DX = DhX + µX K ,

(52)

where for an arbitrary (1,1) tensor A, µA = aA − iA (see [16]). There is, of course, also the S ODE connection associated with the geodesic equations (1) coming from D. Its connection coefficients, according to (27), are given by H

Γki = 21 (γilk + γlik )v l +

1 2

k ∂γlm vl vm, ∂v i

(53)

and it has its own horizontal covariant derivative DH X . From now on, I shall systematically use superscripts h (or subscripts h ) for everything that relates to the derivative Dh , while H (or H ) will refer to the canonical connection of the geodesic S ODE Γ. The difference between the two horizontal distributions determines a type (1,1) tensor, for which I will not introduce a separate notation because it is derived from more fundamental tensors; its components are given by H

Γki − h Γki = 21 (γlik − γilk )v l +

1 2

k ∂γlm vl vm. ∂v i

(54)

The second term on the right is easily seen to be − 21 (T K)ki , while the first term comes from the torsion tensor of D. Indeed, as will be seen in the next section, one can give an intrinsic definition of the concept of torsion of D, which then is found to have the coordinate representation D

T =

1D k 2 T ij

dq i ∧ dq j ⊗

∂ , ∂q k

D

k k T kij = γij − γji .

(55)

So the first term on the right in (54) is 21 (iT D T )ki . The subtle interplay between the different connections which are around can be seen from the following properties. From the general discussion in section 3, it should not come as a surprise that TH is not the geodesic S ODE Γ from which the horizontal lift H is derived. But we do have that Th = Γ. Recall further that an important degree zero derivation associated to a S ODE is the dynamical covariant derivative ∇. But by choosing the vector field X in the different covariant derivative operators we have so far considered to be the

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W. Sarlet

canonical T, there are, so to speak, three more degree zero derivations available which should have some similarity to ∇, namely DT , DhT and DH T . The last one is quite different from the others, in general. Since Th = Γ, ∇, DT and DhT on the other hand all coincide on functions. Yet, they are not the same since, for example, ∇

∂ ∂ = H Γki , i ∂q ∂q k

DT

∂ ∂ = v l γlik k , i ∂q ∂q

(56)

while, in accordance with (52), the difference between DT and DhT is determined by T K. An interesting property of the linear D along τ is that DX T = 0,

∀X ∈ X (τ ).

(57)

This can easily be verified in coordinates. It follows that DhX T = −K(X, T).

(58)

In particular, we have DT T = 0, a property which is in general not shared by the related derivations ∇, DhT and DH T. A different aspect which needs some inspection in this section is the relationship between linear connections along τ and linear connections on the pullback bundle τ ∗ τ : τ ∗T M → T M . This relationship is not a very strict one a priori as, again, there are many different elements one can bring into the picture. But I shall try to develop arguments which bring us to a kind of natural correspondence in the end. At the start of such a discussion, however, one has to make a clear distinction between the defining requirements of a linear connection along τ , as expressed for example by Proposition 1, and the extension to a map D : X (τ ) × X (τ ) → X (τ ) which was needed to arrive at a covariant derivative operator on tensors along τ . The most immediate association between both concepts one can think of, goes as follows. Let D be a linear connection along τ , locally determined by D∂/∂qi

∂ ∂ k = γij (q, v) k . ∂q j ∂q

Define for each ξ ∈ X (T M ) a map ∇ξ : X (τ ) → X (τ ) by ∇∂/∂qi

∂ k ∂ = γij , j ∂q ∂q k

∇∂/∂vi

∇ξ (F X) = F ∇ξ X + ξ(F ) X,

∂ = 0, ∂q j

∇F ξ = F ∇ξ ,

F ∈ C ∞ (T M ), X ∈ X (τ ).

(59) (60)

It is easy to see that ∇ satisfies the requirements for a linear connection on τ ∗ τ , but this association calls for more intrinsic procedures and insights. It seems to me, however, that in view of what precedes, it is appropriate to bring first the availability of an extra horizontal distribution in the discussion. This way, we can make a link also with yet another construction in the literature. Indeed, a pair (∇ξ , PH ), consisting of a linear connection on τ ∗ τ and a horizontal projector on T M is essentially (possibly after identification of the pullback bundle with the bundle of vertical tangent vectors to

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327

T M ) what is called a “Finsler connection” by Matsumoto [17] and in Bejancu’s book [5], for example, and indeed in many other sources. Suppose we have such a pair (∇ξ , PH ), where PH is general here (i.e. not necessarily the S ODE connection of the geodesics), so that every ξ has its decomposition ξ = X H +Y V for some X, Y ∈ X (τ ). Then, for each X ∈ X (τ ) we can define DX : X (M ) → X (τ ) as DX = ∇X H |X (M ) . It is clear that DX has all the right properties (Proposition 1), and if we put ∇∂/∂qi

∂ ∂ = Γkij k , j ∂q ∂q

k = Γk . Observe, however, that the associated h -lift the connection coefficients of D are γij ij of D is not the H we started from. Note also that ∇X V |X (M ) defines a tensorial object which we disregard in this construction. Conversely, suppose the data are a linear D and an arbitrary horizontal lift H (not necessarily the h associated to D). Then, a corresponding ∇ξ can be constructed as follows: for X ∈ X (τ ), Z ∈ X (M ) and F ∈ C ∞ (T M ), put

∇X H Z = DX Z,

∇X V Z = 0,

∇ξ (F Z) = F ∇ξ Z + ξ(F ) Z.

The property ∇X V Z = 0 expresses that ∇ acts on any fibre of T M by so-called complete parallelism (see [7]). The element of caution to mention here is that ∇X H (F Z) 6= DX (F Z). Now, it is clear that in both of the directions of the above construction, one of the data in fact can imply the availability of the other, so let us finally reduce the data again in that sense. This means that in the first construction, we now assume that only a horizontal projector is given. Then, there is a naturally associated linear connection on τ ∗ τ , namely the Berwald type connection (25). The linear D along τ which then follows is determined by DX = DH X |X (M ) , k = and if Γki are the connection coefficients of the given non-linear connection, we have: γij ∂Γki /∂v j . Again, one has to be careful, because the extension of this DX to X (τ ), as discussed in the previous section is not the DH X we started from. For the converse construction, it suffices to have D as only data, because D comes with its own horizontal lift h as discussed before. The above general construction did not depend on the choice of a horizontal lift anyway, so we can carry it out just as well with h . k (and zero), i.e. we recover the The connection coefficients of ∇ξ are given by: Γkij = γij ‘direct association’ we mentioned at the beginning, but in a more elegant way. This time, we do have that ∇X h (F Z) = DX (F Z) for F ∈ C ∞ (T M ). However, the resulting ∇ξ is generally not of Berwald type, which is essentially due to the fact that DX 6= DhX , i.e. to the fundamental tensor K. Nevertheless, this association between D and ∇ξ is the most relevant point in our discussion, and we formalize it, therefore, in the following statement.

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W. Sarlet

Proposition 2. Let D be a linear connection along τ , then there is a linear connection ∇ on τ ∗ τ , which is uniquely determined by the following prescriptions: for X ∈ X (τ ), Z ∈ X (M ) and F ∈ C ∞ (T M ), ∇X h Z = DX Z,

∇X V Z = 0,

∇ξ (F Z) = F ∇ξ Z + ξ(F ) Z,

(61) (62)

where h is the horizontal lift determined by D. We can now give an interesting alternative interpretation of the fundamental tensor K of D. The point is the following: there are five tensors along τ which can be associated to a pair like (∇ξ , Ph ). They can be regarded as torsions of ∇ and were called A, R, B, P and S in [7]. The term torsion can be justified by the fact that any pair of a linear ∇ on τ ∗ τ and a horizontal distribution on T M induces a linear connection on T (T M ) also, whose torsion has five components which (as by now familiar) are lifts of tensor fields along τ (see also [22] and [5] for a discussion about these five torsion tensors). Now, B = S = 0 here (as a result of the second condition in (61)), and P, in particular is defined by P(X, Y ) = ∇X h Y − DhX Y.

(63)

It follows from the first of (61) that P = K. The two other tensors A and R will be encountered in the next section. Incidentally, the association between DX and ∇X h in (61) is a kind of generalization of what are called h-basic covariant derivative operators by Szilasi [23]. The most obvious conclusion one can draw at the end of this section is that one should be extremely careful in comparing or using different types of connections which are around in this area!

6.

Torsion and Curvature of the Linear D Along τ

There are different ways of approaching the concept of torsion of a connection. For a direct definition of the torsion tensor, one needs a bracket of vector fields. In the case of a linear connection D along τ , since D induces a horizontal distribution, it looks natural to think of the associated horizontal bracket, as defined in (24). Definition. The torsion D T of a linear connection D along τ is the vector-valued 2-form along τ , defined by D

T (X, Y ) = DX Y − DY X − [X, Y ]h ,

X, Y ∈ X (τ ).

(64)

It is easy to verify that D T is indeed a tensor. In fact, we have (see [16]), [X, Y ]h = DhX Y − DhY X − h T (X, Y ),

(65)

where h T is the torsion of the non-linear connection h , which according to (14, 49) has components h k h k Tij = Vj (h Γki ) − Vi (h Γkj ), Γi = γilk v l . (66)

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329

This implies that D

T (X, Y ) = K(X, Y ) − K(Y, X) + h T (X, Y ).

(67)

k cancel out, and one Evaluating this expression in coordinates, all derivatives of the γij indeed obtains the previously cited formula (55), which agrees with the expression given by Asanov [4]. Note further that, in terms of the association expressed by Proposition 2, D T is in fact the A-torsion of the linear connection on τ ∗ τ . As for curvature, it is best first of all to make a notational distinction: I shall use the notation curv when talking about curvature of any sort of linear connection and R as in section 2 for the curvature tensor of a non-linear connection. The natural definition of curvature of a linear D along τ would seem to be

D

curv(X, Y )Z = DX DY − DY DX − D[X,Y ]h Z.

(68)

Observe, however, that this is tensorial in X, Y , but not in Z, unless the Z-argument is restricted to X (M ). With F ∈ C ∞ (T M ), it follows from (20) that V

curv(X, Y )(F Z) = F (Dcurv)(X, Y )Z + (hR(X, Y )) (F ) Z,

D

(69)

where hR is the curvature of the non-linear connection associated to the h -lift. In coordinates, with hi referring to the local basis of horizontal vector fields in (43), one readily finds that m m l m l m (Dcurv)m (70) ijk = hi (γjk ) − hj (γik ) + γjk γil − γik γjl . These are effectively (with a transposition of the lower indices in the connection coefficients, as reported before) the curvature components (3) mentioned in [4]. One should keep in mind, however, that they are, for the moment at least, components of a map D curv : X (τ ) × X (τ ) × X (M ) → X (τ ). An interesting side observation is that h i Rjk

= hk (h Γij ) − hj (h Γik ) = (Dcurv)ikjl v l .

(71)

Note also that hR is, up to a sign, the so-called R-torsion of the associated linear connection on τ ∗ τ , determined by Proposition 2. At this point, it is worth referring to the comments in the introduction which follow equation (3), and to illustrate again now that there is a marked advantage in conceiving all objects and operations of interest as living along τ (as opposed to mixing calculations along τ with calculations on T M ). Indeed, the property (69) provides the clue to remedy the non-tensorial aspect of Dcurv. Definition. The curvature of a linear connection D along τ is the type (1, 1) tensor-valued 2-form Dc] urv along τ , defined by

c] urv(X, Y )Z = DX DY − DY DX − D[X,Y ]h Z − DV(hR)(X,Y ) Z.

D

(72)

The components of this tensor, which are obtained by taking coordinate vector fields for the arguments X, Y, Z, are the same as (70), since the extra term does not contribute to this computation. Note again that, as with the torsion (or in fact the covariant derivative

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W. Sarlet

DX itself) the non-linear h -connection is needed to define the curvature of D. It is clear then that one can define a related tensor field of curvature type as follows:

h

curv(X, Y )Z = DhX DhY − DhY DhX − Dh[X,Y ]h Z − DV(hR)(X,Y ) Z.

(73)

One can verify that the relation between both curvature tensors is given by c] urv(X, Y )Z = hcurv(X, Y )Z + DhX K (Y, Z) − DhY K (X, Z)

D

+ K(h T (X, Y ), Z) + K(X, K(Y, Z)) − K(Y, K(X, Z)),

(74)

or equivalently c] urv(X, Y )Z = hcurv(X, Y )Z + DX K (Y, Z) − DY K (X, Z)

D

+ K(D T (X, Y ), Z) − K(X, K(Y, Z)) + K(Y, K(X, Z)).

(75)

Concerning the geometrical interpretation of the tensor hcurv, it is worth observing that this is in fact the horizontal component of the curvature of the Berwald-type connection on τ ∗ τ , associated to the h -lift. Indeed, taking the general formula (20) for brackets of horizontal lifts into account, and denoting the Berwald-type covariant derivative by ∇ξ as in (25), it is easy to see that

h

curv(X, Y )Z = ∇X h ∇Y h − ∇Y h ∇X h − ∇[X h ,Y h ] Z.

Note finally that the property (71) has the following intrinsic content: c] urv(X, Y )T = −hR(X, Y ).

D

(76)

(77)

I now briefly turn to the issue of Bianchi identities. In fact, with the insights we have gained now, we should not expect new features to appear here, because torsion and curvature of D are after all closely related to h T and hR, through (67) and (71) for example, and Bianchi identities for these tensors are known to be compactly represented by properties of the form (16). What looks appealing in this respect, however, is to explore the meaning of an exterior derivative associated to D. For the horizontal exterior derivative associated to the h -lift, we know [16] that on 1V forms α ∈ 1 (τ ), dhα(X, Y ) = DhX α (Y ) − DhY α (X) + α(h T (X, Y )).

(78)

By analogy, therefore, it looks recommended to define the exterior derivative dD on functions F ∈ C ∞ (T M ) by dDF (X) = DX F = X h (F ) and on 1-forms along τ by dDα(X, Y ) = DX α (Y ) − DY α (X) + α(D T (X, Y )).

(79)

But then, in view of (50) and (67), it follows that dD = dh on scalar forms. For the action on vector fields along τ we have dhX(Y ) = DhY X, hence by analogy put dDX(Y ) = DY X. This implies that (dDX − dhX)(Y ) = K(Y, X), meaning that for the full action on V (τ ), we have dD = dh + aK , (80)

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331

V1

where K, in agreement with (12) is regarded here as a tensor in (τ ) ⊗ V 1 (τ ). However, since there is no difference between dD and dh on scalar forms, commutator properties of dD of type (13), which could be regarded as defining torsion and curvature, will actually reproduce h T and hR. As a result, the Bianchi identities essentially remain (see (16)) dh (hR) = 0,

dh (h T )) + dV(hR) = 0,

(81)

and can be re-formulated, using (80), (77) and (67) in terms of corresponding objects related to D if needed.

7.

Variationality versus Metrizability

The two questions to be addressed here in fact have very little in common, but that is not always so clear in the literature. In my opinion, metrizability is a property that can be attributed to a connection, while variationality, in the context of a connection, can only be attributed to its geodesic equations, and as such is common to an equivalence class of connections, namely those which have the same geodesics. A linear D along τ is variational if its geodesic equations (1) are variational. The point to make is that studying this problem subsequently has very little to do with D anymore; instead, the geometric tools of the S ODE connection of the geodesics now enter the scene. And the most comprehensive formulation of the problem is simply the existence of a (non-singular) symmetric g along τ , satisfying the Helmholtz conditions (33, 34, 35). All operations of interest in this problem, such as the dynamical covariant derivative ∇ and Φ, have coordinate expressions in terms of the ‘forces’ f i of the S ODE Γ, which are given by i f i (q, v) = −γjk (q, v) v j v k .

(82)

Hence, as observed in [12], nothing will change if we consider different D, i.e. different i which produce the same f i . In fact, variationality of a given D was defined in [10] also γjk i , possibly different from the given ones but giving rise to as the existence of some set of γjk the same f i , such that the Helmholtz conditions are satisfied. Reference [12] contains another statement which is worth situating within our present analysis. As we discussed in section 5, there is a certain similarity between the dynamical covariant derivative ∇ of the geodesic S ODE and the operator DT (see (56) for example), so it is of some interest to investigate to what extent ∇g = 0 differs from DT g = 0. The answer to this question is the result (8.14) in [12] which, translated into our present notations, states that, provided the linear D along τ is taken to be torsion free, ∇g = 0

⇔

1 ∂γ k ∂γ k DT g = gik lm + gjk lm j 2 ∂v ∂v i

!

vl vm.

(83)

Expressed differently, and in more intrinsic terms, we can say that for a torsion-free D, (∇g = 0 ⇔ DT g = 0)

if and only if

In fact, in view of (52), we then also have DhT g = 0.

µT K g = 0.

(84)

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A discussion of metrizability is more delicate, because here I differ in opinion with some of the references cited so far. In my view, a linear D along τ is metrizable if there exists a (non-singular) symmetric g along τ , such that DX g = 0 for all X ∈ X (τ ) (without any further assumptions on g). The requirements on g here are directly related to the given connection and are quite different from ∇g = 0, for example. In other words, generally speaking, it doesn’t make much sense to expect that metrizability might imply variationality or vice versa, unless of course you modify your definition until that works. There are roughly two aspects one can consider: one is the strict and hard question of studying for a given fixed D the existence of a suitable g; the other and much more tangible question is to start from a given g and try to modify the connection to make it metric with respect to this g. I will argue, however, that even the second question does not have an entirely compatible solution in the case of a linear connection along τ . Inspired by the concept of Cartan tensor which one can find, for example, in the work of Miron and co-workers (see e.g. [20] and [2]), the following looks like a natural concept to introduce here. Definition. The Cartan tensor associated to a given non-singular, symmetric g and a linear connection D along τ , is the symmetric type (1, 2) tensor CD along τ , determined by g(CD (X, Y ), Z) = DX g (Y, Z) + DY g (X, Z) − DZ g (X, Y ),

X, Y, Z ∈ X (τ ). (85)

The coordinate expression of CD is found to be k k CD kij = g kl hi (gjl ) + hj (gil ) − hl (gij ) − (γij + γji )

m + g kl gjm (γlim − γilm ) + gim (γljm − γjl ),

and simplifies somewhat for a torsion-free connection.

(86)

Proposition 3. We have (i) DX + 12 µX CD g = 0, ∀X ∈ X (τ ), (ii) DX g = 0, ∀X ∈ X (τ ) ⇔ CD = 0. Proof. The first property is a straightforward computation: we have

DX + 12 µX

CD

g (Y, Z) = DX g (Y, Z) −

1 2

g(CD (X, Y ), Z) + g(CD (X, Z), Y ) ,

from which the result immediately follows by using (85). Obviously then, if CD = 0 it follows that DX g = 0, ∀X, while the converse trivially follows from the definition of CD . So, metrizability of a given D is (as usual) a matter of a vanishing Cartan tensor, in other k (q, v), words, the hard question then is to study under what circumstances, with given γij the equations glk CD kij = 0 can have a solution for g. It would seem that the statement (i) in the above proposition contains the (expected) answer about how to modify the given connection to make it metric with respect to g. There is a technical problem, however, which makes that this is not quite true! To see this, let’s go back to the original concept of a linear connection along τ , in the interpretation of

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Proposition 1. This shows that the difference between two such connections is a type (1,2) tensorial object indeed, C say, but in the following sense: C : X (τ ) × X (M ) → X (τ ). ′ := D + µ Hence, starting from a DX , putting DX X X C defines a new linear connection ′ to X (τ ) by the rules (45, 46), along τ , but as soon as one wants to extend the action of DX ′ there is a certain incompatibility, because the horizontal lift h induced by D′ is different ′ , without modification, would from the original h -lift, while the defining relation of DX ∞ ′ h imply that on functions F ∈ C (T M ): DX F = X F . As was the case in discussing the notion of curvature (see (72)), one has to correct with a vertical derivative term (which does not modify the connection coefficients) to remedy this deficiency, and the clue on how to do this here comes from the property (57) which every properly extended linear connection along τ should have.

Proposition 4. If C is an arbitrary type (1, 2) tensor along τ , then the following modification of a given D defines a new linear connection along τ which is compatible with its induced horizontal lift: ′ DX = DX + µX C − DVC(X,T) . (87) Proof. Recall that for Y ∈ X (τ ), we have: µX C Y = C(X, Y ) and DVY T = Y . It easily ′ T = 0, and one can verify in coordinates that this is the follows that DX T = 0 implies DX ∞ ′ F = X h′ (F ) where γ ′ k = γ k + C k . same as saying that for F ∈ C (T M ), DX ij ij ij Coming back to the subject of Proposition 3 now, we see that the modified covariant derivative operator of the first statement necessarily has to be taken in its extended sense, since it acts on g (not just on basic vector fields). Therefore, the genuine modified linear e which is at stake here, reads connection D e X = DX + 1 µ D 2 X

CD

− 21 DVCD (X,T) .

(88)

e X g = 0, ∀X, but rather Unfortunately, however, the conclusion then is that we don’t have D e X g + 1 DV D 2 CD (X,T) g = 0.

(89)

One might consider to cover this technicality by defining a linear D along τ to be metrical with respect to some g, if for all X ∈ X (τ ), DX g = 0 modulo vertical derivatives of g. It is interesting to look at the preceding technical problem still from a different perspective. By Proposition 2, we know how to associate with D a linear connection ∇ξ on the pullback bundle τ ∗ τ . In turn, as was done (in a time-dependent set-up) in [18], for example, one can then define a horizontal Cartan tensor in that context by g(Ch (X, Y ), Z) = ∇X h g (Y, Z)+∇Y h g (X, Z)−∇Z h g (X, Y ),

X, Y, Z ∈ X (τ ). (90)

Actually Ch = CD , but we can pose the problem of constructing a modified metrical connection at this level without any complications. Indeed, e h =∇ h+1µ ∇ X X 2 X

Ch

(91)

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e h g = 0, for all X. Since it is only the horizontal component will have the property that ∇ X e of ∇ξ which becomes metrical in this process, one could call this a connection of ChernRund type . The problem we have encountered before comes from the fact (explained in e h to a linear D e X along τ , the horizontal lift detail in section 5) that in going back from ∇ X e induced by this D is not the one we started from. For completeness, coming back to the point I made at the very beginning of the section, I should mention the special case of a standard linear connection on M , where the connection coefficients do not depend on the v i and the geodesics come from a spray. Then, the requirement that for some (quasi-Riemannian) metric g we have gij|k = 0 (for all k) , obviously is equivalent to requiring that gij|k v k = 0, in other words, in that situation we have DX g = 0, ∀X ⇔ DT g = ∇g = 0.

That is why variationality (where ∇g = 0 is a key condition) and metrizability (which is about DX g = 0, ∀X) are closely related then. By the way, it looks like an interesting question to investigate, for the general case of a linear D along τ , is under what circumstances DT g = 0 will imply DX g = 0, ∀X ∈ X (τ ) (some form of homogeneity is probably indispensable for that). There is a final link I should explain to conclude this discussion. A very recent paper [6] carries the title “Metric nonlinear connections”. So what is this about? The author takes a S ODE Γ, plus a non-linear connection with connection coefficients Nji say, to build a covariant derivative operator, ∇ say, by putting (I am identifying vertical tangent vectors to T M with vectors along τ ): ∇F = Γ(F ), F ∈ C ∞ (T M ),

∇

∂ ∂ = Nij j . ∂q i ∂q

(92)

Here, the non-linear connection may or may not be the canonical one coming from Γ (although I don’t see the point really in taking a different one to start with). Anyhow, it is clear that this operator is of the type of a dynamical covariant derivative, and it can be of interest, of course, to study compatibility of ∇ with some metric g, in the sense that ∇g = 0. Whether it is appropriate to classify this question under ‘metrizability’ problems is perhaps debatable here, if a dynamical covariant derivative is all one has. The main problem which is addressed in section 2 of [6] is to construct from ∇ a new ∇′ such that ∇′ g = 0, where g is a given metric along τ . The solution to this problem is in fact quite simple: taking ∇′ and ∇ to be identical on functions, they must be related by a formula of the form: ∇′ = ∇ + µA , for some (1, 1) tensor A, which must be chosen in such a way that g(AX, Y ) + g(X, AY ) = ∇g(X, Y ),

∀X, Y.

It is clear that a solution for A is given by Aij = 21 g il (∇g)lj , which means that the modified Nj′i are determined by gli Nj′i = 12 (∇g)lj + gli Nji = 12 Γ(glj ) + 21 (gli Nji − gji Nli ).

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In the case that the Nji we start from are the canonical Γij (see (27)), so that ∇ is precisely the dynamical covariant derivative of the S ODE Γ, the above relation is exactly equation (12) in [6], and thus explains Theorem 2.2 of that paper.

8.

Hessian Metrics Along τ and Finsler Spaces

Let’s go back to the general question of metrizability of a linear D along τ , meaning that we want a g such that DX g = 0, ∀X. Since solving the vanishing of (48) for g, for example, is a hard problem, one will naturally be led to look at the effect of imposing extra restrictions on g. It seems to me that a good way to proceed would be to work in stages, as follows: first study the effect of imposing that gij be the Hessian matrix of some function (with respect to the fibre coordinates v i ), and secondly assume that g is homogeneous in the v i , for example (but not exclusively) of degree zero. I will not enter into this study in any depth here. Though I think this has not been carried out yet in a systematic way, it is true of course that many aspects of this idea frequently turn up in the literature (sometimes rather related to one of the other types of connections discussed in section 5 though). In the terminology of the Miron-school, for example, the first step would correspond to passing from a “generalized Lagrange space” to a “Lagrange space”. In [11], a linear D along τ is said to be metrizable if there exists a Finsler metric such that DX g = 0, so homogeneity of g is taken to be part of the definition of metrizability (and in fact also of the definition of variationality of D). A general S ODE Γ on the other hand is said to be metrizable if there exists a “variational” metric g, meaning it is a Hessian, such that ∇g = 0. In other words, two of the three Helmholtz conditions (33, 34, 35) are incorporated in the definition of metrizability here, which of course makes life easier. Incidentally, an interesting question which emerges in this context is: under what circumstances is the remaining Helmholtz condition redundant? It is well known that this is the case for Finsler metrics. I claim it is true also as soon as we have homogeneity of any order. It would take me too far away from the subject of this paper, however, to prove this statement here. Instead, to finish, let’s go back to the master we are celebrating in this volume. Specifically, let me return to the paper [10] I started from, because it contains a definition of metrizability, which is very surprising if you see it for the first time and leads to an equally surprising conclusion which is worth explaining in more intrinsic terms. Let me recall first the definition of a Hessian metric in an intrinsic way. Definition. The Hessian of a function L on T M is the symmetric type (0, 2) tensor field g along τ , defined by g(X, Y ) = DVY DVX L − DVDV X L, Y

X, Y ∈ X (τ ).

(93)

Now in [10], a linear D along τ is said to be metrizable if there exists a non-singular, symmetric g along τ , such that DX g = 0, ∀X,

and

∂gij j v = 0. ∂v k

(94)

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In intrinsic terms, the surprise extra condition means that T DVX g = 0,

∀X.

(95)

Lemma. If a symmetric g along τ has the property (95), then it is a Hessian and is homogeneous of degree 0 in the fibre coordinates. Proof. Put L = 12 g(T, T). Then, it follows from (95) and the general property DVX T = X that DVX L = g(X, T). Taking a further vertical derivative with respect to Y and using the intermediate result plus (95) again, we obtain that g satisfies (93), i.e. is the Hessian of L. But a Hessian is characterized by the property (34), from which it follows, taking T as one of the arguments and using (95) again, that DVT g = 0. This precisely means that g is homogeneous of degree zero. This result is far from new: (95) in one form or another is the defining relation for a g along τ to be what is called normal; it is well known as the necessary and sufficient condition for g to be a Finslerian metric (see e.g. [19], [13]) and as such is attributed to Hashiguchi [9]. A side remark: now that Finsler structures come into the picture, I will omit the technicalities about having to pass from the tangent bundle to the slit tangent bundle, and also not go into requirements about positive definiteness. The main theorem in [10] states that under the conditions (94), the connection D is variational and more precisely is the Cartan connection of a Finsler structure. The proof of this theorem leaves the reader a bit startled, however. First of all, it is clear that the k of the given calculations in the proof take for granted that the connection coefficients γij D are symmetric: it is indeed a somewhat hidden assumption throughout the paper that the connection is torsion free. Secondly, what is explicitly shown is an equivalence of connections in the sense of variationality, that is to say: the geodesic S ODE of D is shown to be the same as the one coming from the Euler-Lagrange equations of the Finslerian g. k are There is no explicit verification, however, that the assumptions imply that the given γij effectively those of what is called (at least by some) the Cartan connection in that context. I shall finish by presenting a slight generalization of this theorem which consists in obtaining roughly the same results from somewhat weaker conditions. This will give me a chance to illustrate some of the features discussed in the previous sections, while the details of the Krupka-Sattarov theorem will follow as a special case. Theorem. Let D be a torsion-free, linear connection along τ , for which there exists a non-singular, symmetric g along τ , such that CD (X, T) = 0 and

T DVX g = 0, ∀X.

(96)

Then the following assertions hold true: (i) D is variational: its geodesic S ODE in fact is the canonical spray of g which is a Finsler metric. e along τ which is metric with respect to g: (ii) There exists a variationally equivalent D e DX g = 0, ∀X.

k of D e have an explicit expression in terms of g only (iii) The connection coefficients γeij and as such are those of the Cartan connection of g.

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Proof. (i) From the lemma we know that g is Finslerian: it is the Hessian of L = 21 g(T, T), and L is homogeneous of degree 2 in the v i . From the coordinate expression (86), taking the symmetry of the connection into account, it is clear that CD (T, T) = 0 implies that k i j γij vv

= =

g kl hi (gjl ) + hj (gil ) − hl (gij ) v i v j ∂gil ∂gij set 1 kl ∂gjl + j − v i v j = (γg )kij v i v j , 2g ∂q i ∂q ∂q l 1 2

(97)

where the reduction from the first to the second line follows from the property (95) again. This second line clearly reveals the force terms of the Euler-Lagrange equations of L (and should be read also as defining relations of the functions (γg )kij (q, v)). e reduces to (ii) Since CD (X, T) = 0, the defining equation (88) of D e X = DX + 1 µ D 2 X

CD ,

k are given e X g = 0, ∀X. Moreover, the new γ eij and it follows from Proposition 3 that D k = γ k + 1 C k , so that C (T, T) = 0 implies that γ k v i v j = γ k v i v j , proving the eij by γeij D ij ij 2 D ij variational equivalence of both connections. k from the vanishing of the Cartan tensor of D. e For (iii) We can now compute the γeij simplicity in notations, let us omit the tildes, i.e. assume we are back in the Krupka-Sattarov situation now, so that DX g = 0. Then CD = 0 implies that k γij =

1 2

g kl hi (gjl ) + hj (gil ) − hl (gij ) ,

(98)

but the right-hand sides in this expression contain in the vertical derivatives of the gk v j . Multiplying (98) with v j (in other words, using components factors of the form γij CD (X, T) = 0) and making use of (95) again, we find that k j r j s γij v = (γg )kij v j − 21 g kl γjs v v

∂gil . ∂v r

(99)

Again, we still have γ’s in the right-hand side, but we can eliminate them now by using (97). Substituting these intermediate results back into the expression (98), we finally get k γij

∂gjl ∂gil ∂gij (γg )ris + r (γg )rjs − (γg )rls v s r ∂v ∂v ∂v r ∂gjl ∂git ∂gil ∂gjt ∂gij ∂glt + 14 g kl g rt + − (γg )usp v s v p . ∂v r ∂v u ∂v r ∂v u ∂v r ∂v u

= (γg )kij − 12 g kl

(100)

k in terms of the metric g, which is the same This provides an explicit expression of the γij as equation (4.5) in [10], and is called the Cartan connection there. k , from which the final result can be deduced, Remark. the implicit specification of the γij is also (for a symmetric connection) equation (A.27) in the previously cited Appendix of [4], where it is referred to as the Cartan connection too. The type of computation in part (iii) of the proof in fact is similar also to the way the Cartan connection is set up in [1]. We have seen in section 5, however, that when D is mapped onto a linear connection on the pullback bundle according to Proposition 2, the metric nature of D corresponds to ∇ being

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horizontally metric, so that the more common terminology in that context would be that we are talking about a connection of Chern-Rund type. There are some interesting corollaries of the above theorem. Once we know that the geodesic S ODE of D is the canonical spray of a Finsler metric g, it follows that ∇g = 0. Hence, we are in the situation (84) and know that µT K g = 0. In fact, we can do a bit better and show that T K = 0. Proposition 5. Under the assumptions of the above theorem, the fundamental tensor K of the linear connection D has the property K(T, X) = 0, ∀X ∈ X (τ ). It follows that the horizontal lift h associated to D, and the horizontal lift H of its geodesic S ODE coincide. Proof. For the components of T K, we can write k ∂γij ∂ k j k i j − r v i v j = − r (γij v . v v ) + 2 γrj ∂v ∂v

But we know that CD (X, T) = 0, so that we can use the expressions (97) and (99) to compute the two terms in the right-hand side. It is straightforward to verify that making these substitutions, and using (95), all terms cancel out. The last statement then immediately follows from (54). Finally, taking (52) and (56) into account, plus the fact that Th = Γ always, it follows that under the assumptions of the theorem: DT = DhT = DH T = ∇.

9.

Concluding Remarks

Having arrived in Finsler country at the end of our journey, where there is a vast literature, and where every household seems to foster its own notations, it is possible that there are still other results to which I could or should have compared aspects of what has been discussed in this paper. But I hope the reader will find the road to these results using [3], though this is not an easy navigating system . The general construct of linear connections along τ , in the sense of Rund’s directiondependent connections, has been used strictu senso only occasionally in the literature, but a number of aspects about such operations remained unclear, specifically with respect to the intrinsic foundations of the theory. I hope I have managed to clarify such aspects in this paper. New, potentially interesting questions have come up in my analysis and there are undoubtedly many more one can think of. However, having identified also a number of technicalities and dangers for confusion with related concepts, there is one major question I would like to put forward, namely: “Do we actually need linear connections along τ ?”. Isn’t it possible for example that, with the geometrical calculus offered by linear connections on the pullback bundle τ ∗ τ , we have sufficient tools in our hand to analyse all theoretical questions one might wish to study with linear connections along τ ?

References [1] M. Abate and G. Patrizio, Finsler metrics – A Global Approach (Lecture Notes in Math. 1591, Springer-Verlag, Berlin, 1994).

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[2] M. Anastasiei, Finsler connections in generalized Lagrange spaces, Balkan J. of Geom. and its Appl. 1 (1996) 1–9. [3] P. L. Antonelli (Ed.), Handbook of Finsler Geometry, Vol.1 and Vol.2 (Kluwer Academic Publishers, Dordrecht, 2003). [4] G. S. Asanov, Finsler Geometry, Relativity and Gauge Theories (D. Reidel Publ. Comp. Dordrecht, 1985). [5] A. Bejancu, Finsler Geometry and Applications (Ellis Horwood, 1990). [6] I. Bucataru, Metric nonlinear connections, Diff. Geom. Appl. 25 (2007) 335–343. [7] M. Crampin, Connections of Berwald type, Publ. Math. (Debrecen) 57 (2000) 455– 473. [8] A. Fr¨olicher and A. Nijenhuis, Theory of vector-valued differential forms, Proc. Ned. Acad. Wetensch. S´er. A 59 (1956) 338–359. [9] M. Hashiguchi, On parallel displacements in Finsler spaces, J. Math. Soc. Japan 10 (1958) 365–379. [10] D. Krupka and A. E. Sattarov, The inverse problem of the calculus of variations for Finsler structures, Math. Slovaca 35 (1985) 217–222. [11] O. Krupkov´a, Variational metrics on R × T M and the geometry of nonconservative mechanics, Math. Slovaca 44 (1994) 315–335. [12] O. Krupkov´a, Variational metric structures, Publ. Math. Debrecen 62 (2003) 461–495. [13] R. L. Lovas, J. P´ek and J. Szilasi, Ehresmann connections, metrics and good metric derivatives, In: Finsler Geometry (Sapporo 2005, Advanced Studies in Pure Mathematics, Math. Soc. Japan 48, 2007) 263-308. [14] E. Mart´ınez and J. F. Cari˜nena, Geometric characterization of linearizable secondorder differential equations, Math. Proc. Camb. Phil. Soc. 119 (1996) 373–381. [15] E. Mart´ınez, J. F. Cari˜nena and W. Sarlet, Derivations of differential forms along the tangent bundle projection, Diff. Geom. Appl. 2 (1992) 17–43. [16] E. Mart´ınez, J. F. Cari˜nena and W. Sarlet, Derivations of differential forms along the tangent bundle projection II, Diff. Geom. Appl. 3 (1993) 1–29. [17] M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces (Kaiseisha Press, Japan, 1986). [18] T. Mestdag and W. Sarlet, The Berwald-type connection associated to time-dependent second-order differential equations, Houston J. of Math. 27 (2001) 763–797. [19] T. Mestdag, J. Szilasi and V. T´oth, On the geometry of generalized metrics, Publ. Math. (Debrecen) 62 (2003) 511–545.

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[20] R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications (Fundamental Theories of Physics Vol. 59, Kluwer Academic Publishers, Dordrecht, 1994). [21] H. Rund, Direction-dependent connections and curvature forms, Abh. Math. Sem. Univ. Hamburg 50 (1980) 188–209. [22] J. Szilasi, Notable Finsler connections on a Finsler manifold, Lect. Matem´aticas 19 (1998) 7–34. [23] J. Szilasi, A setting for spray and Finsler geometry, In: Handbook of Finsler Geometry (vol. 2, (P. L. Antonelli, Ed.) Kluwer Academic Publishers, Dordrecht, 2003) 1183– 1426.

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 341-351

Chapter 19

O N THE I NVERSE P ROBLEM FOR AUTOPARALLELS∗ G.E. Prince† Department of Mathematics and Statistics, La Trobe University, Australia

Abstract The inverse problem in the calculus of variations for a given set of second order ordinary differential equations consists of deciding whether their solutions are those of Euler–Lagrange equations and exhibiting the non-uniqueness of the resulting Lagrangians when they occur. In this paper we use the techniques of the inverse problem to examine the conditions under which such a system of equations are the geodesic equations of a Finsler metric.

Key words and phrases. Inverse problem of the calculus of variations, Euler–Lagrange equations, Helmholtz conditions, linear connection, Finsler metrizability.

1.

The Inverse Problem in the Calculus of Variations and Finsler Geometry

This paper addresses the question (loosely described) of when the autoparallels of a linear connection are the geodesics of a Finsler metric. I thank Demeter Krupka for introducing me to this problem through his paper with A.E. Sattarov [14]. This is quite a deep question with many aspects and the work here focuses on rather preliminary stages of an investigation based upon the existing studies of a broader problem known as the inverse problem in the calculus of variations which I now describe. ∗

This paper is dedicated to Demeter Krupka on the occasion of his 65th birthday. It has been my pleasure to work with Professor Krupka over the last few years and to have enjoyed his hospitality. His energy, generosity and mathematical virtuosity are an inspiration to us all. † E-mail address: [email protected]

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The inverse problem in the calculus of variations involves deciding whether the solutions of a given system of second-order ordinary differential equations x ¨a = F a (t, xb x˙ b ), a, b = 1, . . . , n

(1)

are the solutions of a set of Euler-Lagrange equations ∂2L b ∂2L b ∂2L ∂L x ¨ + x ˙ + = a b b a a ∂ x˙ ∂ x˙ ∂x ∂ x˙ ∂t∂ x˙ ∂xa for some Lagrangian function L(t, xb , x˙ b ). (Note that the curve parameter for both systems 2 is t.) Clearly the Hessian matrix ∂ x˙∂a ∂Lx˙ b should be invertible on some domain. When a Lagrangian exists the equations (1) are said to be variational. The problem dates to the end of the 19th century and it still has importance for mathematics and mathematical physics (see [16, 22]). Because the Euler-Lagrange equations are not generally in normal form, the problem is to find a so-called multiplier matrix gab (t, xc , x˙ c ) which is invertible on some domain and such that d ∂L ∂L gab (¨ xb − F b ) ≡ − a. dt ∂ x˙ a ∂ x˙ The most commonly used set of necessary and sufficient conditions for the existence of the gab are the so–called Helmholtz conditions due to Douglas [10] and put in the following form by Sarlet [23]: gab = gba ,

Γ(gab ) = gac Γcb + gbc Γca ,

where Γab := −

1 ∂F a , 2 ∂ x˙ b

and where Γ :=

Φab := −

gac Φcb = gbc Φca ,

∂gab ∂gac = , c ∂ x˙ ∂ x˙ b

∂F a − Γcb Γac − Γ(Γab ), ∂xb

∂ ∂ ∂ + ua a + F a a . ∂t ∂x ∂u 2

When a solution gab exists a corresponding Lagrangian is recovered from ∂ x˙∂a ∂Lx˙ b = gab . A full review of this inverse problem can be found in the recent article by Krupkov´a and Prince [16]. Some key papers in the literature are [3, 5, 6, 8, 20, 19, 24, 25, 26]. There are a number of inverse problems in Finsler geometry (see, for example, section 12.4 of [28]). In [14] Krupka and Sattarov address two questions: “when are the autoparallel equations of a linear connection (depending on both xa and x˙ a ) variational?”, and “when does there exist a Finsler metric tensor covariantly constant under the connection?” A connection having this latter property is called metrisable, and Krupka and Sattarov prove that every metrisable connection is variational using the Cartan connection of the Finsler structure. However, the Berwald connection is arguably a better point of contact between second order differential equations and Finsler geometry (see [9, 12, 18, 19, 28]), so we will use the Berwald connection to answer the question “When are the solutions of (1) the geodesics of a Finsler structure, in the given parametrisation?” Since the Finsler geodesic equations are a x ¨a + γbc (x, x) ˙ x˙ b x˙ c = 0, a = 1, . . . , n,

(2)

On the Inverse Problem for Autoparallels

343 2 2 a := 1 g ad ∂gbd + ∂gcd − ∂gbc , where gab := 12 ∂∂x˙ aF∂ x˙ b for the Finsler function F and γbc c b d 2 ∂x ∂x ∂x this entails applying the inverse problem described above to the case where the functions F a are autonomous and positively homogeneous of degree 2 in the x˙ a (this is explained later). This means that the equations (1) describe the autoparallels of a certain symmetric linear connection whose coefficients are homogeneous of degree zero in the x˙ a , that is, the equations are the autoparallel equations of their own Berwald connection. The multiplier we seek will be the Finsler gab and so of necessity it must be autonomous and positively homogeneous of degree 0 in the x˙ a . The Helmholtz conditions for this Finsler inverse problem will therefore yield identities in Finsler geometry which at best may be new but which at least will provide a new angle on known Finsler facts. In stating this inverse problem we immediately face the dilemma of notation. The inverse problem world and the Finsler world share much common content but not too much notation, but since I will be approaching the problem from the inverse problem perspective I will uncompromisingly use that notation. Occasionally I will identify the corresponding Finsler equivalent using [28] as a standard. There has recently been renewed interest in this and related inverse problems in Finsler geometry and I refer the interested reader to [4, 15, 21, 29].

2.

The Geometry of SODEs

We will provide only enough of the geometric setting of the inverse problem to make the later discussion viable; more complete descriptions and further references can be found in [2, 16]. Suppose that M is some differentiable manifold with generic local co-ordinates (xa ). The slit evolution space is defined as E ◦ := R × T ◦ M , where T ◦ M is the tangent bundle without the zero section, with projection onto the first factor being denoted by t : E ◦ → R and bundle projection π ◦ : E ◦ → R × M . E ◦ has adapted co-ordinates (t, xa , ua ) associated with t and (xa ). We use the slit evolution space to ensure compatibility with our Finsler inverse problem. A system of n second order differential equations (SODEs) with local expression x ¨a = F a (t, xb , x˙ b ) is associated with a smooth vector field Γ on E ◦ given in the same co-ordinates by Γ :=

∂ ∂ ∂ + ua a + F a a . ∂t ∂x ∂u

Γ is called a semispray. It can be thought of as the total derivative operator associated with the differential equations. The integral curves of Γ are just the parametrised and lifted solution curves of the differential equations. When the system admits a Lagrangian as described in section 1, Γ is called the Euler-Lagrange field. The evolution space E ◦ is equipped with the vertical endomorphism S, defined locally by S := Va ⊗ θa (see [7] for an intrinsic characterisation). S combines the contact structure and vertical sub–bundle, V (E ◦ ), of E ◦ , θa being the local contact forms θa := dxa − ua dt

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and Va := ∂u∂ a forming a basis for vector fields tangent to the fibres of π ◦ : E ◦ → R × M (the vertical sub–bundle). It is natural to study the deformation of S produced by the flow of Γ, LΓ S. The eigenspaces of this (1, 1) tensor field produce a direct sum decomposition of each tangent space of E ◦ . It is shown in [7] that LΓ S (acting on vectors) has eigenvalues 0, +1 and −1. The eigenspace at a point of E ◦ corresponding to the eigenvalue 0 is spanned by Γ, while the eigenspace corresponding to +1 is the vertical subspace of the tangent space. The remaining eigenspace (of dimension n) is called the horizontal subspace. Unlike the vertical subspaces these eigenspaces are not integrable; their failure to be so is due to the curvature of this nonlinear connection (induced by Γ) which itself has components Γab := −

1 ∂F b . 2 ∂ua

(In [28] these components are denoted Nba and F a are replaced by −2Ga .) The most useful basis for the horizontal eigenspaces has elements with local expression Ha =

∂ ∂ − Γba b a ∂x ∂u

so that a local basis of vector fields for the direct sum decomposition of the tangent spaces of E ◦ is {Γ, Ha , Va } with corresponding dual basis {dt, θa , ψ a } where ψ a = dua − F a dt + Γab θb . (In [28] the horizontal fields are denoted δxδ a .) The components of the curvature manifest themselves in the expression for the commutators of the horizontal fields: d [Ha , Hb ] = Rab Vd

where the curvature of the connection is defined by 2 d 1 ∂ F ∂2F d 1 ∂F c ∂ 2 F d ∂F c ∂ 2 F d d Rab := − + − . 2 ∂xa ∂ub ∂xb ∂ua 2 ∂ua ∂uc ∂ub ∂ub ∂uc ∂ua It will be useful to have some other commutators: 1 ∂2F c [Ha , Vb ] = − ( a b )Vc = Vb (Γca )Vc = Va (Γcb )Vc = [Hb , Va ], 2 ∂u ∂u [Γ, Ha ] = Γba Hb + Φba Vb ,

[Γ, Va ] = −Ha + Γba Vb ,

and, of course, [Va , Vb ] = 0. We won’t go on to describe the canonical linear connection of Berwald type associated to the semispray, the reader is referred to [12, 17, 19]. Finally, the following will be important Φab = −Hb (F a ) + Γcb Γac − Γ(Γab ), c Ha (Γcb ) − Hb (Γca ) = −Rab , c c c Va (Φb ) − Vb (Φa ) = 3Rab .

(3) (4) (5)

On the Inverse Problem for Autoparallels

345

In the context of our Finsler inverse problem we begin with equations (1) with autonomous and positively homogeneous of degree 2 in the ub . (See [15] for more general, time-dependent considerations.) We immediately make the usual obser∂Γa vation (see, for example, [28]) that Γab is homogeneous of degree 1 in the ua , Γabc := ∂ubc is homogeneous of degree 0 in the ua so that F a = −Γabc ub uc and (1) becomes F a (t, xb , ub )

x ¨a + Γabc (x, x) ˙ x˙ b x˙ c = 0.

(6)

At this point we make the important remark due to Berwald and so well explained in the introduction to Mestdag’s thesis [18], namely that the Γabc transform as the components of a of the Finsler geodesic equation (2) in general do not. In a linear connection while the γbc a ub uc we have fact, for the Finsler forces F a := −γbc Γabc

=

a γbc

+g

ad

e ∂Cbdc e e e ∂Cbdc +F − 2Cbde Γc − 2Cdce Γb , u ∂xe ∂ue

(7)

ab where Cabc := 12 ∂g ∂uc are the coefficients of the Cartan torsion (see [28]: Shen also uses the same notation as us for the Γabc ). Returning to the Finsler inverse problem whose starting point is equation (6), equations (3), (4), and (5) become

a Φab = Rdbc uc ud ,

(8)

c c Rdab ud = −Rab , c c (Va (Rebd ) − Vb (Read ))ue ud

(9) = 0,

(10)

a := H (Γa ) − H (Γa ) + Γa Γe − Γa Γe are homogeneous of degree zero in where Rdbc c db b cd ec bd eb cd a a = −Ra and Ra + Ra + Ra = 0 (compare with [9]). the u and satisfy Rdbc dcb dbc cdb bcd

3.

The Helmholtz Conditions and the Finsler Inverse Problem

We begin with some general observations about the Helmholtz conditions given in section 1.. They are the necessary and sufficient conditions that a two form gab ψ a ∧ θb be closed and of maximal rank on some domain. This can be given an even more geometric phrasing in the following theorem from [7]: Theorem 3.1. Given a semispray Γ, the necessary and sufficient conditions for there to be Lagrangian for which Γ is the Euler–Lagrange field is that there should exist a 2–form Ω such that Ω(V1 , V2 ) = 0, Γ Ω = 0,

∀ V1 , V2 ∈ V (E),

dΩ = 0,

Ω is of maximal rank.

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Following Aldridge [1] the simplest way to see how the Helmholtz conditions arise from Theorem 3.1 is to put Ω := gab ψ a ∧ θb and compute dΩ: dΩ = (Γ(gab ) − gcb Γca − gac Γcb )dt ∧ ψ a ∧ θb + (Hd (gab ) − gcb Va (Γcd ))ψ a ∧ θb ∧ θd

+ Vc (gab )ψ c ∧ ψ a ∧ θb + gab ψ a ∧ ψ b ∧ dt

+ gca Φcb θa ∧ θb ∧ dt + gca Hb (Γcd )θa ∧ θb ∧ θd .

The four Helmholtz conditions are dΩ(Γ, Va , Vb ) = 0,

dΩ(Γ, Va , Hb ) = 0,

dΩ(Γ, Ha , Hb ) = 0,

dΩ(Ha , Vb , Vc ) = 0.

The remaining conditions arising from dΩ = 0, namely dΩ(Ha , Hb , Vc ) = 0

and

dΩ(Ha , Hb , Hc ) = 0,

can be shown to be derivable from the first four (notice that this last condition is void in dimension 2). In the Finsler inverse problem we assume that the multiplier gab , if it exists, does not depend on t, is homogeneous of degree zero in the ua and is symmetric and non-degenerate. The consequences of the remaining Helmholtz conditions are given in following lemmas: Lemma 3.2 (Compare with theorem 2 of [14]). The Helmholtz conditions Γ(gab ) = gac Γcb + gbc Γca and Vc (gab ) = Vb (gac ) imply that ! ˜ ˜ e ∂ Cbdc a a ad e ∂ Cbdc e e u Γbc = γ˜bc + g +F − 2C˜bde Γc − 2C˜dce Γb (11) ∂xe ∂ue and ∂gab d d d ˜ − g Γ − g Γ − 2 C Γ abd c ad bc bd ac ∂xc ∂ C˜abc ∂ C˜abc = −F d + C˜dbc Γda + C˜adc Γdb + C˜abd Γdc − ud d ∂u ∂xd a := 1 g ad ∂gbd + ∂gcd − ∂gbc , and C ˜abc := where F a = − 21 Γabc ub uc , γ˜bc c 2 ∂x ∂xb ∂xd 1 2

(12) 1 ∂gab 2 ∂uc .

Proof. The proof entails using the expression for Γ in the first of the given Helmholtz conditions and then differentiating this condition with respect to uk . Generating a further two conditions by permuting the indices a, b, k and taking the appropriate linear combination gives the result (11) upon using the conditions Vc (gab ) = Vb (gac ) and the homogeneity of the Γabc . The result (12) comes from (11) by permutation of indices and addition. Lemma 3.3. The Helmholtz conditions Γ(gab ) = gac Γcb + gbc Γca and Vc (gab ) = Vb (gac ) imply that Ha (gbc ub uc ) = 0 and Γ(gbc ub uc ) = 0. (13)

On the Inverse Problem for Autoparallels

347

Proof. Following the proof given in section 4.2 of [28] for the Finsler case, we contract (12) on ub uc and use Vc (gab ) = Vb (gac ) and the homogeneity of gab to remove all the C˜ terms giving ∂gbc b c u u = 2gbe ub Γea . ∂xa Hence Ha (gbc ub uc ) =

b c ∂(gbc ub uc ) ∂gbc b c e ∂(gbc u u ) − Γ = u u − 2gbe ub Γea = 0 a ∂xa ∂ue ∂xa

where Vc (gab ) = Vb (gac ) and the homogeneity of gab has been used to establish the last equality. Finally, because gab is assumed independent of t, Γ(gbc ub uc ) = ua Ha (gbc ub uc ) = 0.

Lemma 3.4 (Compare with proposition 8.2.1 of [28]). The Helmholtz conditions Γ(gab ) = gac Γcb + gbc Γca and Vc (gab ) = Vb (gac ) imply that gac Φcb = gbc Φca

d d d and gad Rbc + gcd Rab + gbd Rca = 0.

(14)

Equivalently, c c gac Rdbe ud ue = gbc Rdae ud ue

and

d d d (gad Rebc + gcd Reab + gbd Reca )ue = 0.

(15)

Proof. To see that the gΦ condition is a consequence of the given conditions, consider the commutator [Ha , Hb ] acting on gcd uc ud and use lemma 3.3, the condition Vc (gab ) = Vb (gac ) and the homogeneity of gab : e 0 = [Ha , Hb ](gcd uc ud ) = Rab Ve (gcd uc ud ) e e = 2Rab gce uc = −2Rdab gce ud uc

(16)

Similarly for [Γ, Ha ]: 0 = [Γ, Ha ]((gcd uc ud ) = 2Φba gbc uc . Differentiating this equality vertically gives Vk (Φba )gbc uc = −Φba gbk Interchanging a and k, taking the difference and using (5) gives b 3Rdka gbc ud uc = Φbk gba − Φba gbk .

Applying (16) gives the result. Deriving the gR condition from the gΦ condition (and hence from the given conditions) is a straightforward matter of vertical differentiation of the gΦ condition and the use of (5).

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We point out that for arbitrary SODEs the gΦ condition is generally independent of the differential Helmholtz conditions, although the gR condition is a differential consequence of the gΦ condition. As a result of these lemmas and theorem 3.1 we have Theorem 3.5. Necessary and sufficient conditions for equations (6) to be the geodesic equations of a Finsler metric are that there exist functions gab on E ◦ , symmetric, nondegenerate and homogeneous of degree 0 in the uc satisfying Va (gbc ) = Vb (gac ) and equation (11). The theorem indicates the paramount importance of the Berwald connection in the study of this Finsler inverse problem. However, it does not provide necessary and sufficient conditions on the connection alone for the existence of gab . Such conditions usually follow from the restrictions to specific dimensions or classes of Finsler spaces. Nonetheless the formalism of the inverse problem in the calculus of variations does provide a powerful means of deriving results in Finsler geometry.

4.

Remarks

Since the Riemannian situation is a special case of the Finsler one (once we restore the zero section of T M ) the results so far all apply and they can be refined using the independence of the metric and the connection on the velocities. Our inverse problem in Riemannian geometry is the question of whether a symmetric linear (affine) connection on an manifold M is the metric connection of a Riemannian manifold. This is a known problem of long-standing (see, for example, [11, 27]) but only recently have the techniques of the inverse problem been applied to it: see [4], and we might expect that examination of the Helmholtz conditions will shed light on the extent to which Riemannian curvature determines the metric structure, Riemannian or Finslerian. Following the analysis so far we can immediately state the following theorem Theorem 4.1. Let Γabc be the connection coefficients of a symmetric linear connection ∇ on a manifold M with coordinates xa . If the autoparallels of ∇ are the solutions of EulerLagrange equations with corresponding multiplier gab (xc ), then g is a metric on M with Christoffel symbols Γabc . This result is not useful from a constructive point of view, but the general study of the Helmholtz conditions involves the construction of a maximal rank system of algebraic conditions for the multiplier gab based on the Jordan normal form of the tensor field Φ ([8]). Following Edgar [11], it is fruitful to examine the algebraic conditions (hΦ)T = hΦ for nondegenerate (0, 2) tensor field solutions h. Before stating a minor result for dimension 2 in this regard, I pose the following question: is it possible to derive consequences of the first of equations (14) (equivalently (15)) which are positively homogeneous of order less than 2 in the ua ? Theorem 4.2. Let ∇ be a symmetric linear connection on M with dim(M ) = 2, spray Γ, connection coefficients Γabc , non-degenerate Ricci tensor Rab and Γab := Γabc uc . ∇ is metric

On the Inverse Problem for Autoparallels

349

if and only if the Ricci tensor is recurrent with 1 cd R Γ(Rcd ) − 2Γdd Rab = Γ(Rab ) − Rac Γcd − Rbc Γca . 2 and there exists a function λ on M satisfying 1 ab Γ(λ) = R Γ(Rab ) − 2Γaa . 2

The metric is gab := 2e−λ Rab .

a This result, derived from the Helmholtz conditions and using the fact that Rcdb = a a δb Rcd − δd Rcb in dimension 2, appears to provide a bridge to the holonomy results of Schmidt [27] (see [13] on recurrence and holonomy).

Acknowledgements I thank the Department of Algebra and Geometry, Palack´y University, Olomouc in the Czech Republic for their generous hospitality and support during my sabbatical in 2006.

References [1] J. E. Aldridge, Aspects of the inverse problem in the calculus of variations. Ph.D. thesis, Mathematics Dept., LaTrobe University, Melbourne, Australia. Submitted March 2003. [2] J. Aldridge, G. Prince, W. Sarlet and G. Thompson, The EDS approach to the inverse problem in the calculus of variations, J. Math. Phys. 47 (2006) 103508 pp. 22. [3] I. Anderson and G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Memoirs Amer. Math. Soc. 98 (473) (1992). [4] M. Crampin, On the inverse problem for sprays, Publ. Math. Debrecen 70 (2007) 319–335. [5] M. Crampin, E. Mart´ınez and W. Sarlet, Linear connections for systems of second– order ordinary differential equations, Ann. Inst. H. Poincar´e Phys. Th´eor. 65 (1996) 223–249. [6] M. Crampin, G. E. Prince, W. Sarlet and G. Thompson, The inverse problem of the calculus of variations: separable systems, Acta Appl. Math. 57 (1999) 239–254. [7] M. Crampin, G. E. Prince and G. Thompson, A geometric version of the Helmholtz conditions in time dependent Lagrangian dynamics, J. Phys. A: Math. Gen. 17 (1984) 1437–1447. [8] M. Crampin, W. Sarlet, E. Mart´ınez, G. B. Byrnes and G. E. Prince, Toward a geometrical understanding of Douglas’s solution of the inverse problem in the calculus of variations, Inverse Problems 10 (1994) 245–260.

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[9] M. Crampin and D. J. Saunders, Projective connections, J. Geom. Phys. 57 (2007) 691–727. [10] J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Am. Math. Soc. 50 (1941) 71–128. [11] S. B. Edgar, Conditions for a symmetric connection to be a metric connection, J. Math. Phys. 33 (1992) 3716–3722. [12] M. Jerie and G. E. Prince, Jacobi fields and linear connections for arbitrary second order ODEs, J. Geom. Phys. 43 (2002) 351–370. [13] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I. (John Wiley, New York-London, 1963). [14] D. Krupka and A. E. Sattarov, The inverse problem of the calculus of variations for Finsler structures, Math. Slovaca 35 (1985) 217–222. [15] O. Krupkov´a, Variational metric structures, Publ. Math. Debrecen 62 (2003) 461–495. [16] O. Krupkov´a and G.E. Prince, Second Order Ordinary Differential Equations in Jet Bundles and the Inverse Problem of the Calculus of Variations, In: Handbook of Global Analysis ((D. Krupka and D. Saunders, Eds.) Elsevier, 2008) 841–908. [17] E. Massa and E. Pagani, Jet bundle geometry, dynamical connections, and the inverse problem of Lagrangian mechanics, Ann. Inst. Henri Poincar´e, Phys. Theor. 61 (1994) 17–62. [18] T. Mestdag, Berwald-type connections in time-dependent mechanics and dynamics on affine Lie algebroids Ph.D. thesis, Universiteit Gent, Gent, 2003. [19] T. Mestdag and W. Sarlet, The Berwald-type connection associated to time-dependent second-order differential equations, Houston J. Math. 27 (2001) 763–797. [20] G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Phys. Rep. 188 (1990) 147–284. [21] Z. Muzsnay, The Euler-Lagrange PDE and Finsler metrizability, Houston J. Math. 32 (2006) 79–98. [22] G. E. Prince, The inverse problem in the calculus of variations and its ramifications, In: Geometric Approaches to Differential Equations ((P. Vassiliou and I. Lisle, Eds.) Lecture Notes of the Australian Mathematical Society, CUP, 2000). [23] W. Sarlet, The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics, J. Phys. A: Math. Gen. 15 (1982) 1503–1517. [24] W. Sarlet and M. Crampin, Addendum to: The integrability conditions in the inverse problem of the calculus of variations for second-order ordinary differential equations, Acta Appl. Math. 60 (2000) 213–224.

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[25] W. Sarlet, M. Crampin and E. Mart´ınez, The integrability conditions in the inverse problem of the calculus of variations for second-order ordinary differential equations, Acta Appl. Math. 54 (1998) 233–273. [26] W. Sarlet, G. Thompson and G. E. Prince, The inverse problem of the calculus of variations: the use of geometrical calculus in Douglas’s analysis, Trans. Am. Math. Soc. 354 (2002) 2897–2919. [27] B. G. Schmidt, Conditions on a connection to be a metric connection, Comm. Math. Phys. 29 (1973) 55–59. [28] Z. Shen, Differential Geometry of Spray and Finsler Spaces (Kluwer, Dordrecht, 2001). [29] J. Szilasi and Sz. Vattam´any, On the Finsler-metrizabilities of spray manifolds, Period. Math. Hungar. 44 (1) (2002) 81–100.

Part IV

A PPENDIX

353

In: Variations, Geometry and Physics ISBN 978-1-60456-920-9 c 2009 Nova Science Publishers, Inc. Editors: O. Krupkov´a and D. Saunders, pp. 355-362

D EMETER K RUPKA

List of publications, to December 2007 Research articles and memoirs 1. Lagrange theory in fibered manifolds, Rep. Math. Phys. 2 (1971) 121–133. 2. Some Geometric Aspects of Variational Problems in Fibered Manifolds, Folia Fac. Sci. Nat. Univ. Purk. Brunensis, Physica 14, Brno, Czechoslovakia (1973) pp. 65; arXiv:math-ph/0110005. 3. On the structure of the Euler mapping, Arch. Math. (Brno) 10 (1974) 55–61. 4. On generalized invariant transformations, Rep. Math. Phys. (Torun) 5 (1974) 353– 358. 5. (with A. Trautman) General invariance of Lagrangian structures, Bull. Acad. Polon. Sci., S´er. Sci. Math. Astronom. Phys. 22 (1974) 207–211. 6. A setting for generally invariant Lagrangian structures in tensor bundles, Bull. Acad. Polon. Sci., S´er. Sci. Math. Astronom. Phys. 22 (1974) 967–972. 7. Lagrangians and topology, Scripta Fac. Sci. Nat. UJEP Brunensis, Physica 3–4 (1975) 265–270. 8. A geometric theory of ordinary first order variational problems in fibered manifolds, I. Critical sections, J. Math. Anal. Appl. 49 (1975) 180–206. 9. A geometric theory of ordinary first order variational problems in fibered manifolds, II. Invariance, J. Math. Anal. Appl. 49 (1975) 469–476. 10. A theory of generally invariant Lagrangians for the metric fields II, Internat. J. Theoret. Phys. 15 (1976) 949–959.

356

Demeter Krupka

11. On a class of variational problems defined by polynomial lagrangians, Arch. Math. (Brno) 12 (1976) 99–106. 12. A map associated to the Lepagean forms of the calculus of variations in fibered manifolds, Czech. Math. J. 27 (1977) 114–118. 13. A theory of generally invariant Lagrangians for the metric fields I, Internat. J. Theoret. Phys. 17 (1978) 359–368. 14. Elementary theory of differential invariants, Arch. Math. (Brno) 14 (1978) 207–214. 15. (with M. Hor´ak) On the first order invariant Einstein-Cartan variational structures, Internat. J. Theoret. Phys. 17 (1978) 573–584. 16. Mathematical theory of invariant interaction lagrangians, Czech. J. Phys. B 29 (1979) 300–303. 17. Fundamental vector fields on the type fibres of jet prolongations of tensor bundles, Math. Slovaca 29 (1979) 159–167. 18. Reducibility theorems for differentiable liftings in fibre bundles, Arch. Math. (Brno) 15 (1979) 93–106. 19. On the Lie algebras of higher differential groups, Bull. Acad. Polon. Sci., S´er. Sci. Math. Astronom. Phys. 27 (1979) 235–239. 20. A remark on algebraic identities for the covariant derivatives of the curvature tensor, Arch. Math. (Brno) 16 (1980) 205–211. 21. Finite order liftings in principal fiber bundles, Beitr¨age zur Algebra und Geometrie 11 (1981) 21–27. 22. On the local structure of the Euler-Lagrange mapping of the calculus of variations, In: Proc. Conf. on Diff. Geom. Appl (Nov´e Mˇesto na Moravˇe, Czechoslovakia, Sept. 1980, Charles Univ., Prague, 1982) 181–188. 23. (with M. Francaviglia) The Hamiltonian formalism in higher order variational problems, Ann. Inst. H. Poincar´e, Sec. A 37 (1982) 295–315. 24. Local invariants of a linear connection, In: Differential Geometry (Colloq. Math. Soc. J´anos Bolyai 31, North Holland, 1982) 349–369. 25. Lepagean forms in higher order variational theory, In: Modern Developments in Analytical Mechanics (Proc. IUTAM-ISIMM Sympos., Turin, June 1982, Academy of Sciences of Turin, 1983) 197–238. 26. (with J. Musilov´a) Integrals of motion in higher order mechanics and field theory, Pokroky mat. fyz. astronom. 28 (1983) 259–266 (in Czech). ˇ ep´ankov´a) On the Hamilton form in second order calculus of variations, 27. (with O. Stˇ In: Proc. Internat. Meeting ”Geometry and Physics” (Florence, October 1982, Pitagora, Bologna, 1983) 85–101.

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28. (with J. Musilov´a) Calculus of Odd Base Forms on Differential Manifolds, Folia Fac. Sci. Nat. UJEP Brunensis 24 (1983) pp. 65. 29. Natural Lagrangian structures, In: Differential Geometry (Banach Center Publications 12, Diff. Geom. Semester, Warsaw, Sept.–Dec. 1979, Polish Scientific Publishers, Warsaw, 1984) 185–210. 30. (with J. Musilov´a) Hamilton extremals in higher order mechanics, Arch. Math. (Brno) 20 (1984) 21–30. 31. On the higher order Hamilton theory in fibered spaces, In: Proc. Conf. on Diff. Geom. Appl. (Vol. 2, Nov´e Mˇesto na Moravˇe, Czechoslovakia, (D. Krupka, Ed.) Sept. 1983, J. E. Purkynˇe Univ., Brno, 1984) 167–184. 32. (with V. Mikol´asˇov´a) On the uniqueness of some differential invariants, Czech. Math. J. 34 (1984) 588–597. 33. (with A. Sattarov) The inverse problem of the calculus of variations for Finsler structures, Math. Slovaca 35 (1985) 217–222. 34. Geometry of Lagrangean structures, 1. Odd base forms, Arch. Math. (Brno) 22 (1986) 159–174. 35. Geometry of Lagrangean structures, 2. Differential forms on jet prolongations of fibered manifolds, Arch. Math. (Brno) 22 (1986) 211–228. 36. Geometry of Lagrangean structures, 3. Lepagean forms and the first variation, In: Proc. 14th Winter School on Abstract Analysis (Srn´ı, Czechoslovakia, Jan. 1986 Suppl. Rend. del. Circolo Mat. di Palermo, Ser. II, 1987) 187–224. 37. Regular Lagrangians and Lepagean forms, In: Differential Geometry and its Appliˇ cations (Proc. Conf., Brno, Czechoslovakia, (D. Krupka and A. Svec, Eds.) August 1986, Kluwer Academic Publishers, 1987) 111–148. 38. Variational sequences on finite order jet spaces, In: Differential Geometry and its Applications (Proc. Conf., Brno, Czechoslovakia, (J. Janyˇska and D. Krupka, Eds.) August 1989, World Scientific, Singapore, 1990) 236–254. 39. Topics in the calculus of variations: Finite order variational sequences, In: Differential Geometry and its Applications (Proc. 5th Internat. Conf., Opava, Czechoslovakia, (O. Kowalski and D. Krupka, Eds.) Aug. 1992, Silesian Univ. in Opava, 1993) 473–495. 40. The contact ideal, Diff. Geom. Appl. 5 (1995) 257–276. 41. Lectures on Variational Sequences, Advanced Texts in Mathematics, Open Ed. & Sci., Opava, Czech Republic, 1995, pp. 94. 42. The trace decomposition problem, Beitr¨age zur Algebra und Geometrie 36 (1995) 303–315.

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43. The trace decomposition of tensors of type (1,2) and (1,3), In: New Developments in Differential Geometry (Proc. Colloq. on Diff. Geom., July 1994, Debrecen, (J. Szenthe and L. Tam´assy, Eds.) Kluwer Academic Publishers, Dordrecht, 1996) 243– 253. 44. Variational sequences in mechanics, Calc. Var. 5 (1997) 557–583. 45. (with D. R. Grigore) Invariants of velocities and higher order Grassman bundles, J. Geom. Phys. 24 (1998) 244–264. 46. (with J. Musilov´a) Trivial lagrangians in field theory, Diff. Geom. Appl. 9 (1998) 293–305. 47. (with J. Musilov´a and O. Krupkov´a) The variational sequence in physical theories, ˇ cˇ as. pro fyziku A48 (1998) 330–341 (in Czech). Cs. 48. (with Dao Qui Chau) 3rd order differential invariants of coframes, Math. Slovaca 49 (1999) 563–576. 49. Variational sequences and variational bicomplexes, In: Differential Geometry and Applications (Proc. Conf., (I. Kol´aˇr, O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Brno, August 1998, Masaryk Univ., Brno, Czech Republic, 1999) 525–531. 50. (with J. Musilov´a), Erratum: Trivial lagrangians in field theory, Diff. Geom. Appl. 9 (1998) 293–305; Diff. Geom. Appl. 10 (1999) 303. 51. (with J. Musilov´a) The variational sequence and trivial lagrangians, In: Proc. 13th Conf. of Czech and Slovak Physicists (Zvolen, Slovakia, August 23 - 26, 1999, (M. Reiffers and L. Just, Eds.) The Zvolen Technical Univ., 2000) 328–331 (in Czech). 52. (with M. Krupka) Jets and Contact Elements, In: Proceedings of the Seminar on Differential Geometry ((D. Krupka, Ed.) Mathematical Publications, Silesian Univ. in Opava, Opava, Czechia, 2000) 39–85. 53. (with J. Musilov´a) Recent results in variational sequence theory, In: Steps in Differential Geometry (Proc. Colloq., (L. Kozma, P. T. Nagy and L. T´amassy, Eds.) July 2000, Debrecen, Hungary, University of Debrecen, 2001) 161–186. 54. Global variational functionals on fibered spaces, Nonlinear Analysis 47 (2001) 2633– 2642. 55. Natural projectors in tensor spaces, Beitr¨age zur Algebra und Geometrie 43 (2002) 217–231. 56. Variational principles for energy-momentum tensors, Rep. Math. Phys. 49 (2002) 259–268. 57. The Weyl tensors, Publ. Math. Debrecen 62 (2003) 447–460. 58. (with P. Musilov´a) Differential invariants of immersions of manifolds with metric fields, Rep. Math. Phys. 51 (2003) 307–313.

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59. Global variational principles: Foundations and current problems, In: Global Analysis and Applied Mathematics (AIP Conference Proceedings 729, American Institute of Physics, 2004) 3–18. 60. (with J. Brajerˇc´ık) Variational principles for locally variational forms, J. Math. Phys. 46, 052903 (2005) 1–15. 61. (with J. Brajerˇc´ık) Variational principles on frame bundles, In: Differential Geometry and Its Applications (Proc. Conf., (J. Bureˇs, O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Prague, August 2004, Charles University in Prague, Czech Republic, 2005) 559–569. ˇ enkov´a) Variational sequences and Lepage forms, In: Differential Ge62. (with J. Sedˇ ometry and Its Applications (Proc. Conf., (J. Bureˇs, O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Prague, August 2004, Charles University in Prague, Czech Republic, 2005) 617–627. 63. (with O. Krupkov´a, G. Prince and W. Sarlet) Contact symmetries and variational sequences, In: Differential Geometry and Its Applications (Proc. Conf., (J. Bureˇs, O. Kowalski, D. Krupka and J. Slov´ak) Prague, August 2004, Charles University in Prague, Czech Republic, 2005) 605–615. 64. (with J. Brajerˇc´ık) Noether currents and order reduction on frame bundles, In: Proc. 40th Sympos. on Finsler Geometry ”In the Memory of our Teachers” (Hokkaido Tokai University, Sapporo, Society of Finsler Geometry, 2006) 34–37. 65. Trace decompositions of tensor spaces, Linear and Multilinear Algebra 54 (2006) 235–263. 66. The total divergence equation, Lobachevskii Journal of Mathematics 23 (2006) 71– 93. 67. (with O. Krupkov´a, G. Prince and W. Sarlet) Contact symmetries of the Helmholtz form, Differential Geometry and Applications 25 (2007) 518–542. 68. (with J. Brajerˇc´ık) Cohomology and local variational principles, Proc. of the XV International Workshop on Geometry and Physics (Puerto de la Cruz, Tenerife, Canary Islands, Spain, September 1116, 2006, Publ. de la RSME, 2007) 119–124. 69. The Vainberg-Tonti Lagrangian and the Euler-Lagrange mapping, In: Differential Geometric Methods in Mechanics and Field Theory, Volume in Honour of W. Sarlet ((F. Cantrijn and B. Langerock, Eds.) Gent, Academia Press, 2007) 81–90. 70. Natural variational principles, In: Symmetry and Perturbation Theory (SPT 2007, (G. Gaeta, R. Vitolo and S. Walcher, Eds.) Proc. Conf., Otranto, Italy, June 2-9, 2007, World Scientific, 2007) 116–123. 71. Global variational theory in fibred spaces, In: Handbook of Global Analysis (Elsevier, 2008) 773–836.

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72. The structure of the Euler-Lagrange mapping, paper in honour of N.I. Lobachevskii, Russian Mathematics (Iz. VUZ) 51 (12) (2007), N.I. Lobachevskii Anniversary Volume, 52–70. 73. (with A. Pat´ak) Geometric structure of the Hilbert-Yang-Mills functional, Int. J. Geom. Met. Mod. Phys. 5 (2008) 387–405. 74. (with O. Krupkov´a) Contact symmetries and variational PDE’s, Acta Appl. Math., 101 (1–3) (2008), A special issue dedicated to the 60th anniversary of Valentin Lychagin, 163–176. 75. (with Z. Urban) Differential invariants and higher order Grassmann bundles, In: Differential Geometry and its Applications (Proc. 10th Int. Conf. on Diff. Geom. and Appl., Olomouc 2007, World Scientific, Singapore, 2008, 463–473).

Books 1. D. Krupka, J. Janyˇska, Lectures on Differential Invariants (J. E. Purkynˇe Univ., Faculty of Science, Brno, Czechoslovakia, 1990, pp. 193). 2. D. Krupka, D. Saunders (Eds.), Handbook of Global Analysis (Elsevier, 2008, pp. 1229)

Dissertations 1. A contribution to the theory of weak interactions of strongly interacting particles (in Slovak), MSc thesis, J. E. Purkyne University, Brno, 1965. 2. Geometric Aspects of the Theory of Invariant Lagrange Structures, PhD (CSc.) Dissertation, Faculty of Mathematics and Physics, Charles University, Prague, 1976. 3. Geometric Problems of the Calculus of Variations on Fibered Spaces, Assoc. Prof. (Doc.) Dissertation, J. E. Purkyne University, Brno, 1980. 4. Natural Lagrangean structures, DrSc. Dissertation, Czechoslovak Academy of Sciences, Prague, 1981.

Edited proceedings 1. Differential Geometry and Applications (Proc. Conf. Vol. 2, Nov´e Mˇesto na Moravˇe, Czechoslovakia, Sept. 1983 (D. Krupka, Ed.) J. E. Purkynˇe Univ., Brno, 1984). 2. Differential Geometry and Its Applications (Proc. Conf., Brno, Czechoslovakia, Auˇ gust 1986 (D. Krupka and A. Svec, Eds.) Kluwer Academic Publishers, 1987). 3. Differential Geometry and Its Applications (Proc. Conf., Brno, Czechoslovakia, August 1989 (J. Janyˇska and D. Krupka, Eds.) World Scientific, Singapore, 1990).

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4. Differential Geometry and Its Applications (Proc. 5th Internat. Conf., Opava, Czechoslovakia, August 1992 (O. Kowalski and D. Krupka, Eds.) Silesian Univ. Opava, 1993). 5. Differential Geometry and Its Applications (Proc. 7th Internat. Conf., Brno, Czech Republic, August 1998, (I. Kol´aˇr, O. Kowalski, D. Krupka, and J. Slov´ak, Eds.) Masaryk Univ., Brno, 1999). 6. Proceedings of the Seminar on Differential Geometry ((D. Krupka, Ed.) Silesian University at Opava, Opava, 2000). 7. Differential Geometry and Its Applications (Proc. 8th International Conference on Diff. Geom. and Appl., Opava, Czech Republic, August 27–31, 2001, (O. Kowalski, D. Krupka, and J. Slov´ak, Eds.) Diff. Geom. Appl. 17, 2002, special issue). 8. Differential Geometry and Its Applications (Proc. Conf, August 2001, Opava, Czech Republic, (O. Kowalski, D. Krupka, and J. Slov´ak, Eds.) Silesian University in Opava, Opava, 2003). 9. Global Analysis and Applied Mathematics (AIP Conference Proceedings 729, (K. Tas, D. Krupka, O. Krupkov´a and D. Baleanu, Eds.) American Institute of Physics, 2004). 10. Differential Geometry and Its Applications (Proc. Conf., Prague, August 2004 (J. Bureˇs, O. Kowalski, D. Krupka and J. Slov´ak, Eds.) Charles University in Prague, Czech Republic, 2005). 11. Differential Geometry and its Applications (Proceedings of the 10th Int. Conf., Olomouc, August 2007 (O. Kowalski, D. Krupka, O. Krupkov´a and J. Slov´ak, Eds.) World Scientific, Singapore, 2008).

University textbooks 1. (with F. Klvana) Exercises and Solved Problems in Quantum Mechanics (in Czech), Faculty of Science, Brno University, 1973, pp. 189. 2. Mathematical Foundations of the General Relativity Theory (in Czech), Faculty of Science, Brno University, 1979, pp. 130. 3. (with J. Musilov´a) Integration on Euclidean Spaces and Manifolds (in Czech), SPN Praha, 1982, pp. 320. 4. Introduction to Analysis on Manifolds (in Czech), SPN Praha, 1986, pp. 96. 5. (with J. Musilov´a) Linear and Multilinear Algebra (in Czech), SPN Praha, 1989, pp. 281. 6. (with O. Krupkov´a) Topology and Geometry, Lectures and Solved Problems, I. General Topology (in Czech), SPN Praha, 1990, pp. 404.

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Translations 1. (with. I. Horov´a) V. M. Alexejev, V. M. Tichomirov, S. V. Fomin, Mathematical Theory of Optimal Processes (translation from Russian to Czech), Academia, Praha, 1991. 2. (with I. Horov´a) V. M. Alexejev, E. M. Galejev, V. M. Tichomirov, Problems in Optimization Theory (translation from Russian to Czech), 1988, unpublished.

Unpublished notes, preprints 1. CP-noninvariant weak interactions of strongly interacting particles (in Slovak), Czechoslovak Students Scientific Activity 1965, Bratislava (Slovakia) (paper awarded the first prize in theoretical physics). 2. Differential Invariants (Lecture Notes) (Department of Algebra and Geometry, Faculty of Science, J.E. Purkynˇe University, Brno, 1979, pp. 67). 3. (with L. Klapka and J. Musilov´a) Geometric methods in the Calculus of Variations (Lecture Notes), Poprad 1984, J. E. Purkynˇe University, Brno, 1984 (in Czech). 4. The geometry of Lagrange Structures, Preprint Series in Global Analysis GA7/1997, Silesian Univ. Opava, Czech Republic, pp. 82. 5. Elementary sheaf theory, Preprint Series in Global Analysis GA2/1998, Silesian Univ. Opava, Czech Republic, pp. 64. 6. Smooth Manifolds, Preprint Series in Global Analysis GA14/2000, Silesian Univ. Opava, Czech Republic, pp. 31. 7. (with M. Lenc) The Hilbert variational principle, Preprint 3/2002 GACR 201/00/0724, Masaryk University, Brno, 2002, pp. 75. 8. (with A. Pat´ak) The Hilbert-Yang-Mills functional: Global theory, Preprint Series in Global Analysis, UP Olomouc, 2008.

INDEX 1 1G, 182

boundary conditions, 14 brane, 3, 5 Brno, vii, 3, 24, 25, 26, 27, 50, 51, 52, 53, 54, 75, 113, 143, 165, 166, 167, 186, 187, 204, 205, 206, 232, 251, 355, 356, 357, 358, 360, 361, 362 Bulgaria, 233

A C Aβ, 105, 184, 220, 264, 311 aab, 344 Abelian, 48, 261 achievement, 102 ad hoc, 120, 316, 317 age, 356, 357, 358 AIP, 54, 359, 361 Algebra of differential invariants, 246 Algebraic method, 250 Algebraic methods, 250 algorithm, 110, 139 alternative, vii, 6, 69, 137, 262, 270, 288, 322, 328 alternatives, 221 AMS, 274 Amsterdam, 25, 26, 50, 51, 74, 139, 165, 258, 259 analog, 6 angular momentum, 257, 277, 286, 287 anisotropy, 233 annihilation, 65 anomalous, 213, 216 ants, 246 application, 3, 5, 6, 14, 18, 29, 32, 37, 87, 172, 180, 197, 210, 247, 312, 313 argument, 193, 199, 320, 323 assumptions, 332, 336, 338 Australia, 27, 49, 341, 349 availability, 318, 326, 327

B behavior, 239 Belgium, 261, 315 Bianchi identities, 316, 319, 330, 331 boson, 5 Boston, 51, 73, 232

calculus, vii, 3, 4, 5, 6, 10, 19, 22, 23, 24, 26, 27, 28, 29, 35, 50, 51, 52, 53, 54, 78, 84, 96, 97, 99, 102, 112, 113, 114, 175, 209, 210, 217, 316, 317, 321, 322, 338, 339, 341, 342, 348, 349, 350, 351, 356, 357 CAM, 138 Cartan form, 28, 32, 35, 49, 51, 54, 58, 79, 80, 82, 84, 88, 90, 91, 257 Cartan tensor, 299, 332, 333, 337 casting, vii CCC, 266 Central Europe, 187 charge density, 126 charged particle, 117, 118, 119, 125, 127 Charged string, 125 classes, 7, 22, 48, 49, 212, 303, 304, 348 classical, 3, 5, 14, 23, 26, 28, 35, 42, 49, 51, 57, 58, 59, 78, 83, 84, 85, 86, 117, 120, 124, 128, 129, 130, 137, 139, 144, 147, 149, 150, 151, 152, 157, 158, 159, 160, 161, 162, 176, 183, 184, 189, 190, 191, 194, 195, 197, 198, 200, 210, 211, 217, 219, 221, 224, 225, 228, 238, 275, 277, 279, 290 classical mechanics, 35, 277 classification, 47, 111, 145, 149, 150, 190, 191, 194, 197, 203, 204, 205, 206, 318, 319 closure, 37, 306 Co, 139, 140, 234 Codazzi syzygy, 222 collaboration, 209 communication, 114, 250 community, viii commutativity, 22, 108, 150, 156 compatibility, 238, 247, 249, 250, 334, 343 complement, 101

364

Index

complexity, 210 complications, 212, 333 components, 9, 10, 28, 33, 35, 38, 39, 43, 45, 46, 48, 59, 60, 66, 121, 125, 127, 134, 175, 193, 199, 212, 213, 214, 215, 224, 238, 240, 256, 262, 264, 265, 268, 269, 271, 287, 304, 311, 312, 316, 317, 325, 328, 329, 337, 338, 344, 345 composition, 30, 60, 61, 146, 147, 153, 179, 180, 181, 185, 239, 243, 318 computation, 68, 95, 119, 134, 135, 210, 219, 220, 329, 332, 337 computing, 110 concentrates, 262 concrete, 3, 18, 145, 190 configuration, 86, 87, 118, 120, 121, 122, 123, 124, 126, 130, 132, 262, 277, 281, 282, 284, 287, 288, 304 confusion, 60, 338 conjecture, 69, 73, 100, 249 conjugation, 89, 248 Connection, 320 conservation, 5, 13, 14, 15, 17, 18, 23, 26, 27, 29, 32, 43, 47, 111, 165, 235, 257, 259, 262 Conservation law, 18 constraints, 25, 26, 41, 42, 55, 64, 115, 249 construction, 19, 58, 59, 103, 110, 120, 168, 172, 173, 175, 180, 183, 196, 209, 210, 211, 212, 213, 218, 223, 271, 305, 306, 307, 322, 323, 326, 327, 348 continuity, 112, 124, 145, 153, 156, 294, 297, 299 contractions, 29, 103, 177, 184, 255 control, 210 convex, 227, 231, 232, 294, 295, 300 Copenhagen, 24 cosine, 278, 295 couples, 63, 66, 70 coupling, 157 covering, viii, 144, 145, 153, 237, 258, 304 CRM, 250 C-spectral sequence, 26, 100, 106, 111, 112 curiosity, 6 Curvature, 155, 205, 279, 284, 328 cycles, 287 Czech Republic, 3, 25, 26, 27, 53, 77, 113, 114, 143, 163, 167, 186, 189, 349, 357, 358, 359, 361, 362

D danger, 60 De Donder–Hamilton equations, 39, 40, 41 decomposition, viii, 5, 10, 21, 22, 28, 30, 31, 32, 33, 65, 103, 104, 203, 213, 239, 242, 246, 302, 304, 305, 306, 307, 308, 313, 319, 320, 321, 327, 344, 357, 358 deficiency, 333 definition, 4, 5, 6, 8, 11, 13, 22, 23, 32, 37, 45, 58, 59, 60, 64, 69, 84, 101, 102, 105, 135, 145, 153,

172, 173, 177, 211, 214, 258, 283, 319, 324, 325, 328, 329, 332, 335 deformation, 31, 279, 288, 302, 304, 305, 311, 312, 344 degenerate, 190, 192, 195, 198, 200, 204, 284, 348 degrees of freedom, 57, 58, 59 denoising, 210 density, 117, 119, 121, 122, 123, 124, 125, 126, 127, 257 dependent variable, 77, 99, 211, 219 derivatives, 19, 41, 49, 59, 60, 64, 66, 71, 73, 77, 81, 83, 84, 102, 104, 108, 109, 110, 121, 143, 150, 156, 177, 205, 212, 215, 216, 217, 219, 222, 225, 227, 237, 239, 240, 242, 244, 246, 256, 257, 258, 265, 269, 317, 329, 333, 337, 339, 356 deviation, 295, 300, 301 dichotomy, 238 differential equations, 27, 38, 42, 43, 44, 46, 54, 110, 111, 192, 193, 199, 209, 219, 227, 234, 237, 238, 248, 251, 262, 270, 280, 281, 282, 323, 339, 342, 343, 350 Differential invariant, 114, 166, 206, 232, 234, 237, 358, 360 differentiation, 120, 123, 162, 212, 216, 217, 220, 227, 229, 243, 248, 256, 258, 347 dilation, 320 Discrete mechanics, 97 discrete variable, 94 discretization, 86, 94, 95 distribution, 38, 47, 89, 90, 92, 96, 100, 101, 111, 271, 326, 328 divergence, 43, 113, 222, 359 duality, 86, 203, 231, 244, 320, 321, 324, 325 dynamical properties, 278 dynamical system, 5, 57, 225, 227, 261, 262, 263, 265, 267, 274 dynamical systems, 261, 262, 274

E Education, 113 electromagnetic, 118, 125, 126, 127, 152, 157, 158 electromagnetism, 127 energy, 14, 124, 125, 257, 271, 279, 286, 287, 341 energy-momentum, 128, 257, 358 EP-1, 20, 106 equality, 66, 67, 107, 110, 283, 296, 347 equilibrium, 210 Euclidean space, 190, 238, 277, 278, 281, 284, 287, 294, 296, 297, 299, 309, 310 Euclidean-invariant variational problem, 221, 222 Eulerian, 217, 220, 221, 222, 229 Eulerian operator, 217, 220, 221, 222 Euler-Lagrange equations, 38, 58, 69, 78, 87, 88, 92, 93, 94, 95, 218, 219, 221, 232, 267, 268, 280, 336, 337, 342 evolution, 15, 209, 210, 225, 226, 227, 228, 230, 231, 232, 234, 287, 343

365

Index

F failure, 344 family, 21, 28, 31, 35, 41, 44, 47, 92, 111, 145, 146, 148, 153, 155, 178, 190, 195, 196, 197, 200, 201, 217, 224, 280, 282, 283, 291 fiber, 14, 15, 57, 58, 59, 62, 85, 118, 120, 121, 123, 125, 146, 147, 148, 158, 170, 172, 173, 178, 181, 184, 190, 240, 356 fiber bundles, 85, 190, 356 fibers, 9, 168, 184, 240, 294 field theory, 5, 6, 8, 14, 15, 26, 27, 35, 36, 41, 43, 49, 51, 53, 54, 55, 57, 58, 59, 76, 115, 118, 120, 121, 124, 128, 129, 130, 137, 139, 356, 358 filament, 210, 225, 230, 232, 233 filtration, 212 Finsler arc length, 294 Finsler geometry, 81, 193, 293, 294, 295, 313, 314, 339, 340, 342, 343, 348 Finsler inverse problem, 343, 345, 346, 348 Finsler norm, 293, 294 Finsler space, 293, 294, 295, 297, 299, 300, 301, 302, 303, 304, 305, 306, 307, 309, 310, 311, 312, 313, 316, 317, 339, 348 first principles, 120, 124 flatness, 180, 288 flow, 13, 17, 30, 101, 144, 149, 150, 156, 168, 169, 173, 176, 183, 210, 224, 225, 226, 227, 228, 229, 230, 231, 256, 258, 263, 344 fluid, 96, 124, 127 F-metric, 193, 194, 195, 196 Fourier, 51, 74, 75, 97, 188, 233 France, 23, 26, 114, 233, 234 freedom, 14, 57, 58, 59, 121, 249 friendship, 230

G gas, 200 gauge, 121, 143, 144, 152, 154, 156, 157, 160, 162, 163, 165, 166, 181, 184, 186 gauge group, 144, 152, 156, 157 gauge invariant, 143, 144, 157, 162 Gauge invariant Lagrangian, 162 gauge theory, 152 Gauge-natural bundle, 153, 163, 186 Gauge-natural bundle functor, 153 Gauss-Bonnet, 313 Gaussian, 221, 222, 250 gene, 11, 57, 58, 190 General Relativity, 41, 361 generalization, 18, 22, 50, 57, 58, 85, 152, 158, 163, 181, 186, 270, 278, 293, 296, 328, 336 generalizations, 11, 57, 58, 190 generators, 213, 214, 218, 240, 241, 244, 284 Geodesic, 84, 97 globalization, 37, 85

g-natural metric, 194, 195, 196, 197, 201, 204, 205, 206 google, 275 grants, 49 graph, 248, 250 Grassmannian bundle, 61 gravitation, 257, 259, 288 gravitational field, 158 gravitational stress, 128 gravity, 41 groups, viii, 30, 45, 48, 164, 232, 233, 238, 239, 262, 266, 275, 356 G-structure, 175, 180, 187 guidance, 294, 317

H H1, 223 H2, 108, 223, 278, 284, 285, 287, 288, 290 Hamilton equations, 27, 29, 38, 39, 42, 47, 130, 131, 134, 135, 136, 137 Hamilton extremal, 38, 39, 40, 41, 47, 52, 357 Hamiltonian, 25, 28, 29, 38, 39, 41, 46, 47, 48, 49, 50, 51, 53, 54, 58, 74, 75, 76, 96, 130, 133, 139, 210, 217, 218, 220, 221, 222, 226, 229, 230, 233, 248, 262, 275, 356 Hamiltonian system, 38, 39, 41, 46, 47, 210 Hamilton-Jacobi, 84, 129, 130, 131, 133, 134, 135, 137, 138, 139, 140, 234 hands, 315 heart, 222 heat, 231, 232 Helmholtz conditions, 99, 321, 331, 335, 341, 342, 343, 345, 346, 347, 348, 349, 350 Helmholtz equation, 65, 69, 72, 73 Helmholtz form, 27, 48 hemisphere, 308 heredity, 190, 197 Hermitian connection, 157 Hessian matrix, 335, 342 Higgs, 289 Hilbert, 41, 82, 124, 125, 127, 255, 258, 362 Hilbert form, 82, 124, 125, 127 Holland, 51, 164, 165, 356 homeomorphic, 304 homogeneity, 82, 293, 294, 297, 334, 335, 346, 347 homomorphism, 170, 181, 257, 273, 274 household, 338 Hungary, 24, 25, 164, 165, 358 hydrodynamics, 96 hydrogen, 278, 289 hyperbolic, 277, 278, 281, 283, 284, 290 hypothesis, 58, 92, 171, 172, 174, 175

I IDEA, 3

366

Index

identification, 85, 121, 123, 137, 158, 169, 177, 180, 184, 319, 326 identity, 21, 66, 100, 123, 158, 162, 163, 190, 192, 219, 221, 246, 257, 258, 263, 270, 318, 320, 321 Illinois, 207 IMA, 231 image analysis, 234 images, 81, 111, 224, 311 immersion, 61, 62 inclusion, 102, 103, 106, 107, 108, 180, 185 incompatibility, 333 independence, 71, 215, 348 independent variable, 7, 211, 212 Indiana, 234, 313 indices, 29, 33, 35, 36, 40, 59, 60, 63, 66, 67, 70, 100, 255, 273, 324, 329, 346 induction, 65, 69, 72, 171, 172, 174, 175, 176, 213 inequality, 67, 313 infinite, 5, 100, 105, 107, 111, 115, 124, 146, 165, 225, 227, 238, 242 injection, 168, 170, 174, 180, 181, 182 insight, 123, 320 inspection, 326 inspiration, 230, 341 integration, 19, 29, 46, 47, 119, 219, 220, 228 interaction, 118, 119, 120, 122, 125, 126, 157, 158, 161, 163, 164, 166, 356 interactions, 360, 362 interface, 210 interpretation, 152, 166, 328, 330, 332 interval, 194 intrinsic, vii, 28, 32, 34, 38, 86, 100, 108, 212, 222, 225, 226, 227, 230, 287, 315, 317, 324, 325, 326, 330, 331, 335, 336, 338, 343 Invariant Lagrangian, 151, 152, 157, 162, 275 Invariant Theory, 163 Invariantization, 211, 217 invariants, vii, viii, 49, 114, 165, 166, 190, 191, 192, 193, 199, 203, 206, 209, 210, 212, 213, 214, 215, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 229, 230, 232, 233, 234, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 251, 256, 258, 259, 356, 357, 358, 360 Inverse problem of the calculus of variations, 341 inversion, 211 involution, 168, 179, 180, 183, 186 isomorphism, 30, 42, 107, 108, 169, 176, 181, 182, 191, 258 isotropy, 211, 270 Italy, 50, 52, 99, 359 iteration, 178, 185

J Jacobi endomorphism, 321 Jacobian, 121, 128 January, vii, viii Japan, 189, 314, 339

Jordan, 348 justification, 322

K kernel, 89, 102, 104, 170, 183 kinetic energy, 271, 279 Kirchhoff, 233 Krupka theorem, 32

L L1, 14, 86, 110, 118, 193, 199 L2, 14, 110, 192, 198 Lagrange structure, 10, 14, 16 Lagrangian, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 54, 55, 57, 58, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 109, 110, 113, 115, 117, 118, 119, 120, 122, 123, 124, 125, 127, 143, 144, 145, 147, 149, 150, 151, 152, 153, 155, 157, 159, 161, 162, 163, 165, 219, 220, 221, 222, 223, 229, 233, 235, 248, 261, 262, 267, 268, 269, 270, 271, 272, 273, 274, 279, 280, 281, 282, 342, 343, 345, 349, 350, 355, 357, 359 Lagrangian density, 86, 117, 118, 119, 123, 124, 125 Lagrangian formalism, 26, 65, 69, 115, 235 Lagrangian formulation, 26 language, 18, 78, 210, 212, 248 law, 13, 27, 43, 123, 257 laws, 5, 14, 15, 17, 18, 23, 26, 29, 32, 47, 111, 235, 257, 262 lead, 32, 100, 111, 122, 210, 243, 278, 317 Legendre coordinates, 39, 40, 41 Legendre transformation, 39, 41, 53 Leibniz, 257 Lepage congruences, 85, 86 Lepage equivalent of a Lagrangian, 4, 5, 8, 11, 12, 18, 28, 45, 57, 77 Lepage form, vii, 3, 4, 5, 6, 8, 11, 12, 13, 18, 22, 23, 25, 27, 28, 29, 32, 35, 37, 41, 42, 49, 53, 64, 79, 114, 359 Lie algebra, 154, 156, 210, 239, 256, 257, 258, 259, 263, 273, 274, 284, 285, 356 Lie algebroid, 167, 168, 182, 183, 350 Lie group, 61, 144, 146, 152, 156, 157, 160, 190, 209, 211, 214, 232, 237, 238, 239, 247, 262, 263 Lie-Tresse theorem, 238, 239 linear, vii, 14, 40, 62, 64, 65, 66, 70, 73, 78, 81, 94, 108, 144, 147, 154, 156, 157, 158, 159, 160, 161, 164, 165, 167, 168, 169, 170, 173, 174, 175, 179, 182, 184, 186, 187, 188, 189, 190, 193, 197, 198, 199, 200, 201, 205, 206, 213, 214, 223, 235, 239, 241, 257, 270, 277, 279, 286, 294, 295, 297, 299, 301, 303, 304, 305, 310, 311, 315, 316, 320, 322,

367

Index 323, 324, 325, 326, 327, 328, 329, 331, 332, 333, 334, 335, 336, 337, 338, 341, 342, 343, 344, 345, 346, 348, 350, 356 Linear connection, 158, 316, 349 linear function, 279 links, 320 L-metric, 199, 200 localization, 145, 146 locus, 238, 308, 309 London, 54, 84, 96, 140, 164, 166, 188

M M1, 168, 285, 302, 303, 307, 308 machinery, 118, 212, 321 manifold, 3, 4, 5, 6, 7, 8, 9, 11, 14, 15, 16, 23, 24, 25, 27, 28, 29, 30, 31, 42, 52, 53, 57, 58, 59, 60, 61, 63, 70, 71, 73, 77, 78, 79, 80, 81, 82, 85, 86, 89, 97, 100, 106, 110, 111, 112, 113, 127, 130, 132, 133, 134, 137, 139, 143, 144, 145, 146, 147, 148, 149, 151, 153, 155, 162, 163, 165, 166, 167, 168, 178, 179, 180, 184, 186, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 200, 201, 202, 203, 204, 205, 206, 207, 210, 211, 223, 233, 237, 241, 245, 247, 249, 255, 257, 258, 262, 263, 271, 278, 281, 283, 293, 294, 295, 301, 302, 303, 304, 305, 306, 308, 309, 313, 314, 316, 320, 340, 343, 348, 351, 355, 356, 357, 358 manifolds, 3, 4, 5, 8, 11, 16, 23, 24, 25, 27, 28, 29, 52, 53, 59, 61, 77, 78, 85, 97, 100, 111, 112, 113, 137, 139, 144, 145, 151, 153, 162, 165, 166, 168, 178, 186, 190, 194, 202, 204, 205, 206, 207, 233, 241, 247, 249, 255, 258, 262, 278, 281, 301, 306, 309, 313, 314, 351, 355, 356, 357, 358 manners, 125 mapping, 4, 5, 9, 10, 11, 12, 16, 19, 22, 24, 29, 30, 39, 40, 41, 48, 52, 148, 149, 162, 212, 297, 299, 302, 311, 355, 356, 359, 360 Mathematical Methods, 138 mathematicians, 256 mathematics, vii, viii, 164, 259, 342 matrix, 61, 95, 96, 147, 149, 170, 171, 175, 202, 214, 220, 221, 244, 245, 263, 264, 267, 269, 273, 284, 285, 309, 335, 342 metals, 225 metric, 14, 16, 18, 120, 121, 122, 123, 124, 125, 126, 127, 128, 152, 165, 166, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 207, 240, 241, 244, 248, 257, 271, 272, 279, 280, 281, 284, 285, 293, 295, 296, 297, 299, 301, 310, 316, 322, 332, 334, 335, 336, 337, 338, 339, 341, 342, 348, 349, 350, 351, 355, 356, 358 mines, 288 Ministry of Education, 23, 49, 163, 186 Minkowski space, 5, 295, 299, 302, 304, 305, 311, 312 Minkowski tensor, 124, 125, 127 Minnesota, 209

mixing, 329 models, 225, 288 modules, 99 momentum, 97, 125, 257, 262, 268, 269, 273, 274, 277, 286, 287, 290 monograph, 167, 258, 262 Moscow, 54, 164, 234, 251 motion, 4, 5, 10, 14, 43, 119, 127, 233, 237, 238, 239, 255, 262, 277, 278, 280, 281, 282, 286, 287, 288, 307, 308, 312, 356 motivation, 29, 45, 49, 105, 158 Moving frame, 209, 234, 250 multiples, 97, 178 multiplication, 60, 146, 147, 239, 248, 285 multiplier, 342, 343, 346, 348 multivariate, 220

N natural, vii, 3, 5, 23, 47, 48, 58, 61, 71, 85, 86, 89, 92, 100, 101, 102, 103, 105, 106, 107, 108, 111, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 154, 155, 156, 158, 159, 160, 161, 162, 164, 165, 166, 167, 168, 173, 176, 177, 178, 184, 186, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 237, 240, 255, 257, 258, 265, 280, 282, 288, 293, 326, 328, 329, 332, 344 Natural bundle, viii, 143, 145, 165, 166, 206 Natural differential operator, 145, 150, 152, 156, 207 Natural Lagrangian, 24, 145, 147, 149, 150, 151, 153, 155, 156, 157, 159, 161, 163, 164, 165, 357 Natural transformation, 205, 206 Netherlands, 234 New York, 24, 54, 113, 128, 163, 205, 206, 207, 231, 232, 233, 235, 251, 313, 350 Newtonian, 23, 118 Noether equation, 42 Noether theorem, vii, 27, 28, 262 non-Abelian, 275 nonlinear, 26, 54, 55, 225, 230, 231, 233, 235, 248, 250, 251, 315, 318, 319, 320, 321, 323, 324, 327, 328, 329, 330, 334, 339, 344 nonlinearities, 231 normal, 162, 210, 221, 224, 225, 226, 227, 229, 230, 336, 342, 348 Normal flow, 227 normalization, 209, 211, 217 Norway, 237 numerical analysis, 210

O object recognition, 231 observations, 345 obstruction, 99, 176 operator, 4, 5, 6, 19, 21, 44, 48, 71, 86, 99, 107, 108, 109, 110, 111, 143, 148, 149, 150, 155, 156, 158,

368

Index

159, 160, 161, 164, 167, 168, 172, 176, 177, 184, 213, 217, 218, 220, 221, 222, 223, 225, 226, 227, 228, 229, 230, 241, 247, 248, 255, 315, 316, 323, 324, 326, 331, 333, 334, 343 Operators, 145, 148, 150, 155, 156, 176, 184 Oproiu metric, 196, 197 optics, 210, 225 orbit, 307 ordinary differential equations, 44, 45, 53, 261, 341, 342, 349, 350, 351 orientation, 81, 294, 296 orthogonality, 300 oscillator, 278, 290

P Pacific, 313 packaging, 120, 125 pairing, 86 PAN, 187 paper, vii, 3, 5, 6, 8, 11, 13, 23, 27, 28, 29, 40, 41, 53, 58, 70, 73, 77, 84, 88, 99, 100, 127, 129, 130, 135, 144, 145, 152, 157, 167, 168, 195, 196, 209, 211, 238, 240, 249, 256, 257, 262, 270, 287, 288, 295, 309, 315, 317, 320, 334, 335, 336, 338, 341, 360, 362 parabolic, 278 parallelism, 327 parameter, 14, 15, 31, 92, 226, 227, 228, 278, 281, 288, 294, 307, 308, 342 Paris, 49, 50, 51, 76, 84, 96, 112, 114, 186, 188, 231, 258, 259 partial differential equations, vii, 45, 47, 51, 75, 113, 150, 151, 162, 210, 224, 227, 231, 233, 237, 238, 242, 250, 251, 360 particles, 10, 14, 118, 124, 125, 127, 291, 360, 362 periodic, 228 pH, 212 phase space, 58, 280, 282 physicists, 256, 257 physics, vii, viii, 3, 5, 10, 18, 112, 113, 128, 139, 145, 181, 250, 259, 342, 362 planar, 224, 300 plasma, 127, 128 play, 4, 27, 28, 29, 42, 46, 132, 175, 181, 210, 226, 247, 267, 316 pleasure, 111, 230, 341 Poisson, 233, 256, 262 Poland, 255 polar coordinates, 279, 285, 290 Polyakov action, 18 polynomial, 104, 108, 110, 151, 237, 238, 248, 251, 356 polynomials, 110, 241 pond, 284 potential energy, 14, 271 power, 102, 124, 209, 241 powers, 102

private, 114, 250 program, 247 projector, 60, 89, 133, 134, 326, 327 propagation, 234 property, 6, 11, 19, 23, 32, 36, 39, 41, 42, 45, 47, 59, 78, 79, 105, 107, 108, 149, 156, 197, 278, 282, 284, 287, 288, 294, 295, 300, 302, 303, 308, 309, 320, 326, 327, 329, 330, 331, 332, 333, 334, 336, 337, 338, 342 proposition, 66, 67, 68, 69, 103, 104, 108, 332, 347 pseudo, 162, 237

Q quantum, 157, 158, 164, 210, 278, 289, 290 quantum mechanics, 164, 210 quantum structure, 164 query, 323

R radius, 279, 281, 296, 307, 313 range, 130, 170, 210, 244 real numbers, viii, 194 reasoning, 109, 277, 288 recall, 27, 32, 38, 77, 100, 110, 117, 144, 145, 152, 153, 168, 175, 180, 183, 190, 193, 218, 238, 265, 283, 288, 335 recalling, 28, 191, 197, 203, 318 recognition, 231 reconcile, 120 reconstruction, 261, 263, 264, 267, 270, 271, 274 recurrence, 181, 210, 213, 214, 215, 216, 218, 222, 349 recursion, 225, 228, 230 Reduction, 69, 86, 96, 144, 150, 151, 156, 157, 158, 159, 164, 172, 173, 174, 175, 180, 183, 185, 186, 250, 261, 262, 263, 265, 266, 268, 269, 270, 273, 274, 275, 278, 279, 281, 283, 284, 285, 287, 289, 291, 299, 337, 359 Reduction theorems, 164 reflection, 287, 288 regular, vii, 27, 29, 38, 40, 41, 47, 51, 54, 61, 82, 94, 96, 105, 179, 194, 211, 238, 244, 267, 268, 280, 303, 309, 321 Regular Lagrangian, 38, 52, 357 Regularization, 231 relationship, 59, 111, 324, 326 Relativistic particle, 14 relativity, 152, 210, 247 relevance, 70, 321, 322 repetitions, 308 Representation of the variational sequence, 24, 52, 113 research, vii, viii, 23, 99, 100, 111, 112, 163, 209, 230, 274, 315 resolution, 19, 26, 48, 106

369

Index Ricci bundle, 159, 161 rigidity, 197 Rome, 96 rotations, 286, 287, 302, 307 Routhian, 261, 269, 270, 271, 273 Russian, 53, 54, 112, 113, 164, 204, 251, 316, 360, 362

S Sasaki lift, 194 scalar, 41, 118, 119, 121, 122, 124, 125, 127, 189, 196, 197, 202, 204, 238, 240, 241, 317, 318, 330, 331 scalar field, 118 school, 5, 6, 206 searching, 4 segmentation, 210 selecting, 317 SEM, 118, 119, 120, 123, 124, 125, 126, 127 separation, 289 September 11, 359 shape, 234 SIGMA, 232, 290 sign, 60, 197, 241, 263, 284, 296, 315, 329 similarity, 326, 331 sine, 278, 286 Singapore, 25, 51, 52, 53, 54, 75, 114, 186, 187, 205, 314, 357, 360, 361 singular, 42, 139, 238, 240, 242, 245 singularities, 224 Slovakia, vii, 358, 362 soliton, 210, 225, 232 solutions, 29, 47, 94, 95, 128, 137, 151, 152, 162, 192, 193, 199, 233, 234, 241, 249, 250, 263, 280, 281, 282, 312, 313, 341, 342, 348 Source form, 48 spacetime, 58, 118, 120, 121, 123, 124, 125, 126, 127, 152, 158, 257, 288 Spain, 117, 129, 138, 277, 359 special relativity, 257 spectrum, 241, 247, 248 speed, 234 spheres, 277, 283, 290, 314 spin, 152, 166, 257 SPT, 359 stability, 238 stages, 88, 335, 341 stress, 33, 38, 47, 48, 64, 125, 128 Stress-energy-momentum tensor, 128 students, 6 subgroups, 211 supersymmetric, 23 supervision, vii supply, 212 surface area, 16, 222, 223 surprise, 325, 336 swarm, 118

symbolic, 7, 210 symbols, 147, 149, 193, 199, 246, 348 symmetry, 5, 48, 59, 66, 70, 73, 101, 143, 210, 219, 224, 234, 240, 249, 257, 261, 262, 263, 266, 271, 274, 294, 300, 337 symplectic, 6, 50, 58, 77, 130, 256 systems, 4, 5, 10, 14, 25, 29, 36, 38, 41, 47, 49, 51, 53, 58, 73, 76, 85, 86, 117, 118, 126, 192, 193, 198, 199, 203, 210, 233, 234, 251, 261, 262, 266, 270, 278, 286, 290, 302, 304, 311, 312, 342, 349

T tangible, 332 technology, 211 TEM, 124, 127 tension, 126 tensor field, 158, 161, 176, 204, 255, 257, 315, 316, 317, 320, 322, 324, 325, 328, 330, 335, 344, 348 tensor products, 177 textbooks, 361 theory, vii, 3, 4, 5, 6, 7, 8, 11, 18, 23, 25, 27, 28, 29, 37, 39, 40, 41, 42, 46, 47, 48, 49, 52, 54, 57, 79, 82, 83, 84, 97, 106, 118, 124, 128, 129, 130, 137, 139, 140, 143, 145, 147, 151, 153, 156, 165, 166, 167, 168, 176, 178, 180, 181, 184, 187, 190, 206, 209, 210, 233, 235, 238, 239, 240, 256, 257, 259, 262, 275, 278, 288, 294, 295, 316, 318, 319, 323, 338, 355, 356, 357, 358, 359, 360, 362 third order, 242, 244 three-dimensional, 233, 277, 278, 279, 280, 281, 284, 290 three-dimensional space, 281, 284 time, 11, 14, 15, 28, 86, 94, 95, 96, 118, 120, 123, 124, 130, 152, 164, 224, 227, 256, 268, 287, 316, 324, 327, 335, 349 title, 315, 334 Tokyo, 165, 189, 205, 206, 207 topology, vii, 83, 295, 305, 355 Torsion, 169, 187, 328 tracking, 210, 233 trajectory, 118, 120, 288 trans, 304 transcription, 27 transfer, 29, 30 transformation, 14, 17, 39, 41, 42, 53, 62, 63, 147, 148, 150, 155, 156, 157, 190, 192, 193, 194, 195, 196, 198, 199, 200, 201, 203, 204, 209, 213, 230, 256, 257, 294, 301, 302, 303, 304, 305, 309, 310 transformations, 5, 13, 14, 15, 18, 36, 42, 43, 148, 152, 155, 157, 189, 190, 191, 192, 193, 205, 206, 207, 217, 251, 259, 265, 294, 301, 303, 304, 309, 311, 312, 313, 355 transition, 262, 304, 310, 311 transitions, 305, 311 translation, 14, 112, 113, 171, 174, 185, 204, 221, 228, 257, 304, 316, 362 transport, 316, 323

370

Index

Tresse derivative, 237, 239, 242, 246 TTM, 179, 294, 295 Turkey, 54 two-dimensional, 6, 278, 283, 284, 285, 288, 290, 295, 296 two-dimensional space, 278

U unification, 317 uniform, 95 university education, 28 Utah, 23, 49, 73, 112, 231 Utiyama-like theorem, 158, 160, 164

V valence, 255 values, 78, 80, 83, 86, 90, 119, 123, 126, 147, 151, 158, 162, 169, 170, 171, 180, 249, 268, 278, 281, 282, 283, 284, 286, 287 variable, 121, 123, 195, 196, 279 variables, 58, 63, 66, 70, 71, 73, 93, 121, 128, 200, 201, 212, 269, 286, 289 variation, 5, 6, 8, 10, 25, 27, 28, 31, 32, 34, 37, 42, 79, 85, 86, 96, 256, 357 Variation, 31, 32, 73, 75, 76 Variational equations, 53 Variational sequence, 25, 26, 52, 53, 99, 113, 114, 232, 357, 358, 359 vector, 4, 8, 9, 10, 11, 12, 13, 19, 21, 28, 29, 30, 31, 32, 34, 37, 38, 39, 42, 43, 46, 47, 63, 76, 79, 86,

89, 90, 95, 100, 101, 111, 130, 131, 132, 133, 134, 135, 137, 138, 143, 144, 149, 150, 152, 154, 156, 157, 158, 160, 161, 162, 166, 168, 169, 172, 173, 175, 176, 183, 184, 191, 192, 194, 198, 203, 212, 213, 216, 224, 225, 226, 230, 238, 239, 240, 241, 242, 244, 248, 255, 256, 257, 258, 262, 263, 264, 265, 266, 267, 268, 269, 271, 272, 277, 279, 280, 281, 282, 283, 285, 287, 293, 294, 295, 299, 304, 311, 312, 313, 317, 318, 319, 320, 321, 322, 323, 325, 328, 329, 330, 333, 343, 344, 356 velocity, 60, 280, 288, 312 Victoria, 27 visible, 161, 238 vision, 210, 228, 234 vortex, 210, 225, 230, 232

W Warsaw, vii, 24, 357 wave equations, 250 weak interaction, 360, 362 web, 58 Weierstrass excess function, 77, 78, 83 Weierstrass necessary condition, 83 Weyl tensor, 358 winter, 206 writing, 62, 122, 239

Y Yang-Mills, 162, 163, 275 yield, 100, 102, 104, 105, 123, 215, 300, 302, 343