Global Analysis in Mathematical Physics: Geometric and Stochastic Methods (Applied Mathematical Sciences 122)

Applied Mathematical Sciences Volume 122 Editors J.E. Marsden L. Sirovich E John (deceased) Advisors M. Ghil J.K. Hale T. Kambe J. Keller K. Kirchgassner B.J. Matkowsky C.S. Peskin J.T. Stuart

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(continued following index)

Yuri Gliklikh

Global Analysis in Mathematical Physics Geometric and Stochastic Methods Translated by Viktor L. Ginzburg

Springer

Yuri Gliklikh Department of Mathematics Voronezh State University 3946693 Voronezh Russia

Editors J.E. Marsden Control and Dynamical Systems, 104-44 California Institute of Technology Pasadena, CA 91125 USA

L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA

Mathematics Subject Classification (1991): 5802, 58Z05, 58G32, 58D05, 58D25, 60H10, 70D05, 70G05, 76G05, 76C99

Library of Congress Cataloging-in-Publication Data Gliklikh, Yu. E., 1949Global analysis in mathematical physics: geometric and stochastic methods / Yuri Gliklikh. p. cm. - (Applied mathematical sciences; 122) Includes bibliographical references and index. ISBN 0-387-94867-8 (alk. paper) 1. Geometry, Differential.

2. Stochastic processes.

3. Global

analysis (Mathematics) 4. Mathematical physics. I. Title. II. Series: Applied mathematical sciences (Springer-Verlag New York, Inc.); v. 122. QA1.A647 vol. 122 [QC20.7.D521 510 s - dc20 [530.1'5474]

96-33317

Printed on acid-free paper. © 1997 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even

if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Timothy Taylor; manufacturing supervised by Joe Quatela. Photocomposed copy prepared from the author's TEX files. Printed and bound by Maple-Vail Book Manufacturing Group, York, PA. Printed in the United States of America.

987654321 ISBN 0-387-94867-8 Springer-Verlag New York Berlin Heidelberg

SPIN 10549292

Preface to the English Edition

The first edition of this book entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh University Press in 1989. For its English edition, the book has been substantially revised and expanded. In particular, new material has been added to Sections 19 and 20. I am grateful to Viktor L. Ginzburg for his hard work on the translation and for writing Appendix F, and to Tomasz Zastawniak for his numerous suggestions. My special thanks go to the referee for his valuable remarks on the theory of stochastic processes. Finally, I would like to acknowledge the support of the AMS fSU Aid Fund and the International Science Foundation (Grant NZB000), which made possible my work on some of the new results included in the English edition of the book. Voronezh, Russia September, 1995

Yuri Gliklikh

Preface to the Russian Edition

The present book is apparently the first in monographic literature in which a common treatment is given to three areas of global analysis previously considered quite distant from each other, namely, differential geometry and classical mechanics, stochastic differential geometry and statistical and quantum mechanics, and infinite-dimensional differential geometry of groups of diffeomorphisms and hydrodynamics. The unification of these topics under the cover of one book appears, however, quite natural, since the exposition is based on a geometrically invariant form of the Newton equation and its analogs taken as a fundamental law of motion. Our approach, therefore, depends heavily on differential geometry of finite- or infinite-dimensional manifolds. The monograph is addressed to mathematicians familiar with global analysis and requires certain mathematical culture as well as skills. We assume that the reader is familiar with manifolds, vector and principal bundles, tensors, and differential forms. This material can be found, for example, in the first section of Chap. 6 of [141] and in Chaps. 2 to 4 of [120]. Throughout the book, we also use the notions of connection, covariant derivative, and parallel translation as well as some basic results from differential geometry stated in a coordinate-free form. (See, e.g., [17] and [94].) For the sake of convenience, we recall some results from the theory of connections in Appendix A which, to some extent, can be regarded as a brief introduction to the subject. Appendixes B to E are devoted to the theory of set-valued maps, theory of stochastic processes, the Ito group and principal bundle, and Sobolev spaces. Here our goal is, on the one hand, to give a necessary background and, on the other hand, to provide the reader with some additional information relevant to these topics but omitted in the main text. Being limited by the size of a single book, we deliberately do not include the material covered in a variety of other textbooks and monographs. Thus, here we almost ignore Lagrangian and Hamiltonian systems and just briefly touch upon some other important subjects. The leading role in the selection of material was played, of course, by the author's taste and research interest. The book consists of three parts, which correspond to the aforementioned branches of global analysis. In Chap. 1, we prove some general results concerned with analysis on Riemannian manifolds, which are used in the subsequent chapters and, to the best of our knowledge, have not yet been reviewed in monographs or textbooks. Here we include a necessary and sufficient con-

VIII

Preface to the Russian Edition

dition for the completeness of a vector field, the basic construction of the integral operator with Riemannian parallel translation, etc. In Chap. 2, we describe the differential-geometric approach to classical Newtonian mechanics. The Newton equation is introduced by means of the covariant derivative with respect to the Levi-Civita connection of the Riemannian metric, giving rise to the kinetic energy on the configuration space. First, we deal with the classical cases of potential mechanical systems, a gyroscopic force, systems on Lie groups, etc. In addition to this, however, we study geometric mechanics of systems with discontinuous forces or forces with delay, geometric mechanics with linear constraints in the sense of Vershik-Faddeev,

the integral equations of geometric mechanics (in terms of integral operators with Riemannian parallel translation), the velocity hodograph, etc. In a certain sense, the material of Chap. 2 is a starting point for generalizations studied in the subsequent chapters of the book. Integral operators with Riemannian parallel translation are defined in Chap. 3 and then used to study the qualitative behavior of geometric mechanical systems. In particular, under a very general hypothesis, we prove that a pair of points (or a point and a submanifold) in a nonflat configuration space can be connected by a trajectory of the mechanical system. (This is true, for example, for a system with a discontinuous force, a system with a constraint, etc.) Note, however, that in contrast to the standard case of a flat configuration space, on a manifold we may have inaccessible points even when the configuration space is compact and the force field is smooth and bounded. Stochastic differential geometry studied in Chap. 4, which opens the sec-

ond part of the book, is a currently forming branch of global analysis. It is the discovery of a relationship between differential geometry and the theory of stochastic processes that has led to the invention of this geometrically invariant theory having important applications to mathematical physics. Our exposition is based on the Belopolskaya-Dalecky approach to defining Ito stochastic differential equations on manifolds and on a generalization to the stochastic case

of integrals with Riemannian parallel translation introduced by the author. Taken into account that the theory of stochastic processes and differential geometry are customarily regarded as areas quite distant from each other and that this book is mainly addressed to experts in geometry and global analysis, we give a detailed review of the classical theory of stochastic processes in Sect. 12 (Chap. 4). In addition to this, some basic definitions from the theory can

be found in Appendix C. Throughout the book all the constructions based on probability theory are accompanied by a number of references, yet major attention is paid to the geometric interpretation of the constructions. Chapters 5 and 6 of Part II concern applications of stochastic differential geometry. In Chap. 5, we study the Langevin equation, which describes the "physical" Brownian motion on a nonlinear configuration space and then, in Chap. 6, Nelson's stochastic mechanics, known to be equivalent to quantum mechanics. We emphasize that the Langevin equation and the equation of motion of stochastic mechanics are essentially different generalizations of the Newton equation.

Preface to the Russian Edition

IX

Part III is devoted to the weak differential geometry of groups of diffeomorphisms of a compact manifold and the modern geometric Lagrangian formalism of hydrodynamics proposed and developed by Arnold, Ebin, and Marsden. In Sect. 7, we study the properties of Sobolev H8-diffeomorphisms

of an n-dimensional compact manifold with s > n/2 + 1. The Lagrangian formalism arises naturally from the Newton equation on the group of such diffeomorphisms taken as the configuration space. The basic example of this construction is the so-called system of diffuse matter. The system of an ideal barotropic fluid can then be obtained from the diffuse matter system by introducing a particular force field. (See Sect. 24.) Similarly, the system of an ideal incompressible fluid, studied in Chap. 8, is described by means of a constraint on the configuration space. Since the kinetic energy is a weak Riemannian metric (i.e., it gives rise to the H°-topology, which is weaker than H8), many arguments of finite-dimensional differential geometry fail to apply to systems arising in hydrodynamics, and thus new methods are to be developed to deal with them. Note that the motion of an ideal incompressible fluid is given by a C°°-smooth vector field on the tangent space to the manifold of diffeomorphisms. The passage to the Euler equation, also studied here, leads to the loss of derivatives. We prove the local (in time) existence and regularity of solutions for manifolds with or without boundary. Methods of stochastic differential geometry are applied in Chap. 9 to describe the motion of a viscous incompressible fluid. Our model problem is to study the motion on the n-dimensional flat torus. We define a class of processes (analogous, in a certain sense, to geodesics) on the group of volume-preserving diffeomorphisms of the torus whose mathematical expectations are the curves on the group giving rise to the flow of the fluid. This approach is quite different from the ones used before in the Lagrange formalism of hydrodynamics. A more detailed review of the contents is given in the introductions to the sections and chapters of the book. Throughout the book, theorems, definitions, and formulas are designated by two numbers, where the first one refers to the section; subsections within a section are labeled by capital Roman letters. For example, Subsection A of Section 7 is referred to as Sect. 7.A. Remarks play an important role in the book. Sometimes a remark gives

some extra information, or it contains material (left without proof) to be referred to later on. In the book, we use the standard terminology and notation of modern differential geometry. (See, e.g., [99] and [120].) Note, however, the following exception: derivatives of a change of coordinates given in local charts as 01 o 0 are denoted by primes, e.g., (01 o O)' is the first derivative, (01 o o)" the second, etc. Here 01 o 0 is understood as a map between domains of a Euclidean space and, for example, the second derivative is thought of as a bilinear map to the same space. (See, e.g., [99].) Tensors obtained by lifting or lowering indexes are said to be physically equivalent under the metric (see, [118]). The list of references given in this book is by no means complete. The emphasis is put on textbooks or monographs, rather than original papers.

X

Preface to the Russian Edition

In conclusion, the author would like to express his deep gratitude to his former adviser Yuri G. Borisovich for his constant attention, support, and numerous fruitful discussions. I am in great debt to him for drawing my interest to the subject of this book. I am also grateful to Yuri S. Baranov, Boris D. Gel'dman, and Igor V. Fedorenko, my coauthors in a number of papers cited in this book, for their help and interest in our joint work. My special thanks go to Yana I. Belopolskaya, Yuri L. Dalecky, Vladimir Ya. Gershkovich, Aleksandr I. Shnirel'man, and Anatolii M. Vershik for our useful discussions and to my wife Olga for her infinite patience. Voronezh, Russia April, 1989

Yuri Gliklikh

Table of Contents

Preface to the English Edition

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

V

Preface to the Russian Edition ........................... VII Part I. Finite-Dimensional Differential Geometry and Mechanics

Chapter 1 Some Geometric Constructions in Calculus on Manifolds 1.

....

3

Vector Fields ..........................................

3

Complete Riemannian Metrics and the Completeness of A Necessary and Sufficient Condition for the Completeness of a Vector Field A Way to Construct Complete Riemannian Metrics 1.B Riemannian Manifolds Possessing a Uniform Riemannian Atlas 1.A

2. 3.

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

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

Integral Operators with Parallel Translation ............... 3.A 3.B 3.C

The Operator S ................................. The Operator F ................................. Integral Operators ...............................

Chapter 2 Geometric Formalism of Newtonian Mechanics 4.

6. 7.

7 10 10 12

14

17

Geometric Mechanics: Introduction and

Review of Standard Examples ...........................

................................... ............... Mechanical Systems on Groups .................... Geometric Mechanics with Linear Constraints ............. 5.A Linear Mechanical Constraints .................... 5.B Reduced Connections ............................ 5.C Length Minimizing and Least-Constrained Nonholonomic Geodesics ......................... 4.A 4.B 4.C

5.

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

3 5

Basic Notions Some Special Classes of Force Fields

17 17 19 20 22

22 23 24

Mechanical Systems with Discontinuous Forces and Systems with Control: Differential Inclusions Integral Equations of Geometric Mechanics:

26

The Velocity Hodograph

28

7.A

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

General Constructions

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

29

Table of Contents

XII

7.B

Integral Formalism of Geometric Mechanics with Constraints

8.

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

Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force

31

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

32

Accessible Points of Mechanical Systems .....................

39

Chapter 3 9. 10. 11.

Examples of Points that Cannot Be Connected by a Trajectory

.......................................... Generalizations to Systems with Constraints .............. The Main Result on Accessible Points ....................

40 41 45

Part II. Stochastic Differential Geometry and its Applications to Physics

Chapter 4 Stochastic Differential Equations

on Riemannian Manifolds .................................... 12.

Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces 12.A 12.B 12.C 12.D 12.E

12.F

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

Wiener Processes ................................ TheIto Integral

The Backward Integral and the Stratonovich Integral The Ito and Stratonovich Stochastic Differential

15.

54 56

Approximation by Solutions of Ordinary Differential Equations

57

...................................... Stochastic Differential Equations on Manifolds ............

58

59

Stochastic Parallel Translation and the Integral Formalism

for thelto Equations ...................................

67

Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations

76

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

15.A 15.B 15.C 16.

49 49 50 53

Equations ...................................... Solutions of SDEs ...............................

12.G A Relationship Between SDEs and PDEs ........... 13. 14.

49

Wiener Processes on Riemannian Manifolds .........

Stochastic Equations .............................

Equations with Identity as the Diffusion Coefficient .. Stochastic Differential Equations with Constraints

.........

76 78

80 83

Chapter 5

The Langevin Equation ...................................... 17. 18.

The Langevin Equation of Geometric Mechanics ...........

87 87

Strong Solutions of the Langevin Equation, Ornstein-Uhlenbeck Processes ...........................

91

Table of Contents

XIII

Chapter 6 Mean Derivatives, Nelson's Stochastic Mechanics, and

Quantization ................................................. 19.

More on Stochastic Equations and Stochastic Mechanics in IR" .............................

96

19.A 19.B

Preliminaries .................................... Forward Mean Derivatives ........................

96

19.C

Backward Mean Derivatives and Backward Equations

19.D 19.E

Symmetric and Antisymmetric Derivatives ..........

19.F 20.

95

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

97

98 101

The Derivatives of a Vector Field Along 6(t) and

the Acceleration of 6(t) ........................... 106 Stochastic Mechanics ............................. 107

Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds ............................... 109

20.A Mean Derivatives on Manifolds and

Related Equations ............................... 109

20.B 20.C 21.

Geometric Stochastic Mechanics ................... 114 The Existence of Solutions in Stochastic Mechanics .. 115

Relativistic Stochastic Mechanics

........................ 125

Part III. Infinite-Dimensional Differential Geometry and Hydrodynamics Chapter 7

Geometry of Manifolds of Diffeomorphisms .................. 133 22.

Manifolds of Mappings and Groups of Diffeomorphisms ..... 133 22.A 22.B 22.C 22.D

23.

The Group of H3-Diffeomorphisms ................. 134 Diffeomorphisms of a Manifold with Boundary Some Smooth Operators and

......

136

Vector Bundles over Ds(M) ....................... 137

Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms 23.A The Case of a Closed Manifold

........................ 139 .................... 139

23.B 23.C 24.

Manifolds of Mappings ........................... 133

The Case of a Manifold with Boundary .............

141

The Strong Riemannian Metric .................... 141

Lagrangian Formalism of Hydrodynamics

of an Ideal Barotropic Fluid ............................. 142 24.A Diffuse Matter .................................. 142 24.B A Barotropic Fluid .............................. 143

Table of Contents

XIV

Chapter 8 Lagrangian Formalism of Hydrodynamics of an Ideal

Incompressible Fluid ......................................... 147 25.

Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid 25.A Volume-Preserving Diffeomorphisms

.

of a Closed Manifold .............................

25.B

147 148

Volume-Preserving Diffeomorphisms

of a Manifold with Boundary ...................... 151

LHS's of an Ideal Incompressible Fluid ............. 152 The Flow of an Ideal Incompressible Fluid on a Manifold with Boundary as an LHS with an Infinite-Dimensional Constraint on the Group of Diffeomorphisms of a Closed Manifold ..... 156 27. The Regularity Theorem and a Review of Results 25.C

26.

on the Existence of Solutions ............................ 164

Chapter 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms ............................... 28. 29.

171 .............. 172

Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus

A Viscous Incompressible Fluid .......................... 175

Appendices .................................................. 179

A.

Introduction to the Theory of Connections ................ 179 Connections on Principal Bundles

................. 179

Connections on the Tangent Bundle ................ Covariant Derivatives

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

180 181 183 185

Connection Coefficients and Christoffel Symbols ..... Second-Order Differential Equations and the Spray .. The Exponential Map and Normal Charts .......... 186

B. C.

Introduction to the Theory of Set-Valued Maps ............ 186 Basic Definitions of Probability Theory and

the Theory of Stochastic Processes ....................... 188 Stochastic Processes and Cylinder Sets

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

188

The Conditional Expectation ...................... 188

Markovian Processes ............................. 189 Martingales and Semimartingales .................. 190 D. E. F.

The Ito Group and the Principal Ito Bundle

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

190

Sobolev Spaces ........................................ 191

Accessible Points and Closed Trajectories of

Mechanical Systems (by Viktor L. Ginzburg) .............. 192 Growth of the Force Field and Accessible Points ..... 193

Table of Contents

Accessible Points in Systems with Constraints

.......

Closed Trajectories of Mechanical Systems ..........

XV

197 198

References ............................................... 203 Index .................................................... 211

Part I Finite-Dimensional Differential Geometry and Mechanics

This part consists of three chapters, the last two of which are devoted to studying classical mechanical systems with finitely many degrees of freedom. Here we use freely the notions of modern differential geometry, with which the reader is assumed to be familiar. (See the Preface.) A brief review of this material can be found in Appendix A. In Chap. 1 we deal with some applications, attributed to the author, of finite-dimensional differential geometry

to the analysis on manifolds. These results will be used in the subsequent chapters of the book, in particular, in Chaps. 2 and 3.

Chapter 1. Some Geometric Constructions in Calculus on Manifolds

1. Complete Riemannian Metrics and the Completeness of Vector Fields In this section, we shall study conditions that guarantee that all integral curves of a vector field on a finite-dimensional manifold exist on the interval (-oo, oo). A vector field with this property is called complete. We shall give a criterion for completeness that uses special Riemannian metrics on the phase space.

1.A. A Necessary and Sufficient Condition for the Completeness of a Vector Field Many criteria for the extendability to (-oo, oo) of the solutions of differential equations in vector spaces are known (see, e.g., the Bibliography in [61]). It is easy to see that, under the hypotheses of some of these theorems, one can define a new Riemannian metric on the phase space in such a way that the right-hand side of the equation is bounded by a constant with respect to this metric. Thus, in these cases, the extendability of solutions (the completeness of a vector field) follows from the fact that a solution has bounded length on every finite interval with respect to a complete Riemannian metric and, therefore, is relatively compact. It turns out that the requirement that the vector field should be bounded with respect to a complete Riemannian metric can be modified in such a way that it becomes necessary and sufficient. Let M be a finite-dimensional smooth manifold and X (t, m) a vector field which is smooth jointly in t and m. Denote the direct product M x IR by M+. Obviously, T(,,,,,t)M+ = T,,,M x R. Define a vector field X+ on M+ setting X+ t) = (X(m, t), 1).

Theorem 1.1 ([61]). A field X on M is complete if and only if there exists a complete Riemannian metric on M+ such that X+ is uniformly bounded with respect to it. Proof. Obviously, the completeness of X is equivalent to the completeness of the vector field X+.

4

Chapter 1. Some Geometric Constructions in Calculus on Manifolds

Assume that there exists a complete Riemannian metric on M+, with respect to which the field X+ is bounded. Then every integral curve of X+ has finite length on every finite interval. Since the metric is complete, the last assertion implies the relative compactness of the integral curve on every finite interval. This yields the completeness of the field. Let us prove the "only if" assertion. Let X be complete, then so is X+. Since the field X is smooth by the hypotheses of the theorem, the field X+ is smooth as well. Consider an arbitrary smooth proper real-valued function g on the manifold M. We will call a map proper if the preimage of a compact set is compact, i.e., in our case, the preimage of a compact set in 1R under the map g is compact in M. The function g, satisfying the aforesaid conditions, can be constructed, for example, as follows (see [77]). Choose a countable covering of M by relatively compact open sets; this can be done by virtue of the paracompactness and the local compactness of M. Let us label the elements of the covering with integral numbers, and define a smooth constant function on

each element of the covering to be equal to the number of the element. With the help of a partition of unity, glue these functions to obtain a function g on M, which satisfies the above conditions. Pick an inner product depending smoothly on (m, t) on each tangent space T(m t) (M x {t}) to the submanifold M x {t} of the manifold M+. For example, one can take a Riemannian metric on M and extend it in a natural way. Now we can construct a Riemannian metric (, )1 on M+ by regarding the vectors of the field X+ as being of unit length and orthogonal to the subspaces T(m.,t)(M X {t}). Denote by Ot the diffeomorphism of the manifold M x {0} to the manifold M x {t} along the trajectories of the field X+. The function g can be regarded as given on M x {0}. Since the integral curves of the field X+ are globally extendable, the function f : M+ - 1R, given by the formula f (m, t) = g(,Pt 1(m, t)) +t

is, obviously, smooth and proper. Clearly, X+ f = 1, where X+ f is the derivative of the function f in the direction of the field X+. Let us now choose an arbitrary smooth function 0: M+ -+ ]R such that 5(m, t) > max exp(Y f )2

,

where Y E T(,,,.,,t)(M x {t}) and IIYII1 = 1. Such a function can be defined as follows. For a relatively compact neighborhood of every point (m,', t') E M+, there exists a constant greater than sup max exp(Y f )2, where, as above, Y E T(,.,,,t)(M x {t}) and IIYII1 = 1, and the supremum is taken over all points (m, t) from the neighborhood. Then, using the paracompactness of M+ and,

as a consequence, the existence of a smooth partition of unity, we glue the function 0 defined on the whole of M+. At every point (m, t) E M+, define the inner product on T(,,,,t)M+ by the formula (Y, Z)2 = q52(m, t)

(PmY,PmZ),+PxY PYX

,

1. Complete Riemannian Metrics and the Completeness of Vector Fields

5

where Y, Z E T(,n,t)M+ and p,R, px are orthogonal (in the metric (,)J projections of T(,,,,,t)M+ onto T(, ,t) (M x {t}) and X+, respectively. Obviously, IX+II2=1.

Lemma 1.1. The Riemannian metric (1)2 is complete on M+. Proof of the Lemma. By the Hopf-Rinow theorem (see [17] and [94]), it is enough to prove that every geodesic is extendable to the whole real axis.

Let SM+ be the unit sphere bundle over M+ in the metric (1)2, and it the natural projection SM+ on M+. Since the total space of the bundle SM+ over a compact subset of M+ is compact, the function f = f o it is proper. Also, it is obvious that f is smooth on SM+. Let c(s) be a trajectory of the geodesic flow (for the definition see, e.g., [5]) of the metric M2 on SM+. It is easy to see that df (c(s)) = c(s)f = (pmc(s))f + (pxc(s))f ds

We have Ilpmc(s) II2 < 1 and IIpxc(s) II2 < 1, because c(s) is a unit tangent vector in T,,c181M+. Therefore,

d f (c(s))

pmt(s) IIpmc(s) II2 1

li(ir(c(s)))

f+

pxc(s) Ilpxc(s)

112

pmt(s) f+ IIPmt(s)IIi

I X+ f I< 2

by the definition of 0 and f. Thus, the function f (c(s)) is bounded on any finite interval (a, b), and the set of points c(s) for s E (a, b) is relatively compact, because f is proper. This proves the unlimited extendability of the geodesic. Lemma 1.1 is proved. As mentioned, IIX+II2 = 1, which completes the proof of Theorem 1.1.

Remark 1.1. If the field X (t, m) is Ck-smooth on M+, then the above construction gives a Ck-smooth complete Riemannian metric on M+, with respect to which X+ is bounded.

I.B. A Way to Construct Complete Riemannian Metrics As already mentioned in Sect. 1.A, sufficient conditions for the completeness of a vector field are known, and when they hold, it is easy to find a Riemannian metric such that the vector field is bounded. In this subsection, we construct a complete Riemannian metric using a Wintner-type hypothesis. Further, this

metric will be used to study complicated differential equations (stochastic, with delay, etc.).

Chapter 1. Some Geometric Constructions in Calculus on Manifolds

6

Let us recall the classical Wintner theorem [85]. Consider the following differential equation on the Euclidean space IR':

i = f (t, x(t))

,

(1.1)

where f (x, t) is continuous in (t, x).

Theorem 1.2 (Wintner). Suppose that Il f(x,t)II <- c(t) . L(IIxII)

where the function 0(t) is positive and integrable on any finite interval [0, l], and L: [ 0, oo) -+ (0, oo) is continuous and satisfies the condition 1000 du

L(u) =

oo

(1.2)

Then all solutions of (1.1) are defined on (-oo, oo). The Wintner theorem will be derived from the following result.

Theorem 1.3. Let M be a complete Riemannian manifold with Riemannian metric (,) and L: [ 0, oo) -+ (0, oo) a smooth function satisfying (1.2). Choose a point mo E M and define a Riemannian metric (,) * by

)) (,)m

(1.3)

at a point m E M, where p is the Riemannian distance on M in the metric (,). If (,) is complete, then so is (,) `. Proof.

Lemma 1.2. The minimal geodesics of (,) and (,) * starting at mo coincide up to a parametrization. The lemma can be proved with a straightforward calculation. Namely, it can be shown that the minimal geodesics (beginning at mo) of the first metric satisfy the geodesic equation of the second metric and the cut loci of both metrics coincide. Let us continue the proof of the theorem. It is enough to prove that the

metric ball UT centered at mo of any fixed radius T in the metric (,)- is compact. (By the Hopf-Rinow theorem, this implies completeness.) Suppose the contrary, i.e., for some l the ball U1 is not compact. Since (,) is a complete metric, it follows by Lemma 1.2 that U contains a minimal geodesic u(s) of infinite length in the metric (,) with beginning at mo. Here s is the natural parameter (the length) in the metric (, ). Denote the reparametrization of u(s) with the length in (, )` by u(t). By Lemma 1.2, u(t) is a minimal geodesic of

the metric (,)*.

2. Riemannian Manifolds Possessing a Uniform Riemannian Atlas

7

The definition of the metric (,)* yields ds - L(u(s)) = L(s) dt

and since u(s) lies in llj,

fooLs<`dt=1

.

This contradicts condition (1.2).

0

Corollary. Let M be a complete Riemannian manifold, X (t, m) a vector field continuous in (t, m), and L: [0, oo) -+ (0, oo) a continuous function satisfying (1.2). Suppose there exists a point mo E M such that at every point m E M the inequality IIX (t, m) II < 0(t) L(p(mo, m)) holds, where p is the Riemannian distance on M and 0 is a positive function integrable on every finite interval. Then the field X (t, m) is complete.

Proof. It is obvious that for any constant C > 0, the function L + C satisfies (1.2). There exists a smooth function T/(u) such that L < W < L + C for u E [ 0, oo). Obviously, li satisfies (1.2). Let (,) be a metric on M. Consider (,);11 = W-2 (p(m9, m)) (,),n. By Theorem 1.3, (,)* is complete. Now it is sufficient to observe that the length of every integral curve of the field X(t, m) 0 in the metric is bounded above on any interval [a, b) by fa fi(t) dt. The Wintner theorem is a particular case of the corollary, where M is the Euclidean space IR'.

2. Riemannian Manifolds Possessing a Uniform Riemannian Atlas In this section, we study a special class of complete Riemannian metrics, and prove that such metrics exist on any manifold. These metrics will be used

in Part II to prove the extendability to (-oo, oo) of solutions of stochastic differential equations, in the same way as ordinary Riemannian metrics were used in Sect. 1 in order to prove the extendability of solutions of ordinary differential equations. Let M be a connected finite-dimensional Riemannian manifold with Rie-

mannian metric (,) and let p be the Riemannian distance function on M.

Definition 2.1. An atlas on M is said to be uniform Riemannian if for any point m E M, there exists a chart (U, 0), m E U, of this atlas such that U contains the metric ball Vm (r) centered at m of fixed radius r > 0 independent of m and U, where Vm(r) is taken with respect to the Riemannian distance p.

8

Chapter 1. Some Geometric Constructions in Calculus on Manifolds

Note that, in general, the metric ball Vm(r) = {m' E M I p(m,m') < r} may not be homeomorphic to a ball in the model space and may have a complicated topological structure. Obviously, a Riemannian metric possessing a uniform Riemannian atlas is complete.

Theorem 2.1 ([67]). For any Riemannian metric on M, there exists a Riemannian metric conformal to it that possesses a uniform Riemannian atlas. To prove Theorem 2.1, we refine the methods developed in [82] and [114] for the investigation of convex neighborhoods and complete Riemannian metrics. Note that Theorem 2.1 is a generalization of [114], where it was proved that any Riemannian metric is conformal to a complete Riemannian metric. Pick a Riemannian metric (,) on M, i.e., let (,) m be an inner product in the tangent space T,,,M, and let d be the Riemannian distance on M corre-

sponding to M. It is known (see [94]) that for any point m E M, there exists a number a(m) > 0 such that the d-metric ball V,,,.(a(m)) lies in a normal coordinate neighborhood (chart) of any point m' E V,,,(a(m)). Let r(m) be the least upper bound of such a(m). If r(m*) = oo for a point m* E M, the proof is clear. Assume that r(m) <

oofor all mEM. Lemma 2.1. For any two points m, m' E M, the following inequality holds: Il r(m) - r(m') II < d(m, m,')

.

(2.1)

Proof. First, consider the case where m' E V,,,(r(m)). Then V,n, (r(m) - d(m, m')) C V,,, (r (m))

and, by the definition of r(m'), we have

r(m') > r(m) - d(m, rn') If r(m) > r(m'), then (2.1) follows. On the other hand, m E Vm(r(m')) if r(rn') > r(m), and, therefore, r(m) > r(m) - d(m, m') which proves (2.1). The case where m E V,,,' (r(m')) can be dealt with in the same manner. In the remaining case, the inequalities r(m) < d(m, m') and r(m') < d(m,rn') follow from m' 0 Vm(r(m)) and, at the same time, (r(m')) . Therefore, m0 I r(m) - r(m')I < d(m, m) which completes the proof of the lemma.

0

2. Riemannian Manifolds Possessing a Uniform Riemannian Atlas

9

Without loss of generality, we may assume that the function r(m) is smooth. For if r(m) is not smooth, it can be approximated by a smooth function r*(m) such that 0 < r*(m) < r(m) which satisfies (2.1). Let us introduce a new metric (,) ` on M by the formula

(+)m- r2(m)

(e)m

Denote by p the Riemannian distance on M corresponding to (X-

Lemma 2.2. Suppose that d(m, m') > r(m); then p(m, m') > 1/2. Proof. Let y(t) be an arbitrary piecewise smooth curve such that y(a) = m and y(b) = m'. Denote its length in the metric (,) by L, i.e., L = fa 1I'(t)II dt. The length of y in the metric (, )'" can be found by the formula b

L* = Ja r(y(t)) Ily(t)II dt Using the classical mean value theorem, we obtain L

V = r(y(r)) 1 a b11ry(t)JIdt= r(y(,r)) where r E [a, b]. Then

L` _

L

r(y(r)) - r(m) + r(m)

and, by Lemma 2.1,

L

L` >

r(m) + d(m, y(r)) By assumption, L > r(m). Moreover, d(m, y(r)) is not greater than the length of y on the interval [ a, r), which, in turn, is not greater than L, i.e., L > d(m, -y (r)). Thus L

L > L+L

_

1

2

Since (2.2) holds for an arbitrary y, p(m, m') > 1/2 and the lemma is proved.

Proof of Theorem 2.1. By construction, the metric (, )` is conformal to the

original metric (,). By definition, a normal chart of the metric (,) at m contains the metric ball Vm(r(m)) with respect to the distance d. It follows from Lemma 2.2 that d(m, m') < r(m) when p(m, m') < 1/2. Thus, at any point m E M, the normal chart of the metric (,) contains the ball centered at m of radius 1/2 with respect to the metric p. Therefore, (, )" is the desired metric. The theorem is proved.

Chapter 1. Some Geometric Constructions in Calculus on Manifolds

10

3. Integral Operators with Parallel Translation Ordinary differential equations on a vector space can be turned into equivalent

Volterra-type integral equations. In fact, this method is very often used for investigating ordinary differential equations. For example, one can turn the Cauchy problem

x = .f (t, x(t)) in IR' into the integral equation

,

x(0) = xo

IIt

f (r, x(r)) dr J To do so, we use the fact that for any continuous curve y(t) there exists a unique curve z(t) = fo f (r, y(r)) dr such that the derivative of z(t) at any point t is equal to f (t, y(t)). It is clear that the existence of the curve z(t) is x(t) = xo +

possible only because of global parallelism on the tangent bundle to the vector space. Indeed, the vectors z(t) and f (t, y(t)) belong to the tangent spaces at different points, and the equation z(t) = f (t, y(t)) makes sense only by virtue of the existence of global parallelism, i.e., a canonical isomorphism between all tangent spaces and the vector space itself.

Global parallelism does not exist on an arbitrary manifold. Therefore, classical integrals can be used only locally (in charts); the integrals themselves depend on the choice of the coordinate system. In this section, following [59], [60], and [66], we describe a construction of an analog of the integral operator, in which global parallelism is replaced with parallel translation (with respect

to a connection) along a chosen curve. For the sake of simplicity, we shall consider the case of the Levi-Civita connection of a complete Riemannian metric on a finite-dimensional manifold. Similar notions of absolute and covariant integrals were introduced in a different way in papers by Vujicic. (See, e.g., [138], [139], and [140] and the Bibliography in [140].) Namely, the integral is defined in a local coordinate system with the connection coefficients used in such a way that the integral becomes covariant with respect to changes of coordinates.

3.A. The Operator S Let M be a complete Riemannian manifold, let mo E M, I = [0, l], and let v: I --+ Tm0M be a continuous curve.

Theorem 3.1. There exists a unique C1 -curve y: I -+ M such that y(0) = m0 and the tangent vector %y(t) is parallel to the vector v(t) E Tm0M for every

t E I. Proof. Let bo = (e?, ... , e°) be a basis in the tangent space T,,,o M; bo gives rise

to an isomorphism between IRn and Tm0M by the formula bo(xl, + xne°, where IR" is the model space of M. xleo +

,

x") _

3. Integral Operators with Parallel Translation

11

Consider the time-dependent basic vector field E(bo lv(t)) on the frame bundle B(M). (See Appendix A.) Obviously, this field is locally Lipschitz in b E B(M). Hence, for every point b E B(M), there exists a unique integral curve bo(t) passing through b, b°(0) = b. The curve y(t) = -7rb°(t) is the one we are looking for. (Here 7r is the natural projection of B(M) to M.) Indeed, for any point t* from the domain of y(.), the vectors y(t*) and v(t*) are connected along y(.) by the parallel vector field bo(t)(bolv(t*)). It remains to prove that y(.) is defined on the whole interval [0, l]. Since the metric on M is complete, the metric ball of radius fo JJv(s)JJ ds centered at mo is compact. Now assume that y(t) is defined on [ 0, t*), where t* E [0, l]. The length of y(.) on ( 0,t*) equals ot' IIv(s)Ilds<

f llv(s)Ilds

,

i.e., y(t) belongs to a compact set and, therefore, can be extended to [0, t*]. It is clear that one can extend y(.) to a neighborhood of t*. Thus, the domain of y(.) is open and closed in [0, l], i.e., it coincides with [0, l]. The theorem is proved.

In what follows, we denote by beginning with v.

the curve y constructed as above

Consider the Banach space C°(I, I' 0M) of continuous maps from I to T,,,0M and the Banach manifold C'(I, M) of C'-smooth maps from I to M. As follows from Theorem 3.1, the operator S: C'(I, T,,, M) --+ C' (I, M) is well defined. If M is the Euclidean space, Sv is a primitive of v. It is easy to see that S is a homeomorphism between C°(I, T,,,o M) and its image C,l,,,0 (I, M) in C' (I, M), where the manifold C;,. (I, M) consists of all Cl-curves y with y(0) = m°. We shall need the following property of S.

Theorem 3.2. Let UK be the ball of the radius K centered at the origin of C°(I,Tm,,M). Then, at every point t E I, the inequality IIy(t)II < K holds for all curves y(.) from the set SUK. This assertion is obvious because parallel translation preserves the norm of a vector.

Theorem 3.3. Assume that the point ml E M is not conjugate to mo along some geodesic of the Levi-Civitd connection on M. Then for any geodesic a(.),

a(0) = m°, a(1) = ml, along which mo and ml are not conjugate and for any number K > 0, there exists a constant L(moi m,, K, a) > 0 with the following property: for any t,, 0 < t1 < L(moi m1, K, a), and for any curve E UK C C° ([0, t1], Tm0M), there exists a unique vector Cu E ToM, such that S(u + Cu)(t1) = ml, which belongs to a bounded neighborhood of ti 1 a(0) E Tn0M and depends continuously on u.

12

Chapter 1. Some Geometric Constructions in Calculus on Manifolds

Proof. Consider the mapping

C°([0,1],Tm0M) XTm0M-+M

sending a pair (u,C), where u E C°([0,1],Tm0M) and C E TmoM, to the point S(u+C)(1). Note that the vector field E(bo 1(v(t)) on B(M) is smooth in v(.). (See the proof of Theorem 3.1.) Using this fact, the definition of S, and the classical theorem on the smooth dependence of solutions of differential equations on parameters, one can easily show that this map is smooth jointly

in u and C. Clearly, we have S(C)(1) = exp(C) when u = 0. Thus, by the hypotheses of the theorem, S(c (0)) (1) = m1 and S(C) (1) is a diffeomorphism from a neighborhood of a(0) in TmoM onto a neighborhood of m1 in M. Let us now think of S(u + C)(1) as a perturbation of S(C) (1) = exp(C). Thus, there exists e > 0 such that for any fixed u E UE C Co ([0' 1], TmoM) the operator S(ft + C)(1) is a local diffeomorphism. Therefore, there is a ball

D C Tm0M centered at a(0) such that for any fi E U, there exists a vector Cv, E D solving the equation S(u + C;)(1) = ml. Using the implicit function theorem, one may show that, when D is sufficiently small, the vector C, E D is unique and Cu depends continuously on ft. Let t1 be such that ti 1e > K. For u E UK C C° ([0, t1 ],TmoM) , we define C,, E TmoM by the formula Cu = t1 1Cu, where fi E UE C C° ([0,1], TmoM) is such that u(t) = ti 1

It is easy to see that

S(fi+C°)(t) = S(u+CC)(t t1)

,

i.e., S(u+Cu)(t1) = mi. Now we can take L(mo, m1, K, a) to be the supremum of all such t1.

Note that if M is the Euclidean space, one can take any constant as e in the proof of Theorem 3.3., i.e., the theorem holds for every t1, 0 < t1 < no.

3.B. The Operator I' Let y(t), t E I, be a C1-curve in M and X (y(t)) a continuous vector field along Consider the curve TX (y(t)) in T.. (°)M such that the vector PX (y(t)) is parallel to X (y(t)) E T.y(t)M along

Lemma 3.1 (Compactness Lemma). Let - C C'(1, TM) be such that ir- C C1(I, M), where 7r: TM -+ M is the natural projection. If E is relatively compact in Co (I, TM), then so is FE. Proof. Since the closure 5 of the set 5 is compact in C°(I,TM), the vectors of {e(t) l;(.) E 5"1 are bounded and the set 7r8 is compact in C°(I, M). This implies, in particular, that all curves from ir5 lie in a compact subset of M and the norm of velocity vectors of these C'-curves is bounded by some constant K, since the curves are equicontinuous. I

3. Integral Operators with Parallel Translation

Let

be a limit curve of the set

13

It is clear that the inequality

p (ire* (t),,7rl;* (t')) < Kit - t'l

,

where t,t' E I and p is the Riemannian distance, holds for the curve 7rl;*(). Note that this curve may not be smooth. Parallel translation along such curves was defined in [20] as the limit of parallel translations along their piecewise geodesic approximations. Moreover, it was shown that under the hypothesis above, the procedure of parallel translation converges uniformly on any bounded set of vectors. If the curve is smooth, the new definition of parallel translation is equivalent to the classical one. It was also shown that the parallel translation along a limit curve is just the limit of parallel translations along curves converging to it. Therefore, F sends convergent sequences to convergent ones.

If X (y(t)) = X (t, ry(t)) is the restriction to of a continuous vector field X (t, m), t E I and M E M, then we use the notation T o -y for TX (t, -y (t)). Thus, for a fixed vector field X (t, m), we may consider the oper-

ator r: 61 (1, m) -s Co(I,TM), which is clearly continuous. Let 12K be the set of curves from C' (I, M) satisfying the inequality I1y(t)lI < K, where K > 0 is a real number, at every point t E I and such that E QK} is bounded in M. the set {ry(0) I

Theorem 3.4. The set of curves T(,fQK) is compact in C°(I,TM).

Proof. Because M is complete, it is clear that QK is compact in C°(I,M). Since the field X (t, m) is continuous, the set of curves {X (t, y(t))

I

E .QK }

is compact in C°(I,TM). The theorem follows from Lemma 3.1.

Corollary. The operator F is locally compact. Proof. For every -y E C' (I, M), the continuous function 1V1(t) II assumes its supremum K.y on I. By the definition of the Cl-topology, the inequality 01 11 < K.. + e holds for every -yl(.) from a small neighborhood of y(.).

Remark 3.1. Let us now discuss how the operators F and S are related to the classical Cartan development (see, e.g., [17]). Let m(t), t E I, be a Cl-curve in M. The development 6(m(t)) of m(t) is a curve in Tm(o)M, which can be constructed as follows. First, we take parallel translations of the vectors m(t) to m° along m(.), thus obtaining a curve in Tm(O)M. Then 6(m(t)) is set to be the antiderivative of this curve. In other words,

(m(t)) =

f

T(Th(t)) drr

.

The operator 6: C m10 (I, M) -+ C' (I, TmOM) is invertible: 6-1(u(t)) = Sit(t)

for u E C' (I, Tm0 M). For what follows it is important to emphasize that

14

Chapter 1. Some Geometric Constructions in Calculus on Manifolds

6 1(u(t)) (as well as S(v(t)) for a continuous curve v E C°(I,Tm0M)) is independent of the basis b° used in the proof of Theorem 3.1. To see this, let and replace the basis b° by b1 in Tm0M. us go back to the construction of (See the proof of Theorem 3.1.) Since the connection is invariant, it follows from the definition of a basic vector field that irbl (t) = y(t).

3.C. Integral Operators

Consider the continuous composition operator

Sof:Clmo(I,M) - Cmo(I,M) Theorem 3.5. The fixed points of s o r are precisely the integral curves of the field l;'(t, m) with the initial condition y(0) = mo. P r o o f . Let y(t) be an integral curve of the field X (t, m), i.e., y(t) = X (t, y(t)).

It is easy to see that the operator r on

is equal to S-1, and so y is a

fixed point of So F. Conversely, let y(.) be a fixed point of the operator so T. Using the parallel translation along we transport the vector X (t, y(t)) to y(O) = mo and then back to y(t). The resulting vector coincides with 'y(t) by the definitions of S and T. Therefore, y(t) = X (t, -y (t)). 0

Thus, s o r is a direct analog of the standard Urysohn-Volterra integral operator from the theory of ordinary differential equations on vector spaces.

Theorem 3.6. The operator S o r is locally compact. The assertion of Theorem 3.6 follows from the local compactness of F and the continuity of S. Let 0 be a bounded set in M and the subset in C,,,,0 (I, M)

formed by curves lying in the closure of 0. Consider the second iteration (S o r)' of the operator s o T.

Theorem 3.7. The set (S o -)2C0 (I, 0) is compact in

(I, M).

Proof. Since 0 is bounded and M is complete, 0 is compact. Therefore, II X (t, m) II is bounded on I x 0 by some constant K. Since parallel trans(I, 6) lie in Um c q UK (m) and, by lation preserves the norm, the curves Theorem 3.2, the set s o rCm' o (I, 0) is formed by curves which satisfy the inequality IIy(t) II < K at every point t E I. Now the theorem follows from Theorem 3.4 and the continuity of the operator S.

Composition operators, like S o F, are employed to solve certain problems

in the theory of differential equations on manifolds. (For example, to find periodic solutions for some special classes of differential equations, etc.) The

3. Integral Operators with Parallel Translation

15

constructions of such operators and their applications are described, e.g., in the survey [22]. The theory of topological characteristics is also developed in the survey for a large class of maps of infinite-dimensional manifolds. This theory enables one to prove the existence of fixed points of these operators. Let us discuss one more class of integral operators, which can be used to reduce certain problems on manifolds to problems on vector spaces. Consider the composition r o S. This operator is continuous and acts on the Banach space Co(I,T76OM). If v = T o Sv, then Sv = S o r o Sv = (S o r)Sv is an integral curve of the field X (t, m). Conversely, Sv = S o r(Sv) implies that v = r o Sv because S is one-to-one. Theorem 3.8 The operator .V o S is completely continuous. Proof. Let UK be a ball of radius K in Co(I,TmOM). By Theorem 3.2, SUK C .f2K and, by Theorem 3.4, the set r o SUK is compact.

Remark 3.2. As mentioned in the introduction to this section, we considered the Levi-Civita connection of a complete Riemannian metric only to simplify the presentation of the material. Under certain hypotheses, the constructions of the integral operators may be generalized to other connections. Note, for example, that we never used the fact that the connection has zero torsion, i.e., all of our constructions hold for any Riemannian connection of a complete Riemannian metric. In particular, the construction leads to the classical multiplicative integral for a special choice of the connection on a Lie group. (See, e.g., [52] for matrix groups.)

Chapter 2. Geometric Formalism of Newtonian Mechanics

4. Geometric Mechanics: Introduction and Review of Standard Examples 4.A. Basic Notions The description and analysis of classical mechanical systems by means of differential and, first of all, Riemannian geometry is called geometric mechanics. Here we define a mechanical system to be the collection of the following data:

a configuration space, i.e., a smooth manifold M; a kinetic energy, i.e., a smooth function 1C on the tangent bundle TM; and a force field, i.e., a horizontal 1-form a on TM, which in general is timedependent.

The tangent space TM is called the coordinate-velocity phase space and the cotangent space T*M is called the coordinate-momentum phase space [77]. In what follows, we consider only mechanical systems with quadratic kinetic energy, i.e., )C(X) = (X, X)/2, where X E TX and (,) is a fixed Riemannian metric on M.

Recall that a 1-form on TM is said to be horizontal if it vanishes on vertical vectors. The horizontal 1-form a(t, (m, X)) E T('m X)TM, that is, the force field, gives rise to a 1-form &(t, m, X) E T,nM depending on X E TmM by the formula

a(t,m,X)(Y) = a(t,(m,X))(T1-'YIT(m,x)TM)

.

(4.1)

Since a(t, (m, X)) is horizontal, the form &(t, m, X) is well defined. Formula (4.1) gives a one-to-one correspondence between horizontal 1-forms a(t,(m,X)) on TM and 1-forms &(t,m,X) on M. The latter will also be called a force field.

Let a(t, m, X) be the vector field on M (depending on X E TmM) corresponding to the 1-form &(t, m, X) via the Riemannian metric (, ), which gives

rise to the kinetic energy of the system. In other words, (&(t, m, X), Y) _ &(t, m, X)(Y) for any Y E T,,,,M. We call a(t, m, X) the vector force field.

Chapter 2. Geometric Formalism of Newtonian Mechanics

18

Remark 4.1. Everywhere in what follows, with the exception of Part III, we consider only mechanical systems with finite-dimensional configuration space. The force fields are usually introduced as vector fields and the passage to 1-forms is left to the reader as a simple exercise. The motion of a mechanical system is governed by Newton's equation:

rh(t) = d (t, m(t), m(t))

,

(4.2)

where D/dt is the covariant derivative of the Levi-Civita connection of the metric (,). The definition of D/dt is given in Appendix A. It is also shown therein that a curve m(t) is a solution of (4.2) if and only if its derivative rn(t), as a curve in TM, is an integral curve of the vector field

Z+d(t,m,X)1

,

(4.3)

where Z is the spray of the Levi-Civita connection of (,) and d (t, m, X )l is the vertical lift of d(t, m, X) to the space C T(m,x)TM. A curve m(t) on M is called a trajectory of the mechanical system if it is a solution of (4.2). For any initial conditions m(O) = mo and m(0) = Xo E T,,,,0M, there exists a trajectory m(t) on a sufficiently small interval of time provided that, for example, a(t, m, X) satisfies the Caratheodory condition [21], [49]. To see this, observe that, since Z is smooth, the field (4.3) on TM satisfies the Caratheodory condition and the local existence of a solution follows from the classic existence theorem for ordinary differential equations. The existence of trajectories for more general force fields (e.g., discontinuous) is studied in Sect. 6. This question is related to the passage from (4.2) to differential inclusions. Assume, for example, that d(t, m, X) is locally Lipschitz. Then the trajectory is unique for any given initial conditions. The existence of trajectories on (-oo, oo) may be analyzed using the methods developed in Sect. 1. Let us point out that if the Riemannian metric which gives rise to the kinetic energy is complete, then, by the Hopf-Rinow theorem, the trajectories of the system with zero force field are defined on (-oo, oo). From the physical point of view, this means that a free particle in the system does not go to infinity in finite time. Note that the assumptions, that the Riemannian metric is complete and a solution of the Cauchy problem for trajectories is unique, are easy to justify for systems arising in physics. Here, however, we usually do not require in advance that the solution should be unique.

The cotangent vector p that corresponds to the velocity rh via the metric (i.e., p = (rh, .)) is called the momentum of the system. Consider the operator G: TM T*M sending a vector X E T,,,M to the covector (X, ) E T,;,M. The operator G is called the inertia operator (or the inertia tensor). In particular, p = G(rin). In terms of the inertia operator, the kinetic energy of the system is given by the equation 1C (X) = (G(X))(X)/2. The manifold M is often equipped with a canonical Riemannian metric (, ) (e.g., the standard inner product in 1R3), which is not related to the metric (, )

4. Geometric Mechanics: Introduction and Review of Standard Examples

19

giving rise to K. In this case, we may identify vectors and covectors by means of (,) and view G as an operator from TM to itself. Then (X, Y) = (LX, Y) and G is self-adjoint. Sometimes it is convenient to define the kinetic energy not by the metric (,) but by the self-adjoint operator G: TM -+ TM once the canonical metric (,) is given.

4.B. Some Special Classes of Force Fields If a(t, m, X) is independent of X in T,,,,M and a = -dU, where U is a smooth function on M, then the mechanical system, as well as the force field, are said to be natural or conservative. The function U is called the potential energy or simply the potential. The vector field grad U, such that dU = (grad U, .), is called the gradient of U. For conservative systems, Newton's equation takes the following form: D

th(t) = -grad U

.

(4.4)

The function E: TM --+ IR, E(m, X) = K(X)+U(m), is called the total energy of the conservative system. The total energy E is constant along trajectories. Indeed, taking into account that the Levi-Civita connection is Riemannian, we have dt

E(m, rh)

2 dt

(m, m) + d U(rh)

_ (dt rh, rn) + (grad U, m)

=0. The function L: TM -- IR, L(m, X) = K(X) - U(m), is called the Lagrangian of the conservative system. Using the fact that the Levi-Civita connection is Riemannian and torsion-free, one can easily verify the principle of least action in the Hamiltonian form: the trajectories of a conservative system are precisely the extremals of the action functional fa L (m(t), rh(t)) dt on the space of curves m with fixed endpoints. (See, e.g., [4] and [77].) If m(t) is a trajectory of the conservative system, then the curve p(t) = G(rh(t)) C T*M is a trajectory of the Hamiltonian system with the Hamiltonian H = E(G-1(p)). For conservative systems, we also have the principle of least action in the Maupertius form. According to this principle, trajectories with total energy h are just the geodesics of a new metric (so-called the Jacobi metric):

(h - U) (,)

.

(4.5)

Note that the Jacobi metric (4.5) is Riemannian only when h > U everywhere on M. Hence, in this case, the analysis of trajectories reduces to the description of geodesics on M with the metric given by (4.5). If there are points where the value of U is greater than or equal to h, then the trajectories lie in the domain 12h = {m c M I U(m) < h} called the domain of possible

20

Chapter 2. Geometric Formalism of Newtonian Mechanics

motion (with boundary). The geometry of this domain has been studied by many authors in connection with certain problems of mechanics. Particularly active research in this area started following the works of Kozlov. In [96] the reader may find a review of recent results. A more complex case is where one has the so-called gyroscopic force in addition to the potential one. A gyroscopic force field has the form & (m, X) = w(X, ), where w is a 2-form on M. (Recall that the 1-form al(m, X) takes the value wm(X,Y) on Y E T,,,M.) Usually one assumes that the form w is closed or even exact. An example of a gyroscopic force is the action of a magnetic

field on a charge. A substantial progress in the study of such systems was achieved by Novikov [115]. Another approach was developed in [96].

4.C. Mechanical Systems on Groups Consider a mechanical system which has a Lie group as the configuration space with the kinetic energy given by a right- or left-invariant metric. We shall call such a system a mechanical system on the group. The best-known example of a mechanical system on a group is the one describing the motion of a rigid body which rotates about a stationary point in IR3. It is easy to see that the configuration space of this system, i.e., the set of all possible positions of the rigid body with a stationary point, is the special orthogonal group SO(3). Fixing an orthonormal basis in IR3, we may identify SO(3) with the group of all orthogonal matrices with unit determinant. The

Lie algebra so(3) (i.e., the tangent space to SO(3) at e = id) is formed by skew-symmetric 3 x 3 matrices. Thus so(3) can be identified with W. After this identification, the Killing form (A, B) = tr(A o B) and the commutator [A, B] = AoB-BoA on so(3) become the standard inner and vector products on I6.3, respectively. The Riemannian metric obtained from the Killing form

by left translations turns out to be bi-invariant. This metric is used as the canonical one to express the kinetic energy via the inertia tensor. Let g(t) be a trajectory on SO(3) corresponding to the motion of the rigid body. The velocity vector s(t) E T9(t)SO(3) can be translated to the algebra so(3) = TeSO(3) (identified with IR3) in two different ways. Let a E SO(3).

Consider the left translation La: SO(3) -+ SO(3), Lab = ab, and the right translation Ra: SO(3) -4 SO(3), Rab = ba. The vectors w,(t) = TL9(t)s(t)

and w3(t) =

belong to so(3). The vector we is the angular velocity with respect to the body coordinates (i.e., the coordinate system "attached" to the body and moving along with it); w3 is the angular velocity in the space coordinates (i.e., with respect to a coordinate system fixed in space). Let C: IR3 --+ IR.3 be the inertia operator (tensor) of the rigid body. Recall that G depends on the shape of the body and on the distribution of mass in it. The operator G is self-adjoint with respect to the standard inner product on IR3. Let us define a new inner product (A, B)e = (GA, B) on so(3). This

4. Geometric Mechanics: Introduction and Review of Standard Examples

21

inner product gives rise to the left-invariant Riemannian metric (, ), which determines the kinetic energy by the general formula 1C(X) = (X, X)/2. Note that the choice of the left-invariant metric is dictated by a physical reason: the kinetic energy depends on the angular velocity with respect to the body coordinates, but not on the position of the body in space. Remark 4.2. In Sect. 8, we will consider a mechanical system on the infinitedimensional group of diffeomorphisms. This system describes the hydrodynamics of an ideal incompressible fluid and the energy is given by a rightinvariant (weak) Riemannian metric.

The classical Euler equation of motion of the rigid body (with a stationary point) describes the time variation of the angular velocities w, and w as well as the angular momenta MM = G(w,) and M. = G(w,). Thus, the Euler equation is an equation in the algebra so(3) or in the dual space so(3)*. Throughout this book, we use the following terminology of [86]. The equations in SO(3) are said to describe the motion in the material coordinates or in the Lagrangian representation. The equations for w8 in so(3) are said to

be with respect to the space coordinates or in the Eulerian representation, whereas the equation for w, is the one in the body coordinates or in the convective representation. Similar terminology is retained for the equations for the corresponding angular momenta. Analogous terms are used to describe a mechanical system on an arbitrary Lie group G. Thus, the Newton equation on G is called the equation of motion in the material coordinates (the Lagrangian representation). The corresponding equation on the Lie algebra is called the Euler equation. If the equation is obtained by right translations, then it is said to be with respect to the space coordinates (the Eulerian representation), while for left translations we have it in the body coordinates (the convective representation), etc. This also applies to the so-called Eulerian and Lagrangian specifications in hydrodynamics. Mechanical systems on groups, especially the Euler equations on Lie algebras, have been studied very intensively in the last decades and there is a variety of publications devoted to the subject. A detailed introduction to this field may be found in [4, Appendix 2].

Remark 4.3. Let us point out a well-known result, which will be needed in Part III. Since the kinetic energy is right- (or left-)invariant, the Noether theorem implies a conservation law. For a rigid body with a stationary point this is the well-known angular momentum conservation law in the space coordinates. In Sect. 27 we shall prove its infinite-dimensional version.

22

Chapter 2. Geometric Formalism of Newtonian Mechanics

5. Geometric Mechanics with Linear Constraints This section is an introduction to the modern geometric approach to mechanics with constraints, which goes back to the papers [46] and [135] by Faddeev and Vershik. (See also [136] and [137] and the Bibliography therein.) Here we focus on systems with quadratic kinetic energy as in Sect. 4.A and linear constraints only. Note that the papers mentioned above are devoted to Lagrangian mechanics with more general Lagrangians and constraints that are not required to be linear.

5.A. Linear Mechanical Constraints Consider a mechanical system in the sense of Sect. 4.A on a configuration space M.

Definition 5.1. A linear constraint in the system is a smooth distribution (i.e., a subbundle of the tangent bundle) ,3 on M. In what follows we call a linear constraint simply a constraint.

Definition 5.2. A tangent vector is called admissible if it lies in the distribution /3. A curve in M is admissible if all its tangent vectors are admissible. A constraint /3 imposes a restriction on the motion of the system. Namely, all its trajectories must be admissible. Let P: TM -4 /3 be the operator of orthogonal projection (with respect to the Riemannian metric on M) of tangent spaces on their subspaces /3, i.e., we /3m for every m E M. Let us define the reduced covariant have derivative V on admissible vectors by the formula VXY = PVXY, where V is the covariant derivative of the Levi-Civita connection. Denote by dt

dt

the reduced covariant derivative along a curve. Let a(t, m, X) be the force field of the mechanical system. The equation of motion of the mechanical system with the constraint /3 is the following analog of the Newton equation: d 7h(t) = Pd (t, m, rim)

(5.1)

.

In the same way as for (4.2) and (4.3), one may show that a curve m(t) is a solution of (5.1) if and only if its derivative rh(t), regarded as a curve in the total space of the bundle /3, is an integral curve of the vector field

Y = TP(Z) + (Pa(t, m, rh))1 It is not hard to see that T7rY(m, X) = X E

.

and Y E T(m,X)/3.

(5.2)

5. Geometric Mechanics with Linear Constraints

23

If the distribution 3 is involutive (i.e., integrable, by the Frobenius theorem), then the constraint is said to be holonomic, and (5.1) turns into the Newton equation (1.2) on the integral manifolds of the distribution. Thus, the system with a holonomic constraint reduces to one with no constraint on a manifold of lower dimension.

When the distribution 3 fails to be involutory (i.e., it is not integrable), the constraint is called nonholonomic. In this case, some extra effort is needed

to study the mechanical system. In what follows, we focus on the extreme case, which is opposite, in some sense, to the holonomic one.

Definition 5.3. A constraint /3 is totally nonholonomic if the Lie brackets of admissible vector fields generate the entire Lie algebra of vector fields on M. Remark 5.1. One may introduce the notion of degree of nonholonomity [136], which we do not consider here. Note also that linear constraints are admissible and ideal in the sense of [135].

5.B. Reduced Connections Consider the orthonormal frame bundle ir:OQ(M) -+ M of /3 (i.e., b E Om is an orthonormal frame in /3,,,,). It is clear that Op(M) is a principal bundle with the structural group O(k), k = dim/3m. Theorem 5.1. The reduced covariant derivative V has all four properties of the regular covariant derivative. (See (A.1) of Appendix A.) Proof. Since the operator P is linear on fibers of TM, only the fourth deserves a proof. For admissible X, Y and a smooth function f, we have

Ox(fY) = PVx(fY) = P(fVxY+(Xf)Y) = fVxY+(Xf)Y

,

because PY = Y. Thus, V is the covariant derivative on admissible vectors and, in particular, it gives rise to the parallel translation of admissible vectors along admissible curves. The definition of such a parallel translation is quite similar to the standard one. Since P is orthogonal, it is clear that the parallel translation preserves the inner product on fibers of /3 and (A.2) holds for V. Therefore, on fibers of /3, the parallel translation of orthonormal frames is defined along admissible curves. Consider now the subbundle H in TOO (M), which is as follows. The fiber of H over a point (m, b) E OO(M) is formed by "infinitesimal" parallel translations of the frame e. It is easy to check that the subbundle is invariant with respect to the right action of O(k), and the fibers of H have zero intersection with the vertical subspaces V(m,b). Thus, H can be thought of as an analog of a connection.

24

Chapter 2. Geometric Formalism of Newtonian Mechanics

Definition 5.4. The subbundle H is called a reduced connection.

Remark 5.2. If the constraint is holonomic, the reduced connection is, in fact, the Levi-Civita connection on the integral manifolds with respect to the induced Riemannian metric. (It is interesting to compare the construction of the reduced connection with that of the connection on the adapted frame bundle [94].)

Theorem 5.2. Let X1, X2,. . ., Xk be orthonormal admissible vector fields in a chart U, i.e., at every m C U the vector fields X17... , Xk form an orthonormal basis in Then Vx,X3 = 13X1, where I tl are the tetrad Christoffel symbols (Appendix A) taken for an orthonormal frame in U which contains X1, ... , Xk as a subframe. The result follows from (A.3).

Corollary. The reduced connection of a nonholonomic constraint depends on the Riemannian metric on the entire M, rather than only on the restriction of the metric to 0. Remark 5.3. A variety of open problems concerning reduced connections, as well as their additional properties, is discussed in [135]. Here we only point out that the torsion tensor of a reduced connection cannot be defined in the standard way. The reason is that even though the Christoffel symbols of the reduced connection are symmetric by definition, the difference

VXY - DyX - [X, Y] = P([X, Y]) - [X,

Y]

is zero only if the distribution is involutory.

S.C. Length Minimizing and Least-Constrained Nonholonomic Geodesics Let fl be a nonholonomic constraint on M.

Definition 5.5. An admissible curve m(t) in M is called a least-constrained nonholonomic geodesic if it satisfies the equation

d rh(t) = 0

.

It is clear that least-constrained geodesics are, in fact, trajectories of the mechanical system with the zero force field. Definition 5.5 is analogous to the standard definition of geodesics of a connection.

5. Geometric Mechanics with Linear Constraints

25

Definition 5.6. An admissible curve m(t) is a length-minimizing nonholonomic geodesic if it is an extremum of the action functional

J

(nn(t),?h(t)) dt

on the space of admissible curves with fixed endpoints. For a nonholonomic constraint, the notions of least constrained and lengthminimizing geodesic are not equivalent. Moreover, if the constraint is nonholonomic, the equation of least constrained geodesics is not equivalent to any variational principle, even if the force is conservative (see Sect. 4.B). A more detailed discussion of this matter can be found, for example, in [135].

Remark 5.4. The two notions of geodesics we discuss here are due to Heinrich Hertz, who was apparently the first to notice that the Newton equation and the variational principle become nonequivalent to each other for a system with constraint [136].

Theorem 5.3 (Chow-Rashevsky, see [136] and [137]). Let a constraint /3 be totally nonholonomic. Then for any two points mo, mi E M, there exists an admissible curve which joins mo and ml. Corollary (See [137]). Let the constraint be totally nonholonomic. Then any two points in M can be joined by a length-minimizing nonholonomic geodesic.

Remark 5.5. The differential equation of length-minimizing geodesics (see, e.g, [136]) involves admissible vectors as well as their annihilators (i.e., vectors in TM orthogonal to /3). Therefore, once the beginning mo E M of a lengthminimizing geodesic is fixed, the space of initial conditions has dimension n. Thus, as mentioned above, if the constraint is totally nonholonomic, length-

minimizing geodesics (beginning at mo) fill in the entire manifold M. This question is discussed in more detail in [136] and [137].

Theorem 5.4. On a complete Riemannian manifold, the reduced connection is complete in the sense that all nonholonomic least-constrained geodesics extend to (-oo, oo).

Indeed, since the Riemannian norm of all "velocity vectors" of a leastconstrained geodesic is constant, the Riemannian length of the curve is bounded on every finite interval. Thus, the arc of the geodesic taken over a finite interval is relatively compact because the manifold is complete. This, in turn, means that the geodesic extends to (-oo, oo).

Corollary (See [135]). Let the constraint /3 on M be totally nonholonomic. Then any two points of M can be joined by a piecewise least-constrained geodesic.

26

Chapter 2. Geometric Formalism of Newtonian Mechanics

Remark 5.6. The corollary is sharp: two generic points in M cannot be joined by a least-constrained geodesic even if the constraint is totally nonholonomic. The equation of least-constrained geodesics is a second-order differential equation on the total space of the bundle Q. (The equation is given by the vector field TP(Z) from (5.2).) Thus, the space of initial conditions of least-constrained geodesics starting at a given point mo E M has dimension k = dim)3,, and the geodesics cannot fill in the entire manifold M.

6. Mechanical Systems with Discontinuous Forces and Systems with Control: Differential Inclusions Consider a mechanical system with a discontinuous force field. Such fields appear, for example, in systems with dry friction, switching, or with motion in several media having different resistance forces, etc. When the configuration

space is linear, the following method is often used to study systems with a discontinuous force. First, one extends the discontinuous force field to a set-valued vector field with convex images. Then the second step is to pass from (4.2) to a differential inclusion, whose solutions are trajectories of the system. (See [49].) In this section, we develop a similar method for nonlinear configuration spaces [53]. Note that the equation of motion of a mechanical system with control may also be reduced to a differential inclusion. In this case, the set-valued force, a subset in every tangent space, is formed by all values of the force for all possible values of the controlling parameter. All results on set-valued maps we use here may be found, for example, in [21]. (See also Appendix B for basic definitions.) Consider a locally bounded vector field f on a finite-dimensional manifold M. The vector field f is not assumed continuous, nor even measurable. For any point mo, let us define a subset R(mo) C TmoM as follows. The set R(mo) is formed by the limits of all sequences f (Mk) as Mk --f mo with Mk 34 mo. It is easy to see that

R(mo) = n { cl [( e>Ol

U f (m)) \ mEUE

f(mo)]

}

where UE is the c-neighborhood of the point mo and cl means the closure.

Definition 6.1. The set F(mo) = co R(mo) C Tm0M, where co denotes the convex hull, is called the essential extension of the field f at mo. The essential extension F is a set-valued mapping, which assigns a subset in Tm0M to mo E M. It is natural to call this mapping a set-valued vector field. Note that F = f if f is continuous.

6. Mechanical Systems with Discontinuous Forces

27

Theorem 6.1. The set-valued vector field F is upper semicontinuous.

Proof. Let 6 > 0 be a real number. Fix a metric p on TM, which gives rise to a topology equivalent to that on the tangent bundle. Denote the 6neighborhoods of R(mo) and F(mo) by R6(mo) and F6(mo), respectively. We prove that for any 6 and any m E M, there exists a neighborhood U(m) C M of m such that R(m') C R6(m) for every m' E U(m) and, therefore, F(m') C F6(m).

By the definition of the set R(m), there exists a neighborhood U(m) of

m such that for all m' E U(m) we have p(f (m'), R(m)) < 6. Then there exists an open neighborhood V (m') C U(m) of the point m' such that the inequality p(f(m"),R(m')) < 6 is satisfied for every m" E V(m'). Pick a sequence mk --+ m' in V(m'). We have

limp (f (mk), R(m)) = p lim (f (mk), R(m)) < 6

Hence, R(m') C R6(m) and F(m') C F6(m). Now consider a mechanical system with the configuration space M and the kinetic energy 1C(X) = (X, X)/2, where (,) is the Riemannian metric on M. Let a(t, m, X) be a force field that we require to only be locally bounded in all variables. (Note that, as above, we do not assume that a is continuous or even measurable.) Consider the vector field Z(m, X) + a(t, m, X)1 (i.e., a secondorder differential equation on M; see Appendix A), where Z is the geodesic spray of the Levi-Civita connection of (,) and a(t, m, X) 1 is the vertical lift of a(t, m, X) to the point (m, X) E TM (see (4.3)). It is easy to see that the essential extension (with respect to all variables) of Z(m, X) +a(t, m, X) may be written in the form

Z(m,X)+A(t,m,X)1

(6.1)

,

where A(t, m, X)1 is the vertical lift of the essential extension A(t, M, X) of a(t, m, X) to the point (m, X). Note that A(t, m, X) = co Q(t, m, X), where Q(t, m, X) is the set of limit points of all sequences a(tk, mk, Xk) such that N, mk, Xk) - (t, m, X), Xk E Tm,,M, and N, mk, Xk) Tk,1 (t, m, X). From now on, we focus on the differential inclusion in TM given by the formula (t) E Z(ry(t)) +A(t,ry(t))1 . (6.2)

Definition 6.2. A solution of (6.2) is an absolutely continuous curve y(t) in TM which almost everywhere satisfies (6.2). Alternatively, making use of covariant derivatives, we consider the following differential inclusion on M: Dt

m(t) E A(t, m(t), m(t))

.

(6.3)

28

Chapter 2. Geometric Formalism of Newtonian Mechanics

Definition 6.3. A solution of (6.3) is a C'-curve m(t) in M such that rh(t) is absolutely continuous and (6.3) is almost everywhere satisfied.

Taking into account (6.1) and the definition of D/dt (Appendix A), it is easy to check that (6.2) and (6.3) are equivalent. More precisely, this means that m(t) is a solution of (6.3) if and only if m(t), regarded as a curve in TM, is a solution of (6.2).

Definition 6.4. A solution of (6.3) is called a trajectory of the mechanical system with a discontinuous force field A.

It is easy to see that Definition 6.4 is justified from the physical point of view. As we have mentioned above, for a flat configuration space the reasons supporting the definition are discussed, for example, in [49]. The right-hand side of (6.2) is an upper semicontinuous set-valued vector field with convex images. This implies that locally there exists a solution of the Cauchy problem for (6.2) (e.g., [21] and [491). Thus, for any initial conditions m E M and X E TmM, inclusion (6.3) has a solution on a sufficiently small interval. Note that an interesting question for applications in physics is whether or not the local solution of (6.3) is unique. Certain uniqueness conditions are found in [49]. Another class of mechanical systems involving inclusions like (6.3) are mechanical systems with control. Let the force field a(t, in, X, u) depend on the parameter u E U. We define the set-valued vector field A(t, m, X) on TM as

A(t,m,X) = U a(t,m,X,u) uEU

Now we have to assume that this field is upper semicontinuous and has closed convex images. The solution of (6.3) is a trajectory of the control system for a time-dependent control u(t). Let us point out that, since the configuration

space is nonlinear, we cannot require the control force to be independent of time, coordinates, or velocity as is often the case for linear systems. A very particular example where such an assumption does make sense will be considered in Sect. 8 and Remark 10.1.

7. Integral Equations of Geometric Mechanics: The Velocity Hodograph In this chapter, we use integral operators with parallel translation introduced in Sect. 3 to find integral equations equivalent to the Newton equation of geometric mechanics (Sect. 4). One of these equations describes the velocity hodograph in the sense of [130]. This is an ordinary integral equation in a fixed tangent space. We also introduce analogous integral equations for a system

7. Integral Equations of Geometric Mechanics: The Velocity Hodograph

29

with constraint. Our approach is based on the results obtained in [60], [62], [66], and [71].

Integral versions of the Newton equation and, in particular, the equation of the velocity hodograph turn out to be useful in the study of certain qualitative problems concerning the behavior of mechanical systems, e.g., the existence of special trajectories, etc. It is important to emphasize that the equation of the velocity hodograph is an integral equation in a linear space, and therefore the standard methods may be applied to study it. Integral equations are used in Chap. 3 and also in Part II, where we work with their versions for random force fields. Note that a theory of integral equations on manifolds designed to deal with some other problems of geometric mechanics (e.g., to construct a completely covariant formalism) was developed by Vujicic [140]. This theory

is based on the notion, due to Vujicic, of absolute and covariant integrals mentioned in Sect. 3.

7.A. General Constructions Consider a mechanical system similar to those of Sect. 4. As in Sect. 4, we assume that the Riemannian metric (,) is complete (and so a trajectory of a free particle does not go to infinity in finite time) and that the force field a(t, m.X) is continuous jointly in all variables. The case of discontinuous force fields will be studied in Chap. 3. Since the metric is complete, we can use the operator S introduced in Sect. 3.

Let I'a(t, m(t), rh(t)) denote the curve in T,,,,0M such that the vector for every t. Fix a I'a(t, m(t), rn(t)) is parallel to d(t, m(t), rh(t)) along point mo E M and a vector C in T,,,,(o)M and consider the integral equation m(t) =

(ft

ra (rr, m(r), rh(T)) dr + C)

(7.1)

on I = [0, l].

Theorem 7.1. The solutions of (4.2) with the initial conditions m(0) = mo and rh(0) = C, and only those, are the solutions of (7.1). Proof. It is easy to show that for a given v E Co(I,Tm0M), the C2-curve m(t)

=S(

rt

v(Tr)dT+C)

0

is the only one satisfying the conditions m(0) = mo, rh(0) = C and such that for every t E I the vector Dt(t)/dt is parallel to v(t) along m(.). To see this, let, for some t E I, the curve in T,,,(t)M be obtained by the parallel translation of vectors rh, to the point m(t). Then, by the definition of the covariant derivative, we have

Chapter 2. Geometric Formalism of Newtonian Mechanics

30

=

aTfn(t+ r)I

Dm(t) .

T=0

It is clear that the vector fo v('r) dir + C E Tmo is parallel to m(t) along Let m(.). In other words, the vector Drh(t)/dt is parallel to v(t) along us set v(t) = ra(t, m(t), rh(t)). Then (7.1) means that the vector Drh(t)/dt can be obtained as follows. Namely, we transport a(t, m(t), m(t)) E T..(t)M along first to the point mo = m(0) and then back to m(t). The theorem follows.

Let m(t), t E I, be a trajectory of the mechanical system, i.e., a solution of (4.2).

Definition 7.1. The velocity hodograph of the trajectory m(t) is the curve v: I -+ Tm(°)M such that v(t) is parallel to rh(t) along

It is not hard to see that the velocity hodograph of a solution of (7.1) satisfies the equation v

j

(t)=roaoSv(-r)dr+C

,

(7.2)

where

r o a o Sv(r) = ra (tsv(t),

dt

Sv(t))

It is obvious that if v is a solution of (7.2), then Sv is a solution of (7.1), i.e., by Theorem 7.1, a trajectory of the mechanical system. Remark 7.1. The notion of the velocity hodograph (in the sense of Definition 7.1) was introduced by Synge in [130] where analogs of the standard properties of the hodograph were proved for some mechanical systems. (See also [131].) The hodograph equation (7.2) appeared originally in [60].

Remark 7.2. If we have a mechanical system on a group, then it is natural to pick the initial condition m(0) = e. Thus, (7.2) becomes an equation in the Lie algebra similar to the Euler equation in the body or space coordinates. All three equations are equivalent to the Newton equation on the configuration

space. However, for an arbitrary configuration space, (7.2) is the only one among those three that makes sense. Let us denote the operator which sends v E CO (I, Tmo M) to t

fo

by f roaoSo.

roaoSv(T)dr+C E C°(I,TmoM)

7. Integral Equations of Geometric Mechanics: The Velocity Hodograph

31

Theorem 7.2. The operator o a oSC:C°(I,Tm0M)

Co(I,TmoM)

is completely continuous.

Proof. Since S, a, and F are continuous, so is the operator. Let UK be the ball in C°(I,T,,,.0M) of radius K centered at the origin. Because a is continuous, Theorem 3.2 and Lemma 3.1 imply that (f F o a o Sc) (UK) is compact.

7.B. Integral Formalism of Geometric Mechanics with Constraints Let the configuration space M be a complete Riemannian manifold equipped with a constraint j3, which may be nonholonomic (Sect. 5). To develop an adequate integral formalism, we use parallel translation of admissible vectors along admissible curves. Such a parallel translation arises from the reduced connection H.

Let m(t), t E I, be an admissible Cl-curve and X (t, m) an admissible vector field on M. Denote by FOX (t, m(t)) the curve in such that the vector X (t, m(t)) at m(O) is parallel to FOX (t, m(t)) along under the reduced connection. The properties of the operator rO are quite similar to those of r studied in Sect. 3. Consider the map E: OA X IRk -+ H, k = dim /3,,,, defined by the formula Eb(X) = T7r-1(bX) 1,qb , where b E O,,(M) is regarded as an orthogonal operfrom IRk to (See Appendix A.) It is easy to see that E is smooth and fiberwise linear. Let v(t) be a continuous curve in ,0,,,,a. Fix b° E Omo(M) and consider the time-dependent vector field E(bo 1v(t)) on 00 (M). By definition, this vector field is smooth in b for a fixed t and continuous in t for a fixed b E O'(M). Hence, for the vector field, the Cauchy problem has a solution. As above, it is easy to show that integral curves of E(bo lv(t)) extend to the entire interval I = [0, 1]. Consider the integral curve b°(t) beginning at bo and its projection Se(t) = irbo(t). It is clear that Se(-) is an admissible curve and, in addition, for every t E I the vector dse(t)/dt is parallel to v(t) along Se(-) with respect to the reduced connection H. The following result can be proved in the same way as Theorem 3.5.

Theorem 7.3. Let X (t, m) be an admissible vector field which is continuous jointly in all variables. An admissible curve m(t) is an integral curve of X (t, m) if and only if it satisfies the equation m(t) = SO o FOX (t, m(t))

.

(7.3)

Theorem 7.4. A continuous curve v(t) C /.3,,,o is a solution of the equation

v(t) = FQX(t,Se(t))

(7.4)

32

Chapter 2. Geometric Formalism of Newtonian Mechanics

if and only if Se(t) satisfies (7.3). The operators so, F13 and their compositions have the same compactness and continuity properties as the integral operators from Sect. 3. Now consider the integral equation

rt m(t) = so I

\

r0 Pa (r, m(r), rh(rr)) dr + C I

,

(7.5)

J U0 where C is a vector in Q,,,.o. Taking into account the relationship between parallel translations and covariant derivatives, we get the following result.

Theorem 7.5. An admissible curve with the initial conditions m(O) = mo and rh(O) = C satisfies (5.1) (i.e., is a trajectory of the system with the constraint 0) if and only if it is a solution of (7.5). It is clear that the equation of the velocity hodograph of a solution of (7.5) is

v(t) =

rt

Jo

F Pa I r, Sv(r), d'Sv(T)) dr + C

(7.6)

on the space of continuous curves in #,no.

Remark 7.3. We emphasize that even if M is the Euclidean space IR", the integral operators considered in this subsection (e.g., So, S' oft, etc.) cannot be reduced to their classic analogs (the antiderivatives, the Urysohn-Volterra operator, etc.) unless the distribution ,3 is trivial, i.e., all the spaces /3,n are parallel (in the Euclidean sense) to /3o C TOIR" = R.

8. Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force In this section, we study a certain type of differential equations on Riemannian manifolds. These equations have one of the terms on the right-hand side obtained by parallel translation to the corresponding point along a solution [63]-[66]. The equations are analogous to some differential equations on the Euclidean space, namely, to those with discrete delay or with the righthand side that depends on time only. Our analysis of the equations in terms of geometric mechanics is based on the mechanical interpretation of parallel translation. The mechanical interpretation of Riemannian parallel translation was discovered by Johann Radon and described by Blaschke in [93]. Note that a similar idea was independently used by Synge [130] in order to define the hodograph for a geometric mechanical system. (See Remark 7.1.)

8. Mechanical Interpretation of Parallel Translation

33

Radon proved that for a pendulum moving in the configuration space of a mechanical system, the direction of oscillations is parallelly translated along its trajectory. In other words, the coordinate system attached to a gyroscope (e.g., the stationary one for a flat configuration space) translates parallelly along a trajectory. Consider the motion of a mechanical system with a force field a and a "control" force 0. The latter depends on time, the velocity, and coordinates of the point. Because of some delay present in a real-life situation, 0 acts after time h. The equation of motion of such a system is as follows: Dt

run(t) = a (t, m(t), m(t)) + II (t - h, m(t - h), rh(t - h))

,

(8.1)

where I I means the Riemannian parallel translation along the solution.

Similarly, one may consider the evolution of a system with a velocity field V and delayed control "velocity" W. More precisely, the dynamical system is given by the equation

riz(t) = V (t, m(t)) + II W (t - h, m(t - h))

.

(8.2)

If M is the Euclidean space, (8.1) and (8.2) are quite simple differential equations with discrete constant delay [1]. However, if M is not flat, (8.1) and (8.2) have much more complex properties.

First, since the parallel translation is defined only along Cl-curves and depends on a curve and its derivative, (8.2) is an equation of neutral type [1]. Second, the first-order equation corresponding to (8.1) has distributed delay. The reason is that if M is not flat, the parallel translations of a vertical vector in TM do not necessarily coincide with the lift of the parallel translation in M. (Here the manifold TM is equipped with the standard metric arising from the metric on M.) Finally, note that the first-order equation, which is equivalent to (8.1) and thus, as we have just explained, has distributed delay, is again an equation of neutral type on TM because the righ-hand side is neither continuous in C°-topology nor defined on arbitrary curves. Fix a Cl-curve (k: [-h, 0] --+ M. Definition 8.1. A continuous curve

[-h, c) -+ M, c > 0, is called a solution of (8.1) (respectively, (8.2)) on the interval [-h, e) with initial condition 0 if m(.) is C'-smooth on (0, E), coincides with 0 on [-h, 0], and satisfies (8.1) (respectively, (8.2)) on [0,e)., It is useful to first analyze the particular cases of (8.1) and (8.2), where the control force depends on time only. These mechanical systems are given by the equations

d 7n(t) = a(t, m(t), th(t))+ 11 -P(t) th(t) = V (t, m(t))+ II W(t)

.

,

(8.1a)

(8.2a)

34

Chapter 2. Geometric Formalism of Newtonian Mechanics

In this case

and W take values in the tangent space to M at the initial

condition mo.

It is essential that (8.1) and (8.2) can be reduced to (8.1a) and, respectively, (8.2a). To see this, fix an initial condition 0 E Cl([-h, 0], M) and consider the following T,(o)M-valued functions of 0 E [-h, 0]:

0(0) = II0(0+h, and

W(0) = IIW(e+h,0(0+h))

.

Let t = 0 + h. It is clear that the solutions of (8.1a) and (8.2a) with the control forces fi(t) and W(t) coincide with those of (8.1) and (8.2), respectively. Note that the equations above make sense only if their solutions are C1smooth, for the parallel translation is not defined otherwise.

Theorem 8.1. Let V (t, m) be a continuous vector field on M and W (t) a continuous curve in Tm0M. Then:

(i) equation (8.2a) has locally a Cl-solution; and (ii) the solution is unique provided that for any fixed t E I the field V (t, m) is locally Lipschitz in m.

Proof. Let O(M) be the orthonormal frame bundle over M, and H the Levi-Civita connection on O(M). The tangent map of the natural projecat every point b E O(M). Hence, at every b E O(M), we obtain the vector Tir-1V(t,7rb) E Hb C TbO(M). These vectors form a horizontal (i.e., tangent to H) vector field on O(M). tion 7r: O(M) -* M induces the isomorphism T-7r: Hb -*

Let us fix an orthonormal frame 0 in Tm0M. The frame 0 gives rise to the isomorphism O:IR" --, T,,,0M and therefore, we have the horizontal time-dependent vector field E(0-1W(t)) on O(M). (See Appendix A.) Consider the vector field V (t, b) = Tir-1 V (t, 7rb) + E (O-1 W (t))

on O(M). By the existence theorem for ordinary differential equations, this vector field has an integral curve y*, -y* (0) = 0, defined on the interval [ 0, e).

Since the vector field V is horizontal, the frame y*t is parallel to 0 along y = iry* for every t E [ 0, e). By definition, we have

y* (t) = T7r-1 [v(t, y(t)) + y*(t)

(O-1W (t))

where the last term is the vector with coordinates O-1 W(t) in the basis y* (t). Hence,

y(t) = v(t, y(t)) +

y*O-1(W (t))

Furthermore, by the definition of parallel translation, the last term is parallel

to W(t) along O. This means that y(t) is a solution of (8.2a).

8. Mechanical Interpretation of Parallel Translation

35

Note that the field E(O-'W(t)) is smooth in b for a fixed t. If V(t, m) is locally Lipschitz in m (for every fixed t), then V is locally Lipschitz in b and the resulting solution is unique. 0 Corollary. Assume that the fields V (t, m) and W (t, m) are jointly continuous in all variables. Then for any C1-initial condition ¢: [-h, 0] -+ M, there exists a solution of (8.2). The solution is unique if V is locally Lipschitz in m for any fixed t.

Theorem 8.2. Assume that the field a satisfies the Caratheodory condition (see, e.g., [21] and [49]) and 4i(t) is an integrable function with values in T,noM. Then for any initial condition C E Tm0M: (i) equation (8.1a) has locally a C1-smooth solution; and (ii) the solution is unique if a is locally Lipschitz for every fixed t.

Proof. Consider the direct product O(M) x lRn, n = dim M, equipped with the right action of O(n) given by the formula (b, x)g = (bg, g-1x), where b E O(M), x E IRn, and g E O(n). The quotient space of O(M) x IR" under the right action can be naturally identified with TM. We denote the natural projection O(M) x IRn -+ TM by A. (See Appendix A.)

Let us fix a point (b,x) E O(M) x W. It is easy to see that TA induces an isomorphism of Hb E TbO(M) with the horizontal space in Ta(b,y)TM and an isomorphism of Vx = TXIR" with the vertical space VA(b,x). (Recall that

the latter is the tangent space to T bM, where 7r:O(M) -+ M is the natural projection.)

Pick an orthonormal basis 0 in T,,,M and define the function 0-4(t) with values in 1R" to be the coordinates of 45(t) in the basis O. As in the proof of Theorem 8.1, the function gives rise to the horizontal vector field E(0-4(t)) on O(M). We see that any basis b E O(M) gives rise to a vector

T1Eb(0-4(t)) in the tangent space TbM. Consider the vector fields A, B, and C on O(M) x 1R." such that for any (b, x) E O(M) X IR", A(b,x) = Ta-1Za(b,x) E Hb ,

where Z is the spray of the Levi-Civita connection on M and B(b,x) (t) = TA-1(a(t, rA(b, x), ,\(b, x))) E Vx

Clb,yl(t) = TA-1(T7rEb(0-4(t))) E Vx By definition, A is a smooth field. Since a satisfies the Caratheodory condition, so does B(t). The field C(t) is smooth on O(M) X IR" for every fixed t and measurable in t for any fixed (b, x) due to the hypothesis of the theorem. Therefore, the field A + B(t) + C(t) satisfies the hypothesis of the classic theorem which guarantees the existence of a local solution of the Cauchy problem.

36

Chapter 2. Geometric Formalism of Newtonian Mechanics

Furthermore, if a is locally Lipschitz in t, then the hypothesis of the uniqueness theorem is satisfied as well. (See [49], Theorems 1 and 2 of Sect.l.) Note that local solutions are, by the construction, absolutely continuous curves. Let (b(t), x(t)) be the local solution with the initial condition (0, 0-1C). The curve \(b(t),x(t)) is absolutely continuous in TM and, thus, the tangent vector Y(t) = TA(A+ B(t) + C(t)) = ZA(b(t),x(t)) + TA(B(t) +C(t))A(b(t),x(t))

exists for almost all t. The vector ZA(b(t),x(t)) belongs to the connection and both vectors TA(B(t))A(b(t),x(t)) and T.1(C(t))A(b(t),x(t)) are in the vertical subspace. Hence, T7rY(t) = TirZ. On the other hand, Ti-rZA(b(t),x(t)) = A(b(t), x(t)), because Z is the spray. As one can easily see, this means that the curve A(b(t), x(t)) in TM has the form (y(t), y(t)), where y(t) = 7rA(b(t), x(t)) is a Cl-curve. In particular, the parallel translation is defined along -y. By definition, the projection of (b(t), x(t)) to O(m) is horizontal, and so b(t) is a parallel frame field along y. Hence, for every t the vector TA(C(t)) E T.y(t)M is parallel to fi(t) along -y. Taking into account the definition of the covariant derivative, we see that y satisfies (8.1a). It is clear that y(0) = m,o and y(0) = C. Corollary. Assume that a satisfies the Caratheodory condition and 4p(t, m, X) is jointly measurable in all variables. Then for any C1-curve 0: [-h, 0] -. M, there exists a local solution of (8.1) with the initial condition 0 provided that II -P(t, 0(t), fi(t)) II is integrable on [-h, 0]. If for every fixed t the vector field a is locally Lipschitz in (m, X), then the solution is unique.

Theorem 8.3. Let the Riemannian metric (,) be complete. Assume also that for some point mo E M the following inequalities hold:

IJa(t,m,X)IJ «(t)L(p(mo,m)) and, respectively, IIV (t, m) 11 < `W(t)L(p(mo, m))

,

where the function L is defined in Sect. 1.B and satisfies (1.2), p is the Riemannian distance on M, and li is a positive function integrable on finite intervals. Then the solutions of (8.1) and (8.2), respectively, are defined on [-h, oo).

Proof. Without loss of generality we may assume that inf L = C > 0. (Otherwise, we just replace L by L+C.) Let us rewrite (8.1) and (8.2) in their equivalent forms (8.1a) and (8.2a), respectively, and consider the complete Riemannian metric (,)* introduced in Sect. 1.B. In this metric, JIV(t,m)lI* < W(t).

The norm of W (t) with respect to (,) is bounded on [-h, 0] by a constant K > 0 because it is a continuous function. It is easy to show that any local

8. Mechanical Interpretation of Parallel Translation

37

solution m(t) of (8.2a) satisfies the inequality

K

l'm(t)lI <'I'(t) + C where the norm 11.11 is taken with respect to (,). Thus, m(t) extends to [-h, h]. Covering any given interval I by intervals of length h, one may prove that the solution extends to I. For (8.1) the proof is similar. 0

We conclude this section by noting that the shift operators along solutions of (8.1) and (8.2) were studied in [63] and [64]. The existence of fixed points of these operators (i.e., periodic solutions for a periodic right-hand side) was proved by the methods of [22].

Chapter 3. Accessible Points of Mechanical Systems

In this chapter we study the question of whether or not two points mo and ml in the configuration space of a mechanical system can be connected by a trajectory. It is known (see, e.g., [85]) that for a second-order differential equation on the Euclidean space such a trajectory exists provided that the right-hand side of the differential equation is bounded and continuous. More precisely, for any two points mo and ml and any interval [a, b], there exists a solution m(t) such that m(a) = mo and m(b) = ml. When the right-hand side is linearly bounded, some similar results are known for small intervals of time. The situation becomes much more complex for a nonlinear configuration space. In Sect. 9, we illustrate this by three examples of mechanical systems on the two-dimensional sphere. In the first example, the force field is smooth and independent of time and velocity (and so it is bounded). However, none of the trajectories beginning at the North Pole reaches the South Pole. In the second example, the force field is still bounded, autonomous, and smooth but now depends on the velocity. In this case there is no trajectory connecting

any two antipodal points on the sphere. In the third example, we consider a gyroscopic force on S2. (Hence, the force field is linear in velocity.) The behavior of trajectories in this system turns out to be quite similar to the second example. The difference in the behavior of trajectories on flat and "curved" config-

uration spaces has a deep geometric reason. In Sect. 10, we show that if the force field is bounded (but, maybe, discontinuous), then for any two points mo and ml, there exists a trajectory joining mo and ml provided that the points are not conjugate along some geodesic. Next, we prove that the same is true even if the force field is unbounded, but has linear growth in velocity. Regarding the examples, note that opposite points on the standard sphere in IR3 are conjugate along all geodesics. To prove our results, we use some modifications of the integral operators r and S of Sect. 3 and the hodograph equation of Sect. 7. In Sect. 11, we generalize the results to systems with constraints. A few more results on the existence of trajectories are proved in Appendix F by Ginzburg. In particular, it is shown there that a pair of nonconjugate points can be joined by a trajectory whenever the force field is smooth and has less than quadratic growth in velocity.

40

Chapter 3. Accessible Points of Mechanical Systems

9. Examples of Points that Cannot Be Connected by a Trajectory Example 1 ([621 and [66]). Consider the mechanical system on the unit sphere S2 in JR3 with the force field a(f) = (-y, x, 0), where r = (x, y, z) E S2. The motion of the system is given by the following equations in 1R3:

a(r)-21C or, equivalently,

where 1C

HI

+2 + y2 + .z2

2

2

is the kinetic energy. To obtain these equations, one applies the d'Alembert principle (see, e.g., [119]) to the holonomic constraint F(r) = x2 + y2 + z2.

Denote the North and South Pole of the sphere by N = (0, 0, 1) and S = (0, 0, -1), respectively. Let r(t) = (x(t), y(t), z(t)) be the trajectory of the system such that r(to) = S for some to and T(to) = V 0. Note that if V = 0, then f(t) - S. It is clear that V E TS S2 must have the form (X, Y, 0). We claim that the kinetic energy increases along f(t) until r hits the North or South Pole. By (9.1) we have 1C(r(t)) = thy + yx and

1C(f(t)) = x2 + y2

Note also that 1C(f(to)) = 0. This means that K(f(t)) > 0, unless r(t) = S or r(t) = N. In fact, the derivative 1C(f(t)) is also increasing. Since IC (N) _ JC(S) = 0, we have 1C(f(t)) # N for any t > to. To clarify the geometric picture, consider the function z(t) = z(r(t)). Let

tl > to be such that z(tl) = 0 and z(t) is increasing on [to,tl]. The last equation in (9.1) implies that z(tl) > 0 and, as a consequence, z(ti) < 0, i.e., z(ti) is a local maximum of z(t). Since 1C is increasing along r(t), we see that z(ti) < 1. In the same way, one may show that

sign z(ti) = (-1)i+' and Iz(ti)I > 1z(ts+i)I for all points ti < t2 < . . . such that ,z(ti) = 0. Therefore, the trajectory f(t) goes to the equator of S2 and oscillates near it. In particular, the trajectory never reaches the point N = (0, 0, 1). Example 2. Let us replace the field a in the system of Example 1 by the

9. Examples of Points that Cannot Be Connected by a Trajectory

41

force field

where [, ] is the vector product in 1R3. The equation of motion of the mechanical system is as follows:

r=.R(r,r)-21C-r

(9.2)

A straightforward calculation shows that

1C= (Q(r,r),r) -21C (f, r) =0 along a solution of (9.2) (i.e., the force is always orthogonal to the velocity) and b = 0, where

r

II2 r

+

Ilrll2'[r,r]

Therefore, the kinetic energy 1C = r ll2/2 is constant along the trajectory r(t), and r(t) lies in the plane orthogonal to the constant vector 6. In other words, the trajectory is the circle (f (t), b) = const on S2. Assume that there is a trajectory passing through two antipodal points. Then it must be a great circle on S2 and, therefore, (r(t), b) = 0. Let a be the angle between r(t) and b. A straightforward calculation (based on the explicit formulas for Ilbll and (r(t),b) and on the equality llr(t)II - 1) shows that 11

cos a =

(Ilrll) IIrll2

where j : [ 0, oo) -+ IR is a bounded function. Hence, (f (t), b) goes to zero as 1C -+ no assuming nonzero values only. This means that there is no trajectory in the system passing through two antipodal points. Note also that any

two points which are not antipodal can be connected by a trajectory with sufficiently high kinetic energy.

Example 3. Replace the force Q (f, r) of the preceding example by the gyroscopic force A(r, r) = [r, r] . The equation of motion of the new system is

The analysis of this example is quite similar to that of Example 2. First, we prove that IC = 0 and b = 0, where b = [r, fl. This implies that the trajectory lies in the plane orthogonal to b. If the trajectory were a great circle, so that (r, b) = 0, then this would give us the equality [f, r] = 0, which is impossible. The author is grateful to Evgenii I. Yakovlev for pointing out Example 3.

42

Chapter 3. Accessible Points of Mechanical Systems

10. The Main Result on Accessible Points In the examples of Sect. 9, the points which could not be connected by a trajectory were conjugate along all geodesics. In this section, we prove that if two points are not conjugate along a geodesic and the force field is bounded, then there exists a trajectory joining the points. Following [53], we consider the general case where the force field d(t, m, X) is discontinuous. (See Sect. 6.) Thus the trajectories are, in fact, solutions of the differential inclusion (6.3), where the set-valued vector field A(t, m, X) is upper semicontinuous and has convex images. This general result yields, as a simple corollary, the existence of such a trajectory for a mechanical system with continuous a [60], [62]. Let M be a manifold, let (,) be a complete Riemannian metric on M, and let m° and ml be points that are not conjugate along a geodesic a(t).

Theorem 10.1. Assume that A(t, m, X) is upper semicontinuous, uniformly (in t, m, and X) bounded by a constant C > 0, and A has convex images. Then there exists a constant L = L(m°i ml, C, a) > 0 such that for any to, 0 < to < L, the points m° and ml can be connected by a solution of (6.3) with m(0) = m° and m(to) = ml. Proof. Using the results of the previous sections, we construct an integral operator in order to study the global behavior of solutions of (6.3). Let I = [0, l]. For any curve v(.) E C°(I,T7,M), the set-valued vector field A(t, ry(t),''y(t)) is defined along the curve 7(t) = Sv(t). Fixing v, let us introduce the set-valued map FA o Sv: I --+ Tm0M such that the set TA o Sv(t) is obtained by parallel translation of A(t, -y(t), y(t)) to the point m° along ry. Using the properties of parallel translation and the fact that A is upper semicontinuous, one can show that the map FA o S: C° (I, TmO M) x I -+ M is upper semicontinuous too. Consider the set PEA o S formed by all measurable selections of TA o Sv. Note that such selections do exist [21]. Since the field A is bounded, all elements of PTA o S are integrable. Let us define the set-valued map f PE o S with convex images in C°(I,TmOM) by the formula

f J

PEA o S(V)

{ft

Lemma 10.1. The map f PEA o S sends bounded subsets of C° (I, T,,,, M) to compact ones.

Proof. Since the metric (,) is complete, for any ball UK in Co (I, Tm, M), the union of curves { (ry, -y) ry E UK } lies, by Lemma 3.1, in a compact subset of TM. Then all sets A(t, -y, y), where ry E SUK, are uniformly bounded because the field A is bounded. As a consequence, since parallel translations preserve the norm, the sets (FA o Sv) (t) for v E UK are also uniformly bounded, and so are all their measurable selections PEA o Sv. Thus, continuous curves I

10. The Main Result on Accessible Points

uE

43

U(fPrAos)v vEUK

are uniformly bounded and equicontinuous. The lemma follows.

Lemma 10.2. The map f PrA o S is upper semicontinuous.

Proof. It suffices to prove that the set-valued map f PEA o S has a closed graph. In other words, that vk -+ vo and Uk -+ uo, where uk E (f PEA o S) vk, implies that uo E (f PEA o S) vo, i.e., 'ho E (rA o Svo) (t) for almost all t. Since

the map f PEA o S sends bounded sets to compact ones, the map is upper semicontinuous, provided that it has a closed graph [21]. Recall that the sets (rA o Svo)(t) are convex and the map (rA o Sv)(t) is upper semicontinuous in v and t. As a result, we have uo E (rA o Svo)(t). A detailed proof of a similar (and simpler) inclusion may be found in [21]. The lemma is proved.

Let the constant L(mo, ml, Ctl, a) be defined as in Theorem 3.3. Since A(t, m, X) is bounded by a constant C, the inequality tl < L(mo, ml, Ctl, a) holds for a sufficiently small t1. Let us denote the supremum of all such

tl by L(mo, ml, Ctl, a) and pick to < L(mo, ml, Ctl, a). Without loss of generality, we may assume that the operator f PEA o S acts on the space Co ([0, to], Tmo M). Consider the upper semicontinuous set-valued compact map

Su=

(fPrA0s)(u+c)

on the ball Ucto C CO ((0,to],Tm0M), where the vector Cu is defined in Theorem 3.3. Since parallel translation preserves the norm, B(Ucto) C Ucto. Thus, 13 has a fixed point uo E 8uo [21]. Let us show that m(t) = S(uo(t) +Cuo) is the desired solution. By definition, we have:

m(0) = mo and m(to) = ml; m(t) is a C'-curve; and ran(t) is absolutely continuous.

Note that 'ho is a selection of FA oS(uo +Cuo) because uo is a fixed point of S. In other words, the inclusion do (t) E FA o S(uo + Cu,,) (t) holds for all points t at which the derivative exists. Using the properties of the covariant derivative and the definition of uo, one can show that rio(t) is parallel to Drh(t)/dt along m(.) and FA o S(uo + Cuo)(t) is parallel to A(t, m(t), rh(t)). Therefore, Dt

tin(t) E A(t, m(t), tin(t))

44

Chapter 3. Accessible Points of Mechanical Systems

It is worth noticing that if mo and ml are not conjugate along several geodesics, then any of them can be used in the proof. Naturally, different geodesics can give rise to different solutions and constants L.

Assume that the configuration space M is compact and the metric (, ) has a nonnegative sectional curvature. Then there are no conjugate points on M. As follows from Theorem 10.1, there exists a constant L > 0 such that any two points can be connected by a trajectory m(t) with t E [0, to] for any to > L. In particular, one may take L = oo when M is flat. (See Theorem 3.3.) This means that the corresponding two-point boundary-value problem has a solution on any time interval. Remark 10.1. By definition, Drn(t)/dt is a measurable vector field along the solution. Thus, in the case where A(t, m, X) is the set of possible values of the control force, Theorem 10.1 gives a condition which guarantees the existence of a control sending mo to ml. For example, Theorem 10.1 can be applied to systems with a delayed control force studied in [64]. Consider a mechanical system (in the sense of Sect. 4.A) with a bounded continuous force a(t, m, X) and with a delayed control force. Assume that the possible values of the control force form the set F E Tm0M, where mo is the beginning of the trajectory, and the control starts in time h > 0 (to take into account the delay present in many realistic models). We also assume that BE C F, where BE is the ball of a small enough radius e > 0. As shown in Sect. 6 and 8, the motion of the system can be described by the differential inclusion D

d rn(t) E a(t, m(t), rh(t))+ 11 E (t) where E(t) = 0 for t E [ 0, h), E (t) = BE fort > h, and I I means the parallel translation along the trajectory. It is shown that there exists a measurable control sending mo to ml, provided that mo and ml are subject to the hypothesis of Theorem 10.1. The problem of the existence of the optimal control satisfying this condition is studied in [64].

Theorem 10.2. Assume that the points mo and ml are not conjugate along a geodesic a such that a(0) = mo and a(1) = ml. Let the upper semicontinuous set-valued vector field A(t, m, X) have convex images for all t, m, and X and satisfy the inequalities IIA(t,m,X)II < C + k IIXII where C and k are positive constants, and sup IIylI I I A(t, m, X) I I =

(10.1)

yEA(t,m,X)

Then there exists a constant L(mo, ml, C, k, a) > 0 such that for any 0 < to < L(mo, ml, C, k, a), there exists a solution m(t) of (6.3) with m(0) = mo and m(to) = ml.

11. Generalizations to Systems with Constraints

45

We omit the proof of this result, since it is quite similar to that of Theorem 10.1. (See [73].)

Remark 10.2. It should be noticed that, in contrast to Theorem 10.1, the assertion of Theorem 10.2 is local in time even on a flat configuration space. Let us also point out that an analogous result is obtained in [143] by a different method for a smooth (single-valued) A satisfying certain complex hypotheses and, in particular, for uniformly and linearly bounded smooth force fields.

11. Generalizations to Systems with Constraints In this section, we show how to generalize Theorems 10.1 and 10.2 to systems with constraints considered in Sect. 5. A more detailed account on such a generalization can be found in [71]. In the framework of mechanics with constraints, it is more natural to consider the question of whether or not a submanifold transverse to the union of least-constrained geodesics leaving a fixed point is accessible from the point. The author is grateful to Boris D. Gel'dman for pointing out this problem. In the present section we use the integral operators introduced in Sect. 7.B. Let M be a complete Riemannian manifold equipped with a constraint /3. Fix a point mo c M. The exponential map expmo: ,Q,,o -+ M can be defined in the same manner as for a manifold without constraint. Namely, for X E /3mo, we set expmo (X) = yx(1), where yx(t) is the least-constrained geodesic with yx (0) = m° and 'yx (0) = X. It is clear that expmo is a C°°-smooth map. Definition 11.1. A point ml E expmo(,Qmo) is not conjugate to m° along the geodesic yx (where yx (1) = ml) if the differential d expmo has the maximum rank at X E Qmo .

In particular, this means that the image of expmo is a smooth submanifold in a neighborhood of m1 that is not conjugate to mo. Moreover, expmo is a diffeomorphism of a neighborhood of X E /j,no onto the neighborhood of ml in the submanifold. Assume that mo is not conjugate to m1 along a least-constrained geodesic -yx. Let us fix a submanifold N C M, ml E N, which is transversal to the image of expmo. (In other words, the sum of spaces Tm0N and Tmo expmo (0,,,) coincides with Tm0M.)

Theorem 11.1. Under the hypothesis above, for any K > 0, there exists a constant L(mo, N, K, yx) > 0 such that for 0 < t1 < L(moi N, K, yx) and for any continuous curve u(t) E UK C C°([0,t1],/3,,,,0), there exists a vector Cu E /3mo satisfying the condition S13(u + Cu)(tl) E N. Furthermore, Cu is unique in a neighborhood of ti 1X E p,no and continuous in u.

46

Chapter 3. Accessible Points of Mechanical Systems

The proof is quite similar to that of Theorem 3.3. The only extra argument needed is that the manifold N stays transversal to a Cl-small perturbation of the image of expn,a = Let A be a set-valued vector field on M. As in Theorem 10.1, we assume that A is upper semicontinuous, bounded, and has convex images. Consider the differential inclusion

D

rh(t) E PA(t, m(t), 74(t))

.

(11.1)

It is easy to see that the sets PA(t, m, X) are convex and the set-valued vector field PA is upper semicontinuous and bounded. Such a field can arise, for example, as a discontinuous force acting on the system, or the image of PA can be formed by all possible values of the control force. (See Sect. 6.)

Theorem 11.2. Let PA be upper semicontinuous, bounded, and have convex images. Then there exists a constant L(mo, N, C, yX) > 0 such that for any to, 0 < to < L(mo, N, C, ryx), there exists an admissible solution m(t) of (11.1) which connects mo and N, i.e., m(0) = mo and m(to) E N. The theorem can be proved in the same way as Theorem 10.1. Note that one has to use the operators SO and lQ instead of S and r and apply Theorem 11.1 instead of Theorem 3.3. We leave the proof to the reader as a simple exercise.

Theorem 11.3. Let the field A(t, m, X) be as in Theorem 10.2. There ex-

ists a constant L(mo, N, C, k, 'X) > 0 such that for any to, 0 < to < L(mo, N, C, k, -yx), equation (11.1) has an admissible solution m(t) with the initial conditions m(0) = mo and m(to) E N. The proof of the theorem is analogous to that of Theorem 10.2.

Part II Stochastic Differential Geometry and its Applications to Physics

In recent years, a number of interesting connections between differential geometry and the theory of stochastic processes have been discovered and understood. Moreover, this new understanding led to the birth of a branch of mathematics known as stochastic differential geometry, which has many important applications to certain problems of mathematical physics. (See, e.g., [16], [18], [26], [43], [44], [102], [107], [113], and [122].) In Chap. 4, we discuss some constructions and results from stochastic differential geometry related to stochastic differential equations on Riemannian manifolds. These results enable us to use geometric mechanics in order to study systems with stochastic effects.

In particular, we consider two distinct methods of passing from deterministic equations to stochastic ones, both of which lead to meaningful physical theories. The first method is based on the description of Brownian motion by Langevin's equation (Chap. 5). The second one is Nelson's stochastic mechanics (Chap. 6), known to be equivalent to quantum mechanics.

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

12. Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces This section is devoted to the basic definitions and results of the theory of stochastic equations. Here we introduce integrals needed to understand the geometric properties of stochastic equations, focusing mainly on integrals with

respect to Wiener processes, for they play a central role in the theory. A more complete and detailed exposition may be found in various textbooks and monographs. (See, e.g., [50], [54], [55], [95], [97], [100], [101], and [129].)

However, we should particularly point out the excellent introductory paper [30] illuminating those aspects of the theory which are especially important for our approach. Some basic notions are briefly reviewed in Appendix C.

12.A. Wiener Processes Let (,R, Y, P) be a probability space and Bt, t E [ 0, oo), a nondecreasing family of o-subalgebras of the o,-algebra .F. In what follows, we assume that the v-algebras Bt are complete, i.e., they contain all sets from Y of measure zero. Denote the mathematical expectation and the conditional expectation with respect to Bt by E and E( I Bt), respectively. Here we only consider stochastic processes (random variables) on (J7, Y, P) with values in the Euclidean space with an inner product (,). Fixing a basis, we identify this space with IR'. Definition 12.1. A stochastic process w(t) is called a Wiener process (relative to the family Bt) if:

(1) the trajectories of w(t) are almost surely (a.s.) continuous in t; (2) w(t) is a square integrable martingale with respect to Bt; and (3) w(0) = 0 and E((w(t) w(s))2IB,)= t S for t > s.

-

-

Theorem 12.1 (Levi, see, e.g., [100]). A Wiener process w(t) is a process with stationary independent Gaussian increments. Furthermore, w(t) satisfies

50

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

the following conditions:

E(w(t) - w(s)) = 0 and E((w(t) - w(s))2) = t - s fort > s. In other words, for t > s, the increment w(t) - w(s) is independent of 139 and has the same probability distribution as w(t - s).

Let Pt be the minimal complete a-subalgebra of F such that all random variables w(s), s < t, are measurable with respect to P,. (Recall that completeness means that the a-subalgebra contains all sets of measure zero.) Consider the space [1 = Co ([ 0, oo), IRn) of continuous curves on the semi-

infinite interval and the a-algebra F generated by the cylinder sets in P. (See [124] or Appendix C.) Every Wiener process can be regarded as a map of the measure space (fl, F) into the measure space (Q, F). Thus, P gives rise to a measure v on ((2, f). The measure v, called the Wiener measure, does not depend on the process w, but only on the inner product on IRn. The Wiener measure enables one to introduce the probability distributions of

w(t) in (Q,F), i.e., the conditional probability distributions of the random variables w(tl), ... , w(tk) in IR' for all tl, ... , tk. Let us define a stochastic process w(t) on the probability space (f2' Y, v) as follows. Namely, w(t, w) = w(t), where the elementary event w is by defini-

tion a continuous curve w E fl = C°([0,oo),IRn). Consider the a-algebra Bt generated by the cylinder sets with the base over [0, t], i.e., Bt = P, 7v. Clearly, w(t) is a Wiener process relative to the family 13t. The process w(t) is called a Brownian motion process or a standard Wiener process. To facilitate further references, we summarize here some results on Wiener processes.

Theorem 12.2. Any Wiener process has the following properties:

(1) A trajectory of w(t,w) is a.s. (i.e., with probability 1) nondifferentiable for all t and has unbounded variation on any arbitrarily small interval. (2) The coordinates wi(t) of w(t) are one-dimensional Wiener processes which are mutually independent and the orthogonal projection of w(t) to any k-dimensional subspace of IRn is a k-dimensional Wiener process. (3) Let b be an orthogonal operator in IRn. Then bow(t) is a Wiener process. In particular, if w(t) is a standard Wiener process, then so is bow(t), i.e., the Wiener measure is invariant under the action of orthogonal operators on IRn.

12.B. The Ito Integral Our goal in this section is to define the stochastic integral with respect to a Wiener process. For the sake of simplicity, we restrict our attention to

12. Stochastic Equations and Integrals

51

the construction based on a Riemann-type integral. An approach involving a Lebesgue-type integral can be found in [50), [54], [55], [95), and [100].

Fix a positive constant l < oo. Let A: [0,1) x S? -+ L(1R') be a random operator function, i.e., A(t) is a random linear operator on 1R" for every t E [0, 1]. To define the Ito integral of A(t), pick a partition q = (0 = to < ti < < tq = 1) of the interval [0, 1) and consider the integral sum i=q-1

A(ti)(w(ti+1) - w(ti)) i=o

The limit of such sums as diam q -+ 0 is called the Ito integral of A(t) and denoted by fo A(t) dw(t). (Note that the limit may not exist.) Since the trajectories of w have a.s. unbounded variation, the Ito integral cannot be defined as the Stieltjes integral along every trajectory.

Definition 12.2. A function A(t) is said to be nonanticipative with respect to 13t if A(t) is measurable with respect to fat for every t. It turns out that under certain hypotheses of boundedness, the Ito integral does exist as the L2-limit of the integral sums when A(t) is nonanticipative with respect to tit. In particular, it exists (as a Lebesgue-type integral) if the entries Aid (t) of A(t) satisfy the equality I

1

Pfo A2(t,w)dt
')

i

1

t

A(T) dw(r) = f xtA(T) dw(T) 0

where Xt is the characteristic function of [0, t]. Note that fo A(r) dw(r) is linear in A and dw. Some other important properties of the integral are given in the theorem below.

Theorem 12.3. The process fo A(r) dw(r) has the following properties: (1) it is nonanticipative with respect to Bt; (2) it is a martingale relative to ,tat; and (3) its trajectories are a.s. continuous in t.

Remark 12.1. Actually, to make Theorem 12.3 correct as stated, one must slightly modify the process fo A(r) dw(r) [100]. In what follows it is important to outline the proof of Theorem 12.3(3).

52

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

Consider processes with continuous trajectories of the form i=q-1

Xt(ti+1)A(ti+1)(w(ti+1) - w(ti)) +A(tk)(w(t) - W(tk))

(t) _ i=1

where k = max{i I Xt(ti) = 1}. Under certain hypotheses, the sequence >I(t) contains a subsequence that converges a.s. uniformly to fo A(r) dw(r). This yields assertion (3). Note that one may define the multiple stochastic integral t

1

a(r) dwl (r) .

dwk (T)

of a given stochastic process 'a to be the limit of the integral sums i=q-1

E a(ti)(w'(ti+1) - w1(ti)) ... (wk(ti+1) - wk(ti))

.

i=1

(Note the discrepancy with the definition of the multiple Riemann integral: the multiple stochastic integral on a segment does not have to be zero, unlike the Riemann integral.) Later on, we shall use the following result on the existence and properties of multiple integrals.

Theorem 12.4. Let a be a (scalar) stochastic process and w(t) a Wiener process with values in IV, i.e., w(t) = (wI (t), ... , w" (t)), where wi(t), i = 1, ... , n, are mutually independent one-dimensional Wiener processes. Then

(1) ff a(r) dwi(r) dw (r) = 0 for i # j; (2) .fo a(r) (dwi(r))2 = J a(r) dr; (3) fo a(r) dr dwi (r) = 0; (4) fo a(r) (dwi(r))3 = 0; and (5) all integrals of higher order in dT and dwi(r) exist and are equal to zero. Here we just outline the proof. Assertion (1) follows immediately from the hypothesis that w'(t) and w3 (t) are independent. To prove (2), it suffices to observe that for any Wiener process w(t) we have

E((w(t) - w(s))2) = It - 81 Assertions (3)-(5) result from the fact that the multiple Riemann integral with respect to (dt)k, k > 1, is equal to zero. Remark 12.2. By Theorem 12.2, the derivative w(t) of a Wiener process does not exist (in the standard sense). However, using distributions, one can define this derivative as a distribution-valued process, the so-called white noise. The

white noise is sometimes useful to give a better physical interpretation of solutions of certain equations. In what follows, we avoid dealing with the white noise by using integral (rather than differential) equations and taking

12. Stochastic Equations and Integrals

53

into account that, by definition, t

A(t)w(T) dT =

t A(t) dw(T) fo

Some other versions of stochastic integrals are known. Namely, one may chos

the point t E [ti, ti+1] at which the value of A(t) in the integral sum is taken to be distinct from t = ti (which we took in the definition of the Ito integral). In particular, in Chap. 6, we will use the so-called backward (or anticipative) integral ff A(T) d.w(T) [89]. The backward integral is the limit of the following integral sums q-1

E A(ti+1) (w(ti+1) - w(ti)) 0

if, of course, the limit exists. Note that this integral differs, in general, from the Ito integral. For example, the backward integral with varying upper limit is not a martingale relative to ,Cit. Considering the integral sums q-1

ES = E 2 (A(ti+i - A(ti)) (w(ti+1) - w(ti))

,

(12.1)

0

we arrive at the Stratonovich integral f o A(T) d3w(T) defined as the limit of these sums (if it exists). (See [30] and [129].) It is easy to see that the Stratonovich integral is equal to a half of the sum of the Ito and backward integrals, provided that all three integrals exist. The Stratonovich integral with varying upper limit can be defined in the standard way. Note, however, that this integral (like the anticipative integral and unlike the Ito integral with varying upper limit) is not a martingale with respect to Bt. The differentials dw, d.w, and dsw (appearing in the definitions of the Ito, anticipative, and Stratonovich integrals) are conveniently called the forward, backward, and symmetric differentials, respectively, referring to the location of the point tin [ti, ti+1] at which the value of A is evaluated. Namely, t = ti in the Ito integral, t = ti+1 in the anticipative integral, and, finally, the integral sums for the Stratonovich integral can be written in the form

S = E 1 A(ti)(w(ti+1) - w(ti-1)) Note that one may formally set d3 = (d+d.)/2. The terminology we introduce here is actively used in Chap. 6. Let us now turn to the formulas relating the values of the three integrals.

54

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

By the definition of the Stratonovich integral, we have 9

1 q-1

9

ES = EI +2

>(A(ti+1) - A(ti)) (w(ti+1) - w(ti)) i=o

The limit of the second sum on the right-hand side is a second-order integral in dA and dw, which can be naturally denoted by ft dAdw. Thus,

t A(r) d,w(r) =

A(r) dw(r) + 1 2

0

dA(rr) dw(r)

(12.2)

0

Similarly, oner can show that i

/

A(r) dw(r)

r t

t

dA(r) dw(r)

.

(12.3)

fo

Note that under a certain extra hypothesis, the Stratonovich integral may be defined as the limit of the following integral sums q-1

E

A

(ti+12 ti(w(ti+1) -'w(ti)) ``

l

i.e., the value of A is taken at the middle point of the segment [ti,ti+1] Remark 12.3. The idea to use the so-called "mid point," as in the definition of the Stratonovich integral sums (12.1), is due to Richard Feynman.

12.D. The Ito and Stratonovich Stochastic Differential Equations Consider a time-dependent field A(t, m) of linear operators on IR' (i.e., for every t E IR and m E IR' , a linear operator A(m, t): IR' -+ R') and a timedependent vector field a(t, m). The integral equation

It

g(t) = e(0) + fot a(r, 6(r)) dr + / A(r, 6(r)) dw(r)

(12.4)

,

where the second term on the right-hand side is the Lebesgue integral, is called an Ito stochastic differential equation. (Here we do not discuss the conditions which should be imposed in order to make all integrals in (12.4) exist.) Equation (12.4) is usually written in the differential form

de(t) = a(t, 6(t)) dt + A(t, ¢(t)) dw(t)

(12.5)

meaning the same as (12.4). A similar equation but with the Stratonovich integral, e(t) = 6(0) +

J

t

a(r, 6(r)) dr +

J

t

A(r, e(r)) dyw(r)

,

(12.6)

12. Stochastic Equations and Integrals

55

is said to be a Stratonovich stochastic differential equation. It can be written in the differential form as well:

de(t) = a(t, fi(t)) dt + A(t, .(t)) d,w(t)

(12.7)

.

In order to define stochastic differential equations on manifolds, we need to describe their transformations under the action of smooth maps of 1Rn. To do this, consider the stochastic process

fi(t) =

t a(rr) drr + fo t A(T) dw(r)

I0 on 1R' (usually called an Ito process, see Sect. 6) and let f :1R' -+ IRn be a (at least, C2-) smooth map. Then we have the following identity (see [30], [50], [54], [55], [95], and [100]), called the Ito formula:

.f (fi(t)) = It

t [i'(ar) + 2tr f"(A(r), A(r))J dr

0

+

J0

t

f' (A(r)) dw(r)

(12.8)

.

Here f and f" denote the first and second derivatives of f , and i=n

tr f"(A(r), A(r)) = j:f"(A(T)ez, A(r)et) d=1

where el i . . . , en is an arbitrary orthonormal basis in IRn. Note that the "non-

tensor" term tr f"(A, A) appears in (12.8) as a result of integrating, by Theorem 12.4, the Taylor expansion of f with respect to (dw)2. If f depends on time t, then (12.8) has to be refined. Namely, the term (8f/8t)ja(,) is to be added to the first integrand. A similar result holds when f : IRn ---* 1R'` and

n#k.

For the Stratonovich integral, we have

t

f (77(t)) = f f' (a(r)) dr + 0

where

rt

17 = J a(r) dr + 0

J0

t

f' (A(r)) dsw(r)

,

(12.9)

rt A(T) d,w(r) J0

The "non-tensor" term does not appear in (12.9) because the Taylor expansion at the middle point is free of the second-order term [30]. Therefore, a smooth map f of IRn gives rise to the transformation of an Ito equation by (12.8), while a Stratonovich equation transforms precisely as an ordinary differential equation.

Remark 12.4. Some other stochastic differential equations (SDEs) and their transformations shall be considered in Sect. 19. (In particular, we shall study

56

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

equations with the backward integral.) Although these equations are of some

interest, we emphasize that the central role in the theory is played by the Stratonovich and the Ito equations. Definition 12.3. The function a(t, m) in the Ito equation and the operator A(t, m) o A* (t, m) are called the drift and diffusion coefficients, respectively. A solution of (12.5) is said to be a diffusion process.

Note that the so-called equations of diffusion-type, which are similar to (12.5), often appear in various problems. In diffusion-type equations, which are, in fact, stochastic equations with delay, the coefficients a and A depend on the entire trajectory a(s), 0 < s < t. Sometimes one also needs to consider the equations with random (i.e., depending on w E ,fl) coefficients. If the coefficient A is smooth, then one can pass from the Stratonovich to the Ito equation and vice versa. To see this, pick a solution C(t) of (12.7). Taking into account (12.2), (12.8), and Theorem 12.4, it is easy to show that f (t) satisfies the Ito equation:

doW = la(t,6(t)) + 2trA'(t,6(t))(A(t,£(t))] dt+A(t,6(t)) dw(t)

.

The Ito equations play a more important role in the theory of SDEs than the Stratonovich equations. In particular, the problem of the existence of solutions is much better understood for Ito than for Stratonovich equations.

12.E. Solutions of SDEs In the theory of SDEs, we distinguish between two types of solutions: strong and weak.

Definition 12.4. The Ito equation (12.5) has a strong solution fi(t), if for any Wiener process w(t) there exists a stochastic process fi(t) defined on the same

probability space as w(t), and nonanticipative with respect to Pt such that the processes fi(t) and w(t) a.s. satisfy (12.4) for every t. Definition 12.5. The Ito equation (12.5) has a weak solution 6(t) if there exists a probability space (.R, .F, P), a nondecreasing family Bt of o-subalgebras of F, a stochastic process 6(t) nonanticipative with respect to Bt, and, finally, a Wiener process w(t) relative to Bt such that (12.4) is a.s. satisfied for all t from some interval. We emphasize that a strong solution is defined on any probability space

that admits a Wiener process and is nonanticipative with respect to the Wiener process. On the other hand, a weak solution may be defined on a single probability space only and, in contrast to the strong one, it may fail to be nonanticipative with respect to the Wiener process. (For more details see, e.g.) [100].)

12. Stochastic Equations and Integrals

57

One can introduce the notion of strong uniqueness for strong solutions of an Ito equation. Namely, the equation is said to have a strongly unique solution if any two solutions coincide almost everywhere. A solution is called weakly unique if any two solutions generate the same measure on the space of trajectories. (See Appendix C and the construction of a Wiener process in Sect. 12.A for more details.)

Definition 12.6. The coefficients of (12.5) have linear growth if there exists a constant K > 0 such that for all t c IR and x E IR" the following inequality holds:

Ila(t,x)II + IIA(t,x)II < K(1 + IIxii) where JJAil is the norm of the operator A.

,

(12.10)

Proofs of the existence of strong solutions are usually based on the contraction map principle. For instance, one may require the coefficients to satisfy (12.10) and a Lipschitz-type condition. Results of this kind cover a large class of equations with stochastic coefficients. (See [54], [55], and [95].) In particular, for (12.5) one can prove the existence of a strong solution of the Cauchy problem for t E IR, provided that the coefficients are smooth and (12.10) is satisfied. Yet, there are other ways to prove existence. For example, sometimes one can derive the existence of a strong solution from the existence of a weak

one. However, it is important to point out that some equations have weak solutions, but do not have strong ones. (See, e.g., [55].) The notion of a weak solution is due to Skorokhod. He also proved the

existence of a weak solution under the hypothesis that the coefficients are continuous and satisfy (12.10). A similar result holds for equations of diffusion type. (See [54], [55], and [95].) According to Krylov's theorem [95], [97], (12.5) has a weak solution if the coefficients are measurable and the operator A(t, m)

is nondegenerate for all t and m (i.e., the diffusion coefficient is positivedefinite). The reader interested in more details should consult [95] for the existence theorems for weak and strong solutions. Note that condition (12.10) guarantees the global (in time) existence of a solution of an SDE. More general conditions for global existence can be found, e.g., in [92]. We shall discuss some of these generalizations in Sect. 13. Local existence theorems claim the existence of a solution on a random

interval and in a bounded neighborhood of the initial value, i.e., until the solution hits the boundary of the neighborhood. It is clear that local existence theorems require a less restrictive hypothesis than (12.10).

12.F. Approximation by Solutions of Ordinary Differential Equations A Wiener process can be approximated by a process with smooth (or piecewise-

smooth) trajectories. In other words, one can approximate an SDE by an ordinary differential equation that depends on the parameter w E S2. In fact,

58

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

there are several ways to carry out this approximation. It turns out that, as a rule, solutions of the approximating equations converge to a solution of the Stratonovich (rather than the Ito) equation. Let w be a Wiener process and u: IR -* (0, oo) a C°°-function supported on [0, 1] whose integral is equal to 1. We set ut(s) = u(s/e)/e. The process

wE(t) = fw(t+s)u(s)ds has a.s. C°°-smooth trajectories. Furthermore, its trajectories converge uniformly on every compact set to the trajectories of w as a -+ 0. Consider the following ordinary differential equation for every w E Q: dtxE(t,w) = a(t,xE(t,w)) +A(t,xe(t,w))

dd w

E

(12.11)

t

Theorem 12.5 (P. Malliavin, see, e.g., [1021). Let a(t, m) and A(t, m) be C2 -smooth in an open bounded domain U C IRn and equal to zero outside U. Then there exists a sequence e2 -+ 0 such that the solutions x;n0 (t, w) of (12.11) with the initial condition x;o(0,w) = mo converge to the solution 1;mp (t, w) of (12.7) with the initial condition

(0, w) = mo. The convergence

is a.s. in w and uniform on any interval [0, 1]. Pick an arbitrary partition q of the interval [0, 1] and consider the piecewise approximation w9 of w given by the formula

wq(t) = (ti+1 - Ow(ti) + (t - ti)w(ti+1) ti+1 - ti

--

ti \ t \ ti+1

,

and the ordinary differential equation

dtxq(t,w) = a(t,xq(t,w)) +A(t,xq(t,w))ddtq

,

w E .fl

.

(12.12)

Under certain hypotheses which guarantee the existence of solutions on [0, 1], it can be shown that the solutions of (12.12) with the initial condition xq , (0, w) = mo converge in measure and uniformly on [0,1] to the solution of (12.7) with the initial condition C,,,o (t, w) = mo (w) as diam q -+ 0. It is clear that one can choose a sequence of partitions qi such that xmo converges to a.s. in w and uniformly in t e [0, 11. Various versions of this result due to McShane can be found in [43].

12.G. A Relationship Between SDEs and PDEs First, let us introduce the following notation: (q', ... , q") are the coordinates on IR' ; A* is the matrix of the operator conjugate to A; and 1;,,,(t) is the solution of (12.5) with the initial condition 1;,,,(0) = m. For the sake of simplicity, we assume that the coefficients of (12.5) are Lipschitz continuous and satisfy (12.10). As a consequence, the solution 1;,,,.(t) is strong and strongly

13. Stochastic Differential Equations on Manifolds

59

unique and exists for t E [0, oo). Pick a C2-function f : IR' -+ IR whose first and second derivatives are bounded. Set u(t, m) = E(f (em(t))).

Theorem 12.6 The function u is a solution of the equation au(t, m)

at

= Lu(t, m)

(12.13)

with the initial condition u(0, m) = f (m). Here, the differential operator L is as follows:

Lu= 1>2 o 2

i,2

a2u aqt aqj

+a u ,

(12.14)

where the matrix (vii) = AA* is the diffusion coefficient of (12.5) and a u denotes the derivative of u in the direction of the vector field a. The proof of the theorem is based on the Ito formula and the properties of the Ito integral. The operator L is said to be the generator of the diffusion process l;. If a and A satisfy a certain regularity hypothesis, L determines the diffusion process uniquely, i.e., any two processes with generator L induce the same measure on the space of trajectories. A more detailed account of the relationship between SDEs and PDEs can be found in various monographs and textbooks. The review of SDEs given in [50, Chap.1] is particularly suitable for the first reading. We conclude this section by mentioning that, by Theorem 12.6, the solutions of a stochastic differential equation can be viewed as characteristics of some kind for (12.13). To illustrate this point, we note that the generator of a Wiener process is A/2, where A is the Laplace operator on W.

13. Stochastic Differential Equations on Manifolds Two methods of defining stochastic differential equations on smooth manifolds

are known to us. The first method is based on the Stratonovich equations, while the other relies on the Ito equations. The fact that the Stratonovich equations transform "correctly" under smooth changes of coordinates makes it easier to extend the notion of a Stratonovich equation to smooth manifolds. This approach was developed in a number of papers. (See [18], [43], and [102] and the Bibliography therein.) The study of the Ito equations was started by Ito [87] and soon led to the discovery of some interesting geometric constructions shedding light on the geometry of stochastic differential equations [13], [15], (51]. In the theory of SDEs on smooth manifolds, as well as in the Euclidean space, the Ito equations are more fundamental objects than the Stratonovich equations. This is why we chose to follow the method using the former.

60

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

Let M be a smooth n-dimensional manifold. Denote the group of linear automorphisms of IR" by L(IR").

Definition 13.1 ([15]). The Ito bundle I(M) is a locally trivial fiber bundle over M such that:

(1) its fiber is IR" x L(IR"); in other words, locally, in a coordinate chart (U, 0), the Ito bundle is the direct product U X (IR'n x L(IRn))

(2) if (U, 0) and (Ul, 01) are two coordinate charts, the transition function on U n Ul is (a, A) '-+ 1 (O1 c 0-1)/a + 2 tr (01 o 0-1)"(A, A), (01 o 0-1)'A/ (13.1)

Remark 13.1. The structural group of I (M), which we call the Ito group, and the principal Ito bundle are described in Appendix D. Definition 13.2. The sections of the Ito bundle are called the Ito equations. In what follows we denote an Ito equation by (a, A). Furthermore, (a.., Am)

(or (a(m),A(m))) is the value of the equation at m E M. (If the equation is time-dependent, then we use the notation (a(t, m), A(t, m)).) Note that in every coordinate chart our notation can be understood literally. (See Definition 13.1.) Observe also that A, can be viewed as a map IR" -' TmM, since A,,, transforms as a linear operator under changes of coordinates (see (13.1)). In a given chart, one may regard a,,, as an element of T,"M; however, this identification depends on the chart. Convenient coordinates and an invariant expression for the Ito equations are given below. Let (a, A) be an Ito equation and w(t) a Wiener process in 1R". Consider the Ito stochastic differential equation d£ (t) = a(e(t)) dt + A(C(t)) dw(t)

(13.2)

in a chart (U, 0). Comparing (12.8) and (13.1), it is not hard to see that (13.2) is well defined on the entire manifold M, i.e., (13.2) transforms "correctly" under changes of coordinates. The notions of weak and strong solutions can be extended word-for-word to equations on a manifold and the local existence theorems continue to hold true. Now we are ready to give an example of a global existence result. Let us equip M with a Riemannian metric and consider the local trivialization of I(M) in a coordinate chart. Define the norms Ila,nI1 and IIA,nIl for an Ito equation (a, A) in the chart as follows. Namely, Ila,,,11 is the Riemannian norm of the vector a in T,,,M which is the first component of (a, A) in the trivialization; IIA,,,Il is the norm of the linear operator A,,,:T,,,M --+ T,,,M

13. Stochastic Differential Equations on Manifolds

61

where T,nM is viewed as a Euclidean space with the Riemannian inner product. Recall that any finite-dimensional manifold admits a Riemannian metric with a uniform Riemannian atlas. (See Sect. 2.)

Theorem 13.1. Let an Ito equation (a(t, m), A(t, m)) be smooth in m E M and continuous in t E [0,oo). Assume also that M admits a Riemannian metric with a uniform Riemannian atlas such that for every ball Vm(r) we have II a(t, m') II < C and II A(t, m') II < C for t E [ 0, oo) and m' E Vm (r), where the constant C > 0 is independent of the chart and the ball. Then for any initial condition 6(0) = mo E M, there exists a unique global solution e(t) of (13.2) defined for all t E [ 0, oo).

Remark 13.2. Theorem 13.1 is a refinement of various classic results on the existence of solutions of stochastic differential equations on manifolds [15], [25], [43], [87], [102]. There is, however, a discrepancy between Theorem 13.1 and these results. In Theorem 13.1, we use the charts of a uniform Riemannian atlas and require the norm of the Ito equation to be uniformly bounded relative to the Riemannian metric on balls of fixed Riemannian radius. On the other hand, in [15], [25], [43], [87], and [102], one assumes that there exists an atlas

which is uniform with respect to the Euclidean distance in the charts and requires the Ito equations to be uniformly bounded relative to the Euclidean distance on balls of fixed Euclidean radius in the charts. The precise assertions of the theorems vary, of course, from case to case but proofs follow the same

line. To give an example of the argument, we outline the proof taken from [25]. The local existence of a strong solution implies that there exists a unique

solution of (13.2) in a chart centered at mo. This solution is defined on the random interval [0, Ti], where Ti is the first Markov time when the trajectory 6(t, w) leaves a ball of fixed radius. If the trajectory never leaves the ball, we set Ti = oo. Then we consider a solution with the initial condition 6(Tl, w) on the interval [T1,T2], where r2 is defined similarly to r , etc. We have supTn = no, n = 1, 2. .... This follows from an estimate of the probability that the trajectory reaches the boundary of a fixed ball in a short time t. The estimate, in turn, results from the hypothesis that the equation is uniformly bounded. The proof of Theorem 13.1 is quite similar to the proof of Theorem 2.2 of [15].

The only modification needed is in the proof of the estimates (Propositions 2.1 and 2.2 of [15]). Namely, one replaces the Euclidean norms in the tangent spaces and the Euclidean distances in the charts by the Riemannian ones. The rest of the proof goes through since V,,,,,(r/2) C ,,,(r) for m' E V,n(r/2).

Remark 13.3. Theorem 13.1 is a quite general result on the existence of solutions on the interval [ 0, oo), i.e., on the completeness of the equation. Let us show, for instance, how it can be applied to obtain a version of the completeness theorem for equations in IRn (see Sect. 1.B, where we studied the completeness of ordinary differential equations). Let the Wintner-type condition IIa(t,m)II +IIA(t,m)II
62

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

hold for the Ito equation (12.5), where 11-11 is the Euclidean norm in 1R", t E [ 0, oo), m E IR", and the function L: [ 0, oo) --+ (0, oo) is continuous and satisfies (1.2). For example, if L(u) = K(l+u), K > 0, then (13.3) turns into (12.10), the linear growth condition. Let us make L smooth, while retaining condition (1.2) (see the proof of the corollary of Theorem 1.3) and pass to

the Riemannian metric given by (1.3). In this metric (13.3) means that the equation is uniformly bounded for all t E [ 0, oo) and m E IR". It is clear that IR" with the metric (1.3) possesses a uniform Riemannian atlas. Therefore, it follows from Theorem 13.1 that the equation on IR" is complete, provided that we have (13.3). Employing methods of differential geometry, one can describe the It6 equations in terms of sections of a simpler fiber bundle over M. Denote by L(M) the fiber bundle with the fiber L(IR"). More precisely, in a coordinate chart (U, 0), the bundle L(U) is given as U x L(IR"). On the intersection U fl U1 of two charts (U, ¢) and (U1, 01), the transition function is

A H (01 o q-1) A. In other words, the elements of L(M) transform as tangent vectors with respect to coordinate changes.

Definition 13.3 (See [15]). The sections of the Whitney sum TM ® L(M) are called the Ito vector fields. We denote by (a, A) an ItO vector field and by (am, A,) or (a(m), A(m)) its value at m E M. Similarly, the value is (a(t, m), A(t, m)) when the field is time-dependent. Note that a is a well-defined vector field on M, i.e., am E TmM and A is a field of linear operators Am,:1R" -+ TmM. Once a connection on M is fixed, one can identify the ItO equations and the Ito vector fields making use of the geometry of the "secondary" tangent bundle TTM. To see the identification, consider the vertical subbundle of TTM and the Euclidean connection HO defined in a chart (U, ¢). (See Appendix A, the section on connection coefficients.) Recall that for every Z E T(m,x)TM we have the decomposition Z = Zl + Z2, where Z1 E H X) and Z2 E V m,x) As one passes to another chart (U1, 01), the term Z2 transforms as follows:

Z2 H (01 o

0_1)+Z2

+ (01 o 0-1)" (X, Z1)

.

(13.4)

In particular, if X = Zl = (1/v )Aodw(t), (13.4) turns into the transformation law for the first term of an ItO equation. Let (a, A) be an Ito equation. Using the trivialization in the chart (U, 0), we identify I(M) and TM ® L(M). Therefore, am, for m e U, becomes an element of TmM. Denote its vertical lift to V(,,,,,x) by h1m x). Let us define the random processes pm(t) and Y(t) in TmM and T(m,x)TM, respectively,

13. Stochastic Differential Equations on Manifolds

63

by the formulas dem (t) = 1 Am dw(t)

(0) = 0

and

Y(t)=Y1(t)+Y2(t) where Y 1 (t) in H

>

,

and Y2(t) in V(,,,,,X) are random processes such that

dYi(t) = ±Ttr-1(Am dw(t)) and

dY2(t) = &(,.,X) dt

.

Finally, let us fix a connection H and consider the process Yl,,,,£ltll(t). Note that (t) = YH(t) +Yv(t), where YH is a process in H and Yv is a process in V. By definition, we have Trr dYH = Am dw/s and dYv = a;,i dt, where am. E TmM. Making use of (13.1), (13.4), and Theorem 12.2, it is not hard to show that dY(m E( m,)) transforms as an element of TTM when we pass to another coordinate chart. Hence, a is a vector field on M and thus (a, A) is

an Ito vector field. Applying this construction in every coordinate chart, we obtain an Ito vector field defined on the entire manifold M.

Definition 13.4. An Ito equation (a, A) and the Ito vector field (a, A) constructed above are said to be canonically corresponding to each other with respect to the connection H. Remark 13.4. Recall that the construction of the canonical correspondence involves the secondary tangent bundle TTM. In fact, there is a close relationship, discovered by Laurent Schwartz and having a very general nature, between stochastic processes and second-order geometric objects on manifolds. A more detailed discussion of this question can be found in [45], [106], [107], [121], and [122].

Let us now turn to a convenient construction, due to Belopolskaya and Dalecky [15], [16], [26], describing the Ito equations in terms of the Ito vector fields and connections.

Definition 13.5. The stochastic differential (a,,,,, ^'A,) at m E M given by an Ito vector field (a, A) is the set of stochastic processes in TmM formed by the solutions of all stochastic differential equations

X(s)=Jea(T,X(T))dT+J9A(T,X(r))d'w(r) 0

0

where &(s, X) is a vector field on T,,,,M and A(s, X): 1R' -+ T,,M is a linear operator depending on s E 1R and X E T,.M. We assume a(s, X) and A(s, X)

64

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

to be Lipschitz, vanish outside a neighborhood of the origin in that a(s, 0) = and A(s, 0) =

and such

Remark 13.5. Since &(s, X) and A(s, X) are Lipschitz, X (s) is a strong solution of the equation above, i.e., X(s) exists for any Wiener process in R'.

Let exp be the exponential map of a fixed connection H on M.

Definition 13.6. A process e(t) is said to satisfy the equation (13.5)

4(t) = expE(t) (aE(t), -Af(t))

if for every t there exists a neighborhood of l(t) such that the process (t+s), s > 0, a. s. coincides with a process from the set exp£(t)(a£(t), AE(t)), as long as 1;(t + s) belongs to the neighborhood. If f : M -+ N is a C2-map and exp on N is such that f (exp X) = exp(T f o X) for all X E T..M, then we have (see [15]) df (E(t)) = expf(E(t))(Tf o af(E(t)), T f o A(E(t)))

for any solution l; (t) of (13.5).

Theorem 13.2. Let (a, A) be the Ito vector field canonically corresponding to an Ito equation (a, A). Then 1; satisfies (13.2) if and only if it satisfies (13.5).

Outline of the Proof. Let Fm(. , ) be the local connector of H in a coordinate chart (U, 0). Then we have 1 &,n=am- 2trI...(A,n,A,n)

This follows from the definition of Fn (Appendix A), Definition 13.4, and Theorems 12.2 and 12.4. Therefore, (13.2) can be written as dl; (t) = ae(t) dt - 2 tr f (t) (A, A) dt + A£(t) dw(t)

(13.6)

in the chart (U, 0). Recall that in (U, ¢) the exponential map is given (up to highe-order terms) by the following formula [41]: (13.7)

Theorems 12.2 and 12.4 together with (13.7) imply that 1

expE(t) (af(t), -Af(t)) = e(t) + at(t) dt - 1 tr I''(t)(A, A) dt + Af(t) dw(t) + .

,

(13.8)

13. Stochastic Differential Equations on Manifolds

65

where the dots denote the terms of order o(dt). Now it is easy to complete the proof by comparing (13.6) and (13.8).

Note that Theorem 13.2, as well as (13.5), holds even if the field (a, A) is time-dependent. A more detailed discussion of (13.5) can be found in [15], [16], and [26].

Remark 13.6. The idea to introduce the Ito equations via (13.6) goes back apparently to [13] and [51]. Here we emphasize that the right-hand side of (13.6) is covariant with respect to changes of coordinates. Let us now turn to the existence theorem for (13.5). First, we need the following definition.

Definition 13.7. A connection H is said to be compatible with a metric (,) if the metric admits a uniform atlas such that the local connection coefficient F(X, X), as an operator of X, has a norm bounded by Co > 0 on the balls of the atlas; moreover, the constant Co > 0 is independent of the balls and the charts.

It is clear that any Riemannian metric on a compact manifold is compatible with any connection. Another example of the compatibility is a Lie group G with an arbitrary left-invariant metric and an arbitrary left-invariant connection. To see this, note that such a metric admits a uniform Riemannian atlas, which can be constructed as follows. Namely, we fix a chart in a neighborhood of the unit element and then cover G by left-translations of the chart. It is not hard to show that the norm of the local connector in every chart is independent of the chart and, therefore, the atlas is indeed uniform. The same remains true for right-invariant metrics and connections. As above, let us fix a connection H on M and denote its exponential map by exp.

Theorem 13.3. Let (a(t, m), A(t, m)) be an Ito vector field which is smooth

in m E M and continuous in t E [0,no). Assume that on M there exists a Riemannian metric compatible with H such that 1Ia(t, m)JI < Cl and IIA(t, m) 11 < C1, where C1 > 0 is a constant independent oft and in. Then for

any initial condition (0) = mo, there exists a strong solution 1'(t) of (13.5). The solution is strongly unique and extends to [ 0, no) .

Applying (13.6) and Theorem 13.2, one can derive Theorem 13.3 from Theorem 13.1.

Remark 13.7. Note that the hypotheses of Theorem 13.3 hold for every smooth autonomous Ito vector field on a compact manifold equipped with an arbitrary connection H.

66

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

Let us state one more result, needed in Sect. 14, on the existence of solutions of an equation with stochastic coefficients. Consider a complete probability space (Q, F, P) and the following collection of data: (1) a nondecreasing family 13t of complete o-subalgebras of the a-algebra F, where t E [0, 1] and I > 0; (2) a Wiener process w(t) relative to Bt; (3) stochastic processes a(t) and A(t) with values in 1Rn and L(1Rn), respectively, which are nonanticipative with respect to Bt; (4) a connection H compatible with the metric (,); and (5) a field Em: IRn -p TmM of linear operators which is smooth in m and has a bounded (,)-norm.

Assume that the norms of a(t) and A(t) are a.s. uniformly (in t) bounded. Theorem 13.4. For any initial condition 1= (0) = mo E M, the equation de(t) = expE(t) (EE(t) (a(t)), -EE(t) (A(t)))

has a solution l;(t) on [0, 1]. The solution is strongly unique, nonanticipative with respect to Bt, and has a.s. continuous trajectories. To prove the theorem, one uses (13.6) to reduce the equation to (13.2) and then argues in the same way as in the proof of Theorem 2.2 of [15], taking into account the assertion of Remark 13.3.

Remark 13.8. The method used here to introduce stochastic equations on manifolds by means of the field E,,,, was originally developed for Stratonovich equations in [43], where E,,,, together with a stochastic process on 1Rn were called a stochastic dynamical system. In [43] one can also find various versions of existence theorems implying, in particular, Theorem 13.4.

Remark 13.9. The hypotheses of Theorems 13.3 and 13.4 can be relaxed. Namely, instead of requiring the metric and the connection to be compatible, one may assume the following inequality to hold uniformly on the balls V,,,(r) contained in the charts of a uniform Riemannian atlas: I I tr F,,, (A(t, m ), A(t, M')) II < C2

,

where m' E V,...(r) and the constant C2 is independent of the ball. All the constructions carried out above can be easily extended to the fiber bundle whose fiber is the space L (IR" ,1l ) of linear maps from 1R' to IRn

with k < n. Note that a Wiener process w(t) with values in IRk must then be used in (13.2). The Belopolskaya-Dalecky construction remains unchanged and the existence theorems continue to hold.

14. Stochastic Parallel Translation

67

In conclusion, let us describe a class of stochastic differential equations on infinite-dimensional Hilbert manifolds to be used in Chap. 9. Equations of this type were studied in [15]. (See also [43], for some results on Stratonovich SDEs on infinite-dimensional manifolds.) Let M be a smooth manifold modeled on a separable Hilbert space, H a connection on M, and G(., ) a strong Riemannian metric on M. (The latter means that on every tangent space G(., ) gives rise to a topology equivalent to that of the model space. See, e.g., [99] for definitions.) Definitions 2.1 and 13.7 can be extended word-for-word to this case. Let a(t, m) be a vector field on M and A(t, m): 1R" --* T..M a field of linear operators, where the Euclidean space 1R" carries a Wiener process w(t). In

the same way as in the finite-dimensional case, we call the pair (a, A) an Ito vector field. Equation (13.5) continues to make sense for (a, A) and the exponential map exp of H still exists. To facilitate further references, we state the existence of a solution of (13.5) as a theorem.

Theorem 13.5. Let G, H, a, and A be as above. Under the hypotheses of Theorem 13.3, for any initial condition mo E M, there exists a strong solution e(t) of (13.5) with l;(0) = mo. The solution is strongly unique and extends to the interval [0, oo).

We emphasize that, in particular, the theorem holds if the Hilbert space is just a finite-dimensional vector space IRn (and, thus, M is finite-dimensional). Similarly to Theorem 13.3, Theorem 13.5 can be derived from Theorem 13.1. Recall (see Remark 13.3) that Theorem 13.1 can be verified in a similar way as Theorem 2.2 of [15], originally proved in the infinite-dimensional setting. The assertion of Remark 13.3 remains true in the infinite-dimensional case.

14. Stochastic Parallel Translation and the Integral Formalism for the Ito Equations In this section, we develop an integral formalism needed to deal with stochastic Ito equations on manifolds. Our method, which is analogous to that discussed in Sect. 12, makes a systematic use of stochastic integral operators with parallel translation. In order to define such operators, we refine the constructions

of parallel translation and the Cartan development to take into account the fact that the trajectories need not be smooth, the right-hand side of the Ito equations fails to be a tensor, and other similar phenomena [68], [69].

Let 7r:O(M) --+ M be the orthonormal frame bundle over M, H the Levi-Civita connection on it, and V the vertical distribution on O(M). Recall that the bundles H and V over O(M) are trivial (Appendix A). In effect, V is trivialized by fundamental vector fields, i.e., by the vector bundle isomorphism

0:O(M) x o(n) -+ V. In turn, the trivialization of H is given by means of basic vector fields, i.e., by the vector bundle isomorphism E: 0 (M) x 1R" -. H,

68

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

where Eb(x) = T,7r-1(bx)IHb, b E O(M), and x E W. As a consequence, the tangent bundle TO(M) = H ® V is trivial and, moreover, TO(M) carries a natural trivialization (once the metric on M and a basis in o(n) are fixed).

Definition 14.1. The metric arising from this trivialization is said to be an induced Riemannian metric on the total space O(M). Remark 14.1. It is easy to see that the tangent map Tlrb induces an isometry of Hb and T,(b)M (for any induced metric). The restriction of an induced metric to V comes from an inner product on the Lie algebra o(n). It is known (see, e.g., [17]) that the integral curves of an autonomous basic or fundamental vector field are geodesics of the induced metric on O(M) and, vice versa, all vertical and horizontal geodesics arise this way. Note, however, that integral curves of a constant linear combination of basic and fundamental vector fields may fail to be geodesics. Recall also (see [17] or Appendix A) that the integral curves of autonomous basic vector fields, and only those curves, are the horizontal lifts of geodesics on M. Fix an induced metric on O(M) and denote by e its exponential map.

Lemma 14.1. (i) The restriction el H is independent of the induced metric. (ii) For any Y E H, we have ire(Y) = exp(T7rY), where exp is the exponential map of the metric on M. (iii) Let re be the local connector in a chart on O(M). Then the restriction of 1-+e to H, regarded as the operator X H fe(X, X), is independent of the induced metric. Assertions (i) and (ii) follow from the above-mentioned properties of integral curves of basic vector fields. To prove (iii) observe that f6(X, x) is the second derivative of the horizontal geodesic with initial condition X and this derivative is independent of the induced metric.

Definition 14.2. A Riemannian manifold M is said to be uniformly complete if the following two conditions are satisfied:

(1) there exists an induced metric on O(M) which possesses a uniform Riemannian atlas; and (2) on the balls Vb(r) of the atlas, the norm of the operator X ' r b, (X, X), where X E Hb, and b' E Vb(r), is bounded by a constant C > 0 independent of the chart and the ball. Here are some examples of uniformly complete manifolds: a compact Riemannian manifold (then O(M) is compact), a Lie group with a (right-) left-

14. Stochastic Parallel Translation

69

invariant metric. In the latter case, the desired atlas is obtained by left translations of the charts in O(M) lying over a neighborhood of the unit matrix. Consider a complete probability space (.fl, F, P), a nondecreasing family of complete Q-subalgebras St of F, and a Wiener process w(t) relative to B. Pick a point mo E M. Let a(t) and A(t), t E [0,1], be stochastic processes on (,fl, .77, P) with values in T,,,oM and L(1R',T,,,OM), respectively. We assume that these processes are nonanticipative with respect to 13t and a.s. uniformly bounded by a constant C > Fix an orthonormal basis b in T,,,0M and denote by b the corresponding linear operator IR" -> T,,,0M. Clearly, (E(b-1a(t)), E(b-1A(t))) is a welldefined Ito vector field on O(M).

Theorem 14.1. Let M be uniformly complete. Then, under the hypothesis above, for any initial condition 6(0) = bo, there exists a unique strong solution of the stochastic equation

d6(t) = q(t) o (Ee(t) (b-'a(t)), Ee(t) (b-1A(t)))

.

(14.1)

The solution l; extends to [0, 1].

Note that although (14.1), being written in coordinates, involves the local connection coefficient r,, the equation is independent of the induced metric on O(M) by Lemma 14.1. In particular, one may use the induced metric introduced in Definition 14.2 (see Definition 13.7). Thus, Theorem 14.1 follows from Theorem 13.4.

Throughout the rest of this section, we assume that M is uniformly complete and, therefore, Theorem 14.1 is applicable. Let 1;(t) be a solution of (14.1) with the initial condition e(0) = b.

Definition 14.3 ([68], [69]). The process S(a(r), A(-r)) (t) = 7rl:(t) is called the stochastic Ito integral with parallel translation of (a(t), A(t)). The process fi(t) is said to be the horizontal lift of the process S(a(r), A(r)) (t) with the initial value b.

Lemma 14.2. The stochastic Ito integral S(a(r), A(r)) (t) is independent of the initial value of the horizontal lift.

Proof. Let b E Omo (M). Since 0,,,(M) is isomorphic to O(n), there exists tc E O(n) such that b = b o K. It follows from the definition of E, Theorem 14.1, and the invariance of H with respect to the right O(n)-action that t(t) _ fi(t) o x is the only solution beginning at b of the equation do(t) = et(t) o (EE(t) (b-'a(t)), E{(t) (b-'A(t)))

It is clear that ire = ire, i.e., b(t) is the horizontal lift of S (a(r), A(r)) (t) with the initial value b.

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

70

Lemma 14.3. For every t E [0, l], the process S(a(r), A(T)) (t) a.s. satisfies the equation dS(a(r), A(T)) (t) = exp,,rEltl (T-7rEE(t) (b-la(t)), T7rEE(t) (b-1A(t)))

The lemma follows from Lemma 14.1 and the definition of S (a(T), A(T)) (t).

Remark 14.2. If M is the Euclidean space, then all tangent spaces to it can be canonically identified with M and, as a result, we have S(a(T), A(T)) (t) = I

a(T) d-r + 0

I0

A(T) dr

.

Remark 14.3. Alternatively, the process S(a(T), A(T)) (t) can be defined similarly to the Ito integral with varying upper limit. Namely, for a partition q = (0 = t1 < t2 < . . . < tq), we define a process q(t) beginning at b as follows. On [0,t1], it is formed by the integral curves of the field E[b-1(a(0) + A(0)w(t1))]

with the initial condition b. Then on [tl, t2]-by the integral curves of the field

E[b-1(a(tl)(t2 - t1) +A(tl)(w(t2) - w(tl)))] with the initial condition q(t1), etc. It is clear that this construction leads to a process q(t) defined on [0, l]. The trajectories of are piecewise geodesic. The sequence of processes converges uniformly in t to 7r£ as diam q - 0. In particular, contains an a.s. uniformly convergent subsequence. The detailed proof is omitted, since it is similar to the argument of [51] and to its generalization given in [28]. Observe that the process

z(t) =

J0

t

a(r) dr + J tA(T) dr 0

is nonanticipative with respect to Cit. Retaining the terminology of [54], we call z(t) an Ito process in T,,,0M.

on [0, 1]

Definition 14.4 ([68], [69]). The operator RI sending the process z(t) in T,,,0M to the process Rjz(t) = S(a(r), A(T)) (t) is called the Ito development.

Remark 14.4. This operator is a stochastic version of the classic inverse Cartan development (see Remark 3.1). Note also that the Ito processes in T,,,,oM are stochastic analogs of smooth curves to which the Cartan development applies. It follows from Remark 14.3 that RI is an extension of the

14. Stochastic Parallel `Translation

71

inverse Cartan development from the set of piecewise smooth curves to the set of almost all trajectories of z(t). These trajectories are continuous, but a.s. nonsmooth. (See [51].)

Remark 14.5. A similar construction using the Stratonovich equations on O(M) is known as the Eells-Elworthy development. (See [43] for details.)

Definition 14.5. A stochastic process in M is called an Ito process if it is the development of an Ito process in a tangent space. By analogy with the deterministic case, one can define the parallel translation along an Ito process. Namely, let rt(t) be an Ito process, fi(t) its horizontal lift to O(M), e(0) = b, and v E T,,,0M a random vector.

Definition 14.6. The vector (fi(t) o (b-lv)) E T (t)M is said to be obtained by the parallel translation of v along It is clear that the parallel translation is well defined (i.e., independent of the lift ) and preserves the Riemannian norm and the inner product. By definition, the vector T1rEE(t) (b-la(t)) and the operator T1rEE(t) (b-1A(t)) are parallel to a(t) and A(t), respectively, along 77(t) = S (a(r), A(r)) (t) (see Lemma 15.3).

Remark 14.6. Using the alternative definition of the stochastic Ito integral given in Remark 14.3, one can prove the following results. Namely, there exists a sequence rln of processes with piecewise geodesic trajectories such that the parallel translation along 77 is the limit of parallel translations along r7n. Thus,

the parallel translation in the sense of Definition 14.6 is an extension of the standard parallel translation along piecewise smooth curves to the translation along a.s. all trajectories of 77 (which are continuous but a.s. nonsmooth). Various definitions of parallel translation on manifolds can be found in [16], [36], [43], [88], [89], and [102]. Note that parallel translation in the sense of [88] along local quasi-martingales with continuous trajectories (see [101] for a definition) can also be approximated by parallel translations along processes with piecewise geodesic trajectories. This yields that the same is true for parallel translations along the Ito processes. Let fi(t), t E [0, 1], be an Ito process and (a(t, m), A(t, m)) an Ito vector field on M. Denote by Fa(t, fi(t)) the random vector in TE(o)M that is parallel to a(t, e(t)) along . Similarly, FA(t, e(t)) is the random linear operator IR" TE(o)M parallel to A(t, fi(t)) along .

Assume that a and A are both Borel measurable jointly in all variables and their norms are bounded from above by a constant C > 0. Then the norm of fa(t, fi(t)) and I'A(t, e(t)) is a.s. bounded by C. These processes are nonanticipative with respect to the family Bt of o-subalgebras corresponding

72

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

to fi(t). This follows from the properties of the horizontal lift of a strong solution of (14.1). Therefore, the Ito process S(Fa(T, c(r)), PA(T, 6(T))) (t) exists.

Definition 14.7. The process S(Pa(T, (r)), PA (r, l:(r))) (t) is said to be the line Ito integral with parallel translation of (a, A) along 1;(t).

Remark 14.7. Taking Remark 14.2 into account, it is easy to see that the line Ito integral is analogous to the standard line integral used in the theory of the stochastic Ito equations on IR'. In fact, if M is the Euclidean space, then T is the identity map and 8(Pa(r, l-(r)), PA(r, 1-(T))) (t) = f a(r, £(T)) dT + fot A (r, (r)) dw(r) 0t

Like its classical analog, the line Ito integral is closely related to the Ito equations.

Theorem 14.2. Let (a(t, m), A(t, m)), t E [0, l], be an Ito vector field on a uniformly complete manifold M. Assume that the field is smooth in m, continuous in t, and has a uniformly bounded norm. Then for any mo E M the strong solution 77(t) of the equation

di7(t) = exp,i(t) (a(t, r7(t)), A(t,'7(t)))

,

17(0) = mo

,

(14.2)

is an Ito process satisfying the equation

77(t)=S(Pa(T,77(T)),PA(T,77(T)))(t)

.

(14.3)

Proof. First, observe that the solution 77(t) does exist by Theorem 13.3. Let us construct its horizontal lift as follows. Consider an Ito vector field (a(t, b), A(t, b)) on O(M) such that

a(t, b) = Ta-la(t, irb) E Hb and A(t, b) = T,7r-1A(t, irb)IH, By definition, the field (a, A) is smooth in b E O(M), continuous in t E [0, l], and bounded with respect to the distinguished Riemannian metric on O(M), which exists, since M is uniformly complete. By Theorem 13.3, for any b E 0,,,.(M), there exists a unique strong solution fi(t) of the equation

do(t) = ea(t) o (a(t, fi(t)), A(t, fi(t))) By definition, l(t) is the solution of (14.1) with

a(t) = and

o a(t,

E T,,,M

14. Stochastic Parallel Translation

73

A(t) = 6(1(t)-1 o A(t,ir (t))) E L(IR",TmoM) . Thus, 1(t) is a horizontal lift of q(t). Therefore, a(t) = I'a(t, 77(t)) and A(t) _ I'A(t,77(t)) and, as a consequence, 77 satisfies (14.3).

Equation (14.3) is analogous to the integral form of the Ito equation for the Euclidean space (Sect. 12.D). Similarly to Definitions 12.4 and 12.5, one may introduce strong and weak solutions of (14.3).

Theorem 14.3. Let 77(t) be a weak solution of (14.3). Then 77(t) is also a weak solution of (14.2).

Proof. Let fi(t) be the horizontal lift of 77(t) with &) = b E 0,(M). it follows from the definition of S and (14.1) that for any t E [0, 1] we a.s. have a(t, r!(t)) = T7rEE(t) (b-1I'a(t, rl(t))) and

A(t,71(t)) =TirEE(t)(b-1FA(t,rl(t))) Therefore, by Lemma 14.3, equation (14.2) a.s. holds for every t. The other requirements of Definition 12.5 are fulfilled due to the assumption that 77(t) is a weak solution of (14.3). Corollary. Let 77(t) be a strong solution of (14.3). Then 77(t) is a strong solution of (14.2).

Proof. Arguing in the same way as in the proof of Theorem 14.3, one can show that 77(t) is a solution of (14.2). Furthermore, it is easy to see that all the requirements of Definition 12.4 are satisfied. Thus 77(t) is a strong solution of (14.3).

As for ordinary differential equations, employing integral operators with parallel translation, one can reduce certain problems concerned with SDEs on manifolds to studying SDEs on a single tangent space. Let (a(t, m), A(t, m)), t E [0, 1], be a time-dependent Ito vector field on M. Consider the following stochastic differential equation on T,,,0M:

z(t) =

J0

t

I'a(T, Riz(Tr)) dr + J t FA (7-, Rrz(7 )) dw(T)

.

(14.4)

0

Theorem 14.4. An Ito process z(t) in Tm0M is a strong (weak) solution of (14.4) if and only if 77(t) = Rjz(t) is a strong (respectively, weak) solution of (14.2).

74

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

Proof. Assume that a Wiener process w(t) in IR' and the Ito process z(t) =

I

t a(rr) dr + J t A(rr) dw(r) o

on Tm0M are defined on a probability space ((2, 3:, P) and that they satisfy (14.4). By the definition of E, Definition 14.5, and Lemma 14.3, Riz(t) a.s.

satisfies (14.2) for all t. Conversely, let z(t) and w(t) be such that for all t the development rl(t) = Rrz(t) a.s. satisfies (14.2). By the definition of development, there exists the horizontal lift %(t), l (0) = b E O,no (M), of 77 (t). Hence, the parallel translation along is defined. As follows from Lemma

14.3 and (14.2), we as. have T7rEE(t) (b-la(t)) = a(t, i7(t))

and

T7rEE(t) (b-la(t)) = a(t, ij(t))

for all t. Applying to these equalities the parallel translation to mo along 77(), we see that a(t) = ra(t, ii(t)) and A(t) = PA(t, i(t))

a.s. for all t. Therefore, z(t) and w(t) satisfy (14.4). It is not hard to show that z(t) and w(t) are nonanticipative with respect to the family Pt of Qsubalgebras in the case of a strong solution, and with respect to 3t in the case of a weak solution. To illustrate how (14.4) can be applied, we prove the existence of a weak solution of (14.2).

Theorem 14.5. Let (a(t, m), A(t, m)), t E [0, 1], be an Ito vector field on a uniformly complete Riemannian manifold M. Assume that (a, A) is jointly continuous in all variables and has a uniformly bounded norm (with respect to the Riemannian metric on M). Then equation (14.2) has a weak solution for any initial condition mo E M.

Proof. Here we use the martingale approach to the construction of solutions [54], [55], [95]. Certain preliminary work has to be done to take into account the particular features of (14.2). Let us approximate (a, A) by a sequence (ai, Ai) of smooth Ito fields converging to (a, A) uniformly on [0, 1] x M. Observe that the norms of (ai, Ai) are uniformly bounded by a common constant because (a, A) has a bounded norm. Let rli be the strong solution of the equation

drli(t)=exp,,,(t)(ai(t,rli(t)), Ai(t,rli(t)))

,

71i(0)=m0

.

By Theorem 13.3, the solution 77i exists. It follows from Theorems 14.2 and 14.4 that the process t

t

zi (t) = f rai (T, ryli (T)) dT + fo rAi (T, 74 (T)) dw (T) 0

14. Stochastic Parallel Translation

75

is a strong solution of the equation

zi(t) = Zot rai (r, RIzi(r)) d7- + f TAi (T, RIzi(r)) t o

Set .fl = C°([0,1],T,0M), as above, and let P be the a-algebra in h generated by cylinder sets and Bt, t E [0, 1], the a-algebra generated by cylinder sets with bases over [0, t] (see Sect. 12.A). Recall that all a-algebras are completed by the sets of measure zero. Denote by pi the probability measure on

(fl, ) generated by zi. It is clear that the elementary events of 02, .:, µi) E C°( [0,11,T,,,M) and that zi may be regarded are continuous curves as a coordinate process on (,fl, :, pi), i.e., zi (t, x) = x(t). Furthermore, zi is nonanticipative with respect to Cit. For any x E (, the processes Rix(t) and TY(Rix(t)), where Y E TR,.(t)M, µi-a.s. exist. This follows from Remarks 14.4 and 14.6, since RI is a (µi-a.s.) extension of the inverse development from the space of smooth curves to almost all continuous curves (i.e., to the trajectories of zi) and r is a similar extension of the parallel translation. Note that the latter holds for all i simultaneously. In other words, for all i and j, the processes rah (t, Rix(t)) and TA3 (t, Rjx(t)) are well defined on a set of full pi-measure. By the properties of parallel translation, for every i the sequence Tai (t, Rix(t)) pi-a.s. converges to ra(t, Rix(t)) as j --+ oo. The same is true about the convergence of TAB (t, Rjx(t)) to rA(t, RIx(t)). Both sequences converge uniformly in t. Observe that the set of measures pi is weakly compact. (One can easily prove this using the fact that the fields (ai, Ai) are uniformly bounded by a common constant.) Let p be a limit measure of the µi's. By definition, the coordinate process

z(t, x) = x(t) on (,fl,:, p) is nonanticipative with respect to tat. Applying Prokhorov's theorem (see, e.g., [124]), it is easy to show that Tai (t, Rrx(t)) and TAB (t, Rjx(t)) are well defined on a set of full p-measure and p-a.s. converge to Ta(t, Rjx(t)) and TA(t, Rjx(t)), respectively. As above, the convergence is uniform in t. The argument can be completed just like the proof of the classic result on the existence of a weak solution of an equation with continuous coefficients [54]. Namely, making use of the above convergence, one

constructs a Wiener process w(t) on (,fl,.:, p) such that w(t) is nonanticipative with respect to Bt, and w and z a.s. satisfy (14.4) for all t. By Theorem 14.4, the process Rjz(t) is a weak solution of (14.2).

Remark 14.8. Nowhere in this section have we used the fact that the LeviCivita connection has vanishing torsion. Therefore, all results we have proved remain true for any Riemannian connection if, of course, we replace the hypothesis that the manifold is uniformly bounded by a suitable condition on the connection. In particular, for a certain connection on a Lie group, our construction leads to the so-called multiplicative integral.

Remark 14.9. Let (a, A) be an Ito equation. Different Riemannian metrics and connections lead, in general, to different Ito vector fields corresponding

76

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

to (a, A) in (14.2). The integral operators and the representations of (a, A) in the integral forms (14.3) and (14.4) also depend on the metric. However, the solution of (14.2) or (14.3) is independent of the metric, the connection, etc. In fact, the solution satisfies (13.2), the equation (a, A).

15. Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations We start this section with a construction of a Wiener process on the tangent space to a Riemannian manifold and then use it to define a Wiener process on the manifold itself. In the course of our work, we refine the notion of a stochastic differential equation to make it more suitable for applications in Chaps. 5 and 6. We also refine the constructions of integral operators given in Sect. 14 and prove some existence results. Note that the equations we are going to deal with are more general, i.e., describe a larger class of processes than those of Sect. 13. In particular, this class contains diffusion processes with a nondegenerate diffusion coefficient. It is worth noticing that there is a topological obstruction to describing such processes purely by means of the notions introduced in Sect. 13. Namely, if TM is nontrivial, then for an arbitrary Ito equation (a, A), or an Ito vector field (a, A), the field of operators A degenerates at least at one point mo E M (i.e., the operator Am,,:IR" -* Tm0M fails to be invertible at mo).

15.A. Wiener Processes on Riemannian Manifolds Let M be a Riemannian manifold and H the Levi-Civita connection on O(M).

Consider a Wiener process w(t) in W and the "basic" stochastic process Eb(w(t)) on Hb, b E O(M) (Appendix A). Observe that the field of processes E(w(t)) is smooth on O(M), i.e., obtained from w by means of the smooth

map E:O(M) x IR' -* H. For every m E M, a frame b E b:1R' --+ T bM (Appendix A).

(M) can be regarded as a linear operator

Definition 15.1. The process T-7rEb(w) = b o w is called a realization of the Wiener process w in or simply a Wiener process in T,,.bM.

A realization of w in T,,,M gives rise to the standard Wiener process in T,,,,M, i.e., a measure on the space of continuous curves in T,,,M (see Sect. 12.A).

Theorem 15.1. The standard Wiener process in T nM is independent of the choice of w(t) on ]R7' and b E O,n.(M).

15. Wiener Processes on Riemannian Manifolds

77

Proof. Since b is an orthogonal operator, b o w is a Wiener process in the Euclidean space T,,,M with the inner product given by the Riemannian metric. Thus, the measure determined by b o w on the space of curves in T,,,M is the Wiener measure with respect to this inner product. Let bl, b2 E 0,,,.(M). It is clear that bl and b2 differ by an orthogonal operator on T,,,M. The theorem

follows, since the Wiener measure is invariant with respect to the group of orthogonal transformations. Thus, once a Riemannian metric on M is fixed, we have a well-defined standard Wiener process in every tangent space to M. Furthermore, this field of Wiener processes is smooth, i.e., obtained from the standard Wiener process on IR' by means of a smooth linear transformation, namely, by means of TirE. We denote the Wiener process on T,,,M by w,,, or just by w when no confusion may arise. The realization of w in T,,,M obtained with the use of b is denoted by bow.

Definition 15.2. The development RIw,,,a (t) of a Wiener process w,,,o in T,,,0M is said to be a Wiener process on M beginning at mo E M. Remark 15.1. The definition of a Wiener process on M as the development of a Wiener process in a tangent space is due to Eells and Elworthy. (See [43] and the Bibliography therein.) Definition 15.3. A Riemannian manifold M is said to be stochastically complete if for any mo E M, every Wiener process beginning at mo a.s. extends to [ 0, oo).

On a stochastically complete manifold a Wiener process beginning at mo gives rise to a measure on the space of continuous curves in M which, in turn,

begin at mo. It is not hard to see that this measure is actually independent of the choice of the Wiener process. The coordinate process on the space of such curves is just the standard Wiener process on M beginning at mo. On the other hand, the measure on the space of curves is not uniquely defined if with a;nonzero probability the development RIw goes to infinity in finite time. In other words, the measure depends on the behavior of RIw at infinity, i.e., the geometry of M. (See [43] and [81].) Note that ordinary completeness is insufficient for the stochastic completeness of a Riemannian manifold. Theorem 15.2. If a Riemannian manifold M is uniformly complete, then M is stochastically complete. By definition, for a Riemannian process w(t) on M, we have w(t) = 7r2U(t),

where w is a solution of (14.1) with a = 0, A = b, and the initial condition 6(0) = b. The desired result follows from Theorem 14.1.

78

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

Corollary. Any compact Riemannian manifold is stochastically complete and so is any Lie group with a (left-) right-invariant metric.

Remark 15.2. To avoid confusion with some of the results of [43], we emphasize that here we deal only with positive-definite Riemannian metrics, i.e., the inner product is assumed to be positive-definite on every tangent space. As a consequence, our orthogonal group is compact and, therefore, if M is compact, then so is O(M). This guarantees stochastic completeness. In [43], the reader can find some other sufficient conditions for stochastic completeness. Here we need to point out the following two results:

Yau's Theorem ([43], [146]). A complete Riemannian manifold is stochastically complete provided that its Ricci curvature is bounded from below.

Grigor'yan's Theorem ([81]). Let M be a complete Riemannian manifold. Denote by V(r) the volume of the ball centered at mo of metric radius r. If ODrdr < oo , 10 V(r) then M is stochastically complete.

Remark 15.3. Under the hypothesis of the Yau theorem, we have V (r) < exp(Cr) for a constant C > 0 and, thus, the Grigor'yan theorem applies. Moreover, the hypothesis of the Grigor'yan theorem holds true if V (r) < exp(Cr2), or V(r) < exp(Cr2 logr), etc. Note, however, that for any positive function f such that fo r dr/ f (r) < oc, there exists a complete Riemannian manifold with V (r) < C f (r) that is not stochastically complete. A more detailed discussion of this and related problems can be found in [81].

15.B. Stochastic Equations Let a be a vector field and A a (1, 1)-tensor field on M, i.e., A,,, is a linear TM for every m E M. (Note that the fields may be operator T,,,M time-dependent.) Definition 15.4. A forward stochastic differential a,,, dt+A,,, dw(t) at m E M is the set of stochastic processes in TmM formed by solutions X (s) of all equations

X(s) =

f p s a(rr,X)dr+ J 9A(r,X)dw(T) J0 0

,

where w(t) is the Wiener process in TmM, &(t, x) is a vector field, and A(t, X) is a (1, 1)-tensor field in TmM. The fields & and A are assumed to be Lipschitz,

vanish outside of a neighborhood of the origin, and such that &(t, 0) = am and A(t, 0) = A,,, (see Definition 13.5).

15. Wiener Processes on Riemannian Manifolds

79

Consider the equation dt; (t) = expe(t) (ae(t) dt + Ae(t) dw(t))

(15.1)

,

where exp denotes the exponential map on M. This equation (like those studied in Sect. 13) means that l; (t + s), s > 0, a.s. coincides with a process from the class exp(ae dt + A£ dw) until l;(t + s) leaves a neighborhood of l;(t). (A new notation for stochastic equations and differentials is used here in order to emphasize the difference with Sect. 13.) To deal with (15.1), it is necessary to modify the notion of a solution.

Definition 15.5. We say that equation (15.1) has a strong solution l;(t) if for any Wiener process w(t) in IR' , there is a process e(t) in M defined on the same probability space as w(t) and nonanticipative with respect to Pt such that for every t there is a realization be(t) o w(t) of the Wiener process at l:(t) such that we = be o w and satisfy (15.1).

Definition 15.6. Equation (15.1) has a weak solution if there is: (1) a probability space (,(1,2, P) with a nondecreasing family Bt of complete o-subalgebras of F; (2) a stochastic process l;(t) on M nonanticipative with respect to Bt; (3) a Wiener process w(t) in 1R' relative to Bt; and (4) realizations we(t) = be(t) o w(t) of w(t) in Te(t)M; such that we and C a.s. satisfy (15.1) for every t, as in Definition 13.6.

Using (13.6) and Definitions 15.5 and 15.6, it is easy to show that in a equation (15.1) takes the form

chart (U,

dl; (t) = a(t, e(t)) dt - 1 tr Fe(t) (A, A) dt + Ae(t) o (be(t) o dw(t))

,

(15.2)

where 1r,,,,( , ) is the local connector and t r I ,, (A, A) = t r I ,, (Am b,. , A,,,b,. )

with b E O,n(M) being an orthonormal frame. Observe that trl',,,,(A,A) is independent of b (consistently with the notation), because the trace is invariant

under the action of the orthogonal group. Using this fact, and the results of Sect. 13, it is not hard to show that (15.2) is covariant with respect to changes of coordinates. In what follows be(t) will usually arise as the horizontal lift of (t)

By the definition of a Wiener process, we have:

Theorem 15.3. A Wiener process w on M is a strong solution of the equation dib = exp;,(dw)

.

80

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

In local coordinates, this equation reads dzo(t) = - 2 trT'b(t) (I, I) dt + bw(t) o dw(t)

Corollary. The operator V2/2, where V2 is the Laplace-Beltrami operator, is the generator of a Wiener process on M. Assume that M is uniformly complete. To construct the integral operators needed to study (15.1), we have to alter the construction of Sect. 14. Let the probability space P), the family Bt, the manifold M, and

the functions a(t) in T,,,0M and A(t) in L(T,,,0M) be as in Sect. 14. Fix a realization b o w of the Wiener process w in Tm0M. It is clear that the operator S from Sect. 14 is applicable to the pair (a, A o b). Let a(t, m) and A(t, m) be a vector field and a (1,1)-tensor field on M,

respectively, and let q(t), t E [0,11, be an Ito process on M. Consider the vector and tensor fields T'a(t,r7(t)) and I'A(t,77(t)) obtained by the parallel translation of a(t,77(t)) and, respectively, A(t,77(t)) along 77(.) to 77(0). The operator S can be applied to the pair (Fa(t,77(t)),FA(t,77(t))), provided the fields a(t, m) and A(t, m) are bounded and Borel measurable jointly in t and M.

Therefore, we can define the Ito integral and the line integral with parallel

translation in terms of the field of Wiener processes. To distinguish these integrals from those of Sect. 14, we denote them by S(a(r) dT + A(T) dw(T)) (t) and

S(I'a(7-,77(T)) dT + FA(T,77(T)) dw(T))(t)

,

respectively.

Then (14.3) is to be replaced by the following equation: 1; (t) =

dT +FA

dw(T))(t)

.

(15.3)

Let boow be the initial realization of the Wiener process in Tm0M. Observe that the parallel translation of bo along a solution of (15.3) gives rise to a realization of the Wiener process at l;(t). (See Definitions 15.5 and 15.6.) The equation

z(t) =

J0

t

ra(T, Riz(T)) dT + J FA(r, RIz(T)) dw,,,.o(T) t

(15.4)

0

is an analog of (14.4). Similar results to Theorems 13.3, 14.2, 14.3, and 14.5 hold for equations (15.1), (15.3), and (15.4).

15.C. Equations with Identity as the Diffusion Coefficient Generalizing the classical notion, it is natural to call (15.1) an equation with a nondegenerate diffusion coefficient if the operator A(t ,,,): T,,,M + T,,,M is

15. Wiener Processes on Riemannian Manifolds

81

nondegenerate for all m E M and t E [0, 1] (see Definition 12.3). As we have mentioned above, such equations can hardly be obtained using the formalism developed in Sect. 13. Among equations with nondegenerate diffusion coefficients we are especially interested in those with A = I, i.e., with the diffusion coefficient equal to the identity operator. Then the equation can be written down in the form: de(t) = exp£ltl (a(t, t(t)) dt + dw(t))

.

(15.5)

Solutions of (15.5) will play a crucial role in Chap. 6. Note that for (15.5), the local expression (15.2) turns into the identity

de(t) = a(t, e(t)) dt - 1 tr l'e(t)(I, I) dt + bf(t) a dw(t)

Theorem 15.4. Assume that the Riemannian manifold M is stochastically complete and the vector field a(t, m) is Borel measurable jointly in (t, m) E [0, 1] x M and uniformly bounded. Then a weakly unique weak solution l;(t) of (15.5) exists on [0, 1] for any initial condition l;(0) = m°. Proof. Here, we are using a method based on changing the probability measure [54], [55], [95]. Consider the standard Wiener process w on TnoM, i.e., the

x(t) on the probability space (fl, , v), where coordinate map w(t, generated by cylinder sets, and v ,fl = C°([0,1],T,noM), J-' is the is the Wiener measure. Recall that an elementary event in 11 is a continuous E C° ([0, l], TmOM). Observe that w(t) is nonanticipative with recurve spect to the family of a-subalgebras !3t generated by the cylinder sets with base over [0, t], t E [0, 11. (See Sect 12.A and 14.)

Since M is stochastically complete, the development RIw(t) is well defined. Taking into account Remarks 14.4 and 14.6, we see that RIx(t) and Pa(t, Rix(t)) exist for v almost all x E S7. Furthermore, it follows from the properties of parallel translation and of RI and a(t, m) that the stochastic process Pa (t, Rrw(t)) is uniformly bounded and nonanticipative with respect to Cit. Consider the measure µ on (Q, _'i) such that its density p with respect to v is given by

exp(/ l(Fa(t,Rix(t)),dw(t))

o

\

2 dt I

-2f

.

(15.6)

It is known [54] , [95] that under the hypotheses of the theorem

4 pdv = 1

,

(15.7)

82

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

i.e., Et is a probability measure, and, furthermore,

w(t,xx(t) -

t

fra(r,

drr

is a Wiener process on (12, F, t) relative to Cit. It is not hard to show that p > 0 everywhere, i.e., v is absolutely continuous with respect to a and has density p-1. In other words, the probability measures tt and v are equivalent. The coordinate process z(t, x(.)) = x(t) on (i2, F, p) is nonanticipative with respect to lit and, moreover, Bt = Pt. Since the measures µ and v are equivalent, RI(z(t,x(.)) = Rrx(t) exists µ-a.s. Thus, z(t) and w(t) are related via the equation dz(t) = Fa(t, Riz(t)) dt + dw (15.8) on TmoM. In other words, z(t) is a weak solution of (15.8). It is shown in [54] and 195] that every solution of (15.8) gives rise to a probability measure on (f2, ) with density p. This means that a solution of (15.8) is weakly unique. By definition, the process Rr (t) exists on (12, .P, It). This process is, in fact, a weak solution of (15.5). Hence the solution is weakly unique.

We emphasize that w(t), defined as in the proof of Theorem 15.5, is a Wiener process relative to the family Pt generated by the weak solution z(t). Theorem 15.5. Let M be stochastically complete and a(t, m) Borel measurable in (t, m) E [0, 1] x M. Assume also that the inequality

f

Ila(t,m(t))II2dt < oo

holds for any continuous curve [0, l] -+ M, m(0) = mo E M, and the density p defined by (15.6) satisfies (15.7). Then there exists a weakly unique weak solution C of (15.5) on [0, l] with the initial condition 1; (0) = mo. This result is a simple generalization of Theorem 15.5. The only refinement needed in the proof is as follows. Even though the hypothesis of Theorem 15.6

does not guarantee that Ta(t, Rjx(t)) is uniformly bounded, v almost all E ,fl with x(0) = 0 E T,,,M satisfy the inequality ft

Ilra(t, Rix(t)) 112 dt < no

Arguing in the same way as in the proof of Theorem 15.5, we see that this inequality, together with (15.7), yields the existence and weak uniqueness of a solution of (15.8) [54], [95]. The rest of the proof of Theorem 15.5 remains unchanged.

16. Stochastic Differential Equations with Constraints

83

Corollary. Assume that

Ila(t,m(t))II2dt
l;(0)=mEM. Consider a stochastic process /3(t) on TmoM nonanticipative with respect to Lit and such that

p (J

I IQ(1r) I I2 dr < oo)= 1

.

(15.9)

Define an Ito process z(t) on Tm,,M by the formula

z(t) =

J

t

0(r) dr + w(t)

.

(15.10)

Theorem 15.6. Let M be stochastically complete. Then for any l > 0 the development Riz(t) exists on [0, 1]. The development is unique.

v) and Bt be as in the proof of Theorem 15.5. Denote by Proof. Let pz the probability measure on (Jf , 7) which corresponds to z. Then it follows from (15.9) that jt is absolutely continuous with respect to v. (See 1100, Chap. 7].) Since M is stochastically complete, the development of a standard Wiener process exists. In other words, the development can be a.s. extended from the space of smooth curves to b. Since u,z << v, the same holds true when µZ is replaced by v. Now observe that z(t) coincides with the coordinate process is ). The theorem follows. x(t) on

16. Stochastic Differential Equations with Constraints In this section we study stochastic differential equations with constraints (in the sense of Sect. 5) and generalize the notion of line integral with parallel translation to apply to such equations. The equations we focus on here are similar to those of Sect. 15. However, our method can be generalized to work with the "constrained" version of equations of Sect. 13. We leave this construction to the reader, for it is quite analogous to that given here. Let ,3 be a linear (possibly, nonholonomic) constraint on a Riemannian k < n. First, manifold M (see Sect. 5 for the definition) and let dim we define a Wiener process on /3m. Let the principal bundle O'(M) and the reduced connection H be as in Sect. 5.B. Keeping the notation of Sect. 7.B, we denote by E:00(M) X 1Rk -+ H the vector bundle isomorphism trivializing

Chapter 4. Stochastic Differential Equations on Riemannian Manifolds

84

ft over Op(M). As usual, we identify a frame b E OE(M) with a linear operator b:1Rk --+ ,Q,,, and assume the inner product on ,Q,,, to be induced by the Riemannian metric on M. Let w(t) be a Wiener process on IRk. Following the line of Sect. 15, we set TirEb(w) = bow and call this process a realization of w on /3kb. The properties of the realization are very similar to those discussed in Sect. 15 and here we

omit the exact assertions of these results. Recall, however, that the field of processes E(w) is smooth in the sense that it is obtained from w by means of a smooth map. Since the standard Wiener measure on IRk is invariant with respect to the O(k)-action, the standard Wiener process on /3m is independent of the realization of w. The field of standard Wiener processes on the fibers of ,3 is smooth again, i.e., as before, it is obtained by means of the smooth map T-7rE.

Fix an admissible vector field a(t, m) and a field of linear operators A(t, m): /3m - ,Q,,, on M. Let expm be the exponential map of the reduced connection H (Sect. 11). Consider the following stochastic differential equation:

df (t) = exp£ltl (a(t, C(t)) dt + A(t, e(t)) dw(t))

.

(16.1)

In the holonomic case, (16.1) means that an equation of (15.1)-type holds leafwise. However, if the constraint is nonholonomic, the only way to make sense of (16.1) is to rewrite it in a local coordinate system (U, 0):

a (t, £(t)) dt - 2 trP o F (t) (A o P, A o P) dt + Af(t) o P o q(t) o dw(t)

(16.2)

.

Here P: TM --+ 3 is the orthogonal projection (Sect. 5), is the local Levi-Civita connector, and cf(t) o w(t) denotes the realizations of the ndimensional Wiener process w in the tangent spaces to M at e(t) (see 15.1)). Recall that by Theorem 12.2, the orthogonal projection of IRn onto IR sends n-dimensional Wiener processes to k-dimensional ones. It is clear that (16.2) is covariant with respect to changes of coordinates. It is not very convenient that (16.1) has different interpretations in the holonomic and nonholonomic cases. Nevertheless, this idea does work in some problems. Throughout the rest of this section, we shall deal with nonholonomic constraints only.

Let M be uniformly complete. Fix a point mo E M and a basis b E 013.(M). Pick a realization w on ,3m,, of a Wiener process relative to tat. Assume that we are given processes a(t) on ,Q,,,o and A(t) on L(/3,,,a) like those dealt with in Sect. 15.B. In particular, a(t) and A(t) are nonanticipative with respect to B. As we did in Sect. 14, let us start with the equation <(t) = e o E o b-' (a(t) dt + A(t) dw(t))

,

(16.3)

where the exponential map e:TOO(M) - - + 013 (M) is defined as in Sect. 14. Note that (16.3) is a Belopolskaya-Dalecky equation. In the same way as in Sect. 14, one can prove that on [0, 1] there exists a strongly unique strong

16. Stochastic Differential Equations with Constraints

85

solution of (16.3) for any initial condition (O) = b E OQ(M). (The proof depends on Theorem 13.4.) Let £(t) be such a solution beginning at b. We denote the process

by

S O (a(r) dr + A(r) dw(r)) (t) The operator SO is a nonholonomic analog of S of Sect. 15. In particular, a(t) and A(t) are parallel to TirEE(t) (b-la(t)) and T7rEE(t) (b-1A(t)), respectively, along SO (a dr+Adw). Here the parallelism is meant to arise from the reduced connection. We call SO (a dr + A dw) a nonholonomic Ito development of the process t

z(t) =

J0

a(r) dr + fot A (r) dw(r)

and denote it by RA(z(t)) (see Sect. 14). Frthermore, we adjust the terminology of Sect. 14 calling RA(z) a nonholonomic Ito process. Let 6(t), t E [0, 1], be such a process with £(0) = mo. Consider the processes I'pa(t, e(t)) in 0,,,,o and FOA(t, fi(t)) in L(,3,,,.0) obtained, respectively, by the Then the equation parallel translation of a(t, £(t)) and A(t, fi(t)) along

dz(t) = I'pa(t, RR(z(t))) dt + I'QA(t, R'13 (,(t))) dw(t)

(16.4)

is a nonholonomic analog of (15.4). The nonholonomic versions of Theorems 13.3, 14.2, 14.3, and 14.5 remain true. These results shed light on the relationship between solutions of (16.1) and (16.4) and, in particular, yield the existence of a solution of (16.1) (see Sects. 14 and 15). In conclusion, note that the integral operators defined here (like those of Sect. 7.B) differ from the standard ones even for equations on 1R" unless, of course, the constraint )3 is trivial (see Remark 7.3).

Chapter 5. The Langevin Equation

The generalized Langevin equation describes processes in a mechanical system with both deterministic and random forces which have comparable magnitudes (i.e., neither the deterministic nor random part can be neglected) and the random force is a transformed white noise. Examples of such processes are well

known in physics. In this chapter, we use integral operators with Riemannian parallel translation in order to study the Langevin equations arising in geometric mechanics. Note that in the case under consideration the trajectories of the process are a.s. smooth. This makes the analysis of such systems technically much simpler than that of the general ones studied in Chap. 4. In Sect. 17, we introduce the Langevin equation on a Riemannian manifold

and reduce it to the velocity hodograph equation, which is an equation in a single tangent (i.e., vector) space. This enables us to apply some standard results to carry out a detailed analysis. In Sect. 18 we study an important particular case of the Langevin equation: the equation describing the so-called Ornstein-Uhlenbeck processes arising, for example, in the mathematical model of physical Brownian motion [111]. Sometimes, only the latter is called the Langevin equation, whereas that applicable in a more general context is said to be the generalized Langevin equation.

Throughout this chapter, all Riemannian manifolds are assumed to be complete, although not necessarily uniformly or stochastically complete.

17. The Langevin Equation of Geometric Mechanics Consider a mechanical system in the sense of Sect. 4, i.e., a Riemannian man-

ifold M and a force field on it. As we just mentioned, M is assumed to be complete, i.e., a free particle on M does not go to infinity in finite time. The Riemannian metric enables us to identify differential forms and vector fields on M and henceforth we regard the force field as a vector field. (See Sect. 4.)

Let a(t, m, X) be the force field and A(t, m, X) a (1,1)-tensor field on M. In other words, for every t E [0, 11, m E M, and X E T,,,,M, we have a vector a(t,m,X) E TmM and a linear operator A(t,m,X):T,,,M -+ TAM. Making use of the construction given in Chap. 15, fix a Wiener process w on the tangent spaces to M and denote by w the white noise of w (Remark 12.2). Then the Langevin equation describes the evolution of a system with the force

88

Chapter 5. The Langevin Equation

field:

a(t, m, X) + A(t, m, X)tb

(17.1)

.

More formally, the equation reads Dt

t(t) = a(t, e(t), fi(t)) + A(t, e(t), (t))w(t)

(17.2)

.

Our first goal is to give a rigorous meaning to (17.2) without using distributions. In doing so, we follow [66], [68], [75], and [76] and transform (17.2) into the integral form (more natural for stochastic equations) employing integrals with Riemannian parallel translation. A local coordinate version of the equation is given in [107].

We assume that a(t, in, X) and A(t, m, X) are continuous jointly in all variables and that these fields have linear growth in X. In other words, there exists a constant K > 0 such that

Ila(t,m,X)II +IIA(t,m,X)II
(17.3)

for all t E [0, l], m E M, and X E T,,,,M, where the norm is given by the Riemannian metric (see Definition 12.6). Let w(t) be a Wiener process on a probability space P) relative to the family Cit. (As usual, Cat contains all sets of measure zero.) Let fi(t) be a stochastic process with values in M which is nonanticipative with respect to Cat and such that all the trajectories of 6 are a.s. smooth and l;(0) = mo E M. Thus we can now use the standard Riemannian parallel translation along the trajectories of 6. Keeping the notation of Sections 3 and 7, let

ra(t, £(t), £(t)) and rA(t, 6(t), l (t)) be obtained by the parallel translation of a(t, fi(t), l;(t)) along

and

A(t, C(t), fi(t))

from the point fi(t) to 6(0) = mo. The processes ra(t, c, ) and

M and L(T oM), respectively, and their trajecFA (t, , l;) take values in tories are a.s. continuous, for so are the fields a(t, in, X) and A(t, m, X). Since parallel translation preserves the Riemannian norm, it follows from (17.3) that 11ra(t, fi(t), fi(t)) (I + II rA(t, 6(t), fi(t)) II < K(1 + II ra(t) II)

.

(17.4)

Lemma 17.1. The processes ra(t, l;(t), e(t)) and rA(t, l;(t), 4(t)) are nonanticipative with respect to B. The lemma is a consequence of the fact that the parallel translation operator r is continuous on the space of Cl-curves equipped with the Cl-topology and of our assumptions that C is nonanticipative and the fields a and A are both continuous.

17. The Langevin Equation of Geometric Mechanics

89

By Lemma 17.1, we can define the process z(t) in T,,,M as t

z(t) = f ra(r, l; (r), fi(r)) dr +

t

FA(r, fi(r), fi(r)) dw(r)

(17.5)

,

J0 where the second term on the right-hand side is the Ito integral. It is clear that z(t) given by (17.5) is nonanticipative with respect to Bt and a.s. has 0

continuous trajectories. Let v(t), t E [0, l], be a stochastic process in with the same properties as z(t). Applying to v(t) the operator S from Sect. 3, we obtain a process Sv(t) on M with a.s. smooth trajectories. Recall that S: C° ([0, l], Tm0M) -+ ([0, l], M) is continuous and v is nonanticipative with respect to St. This proves the following:

Lemma 17.2. The process Sv(t) is nonanticipative with respect to Bt. Setting v(t) = z(t), we obtain the Langevin equation in the integral form:

fi(t) = S \(Jt ra(r, l;(r), e(r)) dr+ f t rA(T, e(r), e(r)) dw(r)+C)

.

(17.6)

Taking into account the relationship between the Newton equation (4.2) and (7.1), we see that a solution 6(t) of (17.6) is a stochastic trajectory of the system with the force field given by (17.1) and with the initial condition 6(o) = mo and l; (0) = C E T,,, M.

Remark 17.1. Equation (17.6) involves the Ito integral fo FA(r, l;, £) dw; a similar equation with the Stratonovich integral fo rA(r, d,w is also well defined.

The equation of the velocity hodograph corresponding to (17.6) is

v(t)=10 Foa0S(v(r))dr+ I roA0S(v(r))dw(r)+C ,

(17.7)

0

where (see Sect. 7.A)

F o a o S(v(t)) = ra (tSv(t), dtSv(t)) and

r o A o S(v(t)) = FA (tSv(t) , dtSv(t)) It is clear that the vector field roaoS(x(t)) and the tensor field roAoS(x(t)) are defined along any curve x (.) E CO ([0,11, T,,,0 M) and continuous on the

90

Chapter 5. The Langevin Equation

space IR x Co Q0, 1], T,,,O M). By definition (see Sect. 3.A), we have

dtS(x(t)) = Ilx(t)II

,

and, therefore, due to (17.4),

Ilroaos(x(t))II +IlraAoS(x(t))I1
.

(17.8)

Equation (17.7) on T,,,,0M is an Ito stochastic differential equation with coefficients depending on the past. (See, e.g., [95].) For such an equation, one defines a weak and strong solution by naturally altering Definitions 12.4 and 12.5. (See, e.g., [54] and [95] for further details.) It is clear that v(t) and the Wiener process w(t) in satisfy (17.7) if and only if Sv(t) (taking values in M) and w(t) satisfy (17.6). Observe also that Sv(t) is defined on the same probability space and has the same measurability properties with respect to w(t) as v(t). Thus, we have proved:

Theorem 17.1. The process v(t) is a strong (respectively, weak) solution of (17.7) if and only if Sv(t) is a strong (respectively, weak) solution of (17.6).

Remark 17.2. Let us fix a realization of w(t) in T,n0M. Applying to it the parallel translation along Sv(t), we obtain realizations of w(t) in all spaces Ts (t)M. These realizations give rise to a force field defined along the trajectory. One may also use the realizations to introduce the notion of a solution of (17.6) similarly to Definitions 15.5 and 15.6.

Theorem 17.2. Assume that a(t, m, X) and A(t, m, X) are jointly continuous in all variables and satisfy (17.3). Then on [0, 1], there exists a weak solution of equation (17.6) for any initial conditions l; (0) = m° and 1;'(0) = C E T,,,,o M.

Proof. First, we pass to (17.7), which is equivalent to (17.6). Note that (17.7) is a diffusion-type equation on a vector space. Recall that this means that the coefficients of (17.7) depend on the past, i.e., on the entire trajectory on the interval [0, t]. As has been shown, the coefficients r o a o S and r o A o S are defined and continuous on IR x Co ([0, 1), T, M). Moreover, they satisfy (17.8), the linear growth condition. Thus, the standard theorem that guarantees that a weak solution exists applies also to (17.7). (See, e.g., [54, Chap. 3, Sect. 2] or [95, Sect. 19.3.8].) To complete the proof it suffices to apply Theorem 17.1. The following results can also be proved by passing to (17.7) and applyingd the results of the standard theory of stochastic equations on vector spaces [54], [55], [95].

Theorem 17.3. Let a(t, m, X) and A(t, m, X) be as in Theorem 17.2. Assume that the operator A(t, m, X) is invertible for all t, m, and X. If a solution of

17. The Langevin Equation of Geometric Mechanics

91

the equation

e(t) = S(

dw(r))

t

(17.9)

0

is weakly unique, then so is a solution of (17.6).

Theorem 17.4. Let a(t, m, X) be continuous jointly in all variables, satisfy (17.3), and be such that the solution of the Cauchy problem for (4.2) is unique. Also, let A,(t, m, X), where e E (0, 6) and 6 > 0, be jointly continuous in e, t,

m, and X and satisfy (17.3) with K independent of e. Assume, in addition, that: (i) Ao = 0; (ii) lim,-o A, -> 0 uniformly on every compact in [0,1] x TM; and (iii) a solution of the equation t

C

S(t) =

f ( dr S(J ra1T,S(T),S(T)) I/

+

J

C

t VA, (r, (T), fi(r)) dw(r) +CE

(17.10)

is weakly unique for some CE such that limb CE = C. Then the measures on C,l,t0 ([0,1], M) corresponding to the solutions of (17.10) weakly converge as a -+ 0 to the measure concentrated on the unique solution of (4.2).

Example. Let A = cI, where I is the identity operator. Then it is clear that a solution of (17.9) is unique. Thus, for a as before, the equation

(t) = S

(ft ra (T, e(T), (r)) dT + ew(t) +

C)

has a unique solution. If, for example, a is locally Lipschitz in in and X, then Theorem 17.4 holds true for the latter equation.

Remark 17.3. Let 3 be a (possibly, nonholonomic) constraint on M. Employing the operators SR and rQ defined in Sect. 4.2, and arguing as in Sect. 16, one can extend the notion of the Langevin equation to manifolds with constraints. In this case, the velocity hodograph equation in a fiber of 0 turns out to be very similar to (17.7). As a consequence, analogs of all results of this section hold true for the Langevin equations with constraints.

92

Chapter 5. The Langevin Equation

18. Strong Solutions of the Langevin Equation, Ornstein-Uhlenbeck Processes It is known that equation (17.6) has a strongly unique strong solution, provided that the coefficients T'oaoS and 1'oAoS of (17.6) satisfy a Lipschitz-type condition [54], [95]. However, the existence of a strong solution is rather hard to prove in the general case where the coefficients involve the operators r and

S. The reason is that r and S are defined by means of parallel translation and, as a consequence, we have a condition imposed on the entire mechanical system, rather than just on the force field. On the other hand, the existence can easily be verified for certain particular force fields. Here we consider three examples of such fields: (i) The drag force:

a(t,m,X) ='(t, IIXII) A(t, m, X) ='P(t,

JJXJJ)

. a", (X)

Am(X)

,

where 0 and W are scalar functions, a is a (1, 1)-tensor field with Va = 0, and A is a field of operators Am:TmM -+ L(TmM) parallel along every curve in M. (Note that the equation Vii = 0 means, in fact, a restriction of the same

kind as that imposed on A: the operators &m: TmM -+ TmM are parallel along every curve.) For example, one may take a = ±I or, if M is an oriented two-dimensional manifold, then am may be the rotation by a fixed angle. The

same operators can be taken as examples of A if we assume, in addition, that Am(X) is independent of X (i.e., Am, regarded as a function of X, is constant). (ii) A particular case of (i) involving friction and constant diffusion:

a(t, m, X) = -b(t) X , A(t, m, X) = 0(t) Am

,

where the friction coefficient b > 0 is a real-valued function of time and A is a (1, 1)-tensor field with VA = 0.

(iii) A force given in a "stationary coordinate system." Here we start with amo (t): Tm0 M + Tm0 M and Amo (t): T,no M --+ L(Tm0 M), t E [0, l]. The

operators a and A at other points of the trajectory fi(t) are obtained by the parallel translation of amo and Amo along (See Sect. 8 for a mechanical interpretation of parallelism.) Theorem 18.1. Let a and A be as in (i)-(iii). Assume also that amo and Amo are Lipschitz in X E Tm0M and satisfy (17.3), the linear growth condition. Then (17.6) has a strongly unique strong solution on [0, l].

Proof. Under the hypotheses of the theorem, equation (17.7) on Tm0M is equivalent to the following one:

18. Ornstein-Uhlenbeck Processes

v(t)= I a(T,mo,v(T))dT+fo A(r,mo,v(T))dw(T)+C

.

93

(18.1)

0

This equation has a strongly unique strong solution defined on [0, 1]. The initial

velocity C can be viewed as a random vector measurable with respect to the o-algebra Bo [54], [95]. To finish the proof it suffices to apply Theorem 17.1. 0

Note that if a and A are as in (ii), then the hypotheses of Theorem 18.1 are automatically satisfied, provided that b and 0 are bounded. In this case, the solution v(t) of (18.1) and the solution Sv(t) of the Langevin equation are called the velocity process and the coordinate Ornstein-Uhlenbeck process, respectively.

Note that the assumption that b and 0 are bounded can be omitted in the hodograph equation for Ornstein-Uhlenbeck processes so that the velocity process exists on a random interval up to the so-called explosion time (see Sect. 12.E). Recall that Ornstein-Uhlenbeck processes describe the Brownian motion

in a medium with a drag force. A detailed discussion of this matter can be found in [111].1

Let v(t) be a solution of (18.1). Denote by Ev(t) the mathematical expectation of v(t) in TmoM.

Definition 18.1. The curve S(Ev(t)) on M is said to be the mathematical expectation of the process Sv(t). The function E(Ev(t) - v(t))2 is called the dispersion of Sv(t).

It is easy to see that for a system defined in (i) and, in particular, for (ii) the mathematical expectation of a solution of (17.6) satisfies (4.2). Passing to the hodograph equation and applying the standard results on equations in a vector space, we obtain the following theorem:

Theorem 18.2. Under the assumptions of Theorem 18.1, the solutions of

ft

t'(t) = S(J

\0

dT

IPt

+EJ

I

(18.2)

0

converge as e - 0 to the solution of (7.1) in the topology of the space S (Co ([0, l], L2 (Q, Tmo M)) )

1 Added in proof. Ornstein-Uhlenbeck processes on manifolds are also discussed in: E. Jorgensen, Construction of Brownian motion and Ornstein-Uhlenbeck processes in a Riemannian manifold on the basis of the Gangolli-McKean injection scheme, Z.F.W., 44 (1978), 71-87.

94

Chapter 5. The Langevin Equation

The mathematical expectation of e uniformly converges to the solution of (7.1).

M) is the space of the square integrable maps from ,(l to Here L2 (,R, T,,, M. Note that the convergence means that the dispersion of C converges uniformly to zero. Recall also that equations (7.1) and (4.2) are equivalent.

Remark 18.1. For the Langevin equation with constraint mentioned in Remark 17.3, Ornstein-Uhlenbeck processes remain strong solutions of the hodograph equation.

Chapter 6. Mean Derivatives, Nelson's Stochastic Mechanics, and Quantization

This chapter is entirely devoted to Nelson's stochastic mechanics, a theory that formally belongs to classical physics, but leads to the same results as quantum mechanics. In this theory, the motion of a particle is viewed as a stochastic process satisfying a version of the Newton equation. Starting with the probability density of this process, one may find the quantum mechanical wave function, i.e, a solution of the Schrodinger equation. Apparently, Fenyes [48] was the first to introduce and study such processes. However, stochastic mechanics became well known only after the publication of papers [110] and [111] by Nelson who developed the theory independently and gave it a natural form. A more detailed review of the history of this question can be found in [27], [110], and [113].

Today, stochastic mechanics includes a description of spinning [27], [33] and relativistic [29], [33], [84], 148] particles, the indeterminacy principle [79], quantum fields, etc. (In particular, the Klein-Gordon [33], [84] and the Dirac [29] equation are studied in the framework of stochastic mechanics.) The the-

ory has been extended to cover Riemannian manifolds and, as a result, the geometric nature of the principal notions has been understood and analyzed. The reader interested in a review of modern nonrelativistic stochastic mechanics should consult the book [113] by Nelson published in 1985. (See also [2], [19], [44], and [80].)

Note that when stochastic and quantum mechanics both apply, they lead to the same predictions, although often relying on totally different mathematical models of the physical phenomenon. (For example, this is the case with the electron interference described in [113].) Quoting Nelson [113], we recall that it is still unclear whether stochastic mechanics reflects the true physical nature of phenomena or it is just a convenient mathematical formalism. From another perspective [144], stochastic mechanics is a new method of quantization essentially different from the Hamiltonian or Lagrangian (based on path integrals) approach. In this chapter, primary attention is being given to geometric constructions of stochastic mechanics following the general line of Chap. 4. Note that here the Newton equation is viewed as a second-order stochastic equation, which is completely different from the Langevin equation studied in Chap. 5. To study the Newton equation, we widely use stochastic equations with

Chapter 6. Nelson's Stochastic Mechanics and Quantization

96

various stochastic integrals and the notion, due to Nelson, of the derivative of a diffusion process.

19. More on Stochastic Equations and Stochastic Mechanics in IR't 19.A. Preliminaries Let us consider a stochastic process 6 in 1R' defined on a probability space (.R, .F, P) furnished with a Wiener process w(t) taking values in 1R'. Denote by

E( I B) the conditional mathematical expectation with respect to a a-algebra B (Appendix C). When B is given by a random variable 77 (as preimages of Borel sets) or by a condition U, the conditional mathematical expectation is denoted by E(1 77) or E( I U), respectively. For the sake of simplicity, we always assume that e(t) is defined on a bounded interval [0,1]. For every t E [0,1], the stochastic process C gives rise to the following three a-subalgebras of .77:

"the past," the complete minimal a-algebra Pt with respect to which all

e(s), 0 < s < t, are measurable; "the future" F11 , the complete minimal a-subalgebra with respect to which all c(s), t < s < 1, are measurable; and "the present" Ne, the a-subalgebra generated by the random variable e(t) and all sets of zero probability.

Varying t E [0,1], we obtain families of a-subalgebras, which we assume to be complete (i.e., to contain all sets of measure zero). Clearly, XF C Pt and N£ C Ft . Hereafter, we denote the conditional expectation E(. I Al) by

Et 0. In the framework of stochastic mechanics in IR', a trajectory of a particle is a diffusion process with the diffusion operator a21/2, where a = const and I = id. In other words, a trajectory is a solution of the Ito equation

do (t) = a(t, 6(t)) dt + a dw(t)

,

(19.1)

where a(t, m) is a vector field assumed to be smooth (unless specified otherwise), t c [0,1], and m E M. The constant a2/2 is actually equal to h/(2m), where h = h/(2-7r) is the Plank constant and m is the mass of the particle. In what follows, we always choose a system of units such that m = 1 and, therefore, a -+ 0 means that m goes to infinity.

Remark 19.1. Working with a nonflat metric (,) on 1R' (or even with a Riemannian manifold as in Sect. 20), we should consider a more general equation with a positive-definite diffusion coefficient a2AoA*/2 (Definition 12.3), where

19. More on Stochastic Equations and Stochastic Mechanics in 1R"

97

(X, Y),,,. = (A,,,.A;,,X, Y) and (,) denotes the standard inner product in 1R". Using the results of Sect. 16, one can also deal with equations with a degenerate diffusion coefficient, which arise in stochastic mechanics on manifolds with constraints. Throughout this section, we assume that w(t) is a Wiener process relative to Pt , where C(t) is the solution of (19.1). When e(t) is a strong solution, this assumption is automatically satisfied since Pt = Pt . Applying the existence theorems (e.g., Theorems 14.5 and 15.5), one can prove that this is true even if fi(t) is a weak solution. Below, we refine our approach to stochastic mechanics assuming a trajectory to be an Ito diffusion-type process representable in the form

fi(t) =

J0

t /3(T) d-r + uw(t)

(19.2)

,

where the process ,3(t) is nonanticipative with respect to Pt and has continuous sample paths, w(t) is a Wiener process adapted to Pt, and a is as in (19.1). Note that under our hypotheses, the strong and weak solutions of (19.1) are processes of type (19.2). The major difference between our approach

and the standard one [111], [113] is that a process given by (19.2) may not be Markovian and, thus, we need to show first that it, nevertheless, retains the same relationship to the standard quantum mechanics as the Markovian diffusions of [111] and [113]. (See Sect. 19.F.) Working with a larger class of processes than just Markovian diffusions, we are able to establish the existence of trajectories for an arbitrary force field (Sect. 20.C), while among Markovian diffusions this has been established only for a potential or gyroscopic force.

19.B. Forward Mean Derivatives In this section we define the mean derivative of a stochastic process. This notion, introduced by Nelson, enables us to give a rigorous sense to the velocity and the acceleration of a diffusion process, which we need to find a stochastic analog of the Newton equation.

Note, that since a trajectory of a stochastic process fi(t) is a.s. nondifferentiable, the derivative do(t)/dt does not exist in the standard sense. Thus we define, following Nelson, the forward mean derivative of e(t) as

De(t)

olim o Et f e(t + dt) - fi(t)

I

(19.3)

where Et = E( I N£) and dt - +0 means that At -* 0 and At > 0. Recall that, by definition, Et ( (t+dt) -fi(t)) is an N£-measurable map from (,fl,.F) to 1R". Hence, by [116] and [124], there exists a Borel map Yt,ot:1R' --+ 1R" such that Et (e(t + dt) - e(t)) = Yt,ot (6(t))

98

Chapter 6. Nelson's Stochastic Mechanics and Quantization

This implies that Do(t) = X (t, l: (t)) for some Borel vector field X (t, m) on IR x R', provided that the limit in (19.3) exists in L1. Assume that l;(t) satisfies (19.1). Then w(t) is a martingale with respect

to P. Since a(t,l;(t)) is N4-measurable, we have Df(t) = a(t,E(t)) and, as a consequence, X (t, m) = a(t, m). Similarly, for fi(t) given by (19.2) we have

X(t,l:(t)) = Et (i3(t)). Let us set

De(t)m = alim o E ((t + 'At) - 6(t)

6(t) = m) i

where m E IR' . Then it is clear that D6 (t),, = X (t, m).

19.C. Backward Mean Derivatives and Backward Equations Note that stochastic differential equations like (19.1) and processes like (19.2) are not invariant with respect to time reversal. (See, e.g., [109] for the gen-

eral theory of backward processes.) In particular, the sign of Lt in (19.3) is essential. Let us define the backward mean derivative of 1; (t) at t as

((t) - (

D.1;(t)=olimoEt

(19.4)

We emphasize that in general D6(t) # D.6(t). (However, if the trajectories of l:(t) are a.s. smooth, then the derivatives do coincide.) Note that, as above, Et (e(t) - C(t - at)) is N£-measurable. Using arguments similar to those of Sect. 19.B, one may show that D.6(t) = X. (t, e(t)) for some Borel vector field X. (t, m) on IR x IR, provided that the limit in

,at-+o

(19.4) exists in L1. (Note, however, that X. (t, m) may not be equal to Dl; (t)m.)

For the sake of convenience, we denote X. (t, m) by a. (t, m) when fi(t) is a solution of (19.1). Let

D.l; (t)m = Jim E

(o)

at

- . t)

f (t) = m

Hence, by definition, D. l; (t) m = X. (t, m).

Let x(t) and y(t) be stochastic processes on (0,Y, P) with values in 1R'. We define the y-forward and y-backward derivative of x(t) by the formulas DZx(t)

D;x(t) =

,at-+o o

lim

Et

o Et1

(x(t + Qt) - x(t) )

- Xt))

(x(t) xt

.

(19.5)

The following lemma enables one to calculate the derivatives explicitly.

Lemma 19.1. Let fi(t) be a strong solution of the Ito equation (12.5) and

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99

assume that the operator field A is smooth. Then

D..(t) = a(t, e(t)) -trA'(A(t,t;(t))) +A(t,t:(t)) oD;w(t) fort E (0,l]. Proof. Applying (12.3), we replace the Ito integral in (12.4) by the backward integral. This leads to the equation t

l; (t)=e(0)+

fa(s,c(s))ds-JtrA'(A(s,e(s)))ds+ t fo A (s, 6 (s)) d. w (s) t 0

0

o

(see Sect. 12.C for the Stratonovich case). Then using the properties of con-

ditional expectation, (19.5), and the fact that the second and third terms on the right-hand side of the latter formula are processes with a.s. smooth trajectories, we have:

D.6(t) = D.1 (10t a(s, 6(s)) ds t

J0

t

trA'(A(s, 6(s))) ds

+ f A(s, 6(s)) d.w(s))

= a(t,l:(t)) - trA'(A(s,c(s))) CW(t) - wt - At) )) + ofmo = a(t,l;(t)) - trA'(A(s,£(s))) (w(t) + A(t, fi(t))

o eimo E t

t - at)

)

'At

= a(t, l; (t)) - tr A' (A(s, e(s))) + A(t, le(t)) o D;w(t)

Corollary 1. Let y(t) be a strong solution of (19.1). Then for all t E (0, l ]

D.y(t) = a(t, y(t)) +QD;w(t)

.

Corollary 2. If l;(t) is a process given by (19.2), then D.C(t) = Et (i3(t)) + Lemma 19.2. Let a(t, m) be a time-dependent vector field on 1Rn and y(t) a solution of the following equation: y(t) = yo + J t a(s, y(s)) ds - o I D;w(s) ds + iw(t) JJJ

Then D.y(t) =

fort E (0,1].

.

(19.6)

Chapter 6. Nelson's Stochastic Mechanics and Quantization

100

Lemma 19.2 follows from Corollary 1 of Lemma 19.1 (or can be proved in a similar way). Let, as in Lemma 19.1, y(t) be a strong solution of (19.1).

Definition 19.1. The process w

(t) _ -Dw(s)ds+w(t)

f

is called the backward Wiener process with respect to y(t), provided, of course,

that the integral on the right-hand side exists.

Note that w; (t) is a reverse-time martingale with respect

Xty. This

follows from the fact that D; w; (t) = 0 and the results of [111].

Lemma 19.3. Assume that y(t) is a strong solution of (19.6). Then we a.s. have

y(t) = y+

j

a(s, y(s)) ds + aw(t)

(19.7)

for every t E (0,1 ].

Lemma 19.3 is evident by Lemma 19.2 and Definition 19.1. We call (19.7)

a backward Ito equation. Although this terminology comes naturally in the context of our theory, it should be pointed out that the term we use refers more often to equations of another type. These equations do not appear in the present book, and so no confusion shall arise. Equivalently, (19.7) may be written down in the differential form as follows:

d.y(t) = a(t, y(t)) d.t + d.w.(s)

(19.8)

.

For the sake of convenience, we omit the superscript y at d.w, in (19.7). Note also that no condition which would guarantee the existence of solutions of (19.8) has been found yet. In Sect. 19.D, we give an example of an explicit calculation of a backward Wiener process.

Remark 19.2. One may consider a more general backward equation:

d.e(t) = a. (t, e(t)) d.t + A(t, fi(t)) d.w. (t)

,

which is to be understood as the following stochastic equation

y(t) = yo + J ta. (s, y(s)) ds + J t tr A' (A(s, y(s))) ds 0

0

r

- J t A(s, y(s)) o Dew(s) ds + J t A(s, y(s)) dw(s)

19. More on Stochastic Equations and Stochastic Mechanics in R"

101

Due to (12.3), we have t

J

A(s, y(s)) dw(s) _ -

+

rt

J

trA'(A(s, y(s))) ds

J A(s, y(s)) dw(s)

Applying this identity, one can easily check that D,y(t) = a, (t, y(t)), provided that y(t) is a solution of the generalized backward Ito equation. Note also that for

y(t) =

r

J0

t a (T) dt +

J0 t

A(r) dw(rr)

we have the following "backward Ito formula" :

d. f (y(t)) = at d,t + f'(a,) dt - 2tr f"(A, A) d.t + f'(A) d.w(t) where f (t, m) is a C2-smooth function (see (12.8)). A more detailed account on generalized Ito equations will be published elsewhere.

Remark 19.3. Let x(t) be a solution of an Ito equation. Recall that the forward and backward differentials dx(t) and d,x(t) are not vectors because they do not transform as such under smooth maps or changes of coordinates. (This can be easily seen from the forward and backward Ito formulas giving the transformation rule. See (12.8) and Remark 19.2.) Comparing both the Ito formulas, one may show that the symmetric differential d,x(t) = (dx(t) + d,x(t))/2 is, in fact, a vector, which belongs to a tangent space to IR" (Sect. 12.C). As immediately follows from the definition, Dx(t) and D.x(t) are also vectors. Regarding the forward mean derivative, it is important to emphasize that since Dx(t)m = a(t, m) (Sect. 19.B), the vector is equal to the first term of the Ito vector field, but not of the Ito equation. (See Sect. 13.) It is essential that the "right" choice of a coordinate chart on 1R" which makes the Ito vector field coincide with the Ito equation for the latter is uniquely determined by the field via the Euclidean connection on R.

19.D. Symmetric and Antisymmetric Derivatives Let us consider the symmetric and antisymmetric derivative DS = (D+D,)/2 and DA = (D - D,)/2. Then, following Nelson, we call and DAe(t) the current and, respectively, the osmotic velocity of 6(t). If 6(t) is a solution of (19.1) and (19.8), then DS6(t) = v(t, 6(t))

,

DA6(t) = u(t, 6(t))

where

v= 2(a+a.)

and

u= 2(a-a,)

Chapter 6. Nelson's Stochastic Mechanics and Quantization

102

It is clear that

2(X -X,)

v=

when 6(t) is a process given by (19.2). We also have DAW) = (-1/2)QD;w(t), as is plain to see. The osmotic velocity shows how far the process e(t) is from the deterministic one. Pick a process e(t) of the form of (19.2). It is known that there exists a probability density p(t, x) on [0, l] x IR' such that for any continuous function f (t,

m) on [0, l] x IRf

f

f (t, m) p(t, x) d=

f (t, (t)) o,t] x fl

o,1] x IR^

where p is the Lebesgue measure on [0, 1] x IR'.

Lemma 19.4. The vector field u(t, m) is given by the formula u(t, m) = 0'2 grad In

p(t, m)

.

Proof. We use a method developed by Elliot and Andersen to deal with more complicated processes. An alternative proof can be found, e.g., in [110], [111], and [113], where only Markovian diffusion processes are considered.

Pick an arbitrary smooth compactly supported function f on ]R". It is clear that E[f (((t))EC (w (t) - w(t - 1t))] = E[f (1;(t)) (w (t) - w(t

- at))]

and

E[f

(w (t) - w(t -

f f (e(t)) - f

at))

f +

w(t -

f (e(s))

at

(s) ds

ft 2 Jt-ot tr f"(1;(s)) ds

Q2

t

+0,

t-At

grad f (1; (s)) dw(s)

,

where the dot denotes the inner product in R. Multiplying this equality by w(t) -w(t-At), on the right-hand side we obtain the integrals with respect to ds dw(s), ds dw(s), and (dw(s))2, respectively. Then applying Theorem 12.4,

19. More on Stochastic Equations and Stochastic Mechanics in IR"

103

we have

- f (e(t - at))) (w(t) - w(t -At))]

E[f (e(t))Et (w(t) - w(t - at))] = E[(f (C(t))

Irf t

=E

L t- at

or grad

f

1

ds1

and

-2E [o.f((t)) olim 0Et (w(t) -

E [f

t - At)/

1E [0.2 grad f (fi(t)) 1

2

JO,l]xIR'

1

o.2gradf(m)p(t,m)d,a

a2f(m)gradp(t,m)dp

2 L,i]XIIvI

_

o 2f(M)

1

2 i[O,i]x]R"

grad p(tm) p(t, m) dµ p(t,m)

= 2E [a2f(rn)grad lnp(t,t;(t))] The lemma follows from the latter equality, since f is an arbitrary smooth compactly supported function. We will also need the following result:

Lemma 19.5. The current velocity v and the density p satisfy the continuity equation: Op

at

= -div(pv)

Proof. Let, as in the preceding proof, f (t, m) be a smooth compactly supported function on IRS' x [0, 1]. As follows from the Ito formula (12.8), for t > s we have

E[f(t,e(t)) -

f f tgradf it a2 + f 2 v2 f dr] 11

9

where the dot denotes the inner product in W. Furthermore, by the backward

Chapter 6. Nelson's Stochastic Mechanics and Quantization

104

Ito formula (Remark 19.2),

(,

E [f (t, fi(t)) - f (s, e(s))] = E [ f t of d7-

+

grad f X. (T, £(7-)) dT

jt72] J8

and, therefore,

E [f (t, f(t)) - f (s, 6(s))] = E [ f t ar dT

+

f

t

grad f v (r, 6 (r)) dr]

e

where

X (t, m) + X. (t, m)

v(t, m) =

2

Then

E

[,l e

t aT

d r]

,"

Lt]xlR ///_'

-

af

[s,t] x 1Rn aT

(T, m)

Or

P(r , m) da

[f (r, m)P(r, m)] dp

- J", t]xIRn f(,T m)

)

'P( m) dµ Or

= E [f (t, f (t)) - f (s, (s))]

-

f

8,t] xIR."

m) d f (r m ) 'P(7-' Or

and

t

E Lf grad f v(r, e(r)) dr] 8

= fe,t] x]R" grad f (r, m) v(r, m)p(T, m) dp

f

f (r, m) div(v(r, m)p(r, m)) dµ

s,tl x IR"

Adding up these identities, we see that

E [f (t, fi(t)) - f (s, .(s))] = E [f (t, e(t)) - f (s, £(s))]

I

m) d r m ) Op(r, f(, aT µ - J[s,t]xl."

f

8,t] x IR"

f (r, m) div (v(r, m)p(r, m)) dµ

19. More on Stochastic Equations and Stochastic Mechanics in ]R'

105

Hence,

Js,t]x1R' f (T, m) ap(r, m) dp = - fs,t]x R" f (T, m) div (v(T, m) p(T, m)) dlt where f is an arbitrary compactly supported smooth function. The lemma follows. An alternative proof for a Markovian diffusion e(t) can be found, e.g., in [110], [111], and [113].

Applying Lemma 19.4, we can calculate the backward mean derivative of a Wiener process w(t) in W. It is clear that Dw(t) = 0 for t E [ 0, l) because w(t) is a martingale.

Lemma 19.6. D.w(t) = w(t)/t for every t E (0, 11. Proof. It follows from the definition of the osmotic velocity uw(t,w(t)) that D.w(t) = -2uw (t, w(t)). The density pw(t, X) can be found by the formula (see, e.g., [95]): 2

pw (t' X)

(27rt)n/2 eXp \

t

Thus, by (1.16), we have uw(t, X) = -X/(2t) and, therefore, D.w(t) = w(t)/t.

It is clear that the process w(t)/t does not exist for t = 0. Nevertheless, we have:

Lemma 19.7. The integral fo (w(s)/s) ds a.s. exists for all t E [0, l]. Proof. We start with the estimate

E

(ft

w(s) s

ds)


,

which can be proved by a standard argument using the density pw. Then the lemma follows from the Tchebyshev inequality. It is easily seen by Lemmas 19.6 and 19.7 that

w;(t)=-0t w(ds+w(t)

.

However, it is a rather hard problem to calculate w; (t) for a more general process y. Lemma 19.6 yields that D; w. (t) = 0 and Dww. (t) = -D.w(t). Note that these identities fail if we replace D; by D. and Dw by D. For example, as shown in [90], w; (t) is a Wiener process with respect to its own "past" filtration.

Chapter 6. Nelson's Stochastic Mechanics and Quantization

106

19.E. The Derivatives of a Vector Field Along fi(t) and the Acceleration of C(t) Let Y(t, m), t E [0, l], be a smooth time-dependent vector field on R'. We define the forward derivative DY and the backward derivative D.Y of Y as follows:

DY(t, fi(t)) =

D,10, £(t))

oe m o

Et

= olim o Et

(Y(t +,dt, (t Lit)) - Y(t, C(t)) 'At

dt, e(t - dt))

CY(t, fi(t)) - Y(t

/

.(19.9)

Regarding Y as a map Y:IR' - IR", we may apply to it the forward and backward Ito formulas. (See (12.8) and Remark 19.2.) As a result, we obtain

DY=

(zi+x.v+)Y

D.Y =

2 A + X. 0 + 19 51

19

and z

)

Y

(19.10)

where A is the Laplace operator, V = (a/aql, ... , a/aqn), and the dot denotes the inner product in IR'. By definition, the acceleration of the process e(t) is the vector

a = 2 (DD. + D.D)e , Equivalently, it can be defined as

a= 1(DX.+D.X) or

a = (DsDs - DADA) = Dsv - DAU . It is clear that

Dsv= and

DAU = 2v2Liu + (U. 0)U

Thus, we have

/

)V(26 2

+v

(19.12)

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107

19.F. Stochastic Mechanics Let us consider a classical mechanical system on 1R

with a force field

a(t, m, X). A trajectory of such a system is a solution of the Newton equation dtrrz(t) d

= a(t,m(t),rn(t))

(19.13)

Stochastic mechanics is based on the assumption that a trajectory of the system is a diffusion-type process with the diffusion coefficient 2 a21 (i.e., a process given by (19.2)) which satisfies the condition

a=a(t,c v)

(19.14)

,

the so-called Newton-Nelson equation. It is easy to see that (19.14) is an analog of the Newton equation (19.13). Furthermore, (19.14) turns into (19.13) as a -* 0. The passage to diffusion-type processes and (19.14) from deterministic processes and (19.13) is called stochastic quantization. To illustrate the relationship between stochastic and quantum mechanics, we take as an example a mechanical system with a conservative force field a = -grad U. (See Sect. 4.B.) Assume also that v = a2 grad L with L being the action. Recall from Sect. 19.D that u = a2 grad R, where R = In p/2, and set W = exp(R + iL). Note that L is only defined up to a constant, since we are originally given grad L. Theorem 19.1 (See [110]). For a certain choice of L, the function W satisfies the Schrodinger equation aTI

T=

i

or

2

Alp

uT

(19.15)

where or is as in Sect. 19.A.

Proof. Consider the system of differential equations 2

at = - - grad div v - grad(v u)

at = -grad U - (v 0)v + (u 0)u +

22Du .

(19.16)

Here, the first equation follows from Lemmas 19.4 and 19.5; the second one is a consequence of (19.12), (19.14), and the assumption that a = -grad U. A straightforward calculation shows that (19.16) is equivalent to the single equation a2

aW-

8t =i 2 for some function 6(t), where, as above, W = exp(R + iL). The integral of

p = IM is equal to 1. In particular, the integral is independent of t, and the function 6(t) takes real values. Adding to L a time-dependent constant

108

Chapter 6. Nelson's Stochastic Mechanics and Quantization

function, we can always make b vanish. For such a choice of L, the wave function W satisfies the Schrodinger equation.

The above argument can also be used to prove the converse assertion. Let W be a solution of (19.15) such that f I T1 I2 = 1. Consider R and L defined by the identity W = exp(R + iL) and set u = v2 grad R and v = v2 grad L. Then

a solution E(t) of (19.1) with a = v + u satisfies (19.14) with a = -grad U. As a result, the wave function of coincides with W. For a more complicated force field, the field v and the action L are related to one another in a more complicated way. Consider, for example, a gyroscopic

force (Sect. 4.B), then the 1-form a of the force field is equal to -dU + dw( , rh), where w is a given 1-form. Furthermore, v" + w = a2 dL, where v = ( , v) and (,) is the inner product. The existence of a diffusion process corresponding to the solution W of (19.15) was proved in [24] under a very general assumption. Namely, it suffices to require U to belong to the Rellich class. (See also [19].) Note that certain problems in dealing with fi(t) arise in the domain where the wave function vanishes. These problems were addressed, e.g., in [113].

Remark 19.4. In this section, we quantize the Newton equation, while in other approaches to quantization one starts with the Hamilton or Lagrange equations. Therefore, the domains of various quantization procedures differ essentially. However, in each case where the constructions can be applied, they all lead to equivalent results. Among the publications known to the author, the most general results are proved in [113] and [144] for a velocity-dependent force field. Note also that certain variational methods are developed for systems to which the Lagrangian formalism is applicable. As shown in [83] and [145], using variational methods to quantize a system, one obtains exactly (19.14)

An additional term enters the right-hand side of (19.14) when we deal with purely quantum effects which have no classical analog. For example, the spin force appears in the stochastic quantization of a system with spin [113], but this force vanishes as one passes to the classical system.

Remark 19.5. Defining the acceleration a by (19.11) is more natural than it might first appear. In [111], Nelson studied all possible ways to define the acceleration so as to make the acceleration invariant under time reversal (i.e., physically meaningful) and coincide with the standard one on smooth trajectories. Then, as it turns out, (19.11) is the only possibility to give the correct answer in the particular cases where the acceleration can be found by other methods. It is also shown that the variational methods mentioned in the previous remark lead to the acceleration a given by (19.12).

20. Stochastic Mechanics on Riemannian Manifolds

109

20. Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds It is known that the properties of a stochastic mechanical system are strongly related to the geometry of its configuration space. In fact, this relationship for stochastic mechanical systems is even stronger than for classical ones. In stochastic mechanics, the Riemannian metric not only gives rise to the kinetic energy of the system but also to the field of Wiener processes, which is absolutely essential for describing the motion. Moreover, the curvature of the configuration space is involved in the Newton equation of stochastic mechanics.

As we mentioned in Remark 19.1, a diffusion process with a positivedefinite diffusion coefficient gives rise to the metric. Therefore, one may fix a class of diffusion processes with a given positive-definite diffusion coefficient and recover the metric, which then can be used to define the kinetic energy and other relevant notions. This approach has been developed in [113]. Here, we follow a completely different line. Namely, starting with the metric, we apply the results of Sect. 15 to define the field of Wiener processes. Then diffusion (or diffusion-type) processes arise as solutions of stochastic equations. Throughout this section, M denotes a Riemannian manifold. The exponential map, parallel translation, and other geometric objects on M are assumed to arise from the Levi-Civita connection.

20.A. Mean Derivatives on Manifolds and Related Equations Let, as above, M be a Riemannian manifold and w a Wiener process in W. As we show in Sect. 15, we have a consistent way to define Wiener processes in the tangent spaces to M. Similar to Sect. 19.A, let us start with the following equation, which is analogous to (19.1): dx(t) = expx(t) (a(t, x(t)) dt + o dw(t))

(20.1)

.

Here we keep the notation of Sect. 15. Recall that a(t, m) dt+o dw(t) denotes the forward differential (Definition 15.4), i.e., the class of stochastic processes in TmM formed by solutions of the equation:

X(s) = J td(r,X(T))dT+ J tA(T,X(T))dw(t) 0

,

(20.2)

0

where a(t, x) and A(t, x) are as in Definition 13.5 with the additional assumption that A(t, x) is smooth. Note also that A(t, 0) is the frame bm giving rise to the realization of w(t) in T,,,,M. By analogy with equations (19.1), (15.1), and Definition 15.4, we call (20.1) the forward equation. In what follows, we

110

Chapter 6. Nelson's Stochastic Mechanics and Quantization

suppose that w(t) is a Wiener process with respect to Pt. This is automatically true for a strong solution of (20.1), while for a weak solution, this is guaranteed by our existence results (Theorems 15.5 and 15.6). In local coordinates, equation (20.1) can be written in the form z

dx(t) = a (t, x(t)) dt - 2 tr Tx(t) (I, I) dt + o bx(t) dw(t)

,

(20.3)

is a field of orthonormal where r is the local Levi-Civita connector and frames (Sect. 15.C) taken in the realization of w(t). Let y(t) be a stochastic process. We can apply (19.3) in a local chart to define the forward and backward mean derivatives of y(t) and represent them as compositions of suitable "vector fields" (on the chart) with y(t). (See Sect. 19.B and 19.C.) Note, however, that in general these "vector fields" do not transform as vectors under changes of coordinates. This is the reason why we first take the genuine vectors X° (t, m) and X; (t, m), equal to Dy(t),n and Dsy(t),,,,, respectively, in a normal chart at m E M, and then define the mean derivatives by the following formulas:

Dy(t) = X ° (t, y(t)) and

Dsy(t) = X,° (t, y(t)) . We emphasize that X° and X« are (time-dependent) sections of TM.

Lemma 20.1. Let y(t) be a solution of (20.1). Then X°(t,m) = a(t,m). To prove this, pick m E M and consider a normal chart centered at m. In the chart, the connector vanishes at m, i.e., y(t) is given by (20.3) with Fm (I, I) = 0. Applying the results of Sect. 19.B, we see that Dy(t)m = a(t, m) in the coordinate chart. Since Dy and a are both vector fields, we have Dy(t) = a(t, y(t)). Note that (-072/2) fo trF.,(,)(I, I) ds is the compensating term for o ff bx(t) dw(s) in the local expression for the martingale. Applying the methods of Sect. 19 to the equations in local coordinates, one can easily check that X' (t, m) is well defined, provided that y(t) is a strong solution of (20.1). In this case, keeping the notation of Sect. 19, we replace the symbol X° (t, m) by a.(t, m), so that Dsy(t) = a. (t, y(t)).

Let Ds = (D + D.)/2 and DA = (D - D.)/2 be the symmetric and antisymmetric mean derivatives. Consider the current velocity Dsy(t) = v(t, y(t)) and the osmotic velocity Dsy(t) = u(t, y(t)) of y(t), where

v(t,m) = 2(X°(t,m)+X°(t,m)) and

u(t, m) = 2 (X ° (t, m) - X' (t, m)) The assertions of Lemmas 19.4 and 19.5, proved in Sect. 19 for the osmotic and current velocity on 1R', hold for those on a Riemannian manifold M.

20. Stochastic Mechanics on Riemannian Manifolds

111

Consider the following equation on M:

dy(t) = expy(t)(a(t, y(t)) dt - oD.w(t) + o dw(t))

(20.4)

,

where

a(t, y(t)) dt - aD.w(t) + odw(t) is the class of processes formed by the solutions of the following equation (see Remark 19.2): t+r

X (t + r) = it

a (s, X (s)) ds + it

t

t

-f +

a2 tr A' (A (s, X (s))) ds

t+r

A(s, X(s)) o D;w(s) ds t+r

Jt

A(s, X (s)) dw(s)

(20.5)

.

Here r > 0, a(s, X) and A(s, X) are as in (20.2), and A': TmM x 1R" --* T,mM is the ordinary derivative of A. Applying the same argument as for (20.1)-(20.3), we can write (20.4) in local coordinates. First, observe that

(exp',,A) and (see (13.7)) exp,n(0) = I

,

where the primes denote the derivatives. Thus, the map exp,,,, sends the vector tr A' (A(t, o) - , ) tangent to TmM to the vector

+trl'm(I,I)

,

and the vector A(t, 0) o D; w(t) to bm o D; w(t). Combining these observations with (20.3), we obtain the local coordinate form of (20.4): dy(t) = a(t, y(t)) dt + a2 tr b'Y(t) (by(t)) dt 2

- Qbyltl o D. w(t) dt + 2 tr I'y(t) (I, I) dt + o by(t) dw(t)

.

(20.6)

A straightforward calculation shows that (20.6) transforms covariantly under changes of coordinates. Similarly to (20.1) and (20.2), this means that (20.4) is well defined.

Lemma 20.2. Let y(t) be a solution of (20.4) with y(O) = m. Then D.y(t) _ a(t, y(t)) for t c (0, l ].

112

Chapter 6. Nelson's Stochastic Mechanics and Quantization

Proof. For the sake of simplicity, we prove the lemma only in a coordinate chart of the initial point mo. Observe that there exists a process X (t) on M such that y(t) = expm X (t). Furthermore, X (t) satisfies (20.5). Since Dy(t) is a vector, it suffices to show that D.X(t) = a(t,X(t)), which, in turn, follows immediately from Remark 19.2. Definition 20.1. The backward stochastic differential a. (t, m) d.t + a d.w. (t)

given by the field a(t, m) is the class of stochastic processes formed by the solutions of the equation

X (t - r) =

It J

a(s, X (s)) ds + j t aA(s, X (s)) d.w; (s)

t

,

(20.7)

- r

where r > 0, and a and A are as in (20.2). In this notation, the backward Ito equation takes the form:

d.y(t) = exp,,itl (a(t, y(t)) d.t + a d.w.(t))

(20.8)

.

To be more precise, (20.8) means that for any t from the domain of y(t) the process y(t - r), r > 0, a.s. coincides with a process from the class expYiti (a(t, y(t)) d.t + a d.w. (t))

as long as y(t - r) stays in a neighborhood of y(t). By Remark 19.2, equation (20.7) can equivalently be written in the form: t

X (t - r) = J a(s, X (s)) ds + J t a2 tr A' (A(s, X (s))) ds

t-r

-r

- !-r a A(s, X(s)) o Dew(s) ds t

+

Jt-r

o A(s, X (s)) dw(s)

(20.9)

.

Note the similarity between (20.9) and (20.5). In particular, using the same argument as for (20.4), one may show that in local coordinates (20.8) turns into the following equation: d.y(t) = a(t, 6(t)) d.t + 1 tr l'y(t) (I, I) dt + aby(t) d.w. (t)

.

(20.10)

Remark 20.1. Mean derivatives of diffusion processes on Riemannian manifolds were studied in [113]. The local expressions for the derivatives obtained therein involve the Christoffel symbols (i.e., the local connector). Note, however, that although these formulas are similar to (20.2) and (20.10), the

20. Stochastic Mechanics on Riemannian Manifolds

113

method used in [113] does not rely on the theory of stochastic differential equa-

tions. Yet, as we have shown, the Ito equations in the Belopolskaya-Dalecky form are naturally related to the formalism of mean derivatives.

Let us introduce the mean derivative of a vector field along a stochastic process. To give a definition, we first need to refine some of our notions, as we usually do when passing from vector spaces to manifolds.

Let Y(t, m) be a C2-vector field on M. We define the mean covariant derivative of Y along the process y(t) by the formulas:

DY(t,y(t))=K°olimoEt D.Y(t, y(t)) = K o of

(Y(t + ot, y(t + at)) - Y(t, y(t)) ) (Y(t1t )) - Y(t -

+o

J

Qt

Et

'

'dt,y(t - dt))

at (20.11)

where K: TTM -+ TM is the connector of the connection on M (see the definition of covariant derivative in Appendix A). Note that the limits in (20.11) are evaluated in a local chart. Thus applying to them the connector K, we obtain elements of TM (covariant under changes of coordinates). Now let y(t) = RIz(t), where z(t) is a process (19.2) in a tangent space to M. Then we have the following analog of (19.10): DY (t, y(t)) =

DY(t, y(t)) = where V is the (classical) covariant derivative and V2 is theLaplace-Beltrami operator. To prove (20.12), pick a normal chart and apply the forward and backward Ito formula to Y. (See (12.8) and Remark 19.2.) Note that here Y is regarded as a map M -- TM. Furthermore, in normal coordinates near m the covariant derivative Vz for Z E TmM is just the derivative in the direction Z(m) and V2 is the standard Laplace operator. Therefore, we have (20.12) in normal coordinates, and so in any coordinate chart. Note that V2 corresponds to the trace term in the Ito formulas and V2 differs from the Laplace-de Rham operator A. (See Appendix A.)

Remark 20.2. The derivatives D and D. can also be defined as follows (see Appendix A). Assume that the parallel translation along y(t) in the sense of Sect. 14 is defined. Denote by Fr,.Y the vector obtained by the parallel translation of Y from y(s) to y(r). Then it is not hard to show that

DY(t, y(t)) = olim Et o

(1t+t'(t +,at, y((t

,At))

- Y(t, y(t))

)

,

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Chapter 6. Nelson's Stochastic Mechanics and Quantization

m oEt D.Y(t, y(t)) = At-4

(Y(t, y(t))

- rt,t-oQt(t - At, y(t - At)) (20.13)

Following the line of Sect 19.E, we define the acceleration a of y(t) by the formula a 2 (DD. + D.D)y(t) = 2 (DX +

It is clear that

a = (DsDs - DADA)Y = Dsv - DAU , where Ds = (D +

and DA = (D - D.)/2. Now, by (20.12), we have

Dsv

t

+

and z

DAU = 2 V2U + V u Therefore,

a= I

aV + 51

I- I 12 a2V2U + DuU I.

(20.14)

In conclusion, we note that here Dsv is defined similarly to the covariant derivative of a time-dependent vector field along the field itself.

20.B. Geometric Stochastic Mechanics Consider a mechanical system in the sense of Sect. 4 with a configuration space

M and a force field at, m, X). First, it might appear natural to suppose that the Newton equation of geometric stochastic mechanics is similar (like the Newton equation of Sect. 19) to the classical one: the acceleration a is equal to the force a. However, it turns out that this assumption does not lead to the right correspondence between solutions of the Newton and Schrodinger equations (see Sect. 19). To retain the correspondence, we should replace the Laplace-Beltrami operator V2 in (20.14) by the Laplace-de Rham operator .A, i.e., we have to alter the definition of acceleration. (Apparently, it was originally discovered in [27] that such a modification was necessary. In Remark 20.3, we give more details on one of various ways to carry out this program.) Here, we choose another approach which is based on the Weitzenbock formula for 1-forms:

-aX=V2X-RicoX

,

where Ric(m): TmM -+ T,,,M is the Ricci curvature at m. (See [78] or Appendix A.) Then the Newton-Nelson equation of stochastic mechanics on Rie-

20. Stochastic Mechanics on Riemannian Manifolds

115

mannian manifolds is, by definition, as follows

a = a(t, x(t), vx (t, x(t))) + 2012Ric(x(t)) o u5 (t, x(t))

.

(20.15)

Similarly to Sect. 19.F, stochastic processes satisfying (20.15) with some fixed a(t, rn) are related to solutions of the Schrodinger equation. The reader interested in more details and proofs should consult [27] and [113].

Remark 20.3. If, following [27], [31], [32], and [113], we defined the acceleration of a diffusion process by the formula a

aty+vvv) -

u)

then, as mentioned above, the Newton equation would take the standard form a = a(t, x, v). In [31] and [32], the construction of parallel translation was refined to make the definition of a more natural. (The modified parallel translation takes into account the dispersion of geodesics.) Then one sets a = (DD. + D.D)x, where D and D. are defined by the formulas of Remark 20.2, but using the modified parallel translation. In what follows, we shall search for solutions of (20.15) among diffusion-

type processes (i.e., the Ito processes on a manifold). Note that this class of processes is larger than just diffusion processes and that diffusion-type processes may fail to be Markovian.

20.C. The Existence of Solutions in Stochastic Mechanics Consider a mechanical system in the sense of Sect. 4 on a Riemannian manifold

M. Our main objective in this section is to prove the existence of a stochastic process on M which is a trajectory of the stochastic mechanical system, provided that M and the force field satisfy some natural hypotheses. Recall that if the force field is potential or gyroscopic, then there is a certain connection between processes of stochastic mechanics and wave functions, i.e., solutions of the Schrodinger equation. In [24] (see also [19]), this connection was used to prove, applying the methods of partial differential equations, the existence of trajectories for stochastic mechanical systems on 1Rn under the assumption that the potential belongs to the Rellich class. Here we deal with arbitrary force fields, which we require just to be independent of the velocity. In particular, these fields may be nonpotential or nongyroscopic. Thus, our method enables one to prove the existence of a trajectory for force fields, to which other methods of quantization do not apply. In what follows, we assume that certain measurability and integrability conditions are satisfied. This is a hypothesis similar to that imposed on the potential force in [24]. The Ricci tensor and its covariant derivatives are only supposed to be bounded. Thus, the manifold M may not be flat.

Chapter 6. Nelson's Stochastic Mechanics and Quantization

116

Our argument depends on the methods of stochastic differential geometry and stochastic analysis developed in the preceding sections. In brief, the proof goes as follows. First, using parallel translation along stochastic processes,

we introduce a particular stochastic equation in the tangent space at the initial point of motion (a version of the velocity hodograph equation). Then we prove the existence of solutions for this equation. The next step is to show that starting from a certain (fixed in advance) moment, the development of a solution satisfies the Newton-Nelson equation. Thus, the development is, in fact, the desired trajectory. (Note that the process under consideration begins at a single point (i.e., at a "wrong" distribution) and, therefore, the trajectory cannot begin at t = 0.) Recall also that we are looking for a trajectory in the class Ito processes on M. Thus, the trajectory need not be a diffusion or Markovian process (as in [19] and [24]), but only a diffusion-type process. Let a(t, m), with t E [0, 1] and m E M, be a force field on M, which is assumed to be independent of the velocity. For the sake of simplicity, we set

Q=1. Consider the following tensor fields on M.

(1) the covariant derivative of the Ricci tensor Ric, i.e, the (1, 2)-tensor field T.. M

(2) the trace of the covariant derivative, i.e., the vector field n

tr V Ric(m)(Ric) _

V Ric(m) (Ric (m)eti, ei)

where e1,... , en is an arbitrary orthonormal basis in TmM. Note that the field tr V Ric(Ric) is C°°-smooth because so is Ric. Throughout the rest of this section we assume that M is complete and that M and the force field meet the following conditions: (1) The Ricci tensor Ric(m): TmM -4 TmM is uniformly (in m) bounded with respect to the operator norm given on the tangent spaces by the metric (, ). The vector field tr V Ric(Ric) is uniformly bounded with respect to the metric. (2) The vector field a(t, m) is Borel measurable jointly in t and m, and there

exists a constant C > 0 such that

f Il(t,MW) 11 2 dt < C for any continuous curve m: [0, 1] -+ M, where the norm the metric.

is given by

It follows from condition (1) and the Yau theorem (Sect. 15) that M is stochastically complete. Condition (2) is fulfilled if, for example, a(t, m) is continuous and uniformly bounded.

20. Stochastic Mechanics on Riemannian Manifolds

117

Fix mo E M and consider the probability space ((2°, .77°, µ°), where (1° = Co ([0, l], T,,,,o M), F° is the Q-algebra generated by cylinder sets, and 1L° is the

Wiener measure (see Sect. 12). Recall that µ° is unique, since T,"OM is the Euclidean space with the inner product (1)... Denote by Bt the v-algebra generated by cylinder sets with base over [0, t]. (As usual, all Bt are completed by the sets of measure zero.) Recall that the coordinate process W (t, x(.)) = x(t), where x E ,0l°, is just a Wiener process in T7,,M. Since M is stochastically complete, the Ito development RIW (t) is defined for t E [0, l], and so is the Riemannian parallel translation along RIW(t). (See Sect. 14 and 15.) Denote by Ft,, the operator of parallel translation along a stochastic process y(.) from y(s) to y(t) (see Sect. 14 and Remark 20.2). Thus translating the vectors a (t, RIW (t)), tr V Ric(RIW (t)) (Ric), and

the tensor Ric(RIW(t)) from the random point RIW(t) to RIW(0) = mo along we obtain, respectively, the vectors To,ta(t,RIW(t)) and Fo,t tr V Ric(RIW (t)) (Ric) and the tensor Fo,tRic(RIW (t)) at mo. Recall that RIW (t) is the extension of the inverse Cartan development from the space of piecewise-smooth curves to the space of a°-almost all continuous curves in T,,,oM. The parallel translation along is a similar extension (Remarks 14.4 and 14.6). This implies that Fo,ta, Fo,t tr V Ric(Ric), and To,tRic are defined along p°-almost all continuous curves in Tm0M. The values of these fields (Fo,t tr V Ric(Ric)) (t, E (7° are denoted by (1'o,ta) (t, along and, respectively, (Fo,tRic) (t, x(.)). For to E [0, l], we set to(t) = 1 1/to

if t > to

Let us fix po E T,, ,M and consider the following equation t

3(t) = 3o +J (Fo,9a) (s, W(.)) ds

- 4J 1

+ +

It

(To,, tr V Ric(Ric)) (s, W(.)) ds

It (12 (Fo,,Ric) (s, Jo

1 to(s) I Q(s) ds

f I -(Fo,Ric) (s,W()) +to(s)

- 2 to(t)W(t)

(20.16)

To prove the lemma it suffices to observe that the coefficients of (20.16) are Lipschitz continuous and either independent of 3 or having linear growth in /3.

118

Chapter 6. Nelson's Stochastic Mechanics and Quantization

Being a strong solution, ,3 exists for any Wiener process. Furthermore, it is nonanticipative with respect to Pw. In what follows /3 is always taken for the realization W (t) of the coordinate process in (fl°, F°, p')

Lemma 20.4. The solution /3(t) satisfies (15.9). The lemma follows easily from the aforementioned conditions (1) and (2) and from the properties of parallel translation. Let us define a function 0(l): (2° --p IR as follows:

0(l) = exp \(- 2J/3(s)2 ds + o

Jo

l (Q(s),

dW (s)) I

.

(20.17)

Making use of (1) and (2), it is not hard to show that 0(l) is a probability density on (See [95] and [100].) Let liZ be the measure on (.fl°,Jc'°) which corresponds to 0(l), i.e., d,uz = 6(1)dµ°, and let z(t) be the coordinate process on ((2°,.FO,AZ). Then, as shown in [95] and [100], there exists a Wiener process w(t) on (,(2°,JP, pz) adapted to Pt = PW such that

z(t) =

J

t 3(s) ds + w(t)

(20.18)

.

It is clear that 0(l) > 0 and thus the measures µZ and p° are equivalent. Denote by 6(t) a function which is defined similarly to 0(l) (see (20.17)), but with 1 replaced by t. Applying the Ito formula (12.8), it is not hard to show (see, e.g., [100]) that 6(t) = 1 + J 6(s)(,3(s), dW(s)) t

.

(20.19)

0

Note that z(t) and W(t) coincide as coordinate processes on Therefore, by (20.18), equation (20.16) can be written equivalently as follows: t (s, z(.)) ds NO = 13(0) + f (Fo,sa) 0

a

- 41 f (To,, tr VRic(Ric)) (s, z(.)) ds

+f

t I

-2 (P0,,Ric) (s, z(.)) + 2to(s)J dw(s)

o

- 1 to(t)z(t)

.

(20.20)

Condition (2) guarantees that M is stochastically complete. Thus, by Lemma 20.4 and Theorem 15.7, the Ito development x(t) = Rjz(t) is defined on the entire interval [0,1].

20. Stochastic Mechanics on Riemannian Manifolds

119

Theorem 20.1. For every t E (to, 1), the process x(t) satisfies (20.15), i.e., x(t) is a trajectory of the stochastic mechanical system with the force field a(t, m). Proof. To check that x(t) satisfies (20.15), we need to find the mean derivatives of z(t) and x(t). To do this, let us first recall how to calculate the conditional mathematical expectation under a change of the probability measure. Let (,(1, .F, P) be a probability space and p a probability measure on it. Assume that p is absolutely continuous with respect to P with density p. Let B be a o--subalgebra of .F and 7P a measurable map from (0,T) into a measure space. Denote by E(O I B) the conditional expectation of r1i with respect to B and by E the unconditional expectation of 0 on the space ((2,.'F, P). Also, let E* (0 I B) and E* be the analogous expectations on it). Starting with E(,i I B), we can find E*(,O I .T) using the following method (see e.g., [100]). For any function A which is measurable with respect to B, we have E* (AO) = E* (AE* (t, I B)) = E (AE* (,1 B)p)

= E(AE*(V) I B)E(p I B))

.

On the other hand, E(,\Op) = E(AE(?,bp I B))

and, therefore, 113) E*(t I B) = E(p I which carries the two meaReturning to the measurable space sures µ° and µz, let us denote by E° the (conditional) mathematical expectation on (.(1°,.F°, µ°) and by E' that on (dl°,.F°, µZ). Throughout the remainder of the proof, y(t) denotes the coordinate process on (.fl°,.F°) when the measure is not specified. Thus, y(t) = W(t) if the measure is µ° and y(t) = z(t) if the measure is µZ. Applying (20.19) and (20.21), we have

Dz(t) = Qlim o Ety

(y(t + at) - y(t) )

= olimo E t y(0(l))-1EE = Et y (0(l))

-'

y(y(t+ t) -

y(t)0(l))

At) lim ot-.+o Et y (y(t + At - y(t) )

+Et y(0(l))-' olim oEt y(e+(Ot)) where

e +(at)

y(t + At) - y(t)

at

f

e(t)(a(t),dw(t))

.

120

Chapter 6. Nelson's Stochastic Mechanics and Quantization

Since y(t) = W(t) on ((l°, °, p°),

y(t + At) - y(t)

lim E°

Ate+o

_0

At

and, by Theorem 12.4, lim Et y (O+(L t)) = Et ' (0(t)/3(t))

ot_+o

On the other hand, one may easily show as in Sect 19.B that Dz(t) = Et (/3(t)) Hence

Et

'(0(l))-'EE y(0(t)/3(t))

= Et (/3(t))

(20.22)

.

On the other hand,

a urn o

D.z(t)

Ety

- At)

(y(t)

/

At

(y(t) - y(t - 1t) = Et° y(B(l))-' lim E°t y B(l) ot_+a At Using the same arguments as above, we easily obtain

E° (f((t)) m ee

= olim

0E°

0Et y (Y(t)

At

-At) e(l)))

([f (W(t)) - f (W(t - At))]

+ elim E°

W(t) - W(t -At) 'At

0(l)

(f(w(t_t))W(t)_W(t_t)o(l))

o

As in the calculation for Dz(t), the second summand on the right-hand side is equal to E°(f (y(t))0(t),3(t)), while the first one is equal to E° (9(l) grad f (W (t))) = E°

[f(w(t)) (- rad p'

- grad E'9(l)1 B(l)

pW

0(l)

J

1

J

(The last equality is easy to derive using integration by parts.) Note that by Lemmas 19.4 and 19.6, -grad pi't'/pw = W(t)/t. Thus, taking (20.21) into account, we have

Et (a(t)) +

D:w(t) = ztt) - Et (x(t)) where k(t) = 0(l)-Igrad Et'(0(l)).

t- Et (K (t))

(20.23) (20.24)

0

Lemma 20.5. Let t E [0,1], and let g(t) and h(t) be L1-stochastic processes in IR" defined on the same probability space. Assume that g(t) and h(t) a.s. have continuous trajectories, and Dh(t) and exist. Then:

20. Stochastic Mechanics on Riemannian Manifolds

121

(i) Dhg(t) exists if and only if DhEE g(t) exists and DhEE g(t) = Dhg(t); and (ii) D; g(t) exists if and only if D; Et g(t) exists and D; Et g(t) = D* g(t).

Proof. Fix an arbitrary smooth function f : IR' -+ IR with compact support. First observe that ( Et +°tg(t

-

+ At)) f (h(t + At)) (Et9(t)) f (h(t)) = {Et+°tg(t + At) - Et g(t) If (h(t)) +E+°tg(t + At) { f (h(t + At)) - f (h(t)) }

and, since Dh { (Et g(t)) f (h(t))}

°tlim +o Et

(E+°tg(t + fit)) f (h(t + fit)) - (Et g(t)) f (h(t)) At

we have (see [111] and [113])

EDh { (Et g(t)) f (h(t)) } = E{ (DhEthg(t)) f (h(t)) }

+ E{ (Et g(t)) D; f (h(t)) }

Note that the existence of the second term on the right-hand side follows from the hypotheses of the lemma. Thus, the limit exists if and only if does DhEthg(t) so. On the other hand, it is evident that

E{Et [(E+°tg(t + at)) f (h(t + At))

-

(Et g(t)) f (h(t))] } = E{g(t + At) f (h(t + At)) - g(t) f (h(t)) }

.

A similar argument shows that

EDh{ (Et g(t)) f (h(t)) } = E{ (Dhg(t)) f (h(t)) } + E{g(t)D; f (h(t)) }

if and only if Dhg(t) exists. It is clear that

E{ (Et g(t))D; f (h(t)) } = E{g(t)D; f (h(t)) } Hence,

E{ (DhEthg(t)) f (h(t)) } = E{ (Dhg(t)) f (h(t)) }

,

which proves (i). The proof of (ii) is analogous and based on the equality

E+°tg(t + Lt)) f (h(t + At)) - (Et g(t)) f (h(t)) _ { E+°tg(t + At) - Et g(t) If (h(t + At)) +Et g(t) { f (h(t + At)) - f (h(t)) }

.

Chapter 6. Nelson's Stochastic Mechanics and Quantization

122

Corollary. The assertion of Lemma 20.5 holds when h(t) takes values on an n-dimensional manifold.

The corollary can be proved in the same way as Lemma 20.5. The only refinement needed is to use a compactly supported function on the manifold.

(z(t))

Dz

= EZ /\t

(- --) - z(t)

I'-'I t

\

t

/

t

t2\

t

Lemma 20.7.

DD.z(t) = DZQ(t) + Et (-) - zt2) DDz(t) = DQ(t) .

(20.25) (20.26)

Proof. We apply Lemma 20.5 and formula (20.21) as follows: D£EE (rc(t)) = D£Et [0(1)-1gradEW (0(1))]

= De [0(l)-'grad Er' (0(l))]

=

Et (0(l)-1gr

At-

(EW Ot (0(l)) - EtW (0(l)))

4t

1\

= EW(0(1))-16 where E

=

w (9(l)rad(Et(0(l))

at

t-Et

- Er (0(l))) 0(l) I

(grad(EWat(0(l)) -EW (0(l))))

= Ate+0 lim EW `

'At

= Dw(EW B(l)) = Dw0(l) = 0 , and thus

DCEt

0.

Taking this into account, we derive (20.25) and (20.26) from Lemmas 20.5 and 20.6 and formulas (20.23) and (20.24).

Let us to continue our proof of Theorem 20.1. By definition, x(t) satisfies

20. Stochastic Mechanics on Riemannian Manifolds

123

the following equation (see Lemma 14.3 and Definition 14.6):

dx(t) = expx(t) (ft,o1(t) dt + Ft,o dw(t))

(20.27)

,

where, as above, Ft,0 is the operator of parallel translation along x(.) from x(O) to x(t). Thus, Dx(t) = Et (rt,0i3(t)). To find D.x(t) and the second-order mean derivatives, we need an additional argument. Note that even though the "present" o-algebra Art' in F' differs in general from Nt, the processes z(t) and x(t) have the same probability distribution. This means that to find the probability of the events

z(t) E A C Nt and x(t) E B C Nt, we need to integrate the same density on T,,,,oM over A and B, respectively. (Of course, this is also true for the processes a(t) and Rra(t), when a(t) is regarded as a coordinate process on a suitable probability space. This applies, in particular, to W(t) and RIW(t).) Similar to (20.23), employing the methods of calculating the mean derivative developed above and using the construction of the Ito development, we see that

Et [I't,oi3(t)] + Et

[r0 (zt`)

- ic(t))]

(20.28)

.

In particular,

Dsx(t) = Et [rt,o,3(t)] + 2Et [r',0( z() - r. (t)/ 1

and

DAX(t)

2Et [r0( ztt)

-r.(t))

Now we are in a position to calculate the covariant mean derivatives of x(t) by (20.13). Arguing as before (similar to (20.20), (20.25), (20.28), and the corollary of Lemmas 20.5 and 20.6), we obtain

DD.x(t) = Et [r0 (r0,a) (t, z(.)) - 41't,o(ro,t trVRic(Ric)) (t, z())]

-Et [r0 ( t2)/] +Ef (o(t)) Et' II (Q(t) - ztt)] z 21

Furthermore, using in addition (20.24), (20.26), and Lemma 19.1, one may show that

D.Dx(t) = Et [r0 (ro,ta) (t, z(.)) - 1rt,o(I'o,t tr VRic(Ric)) (t, 1 Et [r,0 (f,o tr VRic(Ric)) (t,

2Et [rt,o(rt,oRic)(tz(.)) oEt (1(-tt) -ic(t)l] + Et [r,o (z (t)

tc tt)

/]

2 Et

[rt,o (6(t)

rc tt)

124

Chapter 6. Nelson's Stochastic Mechanics and Quantization

for t > to, i.e., in the domain where to(t) = 1/t. By the definitions of x(t), z(t), and rt,81 we have the following relationships:

rt,o (I'o,ta) (t, z(.)) = a(t, x(t)) Ft,o (Fo,tRic) (t,z(.)) = Ric(x(t)) rt,o (ro,t tr VRic(Ric) (t, z(.))) = tr VRic(x(t)) (Ric) Therefore, since Ric(x(t)) is measurable with respect to Nt,

2 (DD. + D.D) x(t) = a(t, x(t))

rrt

-

2Ric(x(t)) oEt

t

-r,(t)/J (20.29)

L

On the other hand, as we have shown DAX(t)

2

Et [rt,o

t) - K(t)/

which, combined with (20.29), gives (20.15). Note that when M is the Euclidean space IR' (in particular, Ric = 0), z(t) coincides with x(t) and the proof can be simplified. Then the assertion of the theorem follows directly from formulas (20.20)-(20.26) and we do not need to use any additional argument to deal with x(t).

Remark 20.4. Note that for t E (0, to), the process x(t) does not satisfy (20.15) because

to(t)Dw(t)

Et

#

Lrt'o\ tto)

Et [rt,o (zt2)

to)/J

i--)]

Thus x(t) can only be interpreted as a trajectory whose initial value at to is equal to the random point x(to) and whose initial forward mean derivative is Et (rto,op(to)). It is clear that to can be chosen arbitrarily close to zero and, therefore, we can pick the initial value of x as close to (mo, X3o) as we wish. However, it is impossible to set to = 0, for the integral fo (1/r) dw(T) does not exist. (The integral fo (1/r2) dr diverges. See, e.g., [95].) In other words, equation (20.16) is ill-posed when to = 0. If, to simplify the argument, we picked the nonrandom initial value x(0) = mo for x(t), i.e., a distribution supported at a single point, then such a choice would be unnatural in the framework of our theory.

Remark 20.5. One can easily show that El (Q(t)) is the hodograph of the forward mean derivative of x(t) and (20.20) is an analog of the velocity hodograph equation studied in Sect. 7.

21. Relativistic Stochastic Mechanics

125

Remark 20.6. Let M = 1R' and a = -grad U. Then, as mentioned above, a trajectory of the stochastic mechanical system was obtained by Carlen [24]. Suppose that our and the Carlen trajectories give rise to the same solution W of the Schrodinger equation (Theorem 19.1). Then the trajectories also induce the same probability measure on the space of sample paths (i.e., the trajectory is weakly unique), since the density p is equal to IWI2.

21. Relativistic Stochastic Mechanics Trying to find an adequate mathematical model for relativistic stochastic mechanics, one inevitably faces serious problems, the solution of which requires essentially new ideas. In particular, the techniques of standard stochastic differential equations fail to be applicable and certain approaches to the problem, which may first appear natural, turn out to be misleading. (See, e.g., [84] and [112].) A new approach to relativistic stochastic mechanics, which does not rely on the techniques of stochastic differential equations, is developed in [33] and [84]. The method is based on a modification of stochastic derivatives making the derivatives covariant. In [33] and [84], the method was applied to find a stochastic model describing the motion of a charge in an electromagnetic field on the Minkovsky space. The model turned out to be naturally connected with the Klein-Gordon equation. In this section, we briefly outline the ideas of [33] and [84]. We also modify the notion of a stochastic differential equation to use it in the framework of relativistic stochastic mechanics. Then we make an attempt to describe relativistic stochastic mechanics on the Lorentz manifolds of General Relativity (GR). The reader interested in the basic notions of GR should consult, for example, [23] and [118]. Let M4 be the Minkowski space with signature (1, 3). We split M4 into the orthogonal direct sum IR1 ® IR3, where IR1 is timelike and IR3 is spacelike. Changing the sign of the inner product when necessary, we may regard both IR1 and 1R3 as Euclidean spaces. Thus, there exist independent Wiener processes w°(r) on 1R1 and w(r) on IR3. Here w°(r) is one-dimensional, tii(T) is three-dimensional, and r is a parameter, the proper time. Hence, we are in a position to define the backward Wiener processes w°(r) and w,(r) for a process y(t) (Definition 19.1). For the sake of simplicity, we shall always use a system of units where the speed of light c and the electron charge e are both equal to 1. Let 6(t) be a stochastic process in M4. Following [33] and [84], we modify the definition of stochastic derivatives to make them covariant with respect to the Lorentz group. Namely, let us define the forward derivative D6(r),,, at m E M4 and T E IR by the formula

D6(r)m = lim (E++ + E__) oT-a+o

,

(21.1)

126

Chapter 6. Nelson's Stochastic Mechanics and Quantization

where

_ E++

tt (S(T+AT) -S(T) CC/

E

(T) = m ,

6-T

(S(T + d r) - £(T))2 > 0 1 .AT))2

E--=E(e(T)-S(T- Ar)

S(T) = m ,

QT

(S(T) - S(T -

<

0)

Similarly, the backward stochastic derivative is defined as

D.e(T)m =

lim

.6T-++o

(E+_ + E_+)

(21.2)

,

where

E+_=E(6(T+aT)-S(T)

(T) = m

IT

S(T) = m

DT

,

( (T + &r) - e(T))2 < 0)

,

(S(T) - S(T - QT))2 1 0)

Let a(m) be a timelike vector field on M4. Consider a stochastic process (T) on M4 which satisfies the following "forward" equation: ttT0 ) = (T1) - b(

rTla JTp

t3 ds+a

(S( ))

(

w0 (T1)

-

w0 (T0))

+a(w.(Ti) - tv.(TO))

(21.3)

for all Tl > 7-0. Using the fact that wo and w are independent, it is not hard to show that DS(T),,, = a(m), provided that (T) is a solution of (21.3). Similarly, we see that D.6(T),,, = a.(m) for a solution 6(T) of the "backward" equation: 6(T1)

- SCC(T0) =

fp

a,.(C (S (3)) ds+a (wr (T1)

- w + (T0)\J (21.4)

+ 0, (zn(T1) - w(TO))

where, as above, Ti > To. Furthermore, let

Ds = 2(D+D.) ,

DA

(D-D.)

and

u=DAB As shown in [33] and [84], the field u is related to the density p(T, m) by the formula of Sect. 19.D, where p is taken with respect to the Lebesgue measure on lR x M4. Moreover, u and p satisfy the continuity equation of Sect. 19: ap aT

= -div(p v)

21. Relativistic Stochastic Mechanics

127

For any vector field Y on M4, we define the derivatives DY and D*Y along 6(T) by formulas similar to (19.9), in the same way as (21.1) is obtained from (19.3), and (21.2) from (19.4). Then (see (19.10))

DY= and

z

\\

D*Y

D//// I Y

where is the wave operator (d'Alembertian). Let F(. .) = Fµ" dqµ A dgu/

be the 2-form giving the electromagnetic field on M4 and let

F(

) = Fv aqµ A dq"

It Denote by the 1-form of an electromagnetic potential of F, i.e., F = is well known that the motion of a unit charge in the field F is given by the equation d2

Due to the general concept of stochastic quantization, we assume that a stochastic trajectory is a solution of (21.3) with the field a such that

a = P(., v)

,

where

a = 2 (DD* + D*D) = (DsDs - DADA) = Dsv - DAU . Assume now that there exists a function L = L(T, m) such that v+0 = dL. (Here v is the 1-form corresponding to the vector field v via the metric on M4.) Then we introduce the complex-valued wave function W as follows: T/ (,r, m) =

P(T m) exP ( L(T, m) )

where, as above, p is the density function with respect to the Lebesgue measure. Similarly to what we did in Sect. 19.F for a gyroscopic force, one can show that W satisfies Schrodinger-type equations. Thus, the function

0(m) = exp(

i7-

-')

W(T,m)

is, in turn, a solution of the Klein-Gordon equation. More details on this construction can be found in [33] and [84].

128

Chapter 6. Nelson's Stochastic Mechanics and Quantization

It is not hard to show (see [33] and [84]) that the density p is, in fact, independent of r. Hence, the continuity equation takes a simpler form: div(pv) = 0. Note also that the "time component" (pv)° may not be positive. Negative values of the time component correspond to antiparticles. Let us outline how to generalize this analysis to a nonlinear Lorentz manifold, i.e., how to pass from Special to General Relativity. Let M be a fourdimensional Lorentz manifold with signature (1, 3). For the sake of simplicity

we assume from now on that M is orientable and oriented in time. In other words, the time direction "to future" is fixed in every tangent space T,,,M,

mEM. Consider the principal bundle L(M) with the structural group L±, the proper orthochronous Lorentz group [47]. The action of L± on the Lorentz space preserves the standard orientation and the time orientation. The bundle L(M) is a subbundle of the principal bundle of Lorentz orthonormal frames. Denote by H the restriction of the Levi-Civita connection to L(M). Let w° and w be the Wiener processes on the Minkovsky space M4 defined as above. Every basis b E L(M) gives rise to the decomposition T,rbM ^J IRl ® IR3, where IR3 is spanned by spacelike vectors of b and IR1 by timelike vectors. This enables us to identify M4 and T,rbM, and, thus, to define the processes b o w° and b o w in T,rbM. We call these processes the realizations of w° and w in T,fbM and, when it does not lead to confusion, we denote them simply by w° and w. Note that b o wo = T, Eb(w°) and b o zD = T,.Eb(w), where E is the canonical trivialization of H over L(M) given by the basic vector fields (Appendix A). Recall also that TEb is smooth in b. Let a(m) and a. (m) be arbitrary timelike vector fields on M. Denote by (a(m), z7v°, w.) the class of processes in TmM formed by the solutions of (21.3) with a(0) = a(m). Similarly, let the class (a(m), zD°, w) be formed by the solutions of (21.4) such that a. (0) = a. (m). Consider the following equations:

dl (r) = expc(*)

w°, w*)-

(21.5)

and

d6(r)=exp£(,r)(a.(l;(r))+wo,w) (21.6) where exp is the exponential map of H. (Note the analogy between (21.3), (21.4) and (21.5), (21.6).) Equation (21.5) means that for any r there exist realizations of w° and w in T£(,.)M such that the process £(t+s) belongs to the class exp (a(6), w°, w.)- for a sufficiently small IsI. Similarly, (21.6) means the existence of such a process for some realizations of w° and zD and the class exp£ a. (6), w., i7 j). Only a minor modification of Definitions 15.5 and 15.6 is needed to introduce the solutions of (21.5) and (21.6). Namely, now we have to use two Wiener processes: w° in IR1 and w in 1R3, instead of the process w in IR". Further analysis closely follows the line of Section 20. Note, however, that here we also need to modify the definitions of D and D., as well as the notion of parallel translation along the solutions of (21.5). Let M be the space-time of GR. Recall that under the action of a purely gravitational force, a particle moves along a geodesic of the Lorentz metric

21. Relativistic Stochastic Mechanics

129

on M. By the general concept of stochastic quantization, the motion of a stochastic particle is a solution (T) of (21.5), where a is such that (see (20.8)) (21.7)

In particular, in the Ricci-flat "empty" space-time, we have Dsv = DAU .

(21.8)

Consider Einstein's equation (see, e.g., [118]):

Ricwhere 1C is the scalar curvature, g is the Lorentz metric, and T is the energymomentum tensor. Passing to the (1, 1)-tensors, we obtain the equation (21.9)

Then, combining (21.7) and (21.9), we have (21.10)

Equations (21.8) and (21.10) link stochastic mechanics and GR.

Remark 21.1. A slightly different approach to relativistic stochastic mechanics is developed in [33] and [84]. Namely, instead of working with (24.3) and (24.4), it is assumed that the process (T) describing the evolution of the system has the derivatives DS(T) and D.6(t) given by (21.1) and (21.2). Besides, in every coordinate system on M4, the process must satisfy the equation: z

alim o(E+ - E-) =

2gµ"(m)

where gµ" (m) is the metric tensor at m and E+ and E_ are as follows:

E+ = E where

S(T)=m, (C(T+,dT)-S(T))2 > 0)

(r7+"

+- (6µ(T + QT) - 6µ(T)) (6' (7- + AT) - 6"(T)) LT

and, similarly, E_ = E (7Iµ" where

µ"

1

tt 6(T)

tt

= m , (S(T) - S(T CC

\/C

Q7_))2

G 0)

r))ls"(T) - "(T - AT))

QT

130

Chapter 6. Nelson's Stochastic Mechanics and Quantization

Solutions of (21.3) and (21.4) satisfy the latter equation. It is clear that this alternative method can be applied instead of (21.5) and (21.6). Remark 21.2. We also point out a series of papers by Zastawniak (see, e.g., [148]) which appeared after the Russian edition of this book was published. In these papers, the approach of [33] and [84] to Special Relativity received some further interesting development. Zastawniak's techniques can be generalized to make them apply to General Relativity, similarly to what we did in this section.

Part III Infinite-Dimensional Differential Geometry and Hydrodynamics

The subject of the following chapters is the Lagrangian formalism of modern hydrodynamics developed in Arnold's paper [3] and then by Ebin and Marsden [39]. Here we view a hydrodynamical system as a mechanical system in the sense of Chap. 2 and study it using infinite-dimensional differential geometry. Since the configuration space of the system is infinite-dimensional and it is equipped with a weak Riemannian metric (i.e., the topology arising from the metric is weaker than the original one), the problems we need to solve are essentially different and much more complex than those analyzed in Chap. 2.

This part of the book consists of three chapters. Chap. 7 contains all necessary preliminary material on the (weak) Riemannian geometry of infinite-

dimensional manifolds of maps and, in particular, on the geometry of the group of diffeomorphisms. Here we also consider mathematical models of a barotropic fluid and diffuse matter as the simplest examples of Lagrangian hydrodynamical systems. In Chaps. 8 and 9 we study an ideal fluid and then a viscous incompressible fluid.

Chapter 7. Geometry of Manifolds of Diffeomorphisms

Our main goal in this chapter is to study the geometry of the infinitedimensional manifold formed by the H8-maps of a smooth compact manifold. Here s is a sufficiently large integer and H8 denotes the Sobolev class. More specifically, our attention is primarily paid to the Hilbert manifold of H8- or Ck-diffeomorphisms. Then we use the notions of infinite-dimensional Riemannian geometry to describe the mathematical model of the flow of a barotropic fluid and diffuse matter as infinite-dimensional analogs of the mechanical systems of Chap. 2.

22. Manifolds of Mappings and Groups of Diffeomorphisms In this section, we recall how to introduce the structure of an infinitedimensional manifold on the set of maps from one (finite-dimensional) manifold to another. We also study the group of diffeomorphisms of a given manifold (with or without boundary). For a more detailed discussion of this matter the reader should consult [39], [86], [103], [105], and [117].

22.A. Manifolds of Mappings Let M and N be orientable finite-dimensional manifolds and let 8N = 0. Fix a Riemannian metric on N and denote its exponential map by exp: TN --+ N. The set Ck (M, N) of Ck-maps from M to N can be equipped with the structure of a smooth manifold as follows. Let g E Ck (M, N). Consider the set

T9Ck (M, N) _ If E Ck (M, TN) 17r of = g J

,

where 7r: TN --+ N is the natural projection. Note that T9Ck(M, N), equipped with the Ck-norm, is a Banach space. The map weXp:

TgCk (M, N) -' Ck (M, N)

defined as Wexp f = exp of is one-to-one on a sufficiently small neighborhood of the origin in T9Ck (M, N). Thus, this neighborhood together with weXp can be taken as a chart near g E Ck (M, N). It is not hard to show that transition

134

Chapter 7. Geometry of Manifolds of Diffeomorphisms

maps (changes of coordinates) from one chart to another are C°°-maps in TgCk (M, N). Therefore, we obtain the structure of a C°°-smooth Banach manifold on Ck(M, N). Its tangent space at g is T9Ck(M, N). Note also that this structure is independent of the metric on N. Similarly, one may introduce the structure of a smooth Banach manifold on the space of Holder maps from M to N. However, it is more convenient for us to work with the space of Sobolev maps, since the latter is a Hilbert manifold. In what follows we shall assume that M and N are both compact, oriented,

and dim M = dim N = n. Fix s > n/2. Then H8-Sobolev maps from M to N are well defined. To see this, observe that for such an s the property of a map f : M -+ N to be square integrable in a coordinate chart together with its (generalized) derivatives of order up to s is invariant with respect to changes of coordinates. Let H8 (M, N) be the set of such maps. In the same way as before, for every g E He (M, N), we define the tangent space T9H8 (M, N), which, being equipped with the standard Sobolev inner product, turns out to be a Hilbert space. (See Appendix D.) Hence, H8(M,N) is a C°°-smooth Hilbert manifold. Similarly as in the case of maps of linear spaces, one may show that for s > n/2 + k the manifold H8(M, N) is continuously embedded in Ck (M, N) and the image is everywhere dense. Remark 22.1. Let E be a vector bundle over a compact Riemannian manifold M. Then one can easily define the space H8 (E) of H8-sections of E for any s > 0. In particular, one may take E to be the tangent bundle TM, the vector bundle of k-forms AkT*M, etc.

22.B. The Group of H8-Diffeomorphisms Let M be an oriented compact manifold without boundary. Fix s > n/2 + 1. The manifold H8(M, M) contains the subset De(M) formed by H8-maps which are C'-diffeomorphisms. It is easy to see that D8(M) is an open set and thus a Hilbert manifold. It is easy to see that D8 (M) is a group with respect to the composition with the unit element e = id [39]. The tangent space TeD8(M) is the space of all H8-vector fields on M (i.e., H8-sections of TM). The whole tangent bundle TD8(M) can be identified with the subset of He (M, TM) formed by maps which, when composed with the natural projection to M, give elements of D8(M). In particular, TgD8(M) ={ If E H8(TM) I ?r o f= g} = {X o g I X E TD8(M)}

Note that the composition h o g of two H8-maps may not belong to H8. However, if s > n/2 + 1 and g is a (local) diffeomorphism, hog belongs to H8, provided that h is an H8-map to a compact manifold or, as above, a section of a vector bundle.

22. Manifolds of Mappings and Groups of Diffeomorphisms

135

Let g E D8(M) and h E H8(M, N), where h is as in Sect. 22.A. Consider the maps a9: He (M, N) - He (M, N) and wh: De (M) --+ He (M, N) such that a9(f) = .fog and wh(rl) = h o 77. a-Lemma. The map a9 is COO -smooth and its derivative has the form a9.

w-Lemma. The map wh is continuous. If h E He+k, then wh is Ck-smooth and its derivative has the form wTh. In particular, if h is C°°-smooth, so is wh

Proofs of both lemmas can be found, for example, in [103]. The lemmas are still true if we replace Hs (M, N) by the space He (M, E) of sections of a vector bundle E over M (see Remark 22.1). In particular, h may be a scalar function on M, a map to IR', etc.

Let g E D8(M). Consider the right and left translations R9 and L9 on De (M), where R. (77) = 77 o g and L9 (77) = g o g. Using the a- and w-lemmas, one can easily prove the following results.

Theorem 22.1. The right translation R. is C°O -smooth. Furthermore, we have TR9(X) = X o g for every X E T,1De(M), where TR9:T,7DB(M) T,o9D8(M) is the tangent map of R9.

Theorem 22.2. The left translation L9 is continuous. Let g E Ds+k(M). Then L. is Ck-smooth and, in particular, TL9(X) = Tg o X, where X E TD' (M). Theorem 22.3. The map g '--* g-1 is continuous on D8(M). Its restriction D8+k(M) -+ D3(M) to DB+k(M) is Ck-smooth.

Remark 22.2. Here we do not consider the manifold of COO-maps. This manifold is modeled on a locally convex (but not Banach) space. As a consequence, introducing a smooth structure on it requires some additional analysis. Left translations on the group of COO-diffeomorphisms are COO-smooth.

For a detailed account of the properties of De (M) and the group of C°°diffeomorphisms the reader should consult [39]. Using the fact that R9 is smooth for every g, one may define right-invariant vector fields on D8(M). Then the a- and w-lemmas yield the following result [39], which has no finite-dimensional analog.

Theorem 22.4. Let X E TeD8 (M) be a vector field on M and X the rightinvariant vector field on D8(M) such that X9 = X o g. Then X is Ck-smooth if and only if X E He+k(TM).

Corollary. The field X is C'-smooth if and only if X is so.

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Chapter 7. Geometry of Manifolds of Diffeomorphisms

Let us now turn to the question of the existence of integral curves of a right-invariant vector field on D'(M).

Theorem 22.5. (1) Let s > n/2 + 1 and X be a C'-smooth right-invariant vector field on D'(M). Then for any g E D'(M) there exists a unique integral curve y9(t) defined everywhere in IR with the initial condition y9(0) = g. In particular, y,,(t) is the flow of Xe = X and y9(t) = ye(t) o g. (2) Ifs > n/2 + 2, then the previous assertion holds for any continuous X on D' (M). The first assertion of the theorem follows from the existence and uniqueness theorem for ordinary differential equations with a smooth right-hand side on D'(M) and from the compactness of M. Under the hypothesis of the second assertion, the field X can be C'-smoothly extended to D" (M). To prove

that ye(t) actually belongs to D'(M), one uses the fact that solutions of a differential equation depend smoothly on initial conditions. Theorem 22.5, slightly modified, is still valid for a time-dependent rightinvariant vector field which is continuous in time in the H'-topology. Motivated by the analogy with finite-dimensional groups, we call the space TCD'(M) the Lie algebra of D'(M). The following result gives a partial justification of this terminology.

Theorem 22.6 ([39]). Let .K and Y be right-invariant vector fields on D'(M) with 9, -= X and Ye = Y. Assume also that X and Y belong to the class H'+1 Then [X, Y]e = [X, Y].

Note that TeD' (M) is not a Lie algebra, because the Lie bracket [X, Y] for H'-vector fields X and Y may not be an H'-vector field. On the other hand, TCD'(M) contains a dense linear subspace formed by C°°-vector fields, which is a Lie algebra with respect to the Lie bracket. Theorem 22.5 enables us to introduce the group exponential map from TeD' (M) to D' (M) provided that s > n/2 + 2. To define this map, consider

an integral curve 7,(t) of X. The curve -y(t) is a one-parameter subgroup of D'(M). The group exponential map sends the element X E TeH'(M) to ye(1) E D'(M). The exponential map is continuous, but not smooth. Note also that its image covers no H'-neighborhood of e E D'(M).

22.C. Diffeomorphisms of a Manifold with Boundary Assume, as above, that s > n/2 + 1 and M is a compact manifold possibly with boundary. Let D'(M) be the set of H'-maps from M to itself which are Cl-dffeomorphisms. Consider the subset D' (M) C D' (M) formed by the maps equal to the identity on M. Note that now H' (M, M) is not a smooth manifold. Thus, it cannot be used to introduce a smooth structure on D'(M)

22. Manifolds of Mappings and Groups of Diffeomorphisms

137

and D'(M). As a consequence, we need to modify the construction given in Sect. 22.B.

Fix an n-dimensional closed manifold N with n = dim M that contains M. For example, one may take the double M UaM M.

Theorem 22.7. The sets DI(M) and D(M) are smooth submanifolds of H'(M, N). Let e = id E D'(M) C D'(M). The tangent space TeD'(M) consists of the vector fields on M tangent to 8M. The tangent space TeD' (M) is formed by the vector fields which are zero on 8M.

To introduce an atlas on D' (M) and D' (M), one uses the exponential map of a metric on N chosen so that M is a totally geodesic submanifold. The description of the tangent spaces at e is evident. The rest of the theorem follows from the fact that TeD' (M) and TeD' (M) are closed subspaces in H'(TM). A detailed proof can be found in [39]. The description of tangent spaces, group structures, and the smoothness properties of right and left translations on D'(M) and D'(M) are quite similar to those when 8M = 0. (See Sect. 22.B.) In particular, we have an analog of Theorem 22.5 in the case where M is a manifold with boundary. We finish this discussion by pointing out an important difference between the case under consideration and the material of Sect. 22.B.

Remark 22.3. If X is an

H'+k-vector field on M, then X is Ck-smooth

on D8(M). The converse, however, may not be true. Namely, let X be Cksmooth. Then X is H'+k in int M and in the directions tangent to 8M. On the other hand, X may not be H'+k in the direction normal to the boundary.

22.D. Some Smooth Operators and Vector Bundles over D'(M) Let M and N be compact manifolds, let ON = 0, and let 7r: E -+ N be a vector bundle. Consider a smooth vector bundle w,,: Ha(M, E) -+ Ha (M, N),

where w,r(f) = 7r o f and a > n/2 + 1. Assume that M = N and s > a, and restrict w, to D' (M) C Ha (M, M). Thus, we obtain the vector bundle irr: (E) -- D' (M), where Oe (E)

f c Ha (M, E) 19r of E D(M)}

The fiber -Pe (E)9 of -r over g E D' (M) consists of all compositions f o g with

f E Ha(E). Let 4; (El) and 0e (E2) be two such bundles and P an operator from 0; (Ei)e = Ha(E1) to P; (E2)e = Ha(E2). Denote by (El) -' O; (E2) the right-invariant operator defined as

Pg =R9oPoR91:4;(Ei)9 -'0s(E2)9

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Chapter 7. Geometry of Manifolds of Diffeomorphisms

In general, the operator P is not smooth. Neither is a right-invariant subbundle of 0"(E). Our next goal is to find a condition which guarantees that these objects are smooth. Let IH be a finite-dimensional subspace of P; (E) = He (E) such that all elements of IH are C°O-smooth sections of E and let R: 4; (E)e --+ ]H be a continuous projector. Denote by IH the family of subspaces in the fibers of

0,(E)such that 1Hg={XogI XEIH}. Theorem 22.8. The spaces IH = im R and ker R are smooth subbundles of !P; (E). The map R: ; (E) -+ III is a smooth morphism of vector bundles. Proof [39]. Observe that the right-invariant section X is, by the w-lemma, C°°-smooth for every X E IH. As a consequence, IH is smooth. Let us fix an inner product G(,) on H" (E) so that R is orthogonal with respect to G. Using right translations, we can extend G to all fibers of 0'(E). As a result, we obtain a metric on 41(E), which is smooth by the w-lemma. Then R is just the orthogonal projection on IH and the smoothness of R follows from the smoothness of Ift and the metric.

Let E = Ak = Ak T*M, the bundle of k-forms on M, where k = 0,1, ... , n, and let d denote the differential and 6 the codifferential.

Theorem 22.9. For s > n/2 + 1, the operators 8-1(Ak+1) &,P8(Ak) and 6: 4is (A k)

!Ps-'(A k-1)

are C°°-smooth homomorphisms of vector bundles.

The proof is based on local formulas for these operators and on the observation that the operation of composition of H8-maps is smooth [39]. Properly refined, the theorem remains valid for any differential operator of order q with

s>n/2+q.

Remark 22.4. Note that methods of finite-dimensional geometry often fail to be applicable to prove the smoothness of an operator or of a subbundle of an infinite-dimensional vector bundle. In what follows, we shall use the operators orthogonal with respect to a strong Riemannian metric to prove results of this kind. A strong Riemannian metric has properties similar to those of a metric on a finite-dimensional manifold [99]. Another way to prove smoothness is as follows (Lemma 1 of [39, Appendix A]). Let E, F, and G be vector bundles over a manifold X and let

E

G

23. Weak Riemannian Metrics

139

be an exact sequence of smooth morphisms of vector bundles. (In particular,

the restrictions of f and g to X are equal to id.) Then ker f, im f = ker g, and im g are smooth subbundles of E, F, and G, respectively.

23. Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms In this section, we introduce a Riemannian metric on D'(M), which arises from a Riemannian metric on M. This metric is weak, i.e., it gives rise to the H°-topology, which is weaker than the original H'-topology. In spite of this, the Levi-Civita connection and the geodesic spray of the metric exist and are naturally related to those of the metric on M. We also define a strong Riemannian metric on D' (M) to be used later as a technical tool.

23.A. The Case of a Closed Manifold Let M be a closed Riemannian manifold with a Riemannian metric (, ). Recall

from Sect. 22.A that for g E D'(M) the tangent space TyD°(M) is just the subset of H' (M, TM) formed by X such that 7r o X = g. Let us define an inner product (,) on TyD' (M) by

(X,Y)g = f(X(m)Y(m))g(m)P(dm) ,

(23.1)

where X and Y are vectors in T9D'(M) and p is the Riemannian volume form. Recall that orX (m) = zrY(m) = g(m), and, hence, the vectors X (m) and Y(m) belong both to the tangent space Tg(m)M.

Theorem 23.1. The inner product (,) y is smooth in g E D' (M). To prove the theorem, we recall that (,) is smooth on M. Now it suffices to apply the w-lemma. (In (23.1), the diffeomorphism g is composed with the smooth tensor field (,).) Thus, having the inner product (, )y on every tangent space T9D'(M), we obtain a Riemannian metric on DI(M). By definition, (,) gives rise to the H°-norm or, equivalently, the L2-norm on the tangent space TyD' (M). In other words, the topology on D' (M) arising

from this metric is, in fact, the H°-topology, which is weaker than the H'topology. As a consequence, the metric (,) might fail to have the Levi-Civita connection, geodesics, etc. However, as shown below, all these objects exist for (,) .

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Chapter 7. Geometry of Manifolds of Diffeomorphisms

Denote by 7r1 the natural projection TTM -+ TM. Let K: TTM --+ TM be the connector of the Levi-Civita connection of (,). (See Appendix A.) It is not hard to show that

TTD'(M) _ {Y E H'(M,TTM) I ?r1 oY E TD'(M)} . Consider the map K: TTD' (M) --+ TD' (M) defined as

K(Y) = K o Y

.

(23.2)

Theorem 23.2. The map k gives rise to the Levi-Civitd connection of (,). The connection itself is, by definition, the distribution ker k on TD" (M).

Then the covariant derivative V is given by the standard formula VxY = K o TY(X), where X and Y are vector fields on D'(M). Thus, the proof of Theorem 23.2 has been reduced to checking the characteristic properties of the Levi-Civita connection. The reader interested in a detailed argument should consult [39]. For a C'-curve g(t) C D'(M) and a C1-vector field X(g(t)), we define the covariant derivative by the standard formula:

d X(g(t))=K(dtX(g(t))) Let Z be the spray of the Levi-Civita connection of (,) on M. Recall that, in particular, Z is a vector field on TM.

Theorem 23.3. Let Z be defined as follows: Z(X) = ZoX for X E TD'(M). Then Z is the spray of the Levi-Civitd connection of (,). Corollary. The vector field Z is COO-smooth.

The corollary follows from the definition of Z and the smoothness of Z.

Remark 23.1. By the corollary of Theorem 23.3, the exponential map exp of (,) exists in a neighborhood of the zero section in TD' (M). Note that exp is COO-smooth. To prove this, one uses the standard existence and uniqueness theorems for integral curves of C°°-vector fields on D' (M) together with the fact that an integral curve depends smoothly on its initial condition. Furthermore, the general properties of smooth exponential maps yield that exp sends a neighborhood of the origin in TeD'(M) onto a neighborhood of e E D'(M). Remark 23.2. It is easy to see that our method enables one to define a weak Riemannian metric (along with the connection, the spray, etc.) on the whole space H'(M,M). In this case, exp is defined on the entire tangent bundle. In other words, 2 is complete, i.e., the geodesics extend to (-oo, oo).

23. Weak Riemannian Metrics

141

Theorem 23.4. The map k is invariant under the right translations on D'(M). To prove this, observe that for any U E TTD' (M) Ig, there exists a unique WE such that U = W o g. Hence,

K(U)=K(W og)=KoW og=K(W)og andWog=T2RgoW.

,

Corollary. The covariant derivative V, the spray Z, and the exponential map exp are right-invariant.

23.B. The Case of a Manifold with Boundary Let M be a compact Riemannian manifold with boundary and N a closed manifold introduced in Sect. 22.C. As above, we denote the Riemannian metric

on M by (,). Without loss of generality, we may assume that (,) coincides with the restriction to M of the Riemannian metric on N. Thus, we use the same notation (,) for both metrics. To define a weak Riemannian metric (,) on H'(M, N), we set, keeping the notation of (23.1),

(X,Y)g= fM(X(m), Y( m))()m)

(23.3)

for g E H'(M, N) and X, Y E T9H'(M, N). This is a weak metric on H' (M, N), i.e., it gives rise to the H°-topology. For (,) given by (23.3), we have analogs of Theorems 23.1 to 23.4 and Remark 23.1. The metric (,) can

be restricted to D'(M) C H'(M, N). We can also define the spray Z o X at X E TD' (M), although now the geodesics may not exist (i.e., exp is not defined) because 3M may fail to be a totally geodesic submanifold of N.

23.C. The Strong Riemannian Metric Let M be a closed manifold. For g E D'(M) and X, Y E TeD'(M), we set Xg = X o g, Yg = Y o g E T9D'(M), and define the strong inner product on TgD'(M) as follows:

(Xg,Yg)g = fM(Xg(m),(m))gm) + fM((d + b)'X o g(m), (d + b)'Y o g(m))g p(dm) , (23.4) where d is the de Rham differential, b is the codifferential, and (d + 6)2 = db + bd = A is the Laplace-de Rham operator. Here, as usual, we identify vector fields and 1-forms by means of the Riemannian metric on M.

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Chapter 7. Geometry of Manifolds of Diffeomorphisms

Theorem 23.5. The inner product (, )g is smooth in g E D' (M) . The proof of the theorem goes along the same lines as that of Theorem 23.1. The only refinement needed is to use the smoothness of the operators d and 6 on TD-'(M). It is clear that the Riemannian metric is strong, i.e., it gives rise to the H'-topology on DI(M). Due to general results on strong Riemannian metrics [99], such a metric has the Levi-Civita connection, the spray, the exponential map, etc. On the fibers of the vector bundle 09 (Ak) (Sect. 22.D) one may define inner products similar to those given by (23.1) and (23.4). Namely, let us set (77 o g, (o g)g = f [(?I n *C) o g(m)] M

,

(23.5)

and (77og,(og)9 = fM [(rl A *C) o g(m)} +L [(d+6)'77 A *(d+6)'() o g(m)] (23.6)

where g E D'(M) and 77, ( E Ha(Ak). Formula (23.1) is a special case of (23.5). Also (23.6) turns into (23.4) when k = 1 and a = s. Similarly to Theorems 23.1 and 23.5, one may prove that the products (23.5) and (23.6) are smooth in g.

24. Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid Following the line of Chap. 2, we can apply the results of Sect. 23 to study mechanical systems on the configuration spaces D' (M), H' (M, M), or H' (M, N) with kinetic energy given by the (weak) Riemannian metric. Here we analyze those systems which are naturally related to certain problems of hydrodynam-

ics. Note that according to the Lagrangian formalism, a trajectory of such a system gives the flow of the fluid.

24.A. Diffuse Matter In what follows, we use the notation and the hypotheses of Sect. 22 and 23. In particular, M denotes a compact manifold without boundary. Consider a mechanical system on H8 (M, M) with zero potential energy and kinetic energy 1C(X) = (X, X)12, where (,) is given by (23.1). Then the Newton equation for the system is D

9(t) = 0

,

(24.1)

24. Lagrangian Formalism for an Ideal Barotropic Fluid

143

where D/dt is defined as in Sect. 23.A.

Definition 24.1. The mechanical system defined above by (24.1) is called a Lagrangian hydrodynamical system (LHS) of diffuse matter (with zero external force).

It is clear by (24.1) that the trajectory of every particle of diffuse matter is a geodesic of (,) on M. In other words, the trajectory of the LHS with the initial condition e = id is given by the vector field X E TCD"(M) of the initial velocities and the metric (, ). The kinetic energy is constant along a trajectory of the LHS on H' (M, M) and the trajectory is an extremal of the action functional with the Lagrangian L = K. Similarly, one may define an LHS of diffuse matter on a manifold M with boundary. This time, however, the motion takes place on a larger manifold N without boundary (dim N = dim M). The construction still holds if we take H8 (M, N) as the configuration space. Note that one may also consider an LHS of diffuse matter with an external force.

The motion of diffuse matter is not of much interest for hydrodynamics and we shall use it only as a starting point for our further analysis.

24.B. A Barotropic Fluid Let us now turn to Lagrangian hydrodynamical systems of an ideal barotropic fluid. The major difference between such an LHS and that described in Sect. 24.A is the presence of a force field, the potential of which is called the internal energy of the fluid. Strictly speaking, an LHS of an ideal barotropic fluid is not covered by the general construction of Chap. 2 because the force field looses smoothness, i.e., it is an Hs-1-smooth "vector field" on M. If we considered only CO°-diffeomorphisms, then the force field would be C°°-smooth. However, then the configuration space D°°(M) would be modeled on a locally convex space, rather than on a Banach space. As a consequence, the whole analysis would become much more complicated. Here we briefly outline some definitions and results on LHSs of a barotropic fluid on a closed manifold. A more detailed account can be found in [37], [126][128].

Let M be a compact Riemannian manifold without boundary and let D' (M) be the group of H'-diffeomorphisms of M with s > n/2 + 2. Denote by Va-1 the space of Ha-1-smooth volume forms v on M such that fmV=fmAI

where p is the Riemannian volume form. Following Smolentsev [126], consider

the map W : D(M) --+ Va-1 defined as !111(g) = (g1)*p, where (g 1)*p is the pull-back of µ under the map g-1. Since any two volume forms differ by

144

Chapter 7. Geometry of Manifolds of Diffeomorphisms

a multiplier, which is a scalar function on M, we have il1µ (g) = p(g) p, where

p(g): M -* (0, +oo) is an Hs-1-function called the density of the fluid at g E D'(M). Fix a smooth function U1: (0, oo) --p (0, oo). The composition l4 (p): M -> (0, oo) is called the specific internal energy of the system. Then the internal energy U: D'(M) --k (0, oo) is defined as U(g) =

fM U1(P)PA

=

Im

U1(P)v

where v = pp = W (g). In a true physical system, the function U, depends on the properties of the fluid. Consider also the function p: M -* IR given by 2 dU1

p(P) = P dp

which is known as the state equation in mechanics. The function p is called the pressure of the fluid at g, where Wµ (g) = pp. Remark 24.1. To explain the terminology, we emphasize that the fluid under consideration is compressible, since we have taken the entire group D' (M) as the configuration space. In mechanics a compressible fluid is said to be barotropic if the pressure depends only on the density.

The gradient of U with respect to (,) on D8 (M) might not exist because (,) is just a weak Riemannian metric. However, as the following theorem shows, the gradient exists in the class of Hs-'-vector fields on M.

Theorem 24.1. Let F be the vector field on D'(M) defined by the equation 1

F9 = TR9 (1 grad p(p) p

)

I

where W, (g) = pp. Then for any Y E T9D'(M), we have dU(Y) = (Y, F9), i.e., F = grad U.

Proof [126]. Let g(t) be the flow of Y on M. Differentiating the equation µ = g*(p(t)/), we see that dt

g*(P(t)µ) = 0

or, equivalently,

ji + div(pV) = 0

24. Lagrangian Formalism for an Ideal Barotropic Fluid

145

where V = TR.9(t) YY(t). Therefore, we have

dU(Y)

dtU(9(t))LO B

= fM

=

dt

fm

(-div(PY))pµ

ui(P)Pµl

-

t-0

+ fm Ui (P) (-div(PY)) pµ

f(rad(P Pi ), Y)Pµ + f(gradui,Y)p/i m

= f (grad P, Y) Pµ +

IM ddP

u(grad p, Y) pµ

fM1P, Y)Pp - fM 12

+f r P2 (grad p, Y)Pµ rad p

fM \

= (TR9

VJ

(radP(P))TRV)

_ (F,Y)9 0

Definition 24.2. An LHS of an ideal barotropic fluid without external force is the mechanical system on D' (M) with the kinetic energy 1C(X) = (X, X)/2 given by (23.1) and the potential energy U.

The force field in such an LHS is -grad U, so that the Newton equation takes the form

d g(t) = -grad U .

(24.2)

Using the standard properties of the Levi-Civita connection, it is not hard to show that the total energy E = K + U o 7r is constant along a trajectory of the LHS and that every trajectory is an extremal with fixed endpoints of the action functional with the Lagrangian L = K - U o 7r. Let !P be a vector field on M and the induced right-invariant vector field

on D'(M). Definition 24.3. An LHS of an ideal barotropic fluid with the external force 45 is the mechanical system on D' (M) with 1C as in Definition 23.2 and the total force field -grad U + P. The Newton equation for this system is

d y(t) = -grad U +

(24.3)

Let us now show how to pass to the Euler equation for a barotropic ideal fluid. Let g(t) be a trajectory of the LHS (24.3). Consider the curve u(t) =

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Chapter 7. Geometry of Manifolds of Diffeomorphisms

TR91g(t) in TeD°(M). According to the definitions given in Sect. 4.B, the curve u(t) satisfies the Euler equation, which, in this particular case, coincides with the Euler equation of an ideal barotropic fluid. To see this, we observe that by the definition of i and Theorem 24.1, the right-hand side of (24.3) turns into -(gradp)/p + 0 after being right translated to TD- (M). On the other hand, by the definition of the covariant derivative, we see that the right translation of Dg/dt is equal to au/at +V,u. Similar to the proof of Theorem 24.1, one may show that 8p/8t + div(pu) = 0. As a result, we obtain the system of the Euler equations

T + V u + P grad p ='P 1

(24.4)

at + div(pu) = 0 Remark 24.2. We finish this section by giving some general references on the material presented here. The manifold of C°°-diffeomorphisms was studied in [126]-[128]. In this case the principle of least action in Maupertius' form has been proved and the integrals of motion have been analyzed. The manifold

D8(M) was studied in [37]. Note that, in general, it is harder to work with D8(M) than with D' (M), yet the existence theorems have been proved only for D8(M). In [37], an LHS of an ideal barotropic fluid was regarded as a system with a strong constraint force given by a potential having a minimum on the manifold D'(M) of volume-preserving diffeomorphisms. The latter group is the configuration space for an ideal incompressible fluid. It has been shown

that a trajectory on D8(M) approaches the submanifold Dµ(M) as the parameters of the system go to infinity, so that the fluid becomes incompressible.

Chapter 8. Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid

In this chapter we study Lagrangian hydrodynamical systems (LHSs) of an ideal incompressible fluid. These systems turn out to be systems of diffuse matter with a holonomic constraint in the sense of Sect. 5. (To compare, we recall that an LHS of an ideal barotropic fluid considered in Sect. 24.B is an LHS of diffuse matter with an external force.) In Sect. 25 we define LHSs of an ideal incompressible fluid and study some of their general properties. In particular, we show that such an LHS can be described by a C°°-smooth equation on the integral manifolds of the constraint, even though the Euler equations lose smoothness. This equation is given by the spray of the metric. Sect. 26 contains a construction of a special infinite-dimensional constraint on the group of volume-preserving diffeomorphisms. The LHS of an ideal incompressible fluid on a manifold without boundary subject to this constraint describes the motion of a fluid on a manifold with boundary given beforehand. We continue studying these LHSs in Sect. 27. In particular, applying the results of Sect. 26, we prove a regularity theorem for a flow of an ideal incompressible fluid on a manifold with boundary.

25. Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid We show in this section that the motion of an ideal incompressible fluid on a manifold M can be equivalently described as an LHS of diffuse matter with a constraint. Imposing the constraint means that the vector field generating the flow is divergence-free. We study this constraint using the general approach developed in Sect. 5 and then, following [39], analyze relevant geometric objects on the group of volume-preserving diffeomorphisms, which arises as an integral manifold of the constraint. In many cases, we just briefly outline the proofs, a more detailed argument can be found in [39]. Throughout this section, we

keep the notations and conventions of Chap. 7. In particular, s > n/2 + 1, unless specified otherwise.

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Chapter 8. Hydrodynamics of an Ideal Incompressible Fluid

25.A. Volume-Preserving Diffeomorphisms of a Closed Manifold Let M be a closed oriented Riemannian manifold. Thus all the results of Chap. 7 apply to M. Recall that the divergence of a vector field X is a scalar function div X such that LX p = (div X) µ, where Lx is the Lie derivative and p is the Riemannian volume form. As a result of such a choice of p, we can equivalently define the divergence as follows [120]. Namely, let ax be the 1-form corresponding to X via the Riemannian metric. Then divX = box, where 6 is the codifferential. In what follows we continue to identify differential forms and multivector fields, since the Riemannian metric on M is fixed. Recall also that for H'-differential forms (and thus for multivector fields by our convention), we have the Hodge decomposition (see, e.g., [39] and [141]): H8(Ak) = (25.1) ® 6(Hs+1(Ak+1)) ® kerk.1 , d(H3+1(Ak-1))

where Ai = A2T*M is the vector bundle of i-forms, d(H'+1(Ak-1)) is the space of exact k-forms, 6(Hs+l(Ak+l)) is the space of coexact k-forms, and the last term, kerk A, is the space of harmonic k-forms. The latter space is, by definition, the kernel of the Laplace operator A on k-forms. Recall that kerk A is finite-dimensional, and independent of s, and all its elements are C°°-smooth. Since we have identified vector fields and 1-forms, the space 6(Hs+'(A2)) ®ker1 A should be regarded as the space of divergence-free HsH'+1-functions. vector fields and d(H'+1 (A°)) as the space of gradients of The following result, stated here as a theorem for the sake of convenience, is an immediate consequence of (25.1).

Theorem 25.1. Let X E HI(TM) with s > 0. Then there exists a unique divergence-free H'-vector field Y and an H1+1-function p, unique up to an additive constant, such that

X = Y -gradp .

(25.2)

Remark 25.1. Here we have chosen to take gradp with the negative sign in order to match the standard (in hydrodynamics) choice of signs in the Euler equation.

It is important to point out that for s > 0, all three terms of (25.1) are orthogonal to one another with respect to the H°-inner product (,)e given by (23.1) as well as with respect to the Hs-inner product (,)e given by (23.4). This follows from the facts that d and 6 are (, )e adjoint to each other, d + 6 is self-adjoint, and that d and 6 commute with A = (d + 6)2. Let P: H'(Ak) --4 6(Hs+1(Ak+1)) ®kerk.6 be the operator of the natural projection onto the last two terms in (25.1). Clearly, P is orthogonal with respect to either of the metrics (, )e and (, )e.

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Theorem 25.2. Let r > s. If a is an H'-smooth kform, then so is P(a). Theorem 25.2 follows from the definition of P and (25.1). Let s > n/2+1. Consider the right-invariant operator P: 08(Ak) -4 V (Ak)

on D8 (M) defined as P9 = R9 o P o R9' (Sect. 22.D). It is clear that G; (Ak) = im P is a right-invariant family of subspaces in the fibers of 08 (Ak) over D8(M). In fact, G;(Ak) is obtained by the right translations of G;(Ak)e = b(He+1(Ak+1)) ® kerk 0

Theorem 25.3 ([39]). The family G" (A') is a C'-subbundle of 0;(Ak) and the operator P is a C°°-smooth morphism of vector bundles.

Proof. Let F: H'(Ak) -+ kerk,6 be the orthogonal projection on the last term in (25.1). By Theorem 22.8, we see that Hk = imr is a C'-smooth right-invariant subbundle of 4-i;(Ak) over D'(M). Hence, r is a C'-smooth right-invariant morphism of vector bundles because He = kerk.A is finitedimensional and formed by C°°-smooth forms. Let us now define a family of subspaces W; (Ak) in 0 (Ak) as follows.

Namely, the space W; (Ak)9 is obtained by the right translation R9 of W; (Ak )e = 6 (H3+1(Ak+1)) . Making use of the Hodge decomposition, it is not hard to show that the orthogonal projection Q of H8(Ak) onto 6(H'+1(Ak+1))

can be written as Q

= .6-'d and where I is the identity operator. Taking into account that dr = 0, we have Q = 6,d-ld. Then, in the notation of Sect. 22.D, d,:A-1

Q = &A-ld and Q9 = R9ba-1R91 o R9dR91 By Theorem 22.9, d is a smooth morphism of vector bundles. The properties of 6ii-1 are summarized in the following lemma.

Lemma 25.1. (1) The families imd, kerd, imb, and kerb are C'-smooth subbundles of 0'(Ak). (2) The restriction of 6A-1 to imb is a smooth bijective map onto W. (A); the inverse of this map is d. (3) The family W, (Ak) is a smooth right-invariant subbundle of 0e(Ak). In a slightly different form, the lemma was proved in [39, Appendix A]. To

prove the smoothness for d (assertion 1), we observe that the morphisms in the following exact sequence of vector bundles are smooth (see Remark 22.4): 0y(Ak) -d+ im(I - r) d+ 8_2(Ak+2)

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For 6, we have a similar smooth exact sequence. Assertion 2 follows immediately from (25.1). Since I - I' is inverse to the smooth bundle isomorphism d, the morphism I -T is smooth too. (See Theorem 22.9.) Furthermore, We (Ak) is smooth because it is the image of ker 6 under the smooth map I - T. We are now in a position to complete the proof of Theorem 25.3. Since

P = Q + T, we have G;(Ak) = Hk ® W, (Ak), and the two terms on the right-hand side are both smooth and right-invariant.

Now consider the special case when k = 1. Recall that Al = T*M and TM can be identified by means of the Riemannian metric. Theorem 25.4. The distribution G'(Al) is holonomic on DI(M). Its integral manifold passing through e = id coincides with the group of volume-preserving diffeomorphisms:

Dµ(M)={gED'(M) I g*p=p}

.

Proof.

Lemma 25.2. The set DA(M) is a smooth submanifold of D'(M). A complete proof of Lemma 25.2 can be found in [39]. Here we just outline

the main idea of the proof. Let V be the submanifold of Va-1 formed by volume forms cohomologic to p. (See Sect. 24.B.) It turns out that D'(M) is the preimage of V under the C'-smooth map Oµ: D' (M) -' Va-1

,

c

(g) = g* (p)

and V is regular for q5µ. Therefore, Dµ (M) = 0µ1(V) is a smooth submanifold

of D'(M). The following result is evident:

Lemma 25.3. The set D' (M) is a subgroup of D'(M). Lemma 25.4. The tangent space TCD,'(M) is formed by all divergence-free H'-vector fields on M, i.e., T.D'(M) = Gs(A1)e

To see this, pick a C1-smooth curve g(t) in Dµ(M) with g(0) = e. Since g*(t)p = p by the definition of D'(M), we have LXµ = 0, where X = g(0), i.e., div X = 0. It is clear that this argument applies to any X E TeD' (M) as well.

To finish the proof of Theorem 25.4, observe that since G'(A1) is rightinvariant, we have G,°(A1)9 = TRG'(A1)e = {X o g 1 X E TeD8 (M)} = T9Dµ(M)

for every g E D' (M).

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Remark 25.2. Note that for D8,,,(M), the analogs of many results of Sect. 22.A are valid. For example, the properties of right and left translations and right-invariant vector fields on D- (M) are similar to those studied in Sect. 22.A; every right-invariant vector field has integral curves, and the group exponential map is continuous (but not smooth) on DZ(M). Note also that TTDµ(M) can be called a subalgebra of TCD3(M), meaning that the space of C°°-smooth divergence-free vector fields is closed with respect to the Lie bracket on M (Sect. 22.A).

25.B. Volume-Preserving Diffeomorphisms of a Manifold with Boundary All the results of Sect. 25.A still remain valid when M is a manifold with boundary. However, some proofs need to be altered and the behavior of vector fields on the boundary 3M has to be taken into account. (See [39, Sect. 7 and Appendix A].) First, notice that on a manifold with boundary the operators d and 6 remain H°-conjugate to each other with respect to (, )e only if special boundary conditions are satisfied by the differential forms.

Definition 25.1. A k-form a on M is said to be tangent (respectively, normal) to the boundary 3M if the restriction of *a (respectively, a) to 3M is equal to zero.

It is clear that if a is tangent to 3M, then so is ba. Among various versions of the Hodge decomposition for a manifold with boundary we need the following one (see [39]):

H8(Ak) = d(Hs+1(An-1)) ® 6(He+1(At+1)) ®Hk

,

(25.3)

where An-1 is the space of (k - 1)-forms normal to 3M, Ak+1 is the space of (k - 1)-forms tangent to 3M, and Hk the space of harmonic k-forms on M. Besides this decomposition, we have

H'(Ak) = d(H8+1(Ak-1)) ®C8(At)

,

(25.4)

where ® is the H°-orthogonal direct sum with respect to the metric (, )e given by (23.3), and C8(At) is the space of coclosed H8-smooth k-forms which are tangent to 3M. The following important result is an immediate consequence of (25.4).

Theorem 25.5 ([39], [98]). For every

H8-vector field X on M with 8 > 0, there exists a unique divergence-free vector field Y tangent to 3M and a scalar H'+1 -function p on M, unique up to an additive constant, such that

X = Y - gradp .

(25.5)

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As before (see Remark 25.1), keeping in mind the standard sign convention for the Euler equation, we took gradp in (25.5) with the negative sign. Similar to what we did in Sect. 22.D, one may consider a vector bundle P, (E) and, in particular, ; (Ak) over the manifold H'(M, N). It is important to take the entire manifold H' (M, N) as the base of these vector bundles in order to define a constraint on the LHS of diffuse matter and, thus, to pass to an LHS of an ideal incompressible fluid. To define the constraint, we first

restrict ON ^ to D'(M) C H'(M, N). Then it is not hard to show that the family O; (At) C 0; (Ak) is a smooth subbundle over D'(M). Here ; (At) is obtained by the right translations of O8 (At )e = H'(At ). (To be more precise, one should use a metric on N for which 8M is a totally geodesic submanifold.) Instead of the space kerk A which is too big when we deal with a manifold with boundary, one works with the space Ht of closed and coclosed k-forms tangent to the boundary. This space is finite-dimensional and all its elements

are C°°-smooth. The space 8(Ha}1(Ak+1)) is to be replaced by the space 8(H'+1(At+1)) of codifferentials of (k + 1)-forms tangent to 8M. Now the natural projection P sends O;(Ak) onto !P; (At ), s > n/2 + 1, and F is a projection onto Ht k. After this modification, all the results of Sect. 25.A on the properties of T hold even when M is a manifold with boundary. To define the operator Q, we replace A-1 by the operator At1 introduced in [108].

Note also that if 8M 74 0, the image im P = G; (At) is a C°°-smooth right-invariant subbundle of c (At ). Regarding D,-' (M), we emphasize that Dµ (M) is still a smooth submanifold of D' (M), and so of H' (M, N). The tangent space TeDI (M) is formed by all divergence-free H'-vector fields that

are tangent to W. Remark 25.3. The assertion of Remark 25.2 remains valid for CO°-smooth diffeomorphisms of a manifold with boundary with just one minor refinement (see Remark 22.3). Namely, if X is a right-invariant Ck-vector field on D'(M), then the field X = Xe is H'+k only in int M and in the directions tangent to

the boundary M.

25.C. LHSs of an Ideal Incompressible Fluid Let, as above, M be a compact Riemannian manifold, possibly with boundary. Then Ge (At) (or G; (A1) if 8M = 0) can be regarded as a constraint (in the sense of Sect. 5) on the LHS of diffuse matter. Applying the general method of dealing with such systems, we set

O=Pop, D =Poa

(25.6)

For any H'-smooth force field F on M, we consider the right-invariant field F on D' (M) with Fe = F. Then the Newton equation for the system with

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153

a constraint on D° (M) can be written as

Ltg(t)=PoF .

(25.7)

Remark 25.4. The right-hand side of (25.7) is a right-invariant vector field on D8(M). Evaluating this field at e = id, we obtain PoF. This is the divergence-

free (and tangent to 8M) component of F in the decomposition given by (25.2) (or (25.5) if 8M # 0). This means that we may assume without loss of generality that F is divergence-free (and tangent to eM), i.e., an element of TeDµ(M). Since the distribution Gs (At) (respectively, G s (Al )) is holonomic, we dispose of the constraint by restricting our system to the integral manifold DA(M).

Definition 25.2. An LHS of an ideal incompressible fluid without external force is the mechanical system with the configuration space D"(M), kinetic energy 1C(X) = (X, X)/2, and zero potential energy. Here (,) is the H°Riemannian metric (23.1) (respectively, (23.3)). In particular, the Newton equation for such a system is D Xt)

0

(25.8)

where D/dt is defined as in (25.6). Lemma 25.5. The operator D/dt is the covariant derivative of the Levi-Civitd connection on Dµ(M). Thus, according to the standard results on covariant derivatives (Appendix A), we have dt

X(g(t))

(g(t))) = k (dt X

where k = P o k is the connector and K is given by (23.2) (see Sect. 23). This lemma is a simple corollary of a general result of Riemannian geometry. Note also that every trajectory of the LHS (25.8) is, in fact, a geodesic of the Levi-Civita connection on Dµ(M), i.e., an extremal with fixed end-points of the action functional with L = 1C.

Fix F E TDµ(M) and denote by F the right-invariant vector field on Dµ (M) with Fe = F. Definition 25.3. An LHS of an ideal incompressible fluid, with the external force F, is the mechanical system with the configuration space and kinetic energy as in Definition 25.2 and the external force F. The Newton equation

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154

for this system is (25.9)

Taking into account Remark 25.4, we may assume without loss of gener-

ality that F E TeD'(M). Let us now turn to the elementary geometric properties of the weak Riemannian manifold Dµ (M).

Theorem 25.6. The metric (,) is right-invariant on Dµ(M). The theorem follows immediately from (23.1), (23.3), and the definition

of D'(M).

Theorem 25.7. The spray S of the metric (,) on D'(M) is-a COO-smooth

right-invariant vector field on TDµ(M). Furthermore, S = TP(Z), where Z is the spray of (,) on D° (M) .

Proof. The identity S = TP(Z) is a standard fact of differential geometry. Since P and Z are C°°-smooth and right-invariant, so is S. Theorem 25.8. For every X E TeD, (M), there exists a unique local solution g(t) of (25.8) with g(0) = X.

We call g(t) a local solution, for its existence has been proved only on some interval [0, e).

Proof. Let -7r:TD'(M) - Do" (M) be the natural projection. Observe that if y(t) is an integral curve of S, then 1ry(t) is a solution of (25.8) and, moreover, all solutions can be obtained in this way. The integral curve y exists locally. Furthermore, -y is unique for any initial condition X E TeD1(M). The theorem follows.

Remark 25.5. Since S is right-invariant and every point of Dµ (M) can be right translated to e, Theorem 25.8 remains valid for any initial condition X E TD8 (M).

Theorem 25.9. The image of the exponential map TeDµ(M) - Dµ(M) covers a neighborhood of e c DO (M).

This result, standard in differential geometry, follows from the smoothness of S. It is clear that the same is true for any other point of Dµ(M); the point e is taken only for the sake of convenience. A hydrodynamical interpretation of the theorem is as follows: for the motion of an ideal incompressible fluid without external forces, any configuration which is sufficiently close to the original one can be obtained by suitably choosing initial conditions.

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155

Theorem 25.10. Let F be a divergence free H1 -vector field on M which is tangent to aM. Assume that l > s > n/2 + 1 and 1 > n/2 + 2. Then for any X E TeD,,,(M), there exists a unique local solution g(t) of (25.9) with g(O) = e

and g(0) = X. Like Theorem 25.8, this result establishes existence and uniqueness only on a sufficiently small interval [ 0, e).

Proof. Consider the vector field S + Fi on TDµ(M), where the second term is the vertical lift of F. The projection it sends the integral curves of this field to the solutions of (25.9). If l > s, then the field S + F1 is C'-smooth by Theorem 22.4. This yields the local existence and uniqueness of solutions.

If 1 = s > n/2 + 2, a solution is unique and exists locally on TD' -'(M). Furthermore, by Theorem 22.5, the solution depends smoothly on the initial condition. This completes the proof of the theorem. Let us now turn to the Euler equation in the "space coordinates" (i.e., the Eulerian representation), which arises from the LHS given by (25.9) or (25.8). Pick a solution g(t) of (25.9) and consider the curve

u(t) = R,-' o g(t) E TeDµ(M) Similar to the Euler equation (24.4), it is not hard to show, using the definition

of D/dt, that u(t) satisfies the equation

\

Plat +V u/ =F . Since u E TeD'(M), we have P(au/at) = au/at. Furthermore, taking into account that P(Duu) = Vuu + gradp by (25.2) or (25.5), we obtain the classical Euler equation for an incompressible ideal fluid:

T + Duu + grade = F .

(25.10)

Remark 25.6. In the derivation of (25.10) we assumed that the density p of the fluid was equal to 1. A more general, customary in hydrodynamics, form of (25.10) is

at+V u+Igradp=F P Since the fluid is incompressible, the density is constant and we can always choose a system of units so that p = 1.

Remark 25.7. Recall that besides (25.10), we have one more equation to be satisfied by u, namely, div u = 0, which is equivalent to the condition u E TeDµ(M).

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Chapter 8. Hydrodynamics of an Ideal Incompressible Fluid

Remark 25.8. Proving the existence and uniqueness theorems for the LHS of an ideal incompressible fluid (Theorems 25.8 and 25.10), we used the facts that S was smooth and right-invariant on TDI (M). On the other hand, passing to the Euler equation (i.e., to (25.10)) on TCD- (M), we lost all control of S. We emphasize that, in fact, the Euler equation is a partial differential equation, defined only on TeD'+1(M), a dense subset of TeDµ(M). A direct proof of the existence of solutions of (25.10) independent of Theorems 25.8 and 25.10 is quite complicated.

Remark 25.9. Let V be the spray and V the covariant derivative of a connection on a manifold L. Recall that

V(X)=TXoX-(VxX)°

,

where X is a vector field on L. Taking V = Z on D° (M), we obtaim the following identity:

2(X) = T (X o g-1) o X - (Vxog-iX o g-1)' o g where X E T9D9(M). Similarly, setting V = S on D,,(M), we have

S(X)=T(Xog-1)oX-[P(Vxog-iXog-1)]'og where X E T9Dµ(M).

26. The Flow of an Ideal Incompressible Fluid on a Manifold with Boundary as an LHS with an Infinite-Dimensional Constraint on the Group of Diffeomorphisms of a Closed Manifold In this section we follow [11] and [12] in studying a particularly important infinite-dimensional vector bundle over the group of volume-preserving diffeomorphisms of a closed manifold N. This vector bundle is a subbundle of the tangent bundle and, therefore, can be regarded as a constraint on the LHS of an ideal incompressible fluid on N. An admissible trajectory for this constraint corresponds to the flow of the fluid on a given compact submanifold with boundary, provided that the dimension of the submanifold is equal to dim N. This construction enables us to obtain sharper results on the structure of the flow near the boundary. One result of this kind, namely, the regularity theorem, is proved in Sect. 27. As above, let N be a compact orientable Riemannian manifold without boundary and M a compact submanifold of N which has the same dimension n as N. (Thus M must be a manifold with boundary.) For example, if M is a given Riemannian manifold, we may set N = M UaM M and equip N with

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a Riemannian metric which coincides with the original metric on one of the copies of M C N. Assume that s > n/2 + 1 and denote by D8 (M) and D8 (N) the groups of volume-preserving diffeomorphisms of M and N, respectively. Also, let j: VectN -a VectM be the restriction morphism of vector fields, S the spray on TDµ(N) (Theorem 25.7), and rr: TD-(N) -+ D- (N) the natural A projection. The main results of this section are summarized in the following theorem. JA

Theorem 26.1. There exists a C°°-smooth right-invariant subbundle 58 of TD8(N) and aCOO-smooth fiberwise right-invariant projection R:TD'(N) 5e with the following properties:

(i) The projection j: F' - TeDu(M) is an isomorphism. (Here 5e is the fiber of 58 over e.) (ii) The distribution 58 is nonholonomic. The fibers of 58 have infinite dimension and infinite codimension in the fibers of TD-(N). (iii) Denote by TR: TTD8 (N) -* T58 the tangent map of R. Let X (t) be the integral curve of TR o S with the initial condition X (O) = Y E Fl. Then the curve r)(t) = rrX (t) consists of diffeomorphisms which preserve M, and 77(t)lm is a curve in D'(M). This curve is a trajectory of the LHS of an ideal incompressible fluid (without external force) on M with the initial condition Yo = j(Y). Note that the family of diffeomorphisms rl(t) does not correspond to the motion of the fluid on this set, even though rl(N\M) = N\M. We should also point out that a "free" geodesic on D' (N) with initial condition Y is a flow of the fluid on N that mixes M and N\M.

Let f (t) E 5e be a time-dependent external force and 1(t) the rightinvariant vector field on D-' (N) corresponding to f(t). To obtain the results on the flow of a fluid with external force, we just need to slightly refine (ii). Namely, we replace the field TRoS by the field TRo (S+ f(t) 1), where 1(t)' is the vertical lift of 1(t) to TD-(N). With this modification in mind, we have: Corollary. The curve rl(t) I M in D3 (M) is a trajectory of the LHS of an ideal incompressible fluid on M with the external force fo(t) = j f (t).

It is clear from (i) that f (t) and Y are entirely given by specifying fo(t) and Yo, respectively.

Proof of Theorem 26.1. Consider the vector bundle 09 (A') over Dµ(N) with s > a > n/2 (Sect. 22.4). Recall that the fiber $e (Ak)e over e E Dµ(N) is the space of H8-smooth k-forms on N. Let X be the indicator of M in N, i.e., X(m) = 1 if m E M and X(m) = 0 if m E N\N. In addition to the inner products given by (23.5) and (23.6),

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we also define the following ones on the fibers of ce (Ak): ((7 ° g,

g))y = fM X(g(m)) [('q A *() o g(m)]

+ fM X(g(m)) [((d + d)"r7 A *(d + b)"S) o g(m)] (26.1) and

((ii o g, (° g))9 = f

r X(g(m))

[(,q A *() ° g(m)]

.

(26.2)

Lemma 26.1 ([10]). The inner products (26.1) and (26.2) are right-invariant on D'(N) and C°°-smooth in g. Proof. The inner products are clearly right-invariant by (26.1) and (26.2). To prove smoothness, it suffices to show that the map wX: Dµ (N) - IR ,

wX(g) =

fx((rn))ft

is smooth in g E D' (N). As shown in Sect. 22.A, the map wexp: U -* W C D'(N) ,

wexp(X) = exp oX

is a diffeomorphism on a neighborhood U of the zero section in H8(TN). Fix X and Xl such that X + tX1 E U for t E [ 0, 1]. Then we have 1(X

lim t-0t

o exp(X + tXi)(m) - X o exp(X) (m)) = 0

for all m E N except those which belong to N1 = (exp o X) -1(&M). Note that N1 is a set of measure zero and the limit may not exist at m E N1. Denote by DwX (exp o X, Xl) the Gateaux derivative of wX at exp X. We have

DwX(expoX,X1) = lim 1

t-0 t

=0

f

Xoexp(X+tXl)p -

N

f Xoexp(X)p N

Thus, DwX exists and is independent of X1 and continuous in X. It is easy to see that wX is Frechet C°O-differentiable. Since D' (N) is a smooth submanifold of D9 (N), the restriction of wX to D'(N) is smooth as well. A similar argument

works for the metrics (26.1) and (26.2).

Note that the inner product given by (26.1) is analogous to (23.6), but (26.1) degenerates on the fibers of Oe (Ak). As a consequence, (26.1) does not give rise to a strong Riemannian metric. Similarly, the "weak" product defined by (26.2) is analogous to (23.5) but also degenerate.

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Consider the subbundle F."(^ C (P; (Ak) with the fiber

F, (Ak)g={wogIwEHa(Ak), WIM =O} Lemma 26.2. The vector bundle F, (Ak) is a C°°-smooth right-invariant subbundle of 4i; (Ak).

Proof. To prove the lemma, we will find a surjective morphism of vector bundles A: !P8 (A k) --+ D' (N) x 12 such that ker A = F; (Ak). (Here 12 is the space of square summable sequences.) To define A, we make use of the inner product (26.1). Namely, let us set

A(wog) = (g; ((wOg,wl og)y,...,(wog,w:og)y,...)) are CO°-smooth forms on N defining a basis in Ha (Ak ) where with respect to the inner product (23.6). Now Lemma 26.2 follows from Lemma 1 of [39, Appendix A]. (See Remark 22.4.)

Let Ge (Ak) be the orthogonal complement to F, (Ak) in 0e (Ak) taken with respect to (23.6) and let J: iP; (Ak) --+ G; (Ak) be the fiberwise orthogonal projection.

Remark 26.1. It is clear by definition that the restrictions of w E O9 (Ak)e and Jw E G8 (Ak)e to M are equal to one another. Recall that (23.6) is an Ha-metric (i.e., a strong one). The following lemma is evident.

Lemma 26.3. The subbundle Ga(Ak) of 0e (Ak) is C°-smooth and rightinvariant; the operator J is C°°-smooth and right-invariant. Let 3 be a positive integer.

Lemma 26.4. If w E !P; (Ak)e is Ha+p, then so is Jw. To see this, observe that if w E Ha+Q, then the right-invariant section w of O; (Ak) is Ca-smooth. (This can be proved in the same way as for vector fields.) By Lemma 26.3, Jw is C0-smooth and clearly right-invariant. Therefore, Jw =

(.1 )e is Denote by j: Ha(Ak) -. H'(Ak)IM the operator of restriction of differential forms. It is easy to see that j: G; (A k), -> Ha(Ak)IM is a bijection. Thus, there exists a well-defined extension operator i = j-1: H0(Ak)I M --. 0e (A k), from forms on M to forms on N. Recall that ,3 is a positive integer. Ha+Q.

Lemma 26.5. If w E Ha+O(Ak)IM, then i(w) E Ge+"(Ak)e

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160

Proof. Pick an operator W which extends the forms on M to the forms on N and preserves smoothness. (There are many such operators, one may take, for example, that introduced in [1231.) By definition, JYW(w) = i(w), which completes the proof of the lemma. Observe that (26.1) is nondegenerate on the fibers of G; (Ak) and, as a consequence, (26.1) defines a Riemannian metric on G; (Ak). Let the rightinvariant differential operators d and 5 on c;(Ak) be defined as in Sect 22.D. Consider the right-invariant maps

J o do J: ;(Ak) -+ G',-'(A k+1) and J o b o J: 0;(Ak) -- Gs-1(Ak-1) We set

B = G9(Ak) n ker(J o 6 o J) and Dk = Bk fl ker(J o do J) Finally, let Wk be the orthogonal complement to Dk in Bk taken with respect to the Riemannian metric given by (26.1). By definition, Dk and Bk are C°°smooth right-invariant subbundles of Ge(Ak).

Lemma 26.6. (i) The subbundle WW of G;(Ak) is C°°-smooth and right-invariant. (ii) The operator J o d o J: Wk -p im(J o d o J) is bijective.

(iii) Let Q: im(J o d o J) -+ Wk be the inverse of J o d o J. Then Q is a right-invariant C°°-smooth morphism of vector bundles. (iv) The space j((W,)e) consists of all coexact k-forms on M which are tan-

gent to 8M. The map j is bijective on (W). e Assertion (i) follows from the fact that Wk is the orthogonal complement, taken with respect to a smooth strong Riemannian metric, to a smooth vector bundle. To prove (ii), one applies the Hodge decomposition in the fiber over e. Since J o d o J is smooth, the inverse function theorem and (ii) yield (iii).

Finally, the proof of (iv) is again based on the Hodge decomposition (see (25.3)) and on the observation that AG.(Ak) is bijective. Corollary. The Coo -smooth operator Q o J o d o J is a fiberwise projection of

40(A') onto Wk. The corollary follows from the fact that, by Lemma 26.6,

im(QoJodo J)=Wk and (QoJodoJ)2=QoJodoJ Throughout the rest of this section we deal with the particular case where k = 1. Since the Riemannian metric on N is fixed, we identify 1-forms and vector fields, as usual.

26. An Infinite-Dimensional Constraint on the Group of Diffeomorphisms

161

Let ?Lt be the space of harmonic vector fields tangent to 8M. It is known that 7-Lt is finite-dimensional and every element of 7-(t is C°°-smooth. Therefore, i(Nt) is a finite-dimensional subspace in G9(A')e and, by Lemma 26.5, all its elements are C°O-smooth. Denote by fl the right-invariant subbundle

of Ge(A') obtained by the right translations of i(ft).

Lemma 26.7. The subbundle fl is C°°-smooth. The orthogonal projection 91: G''(A1) -> fl taken with respect to the metric (26.2) is a C°O-smooth rightinvariant morphism of vector bundles.

The proof is obvious: 91 is right-invariant by its very definition; the smoothness of fl and 01 follows from Theorem 22.8.

Consider the right-invariant operator P:V(A') --' TDµ(N) defined as in Section 25.A. By Theorem 25.3, P is a C°O-smooth morphism of vector

bundles. Let 9' be the Whitney sum Wi ® D and let 5' = P(9'). Setting

B=(O1+QoJod)J and R=PoB

,

we have

9' = B(4Ps(A'))

,

s = R(,Ps(A'))

Lemma 26.8. The map B is the C'-smooth right-invariant fiberwise projection of Ve(Al)) onto 5'. The map

joBeoi:H'(Al)IM --,H'(Af)IM is the H°-orthogonal projection of H'(Al)IM onto C'(A') = TD" (M) (see (25.4)).

To prove the lemma, we point out the evident identity B2 = B. The other assertions immediately follow from the definitions.

Lemma 26.9. The operator BoP coincides with B and, therefore, R = RoP. It suffices to prove Lemma 26.9 for the fiber over e. Then one can check that Be(do) = 0 for any scalar function 0. To finish the proof, we apply the Hodge decomposition, i.e., (25.4), together with the definition of P = Pe.

Lemma 26.10. The operator JoP coincides with I ones. Hence, JoR = B. Proof. First, observe that

B(JoPoB-B)=B-B=0

,

and, therefore,

j(PeX - X) E d(H'+1(An-1))IM

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Chapter 8. Hydrodynamics of an Ideal Incompressible Fluid

for X E Oe. On the other hand, we have j(PX - X) E 1-ti (see (25.3)). By the definition of J, this means that J o P = J on ®'. Since J = I on 6', we see that J o P = I.

Lemma 26.11. The subbundle 8' is C°°-smooth and right-invariant in ,P'(A') and in TD'(N) with respect to the right action of D' (N); R is the fiberwise projection of D; (Al) or TD" (N) onto 5' . Furthermore, R is rightinvariant and C°°-smooth.

Proof. By definition, R is right-invariant and C°°-smooth on 0-'(Al). Furthermore, R2 = R and E' is right-invariant on Dµ (N). Thus, R and I - R meet the condition of Lemma 1 of [39, Appendix 1]. This lemma guarantees that 5.' is CO°-smooth on 09(A'). Now it follows from Lemma 26.9 that R(TD'(N)). This completes the proof.

Lemma 26.12. The maps P:0' -+ E'.' and j:

-- TDµ(N) are bijective

and

j o Re o is H'(A')jM --+ H'(Al)lM coincides with j o Be o i.

Proof. The bijectivity of P follows from Lemma 26.10. To prove that j is bijective, one applies Lemmas 26.6 and 26.10, Remark 26.1, and the definition of .(l. Lemma 26.10 and Remark 26.1 imply the last assertion of the lemma. 0

Assertion (i) of Theorem 26.1 follows from Lemmas 26.11 and 26.12. Let us prove (ii).

Lemma 26.13. The distribution 58 in TDI(N) is nonholonomic. The lemma follows from the fact that D.I(M) cannot be smoothly embedded in D,'(N). To complete the proof of (ii), it suffices to observe that the fibers of .' have infinite codimension in the fibers of TD" (N). To prove (iii), consider the vector field TR o S on the manifold S', where S is the spray on TD'(N) defined in Theorem 25.7. By definition, TR o S is C°°-smooth and right-invariant.

Remark 26.2. By Lemma 26.11, the map R is the projection of 9(Al) _ TD'(N)1D,-(N) (and of its subbundle TD,'(N)) onto 58. As follows from Lemma 26.9, TR o S coincides with TR o Z, where Z is the spray on D' (N) introduced in Theorem 23.3. Since the spray S is, in fact, a second-order differential equation on D' (N), we have Tir(TRoS(X)) = X for every X E T5', where 7r: 5' --+ D8 (N) is the

26. An Infinite-Dimensional Constraint on the Group of Diffeomorphisms

163

natural projection. Thus, for any integral curve X(t) of 1? o S, the derivative r)(t) of i7(t) = 7rX(t) is equal to X(t). (See Appendix A.) It is well known from differential geometry that i7(t) must also satisfy the differential equation D (t) = 0 (26.3) dt rl,

.

where

d =Rd =Rd

(26.4)

and the covariant derivatives D/dt and D/dt are defined by (23.2) and (25.6). (The second equality in (26.4) follows from Remark 26.2.) Equation (26.3) is, in fact, the equation of motion for the LHS of an ideal incompressible fluid on D' (N) with the constraint E". Let u(t) = 0)o77-1(t). Then, by (26.3), Re

("U(t) + Pvu(t)u(t)) = 0 .

(26.5)

Since Re o P = Re, we have the following analog of the Euler equation (see (24.4) and (25.10)): Re

Because u(t) E

("U(,) + vu(t)u(t)) = 0

and (8u/at) E e7 equation (26.5) reduces to watt) + Revu(t)u(t) = 0

(26.6)

Lemma 26.14. Equation (26.6) on 8e can be restricted to the space TeDN (M) of divergence-free H8-vector fields on M. On TCD" (M), (26.6) turns into the Euler equation for a free (i.e., without an external force) ideal incompressible fluid on M. Proof. It follows from Lemma 26.12 that ju(t) E TCDµ(M). As a consequence, (8(ju)/at) E TeD- (M). Therefore, by Lemmas 26.8 and 26.12, we have (see Sect. 25.C)

j (&Vu(t)u(t)) = Oju(t)j (u(t)) + grad p E TCD'(M) . 0

Corollary. Let r!(t) be a solution of (26.3) on H' with 77(0) = e and i (0) = X E E". (In other words, 77 (t) is a curve on Ds (N) and ?I (t) = rrX (t), where

X (t) is an integral curve of TR o S.) Then the map 77(t) preserves M; the curve rl(t)IM is the flow of an ideal incompressible fluid on M (without an external force) satisfying the initial conditions 77(0)lM = e and (0)JM = jX. Lemma 26.14 and the corollary complete the proof of Theorem 26.1.

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Chapter 8. Hydrodynamics of an Ideal Incompressible Fluid

Now we are ready to prove the corollary of the theorem. Let X (t) be an integral curve of TR(S + 1(t)') and let i(t) = 7rX (t). It is clear that i (t) = X (t) and ij(t) satisfies the equation

d fl (t) = .f (t)

.

(26.7)

As before, we may pass to the Euler equation for u(t) = fi(t) o i7(t)-' which reads:

a+ReVu(t)u(t)=.f(t) . wtt)

(26.8)

Similar to the proof of Lemma 26.14, one can show that (26.8) can be restricted to TCD'(M), where it turns into the standard Euler equation with the external force f (t). Therefore, Theorem 26.1 (suitably modified) still holds for i1(t).

Remark 26.3. It is not hard to show that if ii(t) is a solution of the Euler equation on TCD'(M) (with or without an external force), then ifi(t) is a solution of (26.6) or (26.8). Hence, we have two equivalent descriptions of the

flow of an ideal incompressible fluid on a manifold M with boundary. The first one uses an LHS on D" (M), while the second one is in terms of an LHS on Dµ (N) with a constraint. This means that the solutions of flow equations in both descriptions must exist for the same values of t simultaneously. It turns out that the use of the second description may sometimes simplify the argument. In the next section, we show how this actually happens. Remark 26.4. It is clear by definition that the vector bundle 5k defined for all k > s is the intersection of S.8 with TDkk (N) C TD' (N) and R is a vector bundle morphism TDA(N) --+ =k. The same is true for ek and B, and Wi and Q o J o d o J. Moreover, 1? and Cl are independent of s.

27. The Regularity Theorem and a Review of Results on the Existence of Solutions As we have shown, a trajectory of an LHS of an ideal incompressible fluid exists

locally on D' (M) provided that s > n/2 + 1. (See Theorem 25.8 and, for a system with external force, Theorem 25.10.) Passing from the LHS to (25.10), we obtain the local existence of solutions of the Euler equation. The solutions belong to H8 with s > n/2 + 1, and so are smooth in the standard sense. The existence of solutions on the interval (-oo, oo) has so far been proved only for a two-dimensional manifold M. For various two-dimensional problems, results of this kind were obtained in [91], (142], and [147]. In higher dimensions, proving the global existence is an important and still unsolved problem. This difference between hydrodynamics on two- and three-dimensional manifolds has roots

27. The Regularity Theorem and the Existence of Solutions

165

in the fact that the geometric properties of the group of volume-preserving diffeomorphisms D8 (M) change drastically as we pass from dimM = 2 to higher dimensions. (See [4, Appendix 2] and [105] for a detailed discussion of

this matter.) Consider an LHS of an ideal incompressible fluid without external force on a manifold M. Geometrically, the existence of trajectories on (-oo, oo) means that the weak Riemannian manifold Dµ (M) with the metric given by (23.1) or (23.3) is geodesically complete. Note that here we have no analog of the Hopf-Rinow theorem, i.e., geodesic completeness does not mean that any two points of D-' (M) can be connected by a geodesic. Later in this section we discuss the problem of whether or not any element of Dµ (M) can be obtained as a flow of an incompressible fluid with the initial condition e E D' (M). Now let us turn to the regularity theorem, which is a very important result in higher-dimensional hydrodynamics. This theorem claims that on the interval where the flow of the fluid exists, the diffeomorphisms forming the flow

are as smooth as the initial conditions. (In two-dimensional hydrodynamics the regularity theorem follows from the existence and uniqueness theorem.)

Let M be a compact orientable Riemannian manifold, possibly with boundary, and let n = dim M. Assume also that s > n/2 + 1, q > 0, and the external force fo(t) E TDµ+4(M) C TeDµ (M) is continuous in t in the topology of TeDµ+9(M). Pick the initial condition Xo E TeD' (M) to be an H'+k-smooth vector field on M with 0 < k < q. Denote by ij(t) E D' (M) the flow of an ideal fluid on M with external force fo such that i(0) = e and

,(0)=X0. Theorem 27.1. The diffeomorphism t7(t) belongs to DA-,+k(M) for all t for which the flow exists in DA -(M). . Equivalently, the solution X (t), X (O) = Xo, of the Euler equation with external force fo is an H'+k-smooth vector field on

M (i. e., X (t) E TD-+k (M)) for all t such that X (t) exists as an element of

H' . Proof. Following [8] and [9], we first analyze the more complex case where 8M # 0, and then conclude the proof by indicating what modifications can be made when 8M = 0. Let, as before, N be a closed Riemannian n-manifold and let M be isometrically embedded in N. Recall that one may, for example, take the double of M as N. Then the metric on N is chosen to coincide with the original metric on one of the copies of M C N. First, let us use the constraint H' introduced in Sect. 26. Let Yo = Re o i(Xo) and f = Re o i(fo) . By definition, we have (see Remark 26.4) Yo E ^e fl TeDµ+k(N) _

+k

and f E 27 e fl TeDµ+q(N) =e+a

Denote by Yo the right-invariant vector field on Dµ(N) such that (Po)e = Yo. Clearly, Y0 is a Ck-section of 5" which is Ck-smooth as a vector field on Dµ (N).

166

Chapter 8. Hydrodynamics of an Ideal Incompressible Fluid

Applying the methods of Sect. 26, consider the mechanical system with

the constraint E' and the vector field TR(S + ft) on 5' introduced in the corollary of Theorem 26.1. (Recall that f i is the vertical lift of the rightinvariant vector field f (t).) By definition, TR(S + P) is right-invariant on Dµ (N). This field is also C9-smooth, since f (t) is C4-smooth on D' (N) and TR and S are both C°°-smooth. Denote by Ot the flow of TR(S + f t) on 5'. The local existence of
Ot. For a fixed t, the diffeomorphism Ot is a C9-smooth map 5' -- 5'. Since Yo and Ot are both right-invariant, the domain of the function t F-+ /t(V) is independent of V E Yo. For the same reason, the field Y(t) = q5t(Yo) is rightinvariant for every t. Note also that because Yo is a Ck-submanifold of 5', the field Y(t) is Ck-smooth on D',(N) for every t. Denote by Ye(t) the vector of Y(t) that belongs to 5e C TeD(N). The results of Sect. 22.B yield that H'+k-smooth vector field on N for all t such that Ot(1') exists, Ye(t) is an i.e., Ye(t) E TeDµ(N).

Let i7(t) be the flow of Ye(t) on N. It follows from the results of Sect. Furthermore, i(t) is a 22 that i (t) E D'+k(N) as long as Ye(t) E Ck-smooth curve on DZ(N) (Sect. 25). In particular, dt o) = Ye(t) o o) = Y(t) k(t)

By the corollary of Theorem 26.1, we see that i7(t) = (t)IM is a trajectory of the LHS of an ideal incompressible fluid on M and X (t) = Ye (t) I M is a solution

of the Euler equation. It is clear that i (t) E Ds+k(M) and X(t) E TeDµ+k(M) for all t for which q(t) and X (t) exist and belong to H'. If aM = 0, the proof is easier. Namely, there is no need to pass to Yo, f,

and the fields on 5'. Instead, dealing with Xo, fo, and S +.P on TD-(M), one can apply the same argument.

Corollary. Let the force fo(t) be a continuous curve in TeDµ (M) and let Xo E TeD°(M). Then ?7(t) E D°(M) as long as i1(t) E TeD,(M). Equivalently, X(t) is C°°-smooth as long as it is H'-smooth. Remark 27.1. The idea of our proof of Theorem 27.1 was originally used in [39] to prove the regularity theorem on a closed manifold or on M\aM. It is essential that in our method (developed in [8] and [9]) we deal with vector fields on 5' right-invariant under D" (N) and the latter group can "move" the boundary W. On the other hand, the group D' (M) used in [39] preserves 9M. As a consequence, one cannot obtain the regularity in the normal directions to aM by working only with the fields on TD'(M).

Remark 27.2. In [105], the regularity for a manifold with boundary was proved in the particular case of a potential external force. The proof of this

27. The Regularity Theorem and the Existence of Solutions

167

result can be reduced to studying the flow of a free fluid. In our proof the force is only assumed to be divergence-free. As pointed out in Remark 25.4, the general case can be formally derived from that analyzed by us. Note also that the regularity theorem for a general external force on a bounded domain in 1R' was announced in [132] and [133]. However, as pointed out in, e.g., [134], the proof had been incomplete.

Let us now turn to the problem of whether or not two given elements of Dµ(M) can be connected by a flow of an ideal incompressible fluid without external force, i.e., by a geodesic of the weak Riemannian metric. For dim M =

2 and dim M = 3, this problem was studied by Shnirel'man [125] in the following context. Let 77 E Dµ(M) and let there exist a piecewise smooth curve 77(t), t E [0, 1], in Dµ(M) which joins id with r) (i.e., 77(0) = id and r7(1) = 77). In other words, we assume that 77 belongs to the path-connected component of id. Denote by the length of the curve 17(t) evaluated with respect to the H°-metric defined by (23.1) or (23.3). Thus, 1o (11(')) Io = j 6 (t) fl(t)) dt l

.

Taking into account that the flow of an incompressible fluid is a geodesic of (23.1) (or (23.3)), we see that the question is whether or not there exists a smooth extremal of to with fixed endpoints id and 77. The main result of [125] (Theorem 1.1) is as follows. Let M be the threedimensional cube. Then there is a diffeomorphism i7 in the path-connected component of id such that for any piecewise smooth path with 77(0) = id with the same endpoints and strictly and i7(1) = 17, there exists a path As a consequence, 77 cannot be joined smaller length: 1< with id by a flow of the fluid. For a two-dimensional M, it is still unknown whether or not a given diffeomorphism 77 from the connected component of id in DZ(M) can be connected with id by a flow of the ideal fluid. However, it was conjectured in [125] that such a flow always exists. The proof of the main theorem of [125] is based on the following important, even though technical, results. Let dist(61i b2) be the infimum of lo-lengths over all curves in Dµ(M) which connect 61 and 62. Similar to the finite-dimensional

case, dist is a metric (i.e., a Riemannian distance) on D" (M). This metric induces the weak (i.e., H°-) topology on D" (M). (Note that, according to a result of [125], the closure of D'(M) with respect to dist contains no interior points.) It is shown in [125] that the diameter of Du(M) with respect to dist is finite if M is three-dimensional and contractible, and infinite if M is a twodimensional domain. When M is three-dimensional, we have the following

estimate for dirt. Let L£(x) = p(x,l;(x)) be the distance from x to £(x) on M. It is not hard to see that z1 E L2(M). As shown in [125], there exist constants a > 0 and C > 0, which depend on M only, such that for every

168

Chapter 8. Hydrodynamics of an Ideal Incompressible Fluid

element l; E D' (M) we have

dist(id,.) < C(1IB£IIL,)"

Let n = dim M and s > n/2 + 1. Recall that, by Theorem 25.5 (proved originally in [39]), for any manifold M, there exists an Hs-neighborhood W

of id E Dµ(M) such that every element of W belongs to a flow of a free ideal incompressible fluid starting at id. It is shown by Baranov [6] that W is also filled out by flows of the fluid with an external force, provided that the following smoothness hypothesis holds. Namely, we assume first that s >

n/2 + 1 and either the external force f is independent of time and smooth (i.e., f E TeD-'+1(M)) or, if f is time-dependent, then f (t) is smooth for every t, continuous in t in TeDµ+1(M) and Cl-smooth in t in H'+1-

H'+1-

TeD' (M). Furthermore, if s > n/2 + 2, then this assumption can be relaxed: f E TeDµ(M) for an autonomous f; or, otherwise, f(t) is continuous in t in TeDµ (M).

To prove the latter assertion one passes to the group Dµ-1(M) and then applies the regularity theorem (Theorem 27.1) together with the relative version of the theory of topological degree [6]. Remark 27.3. As shown by Baranov [7], the neighborhood W is also covered by flows of a viscous incompressible fluid. The problem of determining the size of W seems to be interesting and important. The aforementioned results of [125] show that in the three-dimensional case W is strictly smaller than the connected component of id in DA-(M).

In conclusion, let us prove that the flow of a free incompressible fluid has a first integral. This result is analogous to the angular momentum conservation law for the motion of a rigid body with a stationary point. (See Section 4.C and, in particular, Remark 4.2.) The existence of such a first integral is known in hydrodynamics as the circulation conservation law [4]. Apparently, this integral was originally considered in [104] by means of the Lagrangian approach for the group of COO-diffeomorphisms. Similarly to the finite-dimensional case, the existence of a first integral follows from the fact that the metric (the Lagrangian) is invariant with respect to the group structure (the Noether theorem). The proof below is obtained by adapting the standard finite-dimensional argument to the infinite-dimensional setting (see [4]). An essentially new point in the proof is that now we have to apply the regularity theorem (Theorem 27.1). Let s > n/2 + 1. Consider the set G'-1(Al) of Ha-l-smooth divergencefree vector fields on M which are tangent to the boundary. The fields from Gs-1(Al) are continuous on M, but may not be C1-smooth. Thus, vanishing of the divergence means only that Ga-1(Al) is orthogonal to the space of exact forms in the Hodge decomposition. (See (25.1) and (25.4).) Denote by Gsa -'(A') the subbundle of Ve-1(A') over D' (M) obtained by right translations of Gs-1(A1) = Gsa-1(A1)e C 4s;-1(Al)e (Sect. 22.D). For JA

27. The Regularity Theorem and the Existence of Solutions

169

V E Ga-1(A1), let V be the right-invariant section of G'-1(A') such that Vg = Tg o V. Observe that the fibers of G,- '(A') inherit the right-invariant inner product given by (23.1) or (23.3). Since TeD,(M) C G-1(A'), the inner product (V, X)g, where X E T9Dµ(M), is defined. We emphasize that the metric given by (23.1) or (23.3) is just the restriction of the metric on the fibers of Ge-1(A') to the fibers of TDI (M). Fix a vector Xo E TeD- (M). Let i7(t) be the geodesic on Dµ(M) (the flow of the fluid) with the initial conditions 77(0) = id and f7(0) = Xo.

Theorem 27.2. For any V E G"-' (M), the inner product (V, i) is constant along 77.

Proof. First, let us assume that V E G- (Al) = TeD"11 (M)

and

Xo E TCDµ+l (M) C TCDµ (M)

Then by Theorem 27.1, we have 77(t) E Dµ 1(M) as long as 77(t) exists as an element of DA-' (M). Hence, Vg = Tg o V E T9Dµ(M). Denote the flow of V by g(T). In what follows we regard g(r) as a one-parameter subgroup of D' (M) and consider the right action Rg(T) on D" (M). Since 77 E D,"(M), we have

,d Rg(r)orl d-r

dT(17-g(T))=T77

odTg(T)

Tr7oV=V,7

Thus, V is the generator of R9(,.). Since the spray S on TDµ(M) is rightinvariant under the action of D5(M) (Sect. 25), the curve t'-+ Rg(T)r7(t) is a geodesic for every fixed r. Let rl (t, r) = Rg(T)91(t)

and f7(t, r) = dt rl (t, r)

By definition, we have dr7(t, T)/ dr = V. It is easy to see that the fields f7(t, r) and V commute on the intersection of their domains, i.e., [i (t, T), V] = 0. Clearly,

V (rl(t,T)>rl(t,T)) where V

dr (i (t,r)+rl(t,r)) = 0

r), r7(t, r)) is the derivative of the scalar function

r), f7(t, r))

in the direction of V. Because f7(t, T) and V commute and the Levi-Civita connection is torsion-free, we have

Di/(t,T)V = OV0, r) Observe that V,r(t,T)i (t, r)

_ D

dti7(t, ,r= 0

Then, by the definition of the Riemannian connection (see Appendix A,

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Chapter 8. Hydrodynamics of an Ideal Incompressible Fluid

formula (A.2)),

0 = V (il, ) = 2 (V vi , i) = 2 (vn V,'t) =2

+ 2 (Onr7, V) = 2 dt

T))

along 77(t, r). Therefore, d(V, f7(t)) / dt = 0 along the geodesic 77(t) = 77(t, 0). This completes the proof of the particular case.

Now let us turn to the general case where V E Gs-'(Al) and X° E TeDµ(M). Recall that G'(A1) is dense in Gs-'(Al) and TDµ+1(M) is dense in TeDµ(M). Thus, there exists a sequence Vj E Ga-1(A1) converging to V in the H'-1-topology and a sequence Xi E TeDµ+1(M) converging to X° in the H'-topology. We pass to the limit as follows. Let 77i(t) be the geodesic in Dµ(M) with r)a(0) = Xi. Since the solution of a differential equation depends continuously on its initial condition, 77i(t) converges to 77(t) uniformly on every finite interval. Note that for 77ti(t), Xt, and VV, the theorem has already been proved. It is not hard to see that the linear map V --+ (V, ), where (,) is the weak inner product (see (23.1) or (23.3)), is a continuous embedding of G'(A1) into Te DA' (M). (The convergence of the vector fields in the Ha-'-topology implies the convergence in the space H-'. The latter space can be identified with the dual to H' by means of the H°-inner product (Appendix E).) It is clear that the sequence (Vj, i) converges to (V, i7) as i, j -+ oo. Since (Vj, i7;,) is constant 0 along i7zf the function (V, r7) must also be constant along 17.

Chapter 9. Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms

In this chapter, we study the hydrodynamics of a viscous incompressible fluid. The Lagrangian hydrodynamical systems (LHSs) of a viscous incompressible fluid were introduced in [39] as generalizations of those of an ideal incompressible fluid. Namely, these LHSs were defined as systems on the group of volume-preserving diffeomorphisms with an additional right-invariant force

field which depends on the velocity of the fluid. In the tangent space at id, the force field is v A where A is the Laplace-de Rham operator and u is the viscosity coefficient. Note, however, that the operator A does not preserve the space of Hs-smooth vector fields. In fact, a sends H'-smooth vector fields to fields which belong to a broader Sobolev class. As a consequence, the method relies heavily on the theory of partial differential equations, leading to the loss of many natural geometric properties of the LHSs of an ideal incompressible fluid (Chap. 8) in the passage to a viscous incompressible fluid. Here, we develop another, more natural as it seems to us, approach to generalizing the LHSs of Chap. 8. Our method is based on stochastic differential

geometry that enables us to avoid using the force field vL. As a result, our LHS is defined on the group of diffeomorphisms of a fixed Sobolev class, and similar to the case of an ideal incompressible fluid, we deal only with smooth vector fields on the group of diffeomorphisms. Our model problem is to study the flow of a fluid on the n-dimensional torus. To do so, we describe the class of stochastic processes on the group of volume-preserving diffeomorphisms of the torus, such that the mathematical expectation of a process from this class is a curve on the group and this curve turns out to be a flow of the fluid. The processes satisfying this assumption are natural stochastic analogs of solutions of ordinary differential equations

studied in Chap. 8 (e.g., the Newton equation). In particular, the processes correspond to geodesics when the external force is zero. The chapter is organized as follows. Sect. 28 is devoted to stochastic differential equations on the group of all diffeomorphisms of the torus and equations on the group of volume-preserving diffeomorphisms. In Sect. 29, we restrict our attention to the latter group and define the aforementioned class of stochastic processes, for which we study the relationship with the fluid flows.

172

Chapter 9. Hydrodynamics of a Viscous Incompressible Fluid

28. Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus Let Tn be the flat torus, i.e., Tn = 1Rn/7Ln, the metric on Tn being inherited from the Euclidean metric on 1Rn. Denote by 51: TTn -+ Tn X IRn the natural trivialization of TTn obtained from that of T1Rn. The trivialization 45 can be described in terms of frames (Appendix A) as follows. First, we observe that Tn = Si x ... X Sn, where the circle Si' is the quotient of the ith coordinate axis in 1Rn over its integer lattice.

Let 0 be the standard angle coordinate on S1 and let qt = 0/2ir. Thus, we obtain a coordinate system q1, ... , qn on T'. For every m E Tn, the vectors a/aql, ... , a/aqn form an orthonormal basis in TmTn. Therefore, for (Xta/aq`),n E TmTn, we have

(Xi a

aqt

= (m, (X1,...,Xn)) E Tn X 1Rn m

Consider the map A = 0-1: T' x 1R' -+ TTn. It is clear that Am(x) _ (Xta/agt)m form E Tn and x = (X1,..., Xn) E 1Rn. The map A is smooth jointly in all variables and, f o r a fixed x, the vector field A(x) is constant on n

(i.e., the components of A(x) in the basis a/aq', ... , a/aqn are independent of m). In particular, the field A(x) is C°°-smooth. Let D'(Tn) be the group of H'-diffeomorphisms of Tn with s > n/2 + 1.

The flat metric (,) on Tn gives rise to a weak Riemannian metric (,) on D'(Tn) (see (23.1)), and, as a consequence, we have all geometric objects of Sect. 23 arising from the metric. In particular, the Levi-Civita connection on D'(Tn) given by (23.2), the covariant derivative V, and the exponential map e5p are among these objects. Consider a map A: D'(Tn) x 1Rn -* TD'(Tn) such that:

(1) the map Ae:1Rn -- TeD'(Tn) is, by definition, equal to A (i.e., Ae(x) _ A(x)); and (2) for every g E D'(Tn) the map A9:IRn -+ T9D'(Tn) is obtained from Ae by the right translation: A. (x) = R9 o Ae (x) = (A o g) (x). Since A is C°°-smooth, the u.-lemma implies that A is C'-smooth jointly in x E 1Rn and g E D'(Tn). In particular, the right-invariant vector field A(x) is C°°-smooth on D'(Tn) for every x E 1Rn. Let a > 0 be a fixed real number and a(t, m) a time-dependent Ha-vector

field on Tn, where t E [0,11 and a > s is an integer. Denote by a(t, m) the right-invariant vector field on D'(Tn) induced by a(t,m). Since a > s, the field a is at least C1-smooth. The pair (a, A) is an Ito vector field on D'(Tn). Pick a Wiener process w on a probability space P) such that w takes values in 1Rn. Applying the Belopolskaya-Dalecky method (Sect. 13), we start with the following stochastic differential equation on D'(Tn): (t)))

(28.1)

28. Stochastic Differential Geometry on Di f f ('T")

173

Note that this equation is a particular case of (13.5). Let, as above, l be a given positive real number.

Theorem 28.1. For any g E D' (T" ), there exists a unique solution 69(t) of (28.1) on (0, 1] with the initial condition t:y(0) = g.

Proof. Consider a normal chart on a neighborhood of e E D' (T") defined by means of exp. Note that all local connectors (see (23.2) and Appendix A) vanish in this chart, since the connection arises from the flat Euclidean metric on V. Fix a strong right-invariant Riemannian metric on D'(T"), for example, that given by (23.4). Let Ve(r) C D'(T") be the ball of radius r centered at e taken with respect to the strong metric. Pick a sufficiently small

r > 0, so that Ve(r) is contained in the chart. (Note that such an r does exist, for the chart is open.) For every g E D'(T" ), we define a chart on a neighborhood of g by means of the right translation of Ve(r). Thus, we obtain

a uniform Riemannian atlas on D' (T") with respect to the strong metric. The local connectors of (23.2) vanish in this atlas because the connection is right-invariant (Theorem 23.4). Since the Ito vector field (a, A) is bounded with respect to the strong metric, Theorem 13.5 is applicable to (28.1). The desired theorem follows.

Let fi(t) = &(t). We have £9 (t) = l; (t)og because (28.1) is right-invariant.

It is easy to see that C(t) is a general solution of the following stochastic differential equation on T"`:

d£(t) = expf(t)(a(t,C(t))-,o,A(l;(t)))

.

(28.2)

In other words, for every m E T", the process e(t)(m) is a solution of (28.2) with the initial condition l; (0) = m. In particular, we see that (28.2) has a general solution. This also follows from Theorem 13.3, which applies because T' is compact and (a, A) is smooth. However, working with the infinite-dimensional equation (28.1), we gained some additional information about l:(t). Namely, we proved that for every T" -. T" is an H'-diffeomorphism of w E 11 and t E [ 0, 1], the map f (t,

the torus V. Remark 28.1. This construction can be easily generalized to the group of diffeomorphisms D' (M) of an arbitrary compact manifold M and an Ito vector

field (a', A) on M. Here, as before, we assume that a' and A' are both H"smooth with ca > s. Similar to the case of V, the field (a', A') gives rise to an Ito vector field (a', A') which is right-invariant and Cl-smooth. Then we get the solution l;(t) = &(t) of (28.2) from the solution £(t) of (28.1). This relationship between the solutions of (28.1) and (28.2) is a special case of a result of Elworthy [43]. Note that here we employ the Belopolskaya-Dalecky

method to analyze the Ito stochastic equations on manifolds, whereas the original proof [43] is obtained using the Stratonovich equations. (See Remark 13.11.)

Chapter 9. Hydrodynamics of a Viscous Incompressible Fluid

174

Consider the following stochastic differential equation on D'(TT): q(t) = exp,n(t) (0,, A(q(t)))

(28.3)

Theorem 28.2. Let w be a Wiener process on IRn and qe(t) the solution of (28.3) with rl(0) = e. Then, for any fixed w E (1 and t E [ 0, l],

(i) the field A(w(t, w)) is divergence free on Tn, i.e., A(w(t)) is a stochastic process in TeD'µ(T"); and (ii) the map 1]e(t,w):Tn --+ Tn is a volume-preserving diffeomorphism of Tn, i.e., rie (t) is a stochastic process in Dµ (T'). Proof. Fix w E 17 and t E [0, 1], and consider the vector w(t) E IRn. By definition, A(w(t, w)) is constant with respect to the trivialization on Tn. (In other

words, A(w(t,w)) has constant components in the basis a/aql, ... , a/aqn.) This means that A(w(t, w)) is divergence-free. The map ie (t, w) sends m E Tn to exp,n A(w(t, w)), where exp,,,,: TnT" -+ T" is the exponential map on the flat torus. Thus, 7le(t,w) is just the translation by w(t,w) on Tn. It is clear 0 that the translation is volume-preserving.

Note that i e(t) can be called the development of w(t) on D'(Tn). Let a(t, m) be a time-dependent divergence-free vector field on Tn. Unless specified otherwise, in what follows we assume that a(t, m) is He-smooth with a > s being integer. As a consequence, the right-invariant field a on Dµ(T") is at least Cl-smooth (Remark 25.2). Assertion (i) of Theorem 28.2 implies that A has a well-defined restriction A: Dµ(T") X IRn -* TD'(Tn). Consider the following stochastic differential equation on DS(T' ): dl; (t) = ezp£(t) (a (t, l; (t)) , QA(l (t)))

(28.4)

where ezp is the exponential map of the spray S on D,,,(Tn) (Sect. 25) and

o>0.

Theorem 28.3. For any g E Dµ(Tn), there exists a unique solution 1'9(t) of (28.4) on [0, 1] with the initial condition g(0) = g. Proof. Consider the strong Riemannian metric given by (23.4). This metric is obviously right-invariant. Let W be a neighborhood of e in D", (Tn) which lies in the image of ezp,. Such a neighborhood exists by Theorem 25.9. Furthermore, ezp, gives rise to a normal chart on W. In this chart, the strong norm of the connector rg(., ) of (23.1) (the weak metric) is a continuous function

of m E W. (Note that here the connector is regarded as a quadratic operator.) In addition, we have F ,,( .,- ) = 0. Therefore, there exists an open subset is less than a fixed constant C > 0 U C W such that the norm of at every point of U. Denote by Ve(r) the ball of radius r > 0 centered at e, where the distance is taken with respect to the strong metric. Since U is

29. A Viscous Incompressible Fluid

175

open, it contains the ball VV(r) for some r > 0. Now we can define a chart on a neighborhood of every point g E Dµ(T") as the right translation of U. As a result, we obtain a uniform Riemannian atlas for the strong metric. Since the strong metric given by (23.4) is right-invariant, the local connector r is bounded by C with respect to the strong norm on the balls V9(r) (Theorem 23.4). It is clear that the right-invariant Ito vector field (a, A) on D" (T') is bounded with respect to the strong metric. Thus the assumptions of Theorem 13.5 are satisfied for (28.4) and the desired theorem follows.

29. A Viscous Incompressible Fluid In this section we continue the study of stochastic processes on D8(T') and D' (T") started in Sect. 28, and show that there exists a deep connection between hydrodynamics of a viscous incompressible fluid on T' and stochastic processes of a certain type. In particular, it turns out that for these processes the Navier-Stokes equation (see, e.g., [981) is an analog of the Euler equation studied in Chap. 8.

In the same way as in Sect. 19 and 20, one can define the forward and backward mean derivatives D and D.te of a solution t; of (28.1) on D8(T' ) (See (19.2) and (19.4).) Note that since t; is a strong solution of (28.1), we have Pi = Pt . Following the line of Sect. 19 and 20, it is easy to show that D.t; (t) = a. (t, t; (t)), where a. (t, g) is a certain right-invariant vector field on De (T"). Denote by a. (t, m) its value at id, which is a vector field on T". Let X (t, m) be an H'+2-vector field on T' and X (t, g) the right-invariant

vector field on De (T") such that .(t, e) = X (t, m). Recall that X (t, g) is C2-smooth (Sect. 22.B). Let k be the connector on D8(T") given by (23.2). Now we are in a position to define the forward and backward covariant mean derivatives DX and D.X along t(t) by (20.11) with the above K. Let F(t, g) be a right-invariant vector field on De (T') and let Fe = F(t, m) be the field induced on T'. Denote by V2 the Laplace-Beltrami operator on Ti'. (Since T' is a flat manifold, Ric = 0, which means that the operator -V2 coincides with the Laplace-de Rham operator A.)

Theorem 29.1. (i) Assume that DX = F along t . Then X (t, m) satisfies the following equation on TeD3(T" ): 2

aX+2 (ii) Similarly, if D.X = F along

V2X+VaX=F. then X (t, m) is a solution of the equation

z

8tX 2 V2X+Va,X=F on TeD8(T' ).

176

Chapter 9. Hydrodynamics of a Viscous Incompressible Fluid

Proof. Consider the equation obtained from DX = F by the action of TR£ 1 Taking into account that Nf = N{, it is easy to see that the right translation TR 1DX coincides with the forward mean derivative of X along the solution C of (28.2), just like the standard covariant derivative D/dt on De(T") (Sect. 24). Now assertion (i) follows from the fact that F is right-invariant and from the first formula of (20.12). Similarly, to prove (ii) one applies the second formula of (20.12).

The forward and backward derivatives are also defined along a solution of (28.4) on Dµ(T"). We denote these derivatives by D and D., respectively. As above, Pt = PL because we deal with a strong solution. Hence, due to the general properties of stochastic equations, we have Dl = a(t, li). It is not hard to show that there exists a right-invariant field a. (t, g) on Dµ (T") such that D. t; = d. (t, l). Throughout this section, we keep the notation a(t, m) and a. (t, m) for a(t, e) and a. (t, e), respectively. Let k be the connector introduced in Lemma 25.5. Denote by D and, respectively, D. the forward (respectively, backward) covariant mean derivative on DA" (T") along e(t). These derivatives are defined by (20.11) with the aforementioned K. Remark 29.1. Here the new symbols b£ and D.t; are introduced only for the sake of convenience. It is easy to see that, in fact, Dl; = Dt and D.1; = D.l;,

where ti on the right-hand side is regarded as a process in D'(T"). Note, however, that all four operators b, D., D, and D., regarded as differentiations of vector fields along , are distinct.

Let X (t, m) and F(t, m) be divergence-free vector fields on T" and let X (t, g) and F(t, g) be the right-invariant vector fields on Dµ(T") with X (t, e) = X and F(t, e) = F. Assume that X is H'+2-smooth and that F is H8-smooth. Then it is clear that X is C2-smooth on DA-(T').

Theorem 29.2. (i) Assume that DX = F along the solution lie of (28.4). Then X (t, m) satisfies the following equation on TeD,(T"):

X + 2 OzX + DaX + grade = F (ii) Similarly, if D.X = F along the solution tie of (28.4), then X (t, m) is a solution of the following equation on TeD"(T"): 2

tX-20zX+Da,X+gradp=F

(29.1)

where a(t, m) and a. (t, m) are as before and p is an H8+1-function on the torus T".

29. A Viscous Incompressible Fluid

177

Proof. By definition, we have

DX=PoDX and D,X=PoPD.X

,

where P is the orthogonal projector defined in Sect. 25.A. The theorem follows from Theorems 29.1 and 25.1. Let be a solution of (28.4) on Dµ (T") satisfying, in addition, the equation

D. D.6 = F is the right-invariant vector field on Dµ(T") with Fe = F. Let us find the backward mean derivative Y(t,g) = D. on DA-(T'). Note that our notation is consistent: Y is right-invariant and, therefore, there exists a field Y on T' such that Ye = Y.

Theorem 29.3. Assume that l; satisfies (29.2). Then Y(t) is a solution of the classical Navier-Stokes equation (see [39] and [98]) on TeDµ(T"):

0 Y - v02Y + V yY + grad p = F

(29.3)

with the viscosity coefficient v = a2/2 and external force F.

F. Proof. In this particular case, (29.2) is equivalent to the equation Thus, the theorem follows from (29.1), since the backward velocity a. is equal

toYfore. Corollary. The integral curve q(t) of Y(t, g) on Dµ (T") is a flow of a viscous incompressible fluid on T" with external force F and viscosity v.

The corollary follows from the observation that, by the definition of the integral curve, the field Y(t, m) satisfies (29.3). (See [39] for more details.) Borrowing the terminology from the theory of stochastic differential equations on vector spaces, we call q(t) the mathematical expectation of l;(t). If F = 0, then (29.2) is easily seen to be analogous to the equation of geodesics (Appendix A). In other words, the diffusion process l;'(t) is a stochastic analog of a geodesic on Dµ (T"). In particular, if we assume in addition that

a = 0, then (29.2) turns into the equation of geodesics on D'(T'), and (t) becomes just a geodesic. We emphasize that using the forward mean derivative one can introduce another stochastic analog of geodesics which, apparently, has no applications to hydrodynamics. A more detailed analysis of (29.2) goes far beyond the scope of this book.

However, it is worth pointing out that the processes 0) converge as a --> oo to the curve q(t) on Dµ(T") satisfying (25.9), i.e., to a flow of an ideal incompressible fluid. In the framework of the Lagrangian approach such a convergence was proved in [39] for a closed manifold and, in [38], for a manifold

with a "sliding" boundary.

Appendices

Appendix A. Introduction to the Theory of Connections To make our exposition self-contained, here we recall some basic notions and results concerning connections on manifolds. For a more detailed account on the theory of connections, the reader should consult [17], [35], [82], and [94].

Connections on Principal Bundles Let ir: L -+ M be a principal bundle with a structural group G over an ndimensional manifold M. Recall that G acts on L from the right; the fibers of L are diffeomorphic to G. For 1 E L, the so-called vertical subspace V1 in T1L is formed by vectors tangent to the fiber through 1. The collection V of subspaces V1 is a subbundle of TL (or, equivalently, a distribution on L). Furthermore, V is trivial as a vector bundle over L. To show this, denote the Lie algebra of G by g. The canonical isomorphism 01: g -+ V is defined as 01(X) = Tl(X), where l E L is regarded as the identification g H 1 o g of G with the fiber through 1. The map 0: L x g --+ TL is smooth jointly in both variables [17]. As a result, we obtain the desired isomorphism between L x g and V. A connection H on L is a C°°-smooth distribution on L which is complementary to V (i.e., V ® H = TL, where (D is the Whitney sum) and rightinvariant with respect to G (i.e., TR9 o Ht = Hl°9 for all l E L and g E G). The spaces HI are said to be horizontal; they form the so-called horizontal bundle over L. Note that the projection T7r: HI --+ T,(I)M is evidently an isomorphism. In other words, for any given vector field X on M, there exists a unique field X on L tangent to H and such that T-7rXj = X,r(L). The field X is called the horizontal lift of X to L. If a connection on L is given, one may define the parallel translation of l E L along a smooth curve m(.) on M as the integral curve through 1 of the horizontal lift of the velocity field rh(t). It is clear that any horizontal (i.e., tangent to H) curve 1(t) on L is the parallel translation of any point of 1(t) along 7rl(t).

Among various principal bundles over M there are two that are particularly important to us. These are the frame bundle B(M) and, if M carries a Riemannian metric, the bundle O(M) of the orthonormal frames. It is clear

180

Appendices

that the structural group of B(M) is the group of nonsingular matrices GL(n); and the structural group of O(M) is the orthogonal group O(n).

Let H be a connection on B(M). It turns out that the distribution H is trivial as a vector bundle over B(M). Moreover, H has a canonical trivialization which is the smooth map

E:B(M) xIRn -+H, Eb=T1r-1(bx)IHb

,

where x = (x1, ... , x") E IRn and b E B(M) is regarded as the linear operator b: 1R" --+ Thus, bx is the vector in TbM with components (x',.. . , x") in the basis b. It is clear that we have a completely similar trivialization for a connection on O(M). Following [17], we call E(x) for x E 1R" a basic vector field on B(M) or

O(M), whereas the field q5(X), where X E gl(n) (or X E o(n) for O(M)), is said to be fundamental. Evidently, a basic field belongs to H (i.e., it is horizontal) and a fundamental field is vertical.

Connections on the Tangent Bundle Let, as above, -7r: L -+ M be a principal bundle with a structural group G and let F be a manifold equipped with a left action of G. The associated bundle over M with fiber F is defined as follows. Consider the product L x F with

the right G-action (1, f) o g = (1 o g, g-1 o f). Then the quotient of L x F by this action is the desired associated bundle. Denote by A the projection of L x F onto the quotient space. A connection on the associated bundle is by definition the image of a connection on L under the map TA. Since the connection on L is G-invariant, the image is well defined. (See, e.g., [17].) Let us take B(M) or O(M) as L. Then the associated bundle with the fiber IR' (equipped with the standard action of GL(n) or 0(n)) coincides with the tangent bundle TM. Consider the so-called vertical subbundle V in TTM formed by vectors the fiber tangent to the fibers of TM. Thus, for m E M and X E V(.,,X) C Tim X)TM is just the space of vectors tangent to TmM. It is not hard to show that a connection on TM, which we still denote by H, has the following properties:

the distribution H is CO°-smooth and invariant with respect to multiplication by real numbers, i.e., Ta(H) = H, where a: TM -+ TM is the map X -+ aX of multiplication by a E IR; and

H is complementary to V as a subbundle of TTM, i.e., TTM is the Whitney sum V ® H. Parallel translation on M can be defined in two equivalent ways. The first and b(t) be the parallel translation of definition is as follows. Let X E the basis bo E B,,,o(M) along a curve in M. Then the parallel translation of X along is the curve b(t) o (bo 1X) in TM. In other words, parallel

vectors have constant coordinates in parallel bases. It is easy to show that

Appendix A. Introduction to the Theory of Connections

181

the parallel translation is independent of the initial value bo. For O(M), the definition is completely similar.

Since V = kerTrr, where 7r: TM - M is the natural projection, Trr induces an isomorphism H(,,,x) -+ for every m E M and X E TmM. Hence, the horizontal lift to TM of a tangent vector to M is well defined (see the previous section). Another way to define the parallel translation is to use the horizontal lift of the velocity field similar to what we did when introducing the parallel translation in the principal bundle. Since V(m x) is a tangent space to the vector space TmM, there exists a canonical isomorphism p: V(,,,,,x) Y1

+ T,,,.M. Let Y E TmM. Then

= t' 1Y E V(m,X) C T(m,x)TM

is called the vertical lift of Y to (m, X). A connection H gives rise to the so-called connector (or the connection map)

K: TTM -+ TM ,

K(Y) = p o PH(Y)

,

is the projection along TmM is an isomorphism. Conversely, if we are given a map K that satisfies this condition and is homogeneous with respect to multiplication by real numbers, then the distribution H = ker K is a connection. (The homogeneity of K implies that H is invariant under multiplication by real numbers in TM.) In what follows, when no ambiguity can arise, we use the same notation H for connections on the tangent bundle and those on the principal bundle B(M) (or O(M) if M is a Riemannian manifold) and call H just a connection on M. where Y E T(m,X)TM and

H(,n,x). It is easy to see that H = ker K, and so K:

Covariant Derivatives Let M be equipped with a connection. The covariant derivative of a vector field Y along a vector field X is defined as VxY = K o TY(X), where Y is regarded as a map Y: M -+ TM and, thus, TY is a map TM --+ TTM. One can easily check that V has the following four properties:

(1) Vxi+xaY = Vx1Y+Vx2Y; (2) V fxY = fVxY; (3) Vx(Yi + Y2) = VxY1 + VxY2; and (A.1)

(4) Vx(fY) = (X f )Y + fVxY;

where X, Y, Xi, and Yi for i = 1, 2 are vector fields, f is a smooth function on M, and X f is the derivative of f along X. Let m(t) be a smooth curve on M and Y(t) a smooth vector field along m(t). Define the covariant derivative DY(t)/dt along m(t) by the formula D

Y(t)

K (dt Y(t) )

'

182

Appendices

where Y(t) is viewed as a curve in TM and (dY(t)/dt) E Ty(t)TM is its velocity vector. To give an alternative definition of DY(t)/dt, consider the curve in Tm(t)M

obtained by the parallel translation of Y(s) to m(t) along It is easy to show that DY(t)/dt is the derivative of this curve at s = t. As follows immediately from this definition, a vector field X (t) is parallel along m(t) if and only if DX (t)/dt = 0. A curve m(.) is called a geodesic of the connection H if the velocity field rh(t) is parallel along m(.). Equivalently, m(.) is a geodesic if and only if

Dtrh(t)=0. Note that geodesics on M may also be defined as follows. Let -Y(t) be an integral curve of a basic vector field E(x) on B(M). Then the curve -7rry(t) is a geodesic on M. Conversely, every geodesic can be obtained by this construction from an integral curve of a basic vector field. The curvature tensor R of H is a (1, 3)-tensor field on M defined by the formula

R(X, Y)W = VxVyW - VyVxY - V[X,Y]W , where X, Y, and W are vector fields on M. The contraction (or trace) Ric = tr [v H R(u, v)u]

of R is a symmetric (0, 2)-tensor called the Ricci tensor (or curvature) of H. If M is a Riemannian manifold, one may pass from Ric to the (1, 1)-tensor Ric by lifting one index with the aid of the metric. The contraction 1C of Ric is called the scalar curvature. The torsion tensor T of H is defined as

T(X,Y) = VxY - VyX - [X,Y]

.

Let (,) be a Riemannian metric on M. A connection is said to be Riemannian if

X(Y,W) = (VxY,Y) + (Y,VxW) for arbitrary smooth vector fields X, Y, and W on M.

(A.2)

It is not hard to show that parallel translations with respect to a Riemannian connection preserve the Riemannian inner product. Among various Riemannian connections, there exists a unique connection with identically vanishing torsion. This connection is called the Levi-Civita connection of the metric. Its geodesics are extremals with fixed endpoints of the action functional with the Lagrangian L(X) = (X, X)/2. Let V denote the covariant derivative of the Levi-Civita connection. Consider the Laplace-Beltrami operator V2 = V o V. Then, in local coordinates, we have V2 = gz' V V , where g2" and g2j are the metric tensors and Vk is the covariant derivative in the direction of O/8qk.

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183

In general, V2 does not coincide with the Laplace-de Rham operator de-

fined as A = (d + A)2 = d6 + 6d, where d is the differential and 6 is the codifferential with respect to (,). (Sometimes L is also called the de RhamKodaira-Hodge Laplacian.) The operators V2 and L are related via the socalled Weitzenbock formulas. In particular, for 1-forms (or vector fields) we have [78]:

AX=-V2X+RicoX

.

Connection Coefficients and Christoffel Symbols There is a standard way to describe a connection by identifying the decomposition TTM = V ® H with the decomposition of TTM into the sum of V and a fixed complementary vector bundle. In particular, once a local chart is chosen the complementary bundle arises from the trivialization of TM. Let us analyze this construction in detail. Let (U, 0) be a local chart on M and 0:TMIu -+ U X IR" a trivialization of the tangent bundle to U. For every m E U and X E TmM, we have T(m,,x)TM

V(., x) + H fl,x)

,

where H"m x) is the tangent space to the manifold {Y E TMI u I P2 (4i(Y)) = P2 (1i(X ))

with P2: U x 1R" - IR" being the natural projection. We emphasize that V(m x) is the vertical space defined above, while, in general, H0 'X) does not coincide with the connection space H(m,x), nor with the space H'm X) for another trivialization W. Define the map F,m: TmM x TmM -+ TmM as follows. Fix X E TmM. For W in TmM, we pick Wl E HO (,,,,X)

and

WH E H(m,x)

such that

TirW1 = T7rWH = W .

These vectors do exist and are unique because H(,,.i x) and H(m,x) are complementary to V(m,x) and, thus, W1 - WH E V(m,x). We set

rm(X,W) =p(W1 - WH) and call rm the connection coefficient with respect to the trivialization 4i. The operator I'm is linear in X and W [34]. A formula for the transformations of 1 under changes of coordinates or changes of the trivialization 1P can be found, for example, in [42]. When a trivialization 45 is fixed, a vector Y E T(m,x)TM can be given as the quadruple Y(m,x) = (m, X, Y1, Y2), where Y1 E Ht,m x) and Y2 E V(m,x)

184

Appendices

In this notation it is easy to obtain the following formula for the local connector K:

K(m,X,Y1,Y2) = (m,Y2+F,(X,Y1))

.

Just to compare, we point out that in this notation we have Tir(m,X,Y1,Y2) = (m,Yi)

and 7rl(m,X,Yi,Y2) = (m,X)

,

where,7rl is the projection TTM onto TM. Now we turn to a detailed analysis of P for two trivializations 43 which are particularly important to us. (i) The trivialization 0 given by the coordinate frames a/aql, ... , a/agn, where (q1, ... , qn) are the coordinates on U. In other words, 0:

(M'Xi= aq

-- (m, (X1,...,Xn)) E U x Rn .

In this case F,n is called the local connection coefficient and is denoted simply by Fn. Since Fm is linear, it can be given by n3 functions f . (m) on U satisfying the following identities:

° eq. age = r i aqk The functions T (m) are called the Christoffel symbols. It is easy to see that the operator r,,, (X, W) is symmetric in X and W (i.e., F! (m) is symmetric

in i and j) if and only if the torsion tensor vanishes. For the Levi-Civita connection, T. can be expressed by the standard formula:

I, 23

-1 2

a a a (aqigjl + aq.i gtit - agt g'j

9kl

where gjj and gtj are the metric tensors. When 0 is the trivialization in a local chart (U, ¢), we use the notation HO

for H(,n

X).

(ii) The trivialization c over U is given by an orthonormal frame field (el,... , en) on U. In other words, we have

45: (m,Xieti) - (m, (Xl,...,Xn)) E U x 1Rn For such a 0, we denote the connection coefficient by I'm (X, W). Following [35], we call the coefficients P k of F the tetrad Christoffel symbols. It is easy For the Levi-Civita connection, we have to see that °e;ej =

I' = 2 (Ckj +Ck{+4) where C,q are such that [er, e9] = cpget.

(A.3)

Appendix A. Introduction to the Theory of Connections

185

Formula (A.3) holds only when (,) is positive definite; for a pseudoRiemannian metric one has to take into account the signs of (ei,ej) [35]. For a trivialization by coordinate frames, when we pass from a coordinate system 0 to 01, the aforementioned quadruple (m, X, Y1, Y2) transforms into the following one:

(X, Y1))

This transformation rule yields (13.4). The second summand in the transformation formula for Y2 arises from the fact that HO 0 H41 because these connections are given by different trivializations. Note also that HO is a connection over U. In what follows we call Hk the Euclidean connection over the chart (U, 0).

Second-Order Differential Equations and the Spray A second-order differential equation on M or a special vector field on TM (we use both terms) is, by definition, a vector field Y on TM such that

T7rY(,,, X) = X E TM

for all (m, X) E TM. Let -y(t) be an integral curve of Y. The curve m(t) = 7r-y(t) is called a solution of the second-order differential equation Y. It is easy to see that -y(t) = rh(t), and m(t) is indeed a solution of the equation ti(t) = pY2 (t, m(t), rh(t))

,

where Y = (m, X, Y1, Y2) in a given chart and p is as above. (See the second section of this appendix.) Let H be a connection on TM. Since Trr: H(,,, X) -* T,,,M is an isomorphism, there exists a unique special vector field Z that belongs to H, namely, Z(m,X) = TIr-1X E H(,,,,X). The field Z is called the spray of H. Note that Z is C°°-smooth if H is so. Consider a special vector field Y on TM. The field Y can be canonically decomposed as the sum of its vertical and horizontal components: Y = Yv + YH, where Yv E V and YH E H. It is clear that TirY = TzrYH, and so YH is a special vector field. Since YH E H, it must coincide with Z. As a consequence, we obtain the decomposition

Y=Z+Yv

,

where Yv is vertical. Using the definition of D/dt, one can show that the solutions of the second-order differential equation Y are exactly those of the

186

Appendices

equation

D

d m(t) = pYv (t, m(t), rh(t)) In particular, the solutions of Z are geodesics, i.e., they satisfy the equation Drh(t)/dt = 0.

The Exponential Map and Normal Charts Let M be a Riemannian manifold, mo E M, and X E T,,,. M. Denote by 7x(t) the geodesic with initial conditions y(0) = mo and y(O) = X. If 11XII is sufficiently small, yx(t) exists for t E [0,1]. We define the exponential map T,,, M - M as exp,.,io (X) = yx (1). Taking together the maps exp,n

for all m E M, we obtain the map exp: TM -+ M x M with the following properties:

(i) exp is C°°-smooth; and (ii) exp is a diffeomorphism of a neighborhood of the zero section in TM onto a neighborhood of the diagonal in M x M.

A formula expressing expm in local coordinates is given in the proof of Theorem 13.2. (See also [41].) As follows from (ii), for every mo E M, there exists a neighborhood N of the origin in Tm0 M which lies in the domain of exp,no and such that exp,,,, 1N

is a diffeomorphism onto a neighborhood U of mo in M. Thus we obtain the so-called normal chart on U. In this chart the local connection coefficient vanishes at mo. This construction, as well as the basic properties of exp, remain valid when we replace the Levi-Civita connection by an arbitrary connection on M.

Appendix B. Introduction to the Theory of Set-Valued Maps Here we give a brief review of the basic notions and results on set-valued maps. A more detailed introduction and proofs can be found, for example, in [21].

A set-valued map F from a set X to a set Y is, by definition, a correspondence between the points of X and nonempty subsets of Y. The subset F(x) C Y, where x E X, is called the image of x. Let F: X -+ Y be a set-valued map. A single-valued map f : X -+ Y is called a selection of F if f (x) E F(x) for all x E X. When X and Y are metric spaces, there are a few nonequivalent ways to extend the notion of continuity to set-valued maps X -+ Y. (Here we do not consider the continuity of set-valued maps of topological spaces. The reader interested in this notion should consult [21].) A set-valued map F is called upper semicontinuous at x E X if for any e > 0, there exists a neighborhood U of x such that for any x' E U the subset

Appendix B. Introduction to the Theory of Set-Valued Maps

187

F(x') lies in the e-neighborhood of F(x). The map F is upper semicontinuous on X if it is upper semicontinuous at every point of X. A set-valued map is called lower semicontinuous at x E X if for any e > 0, there exists a neighborhood U of x such that for any x' E U the subset F(x) lies in the e-neighborhood of F(x'). The map is lower semicontinuous on X if it is lower semicontinuous at every point of X. The map is said to be Hausdorff continuous (or just continuous) if it is lower and upper semicontinuous simultaneously. Assume, in addition, that the image F(x) is bounded and closed for every x. Then a Hausdorff continuous map F is continuous with respect to the so-called Hausdorff metric. The latter is defined on the collection of nonempty closed bounded subsets of Y as follows.

Namely, for two such sets A and B we put p. (A, B) = sup p(a, B) aEA

where p is the distance on Y. Then the Hausdorff distance is h(A, B) = max(p. (A, B), p. (B, A))

.

A set-valued map is called closed if its graph is closed in X x Y. If F is closed and for every x E X there exists a neighborhood U such that F(U) is relatively compact, then F is upper semicontinuous. Now let Y be a Banach space, while X is still an arbitrary metric space. Assume that F is lower semicontinuous and the image F(x) is convex and closed for every x E X (that is to say, F has closed convex images). Then F admits a continuous selection. Upper semicontinuous maps may not admit continuous selections but they always admit measurable ones. Note that upper semicontinuous maps arise in applications more often than Hausdorff continuous maps. Let X be a Banach space and let F: X -+ X be an upper semicontinuous map with closed convex images. Assume also that for every bounded set Q C X, the image F(Q) is relatively compact and that F(B) C B, where B is a ball in X. Then F has a fixed point on B, i.e., there exists x E B such that x c F(x) (the Schauder principle). Let F: 1R x IR' - 1R' be a set-valued map. The differential inclusion E F(t, x) is an analog of a differential equation. (It turns into a differential equation if F is single-valued.) If F is upper semicontinuous and has closed convex bounded images, then for any xo E IR" and to E 1R', the inclusion has a local solution x with x(to) = xo.

188

Appendices

Appendix C. Basic Definitions of Probability Theory and the Theory of Stochastic Processes Throughout this appendix the basic notions of probability theory (including the notion of independence) are assumed to be familiar to the reader. A more detailed account of the material discussed here can be found [100], [101], [116], [121], and [124].

For the sake of simplicity, we only consider random variables (i.e., measurable maps) on a probability space ((2, .7 , P) with values in 1R" equipped with the a-algebra of Borel sets. Note that many of the notions we discuss below can be extended word-for-word to random variables with values in a manifold or an infinite-dimensional vector space. Particular attention is given to the notions of a martingale and semimartingale on a manifold, since their definition involves a special construction.

A a-subalgebra BF of .P is said to be generated by a random variable ,R --+ 1R" if Be is the minimal a-subalgebra that contains all preimages of Borel sets under . In other words, B£ is the minimal o -subalgebra with respect to which C is measurable.

Stochastic Processes and Cylinder Sets A stochastic process is a time-dependent random variable. A process 77(t), where t E [0, oo), is said to have a.s. (almost surely) continuous trajectories if for P-almost all w E ,f1, the trajectory ?7(t, w) is continuous. In this case CO ([ 0, oo),1R") is called the space of (sample) trajectories. Fix a finite number of points t1, ... , tk E [ 0, oo) and a finite collection of Borel sets B1, . . . , Bk C R. A cylinder set is a subset of the form

Jtl,...,tk(Bi) ...,Bk) =

E C°([0,no),IR") I x(ti) E Bi}

of CO ([ 0, oo), W). In other words, the elements of such a cylinder set are those curves which take a value in Bi at ti for i = 1, . . . , k. A stochastic process with a.s. continuous trajectories can be regarded as a random variable with values in CO ([ 0, oo), IR"), where the latter space is equipped with the a-algebra generated by cylinder sets.

The Conditional Expectation Consider the Hilbert space L2 (,R, .F, P) of square integrable random variables. (See Appendix E.) Let FO be a a-subalgebra of F and L2(Q, Fo, P) the space

of square integrable random variables which are It is clear that L2 ((, Fo, P) is a closed subspace of L2 (,fl, F, P). Denote by 'Fo-measurable.

Q: L2 (0, Fo, P) - L2(Q, F, P)

the orthogonal projection. For any 6 E L2 (,(l, .P, P), the random variable Q6 E L2 (.R, YO, P) is called the conditional expectation of e with respect to

Appendix C. Stochastic Processes

189

.To; it is denoted by E( I .Fo). One may show that Q can be extended to a continuous projection L1(.fl, .F, P) -4 Ll (.(l, Fo, P). This means that the conditional mathematical expectation E(C I -1 o) exists even for E E L, (J7, 97, P). It is important to point out that E(C I Fo) is a random variable in L1(.fl, F'o, P), unique up to its values on the sets of measure zero, such that

fdP=fE(eI.Fo)dP for any A E .F0. Thus, this identity can be viewed as another definition of conditional expectation. The existence of a random variable with this property follows from the Radon-Nikodym theorem. General properties of conditional expectation (see (116] and [124]) follow from those of the projection Q. Let A E .F and let XA be the indicator of A. The random variable P(A .Fo) = E(XA I Yo) is called the conditional probability.

Markovian Processes In our introduction to Markovian processes we follow [100]. Throughout this section all a-algebras are assumed to be complete, i.e., to contain all sets of P-measure zero. Let Bt, where t E [ 0, oo), be a nondecreasing family of vsubalgebras of F and fi(t) a stochastic process. Denote by 11 the o-algebra generated by £(s) for s > t, and by NC the v-algebra generated by fi(t). The process fi(t) is called Markovian relative to Bt if

P(B n F I N£) = P(B I .N)P(F I .N ) P-a.s. for any t E [ 0, oo), B E Bt, and F E .. The process is said to be Markovian if it is Markovian relative to the a-subalgebra Pt generated by a(s) for 0 < s < t. A process f is Markovian if and only if one of the following two conditions holds: (1)

E(q5IBt)=E(cbIN{) for any t E [ 0, oo) and any bounded .-measurable random variable 0 with values in 1R; (2)

E(f (fi(t)) I B8) = E(f (fi(t)) I N; ) for any t > s > 0 and any measurable bounded function f on R. A random variable r(w) with values in [ 0, co) is called a random time. It is called a Markov time if {w I T(w) < t} E Bt for any t > 0. If, in addition, P(r(w) < oo) = 1, then it is said to be a stopping time.

190

Appendices

Martingales and Semimartingales A random process i7(t) is a martingale relative to a nondecreasing family of o-subalgebras St, t c [0, oo), if 71(t) is Bt-measurable for every t and E(r1(t)

I

B.,) = 77(s)

for any t and s such that t > s > 0. A process i7(t) is called a local martingale if there exists a nondecreasing sequence of Markov times T,a such that lim r,a = oo and the process r7(t A r,,,)

such that r7(t n r,,,w) _ i7(min(t,rn(w)),w) is a martingale for every n. A process 77(t) is a semimartingale if there exists a local martingale M(t) and a processes A(t) such that q7(t) = M(t) + A(t) and the trajectories of A(t) have bounded variation in t for almost all w. (The latter condition implies that an integrable function is a.s. Stieltjes-integrable along a trajectory of A(t).) It is clear that a martingale is a local martingale and a local martingale is a semimartingale. Note that one may define the integral of a random variable over a semimartingale. A particular case of such an integral is the integral over a Wiener process [101]. It is important that (local) martingales need not transform to (local) martingales under smooth changes of coordinates, while semimartingales transform to semimartingales. (The decomposition q = M ± A does not transform termwise under a smooth change of coordinates.) As a consequence, only semimartingales are well defined on smooth manifolds. However, given a connection on a manifold, one may define a martingale on it as a semimartingale with certain special properties with respect the connection. This definition is due to Laurent Schwartz. (See [107] and[ 122] for more details.)

Appendix D. The Ito Group and the Principal Ito Bundle In Sect. 13 we introduced the Ito bundle I(M). However, the definition of I (M) as a Stinrod bundle has not been completed, since we have not specified the structural group, nor the principal bundle to which I (M) is associated. In this appendix, which is prepared jointly with Marina Tonkikh, we fill in the gap. Let L2(IR') be the set of bilinear homomorphisms IR' x IR' -+ IR'.

Definition D.1. The Ito group Gr is the set of pairs (B, 3), where B E GL(n) and 3 E L2 (1R'), with multiplication (B,)3) ' (G, y) = (B o G, Boy('

,

) + a(G('), G(')))

(D.1)

Appendix E. Sobolev Spaces

191

Theorem D.1. GI is a group. Proof. The associativity of (D.1) can be checked by a straightforward calculation. The unit in GI is the pair (I, 0), where I E GL(n) is the unit matrix and 0 E L2 (1R") is the zero homomorphism. The inverse to (B, /3) is the pair 1(.),B-1(.))) . o (B,/3)-1 = (B-1, -B-1 °Q(B Define a left action of GI on the fiber IRn x L(1R") of I(M) as follows:

(B, /3) o (X, A) _ (Bx + 1 tr f

A(.)), B o A I

.

(D.2)

Theorem D.2. The action of GI on the fibers of I(M) given by (D.2) makes I(M) into a bundle with the structural group GI. The proof of the theorem is standard and we omit it. Consider a locally trivial bundle GI over M such that GI over a local chart (U, 0) is isomorphic to U x GI and (m, (B,Q)) H [(01 0.0-1) m,

((01 0 0-1)'B, (01 o

) + (.01 0

0-1)ii

(B(.), B(.)))]

is the transformation map as we pass to another chart (U,, q51)

Definition D.2. The bundle GI(M) is called the principal Ito bundle over M.

It is clear that GI is indeed a principal bundle (see, e.g., [17]), and I(M) is its associated bundle with the action of GI on the fiber given by (D.2).

Appendix E. Sobolev Spaces In this appendix we briefly recall the definition of Sobolev spaces needed in Part III. For the sake of simplicity, we only consider scalar functions on a domain E C 1R' with smooth boundary. The case of R'-valued functions can be dealt with in a similar way or reduced to the case of scalar functions by taking the components. Naturally, when working with vector-valued functions, one should replace the multiplication in 1R by the inner product on W.

Let E be a domain in IR" with smooth boundary. Consider the space L2(E) of square integrable scalar functions. To define the Sobolev spaces H'(E) for all s E Z, we first set H°(E) = L2(E). For s > 0, the space H'(E) is formed by square integrable functions on E whose generalized derivatives up to order s exist and belong to L2(E). The

192

Appendices

space HI(E) is, in fact, a Hilberrt space with the inner product

(u, v)(') = L(8Dx)Dv(x)) dx where a is a multi-index. It is easy to see that this inner product coincides with that on L2 (E) when s = 0. The norm on H' (E) is given by

Hull(', =[ f, (

\

1

211

IDau(x)l I dx] II4S8

//

The spaces H'(IR) are defined in a similar way. The space H-8(IR") for s > 0 is formed by the distributions v on IR." such that the integral

f1V every continuu ous linear functional on H3 (IR'n) can be written as the above integral for some

v E H-'(IR"), i.e., H-'(IR") = (H'(IR"))`. The space H-'(E) is formed by functions that can be extended to IR" as elements of H-' (an). When sl > 32, the space H11 is continuously embedded into H'2. Furthermore, HI(E) is embedded into the space of smooth functions for a large

enough s. More precisely, let s < n/2 + k where k is an integer. Then u E H'(E) coincides almost everywhere with a function u E Ck(E). Here Ck (E) is the space of functions on E which have continuous derivatives up to order k. In addition, we have JIullck < AllullO

where the constant A is independent of u. A more detailed account of Sobolev spaces can be found in [40].

Appendix F. Accessible Points and Closed Trajectories of Mechanical Systems by Viktor L. Ginzburg2 In this appendix, we refine some of the results proved in Chap. 3. In particular, we prove that two nonconjugate points on a complete Riemannian manifold

can be connected by a trajectory of a mechanical system provided that the force field has less than quadratic growth in velocity. As shown below, this 2 Department of Mathematics, Stanford University, Stanford, CA 94305, USA.

Appendix F. Accessible Points and Closed Trajectories

193

result is sharp. Namely, we give an example of a mechanical system on the Euclidean plane with a force field quadratic in velocity such that no two points sufficiently distant from each other can be connected by a trajectory. The proof of the theorem is based on a simple perturbation argument applied to the Newton equation. Then we turn to systems with constraints and to the existence of closed trajectories of mechanical systems because, to some extent, these problems can be dealt with by the same method. In conclusion, we briefly discuss a problem, appearing to be similar but actually much harder, where the absolute value of velocity is bounded from above. An example of such a problem is the existence of a closed trajectory of a charge with prescribed energy in a magnetic field on a surface. For the sake of simplicity, we restrict our attention to single-valued smooth force fields only. The generalization of our results to multivalued forces needs some additional consideration and can be proved by the methods developed in Chap. 3.

Growth of the Force Field and Accessible Points As in Sect. 10, let M be a complete Riemannian manifold. Consider a mechanical system on M with a smooth force field a, which is assumed to be dependent on time, velocity, and coordinates. We recall that, in fact, a is just a time-dependent vector field on TM tangent to the fibers. In particular, the metric on M enables us to define JIa(t, m, X) II for all t, m, and X E TmM. Recall also that the motion of the system is then given by the Newton equation:

It is important for what follows that the Newton equation can be thought of as a first order differential equation on TM. Then the solutions of the Newton equation are just integral curves of the vector field Z + a on TM, where Z is the spray of the metric on M. The field a is said to have less than quadratic growth in velocity if for any compact K in M and any interval I = [0, 1], we have lim

IIXII--

1ja(t, m, X) II= 0

(F. 1)

11X11 2

uniformly on I x K. Note that the property of a to have less than quadratic growth is independent of the metric on M. Let mo and ml be two points on M which are not conjugate along a geodesic y with y(0) = mo and 'y(1) = ml. The following result sharpens Theorems 10.1 and 10.2.

Theorem F.1. Let a have less than quadratic growth in velocity. Then for any sufficiently small to > 0, there exists a solution m(t) of the Newton equation such that m(0) = mo and m(to) = ml.

194

Appendices

Proof. For the sake of simplicity, we prove the theorem for an autonomous force field. Only a minor refinement is required to deal with the general case. First, let us introduce some notation. Fix an arbitrary interval I = [0, 1]. Let y be the geodesic on M with the initial conditions mo and y' (0) = v, and let z/, be the solution of the Newton equation with the same initial conditions. Since the metric is complete, yv extends to I.

Let c > 0 and let v be a unit vector. The key step of the proof is the following lemma, which, roughly speaking, says that after rescaling t -+ t/c the solutions ? C converge to y., as c --+ oc.

Lemma F.1. (i) For a sufficiently large c, the map q5,,,:t --+ Vv (t/c) is defined on the interval I for every unit vector v in Tm0M. (ii) 0cvII Cl-converges to -y,11 uniformly in v as c The role of assertion (i) in our proof is rather technical: we need it only to make sure that (ii) makes sense. Equivalently, (i) can be restated as follows:

for any c > 0, there exists an interval [0, l'] C I such that the solution Vi, extends to [0, l'] for every unit vector v E T,,,oM.

Proof of the Lemma. Before we go into technical details, let us explain the idea of the proof of (ii). A free particle moving with the initial velocity cv along yv traverses the arc y,(I) in time ro = l/c. For a large c, the time ro is small. Then, as follows from the Newton equation, the effect of the force field a is approximately proportional to 7- 02. (Example: S = vt + gt2/2, where g = 9.81 m/sect.) In other words, for any r such that 0 < T < To, 0...(r)) < const T2 . 11all < const . 12 max

11a(m, cX)II C2

where the maximum is taken over all m in a compact neighborhood of 'y (I) and the unit vectors X E T,nM. Due to (F.1), the ratio on the right-hand side goes to zero as c --+ oc. Let t E I and r = t/c. Then 0cv(t) _ v(r)

and

y,,(t) = y v(r)

Therefore, lim C00

711 W) = 0

This shows that we have at least Co-convergence in (ii). A rigorous proof goes as follows. First, let us pass to vector fields on TM. As we have already mentioned, the solutions ?,b of the Newton equation are in a one-to-one correspondence with the integral curves W = (0, ) of the vector field Z + a on TM, where Z is the spray of the Riemannian metric on M. Let

Appendix F. Accessible Points and Closed Trajectories

h be the dilatation h: X '--* X/c. Then 0..v = the vector field

195

is an integral curve of

F=Z+h.a

c

The relationship between the integral curves of F and Z + a is very easy to figure out: 45, (t) = rev (t/c), where (mo, v) and T1,, (0) = (mo, cv). Since a is vertical, it follows from (F.1) that (h.a)(m, X)

a(m, cX)

c

C2

-> 0 as

c

oo

In other words, if we set e = 1/c, then F = F(e) is a family of vector fields continuous in e > 0. By (F.1), F(e) converges uniformly on any compact set

to F(0)=Zas E-*0. Since Z is complete, this readily implies that for a sufficiently large c, the solutions 0,ti extend to I for every unit vector v in T.,,,0M. Assertion (i) is proved.

For brevity, from now on we shall use the same notation for a curve and its restriction to I. Since the solutions of a differential equation depend continuously on parameters, cPav is continuous in c, and 4i, C°-converges to (y,,, as c --- oo. As a consequence, 0, C1-converges to 'y,,. An argument similar to that we used to show the extendibility to I proves that the convergence is uniform in v. This completes the proof of (ii).

Let U be a germ of a hypersurface through ml meeting the geodesic y transversely. For any unit vector v in T1,,M that is sufficiently close to v° = 'Y(0)/II'Y(0)II, the geodesic y with the initial condition (mo,v) meets U. Let V be a small neighborhood of vo in the unit sphere in Tm0M. The "Poincare map" P: V --+ U sending v to the first intersection of yv with U is well defined, smooth, and P(vo) = ml. Furthermore, P is a diffeomorphism on its image, since mo and ml are not conjugate along y. Denote by T(v) the arc length on y,, from mo to P(v). Then T: V -+ 1R is a smooth positive function and P(v). Shrinking V if necessary, it is easy to find an interval I = [0, 1) such that for all v E V, we have T (v) < l and the geodesics 7v(I) pierce U exactly once. Let c be a sufficiently large positive constant and let v E V be close to vo. Then, by Lemma F.1, for every v E V the curve (or rather the one-dimensional ([0,1/c]) = 0ev (I) is transversal to U and pierces U only once. As a consequence, shrinking V again if necessary, we obtain a smooth map PP: V -+ U defined for an essentially large c. The map Pc sends v to the unique with U: intersection of submanifold) i,L

P,,(v) = &w(I) fl U It follows from Lemma F.1 that P,, C°-converges uniformly to P as c -+ 00.

Without loss of generality we may assume that V is a topological ball and all P,, as well as P, are defined on the boundary V. Since for a large

196

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c the maps P° and P are close to each other, so are their restrictions to the boundary. This means that P°Iav and Plav are homotopic as maps to U\ml. The standard degree argument shows that the equation P°(v) = ml has a solution vl for every large c. Namely, being homotopic to PI av, the map P°Iav gives rise to the generator of H.n_2(U\m1;Z), where n = dim M. As a consequence, ml has a preimage under P. The solution of the Newton equation with the initial conditions (mo, cvl ) passes through ml for every large c. The theorem is proved. 11

Remark F.1. Theorem F.1 is sharp in the sense that if the force field has quadratic growth in velocity, then it may be impossible to join two distant points by a trajectory of the system. Let, for instance, M be the Euclidean plane and a(t, m, X) = II X II JX, where J is the counterclockwise rotation by it/2. Then the Newton equation reads m= IImIIJrn .

The force field a has quadratic growth in X. It is easy to see that all the solutions of the Newton equation are circles of radii less than Rmax = 1/0. This means that the distance between mo and ml cannot exceed if there exists a trajectory connecting mo and ml. The field a fails to be smooth along the zero section of TM. This defect can be easily eliminated by altering slightly the force field. Pick a positive even C°°-function f such that for every x E IR, we have f (x) > Ixi and f (x) = Ixl

when Ixi > 1. To make a smooth, let us replace IIXI! by f (IIXII) in the definition of a. Thus, we obtain the force field f (II X II )JX. The trajectories of the system are still circles whose radii do not exceed Rma.. Therefore, if

dist(mo, ml) > 2Rm. = vF2, then mo and ml cannot be connected by a trajectory.

Remark F.2. When mo and ml are conjugate, a trajectory connecting these points may fail to exist even if the force field a is bounded. (See Example 1 of Sect. 9.) A simpler example with linear growth: a charge in a constant (nonzero) magnetic field on S2. (See Sect. 4.B and Example 3 of Sect. 9.) It is also essential that we put no restrictions on the norm of the initial velocity of a trajectory: a low-velocity trajectory connecting mo and ml, in contrast to a low-velocity geodesic, might not exist even for a bounded a. Remark F.3. The force field a is allowed to grow faster in X than the spray Z (which has only linear growth) because we deal with the Newton equation, i.e., a is vertical. Otherwise, we would have to require a to grow slower than the norm of Z.

Appendix F. Accessible Points and Closed Trajectories

197

Accessible Points in Systems with Constraints Let /3 be a (linear) constraint on M, possibly nonholonomic. Denote by D/dt the reduced covariant derivative of the Levi-Civiti connection on M. In "constrained" Newtonian mechanics an admissible (i.e., tangent to Q) curve m(t) is a trajectory of the mechanical system if and only if m(t) satisfies the following version of the Newton equation:

D.m = pa (t, rn, m)

(F.2)

where a is the force field on M, and P denotes the orthogonal projection of TM onto /3. We recall that (F.2) can be regarded as a vector field on the total space of the subbundle 0 C TM (Sect. 5.A). Two generic points mo and ml of M cannot be connected by a solution of (F.2) even when a = 0. The reason is that the space of solutions of (F.2) beginning at mo is parametrized by Qmo. As a consequence, such solutions fill out only a k-dimensional subset of M, where k = dim /3, provided that a = 0. (For a generic a, however, this set is (k+l)-dimensional.) Thus, if dim,o < dimM and a = 0 or dim/3 < dimM-1, the points inaccessible from mo form a full-measure set. One of the ways to handle this problem is to modify the very question addressed as suggested in Sect. 11. Namely, let us try to find a trajectory joining mo with a "large enough" submanifold N of M. To state our result, we need first to recall some notation and definitions of Sect. 5.C and 11. Let expO o: Qmo -+ M be the exponential map of the reduced connection. This map sends X E /3mo to the point ryX(1), where ryX is the solution of the homogeneous Newton equation (a = 0) such that yx (0) = mo and -' x (O) = X. In Sect. 5.C, ryX was called a least-constrained geodesic.

Also, let N C M be a smooth submanifold containing a point ml which can be connected with mo by a least-constrained geodesic and such that expoo is transversal to N at ml. In other words, we assume that there exists X E /3,no with ml = expmo (X) E N and

im (dx exp0o) + T,n, N = T,,,,, M

.

(F.3)

In fact, it is sufficient to deal only with the germ of N at ml. Now we are ready to sharpen Theorems 11.2 and 11.3. Theorem F.2. Let a have less than quadratic growth in velocity. Then, for a sufficiently small to > 0, there exists a solution m(t) of the Newton equation (F.2) with m(0) = mo and m(to) E N. The proof of this result repeats word-for-word that of Theorem F.1.

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Remark F.4. Note that, being forced anyway to replace the point ml by a submanifold, we omitted the assumption dim N + dim /3 = dim M from the hypotheses of Theorem F.2 (see Theorems 11.2 and 11.3). In fact, this assumption is irrelevant; what matters is the transversality condition (F.3). In particular, one may "generalize" Theorem F.1 even further: the point mo can be joined with a submanifold N through ml even when mo and ml are conjugate but N satisfies (F.3). For example, the transversality condition holds

automatically if dim N = dim M, and mo can be connected with a point arbitrarily close to ml (Lemma F.1). Furthermore, it is sufficient to assume in Theorem F.2 that the force field a has less than quadratic growth only in X E /3.

Remark F.5. Regarding possible applications of Theorems F.1 and F.2 to control theory, it is important to point out that, besides the constraint /3, in real life one always has an upper bound for values of Ifth due to physical, technical, or other limitations (e.g., a speed limit). As a consequence, although the desired trajectory must exist by the theorem, there may not be any acceptable in practice (see Remark F.2). It is also interesting to notice that the assertions and hypotheses of Theorems F.1 and F.2 are somewhat excessive for a "real-life" application. In fact, our ability to join mo with a point close enough to ml, which is always possible (Remark F.4), would be absolutely sufficient.

Closed Trajectories of Mechanical Systems The problem of existence of closed trajectories is considerably more subtle than the boundary-value problem, yet the method developed here enables us to prove the existence under quite general hypotheses. A trajectory t H m(t) of the mechanical system (F.1) is said to be closed if there exists T > 0 such that m(0) = m(T) and m(0) = rh(T). The infimum of all such T > 0 is called the period of m. To ensure the existence of a (nonconstant) closed trajectory, we have to impose some additional restriction on M and a. Namely, throughout this section we assume that M is compact, a is independent of time and, furthermore, a(m, X) is orthogonal to X for all (m,X) E TM. While the reason for working with an autonomous force field on a compact manifold is evident, the orthogonality condition deserves a discussion, which we put off until later. (See Remark F.6.) Note, however, that the orthogonality condition yields the energy conservation law H(7h) = const

along every solution m of the Newton equation (F.1), where H: TM -+ IR, H(X) = (X,X)/2, is the kinetic energy. A basic example of a with these features is a magnetic field perpendicular

Appendix F. Accessible Points and Closed TYajectories

199

to a surface M. Here we have (Sect. 4.B)

a(rn,X) = k(rn)JmX

,

(F.4)

where k: M --+ IR is a smooth function and Jm: TmM --+ Tm,M is the rotation

by 7r/2. A trajectory of this system has geodesic curvature k(m) at in E M. More generally, one considers a manifold M with a closed 2-form w (the electromagnetic field). Then the force field a is given by the condition that the contraction of a(m, X) with the metric coincides with the contraction of X with w: (a(m, X), .) = w(X, ) .

For example, if M is a surface, one takes w = k dS, where dS is the area form on M. It is clear that a has linear growth in X.

To state the existence result, we fix a free homotopy class u of maps Sl -+ M.

Theorem F.3. Let the metric on M be generic and either u # 0 or M be simply connected. Assume also that a has less than quadratic growth in velocity. Then in the homotopy class u there exists a nonconstant closed trajectory of (F.1) with any sufficiently short period T > 0.

Corollary. For a generic metric on M, there exists a closed trajectory of (F.1) with any sufficiently small period T > 0. In other words, every energy level {H = c} contains a closed trajectory for a large enough c. The hypotheses of the theorem deserve some discussion. Here the metric is generic if it belongs to a certain open and dense (in the C2-topology) subset of the space of all metrics on M. This subset depends on the class u only. Once u and the metric are fixed, a can be an arbitrary field with less than quadratic growth which satisfies the orthogonality condition. (We continue to discuss this hypothesis in Remark F.7.)

Proof. The argument follows the same line as in the proof of Theorem F.1.

First, observe that under our condition on u, for any metric on M, there exists a closed nonconstant geodesic -y in the homotopy class u. Let E. denote

the energy level {H = c} and U a hypersurface germ in Ec transversal to F = (y, %y) at an arbitrary point m0 on F. Let V be a small neighborhood of mo in U. Then we have a well-defined Poincare return map P: V -+ U, which sends a point v E V to the first intersection of the geodesic through v with U.

It is evident that P(mo) = mo and P is independent of c > 0. For a generic metric, id - P is a diffeomorphism of V on its image. (To be more accurate: for a generic metric, there exists ry E u such that id - P is a diffeomorphism on its image.) Now acting as in the proof of Theorem F.1, we define the Poincare map PP: V -+ U in the energy level E, for the mechanical system with the force field a. An argument similar to Lemma F.1 shows that PP exists for a sufficiently

200

Appendices

large c and lim Pc = P as c , oo. One finishes the proof by applying the degree argument and showing that Pc has a fixed point.

Remark F.6. To see why the assumption that a(m, X) is orthogonal to X is essential, let us consider a slow charge in a magnetic field in a homogeneous isotropic medium. For example, let M be a flat torus and

a(m, X) = kJX - eX , where k and a are positive constants. (The second term on the right-hand side is the drag force.) It is easy to check that every trajectory of this system is a spiral. In particular, the system has no closed (nonconstant) trajectories although a has only linear growth in X. A similar phenomenon can also occur for a bounded force field. Remark F.7. It is essential in the theorem that the metric is generic. Namely, for a flat metric on the torus T2 and a constant magnetic field (i.e., k = const) the lifts of trajectories to the universal covering 1R2 of T2 are just circles. As a consequence, every trajectory on T2 is closed and contractible. Thus, although closed trajectories exist, the theorem is not applicable to this system. The metric is not generic: it has no stable closed geodesic to which we could apply our perturbation argument. The hypothesis of the theorem can be slightly altered. Namely, at least when M is two-dimensional, one may first fix the class u and the field a having less than quadratic growth in X. Now the metric is to belong to a dense open subset in the space of metrics satisfying the orthogonality condition for a. The only modification needed in the proof is quite straightforward: one simply shows that P, - id can be turned into an isomorphism by varying the metric within this space. In the rest of this appendix let us restrict our attention to a particular class of force fields a. Namely, the class of magnetic fields (Sect 4.B and (F.4)). For such a, the corollary is unlikely to be sharp. There is a variety of indications [58] that a closed trajectory exists for any metric and, perhaps, on every high energy level. The existence of a closed trajectory on a prescribed energy level is a very subtle problem, which, for certain classes of force fields (including magnetic fields), appears to be tied to symplectic geometry and the Novikov multivalued Morse theory. Here we discuss a particular example, where it is not very hard to show that a closed trajectory does exist on a low-energy level. The following theorem concerning the motion governed by a "nondegenerate" magnetic field can be proved by the methods of symplectic geometry [56].

Theorem F.4. Let M be an orientable closed surface and a a nonvanishing magnetic field, i.e., a is given by (F.4) with a nonvanishing k: M -# R. For any sufficiently small c > 0, there exists a closed trajectory of the system on the level EE = {H = c}.

Appendix F. Accessible Points and Closed Trajectories

201

Remark F.8. Theorem F.4 can be further refined by giving a kind of "Morse inequalities" linking the number of closed trajectories with the topology of M. Here we just state a result of [56]. Denote the genus of M by g. Under the hypothesis of Theorem F.4, there are at least 2g+1 distinct closed trajectories on E. if g > 1, and at least two if M = S2. Counting with multiplicities, we have no less than 2g + 2 closed trajectories. All the trajectories counted here have contractible projections on M. Another direction in which this result can be improved is to replace the varying level Er by a fixed one El under the assumption that the magnetic field is strong. Depending on what is meant by "strength," this may be quite a delicate problem [57], [58]. Combining Theorems F.3 and F.4, we see that for a nonvanishing magnetic

field, a closed trajectory exists on every level E, with a small c > 0 as well as with a large c (provided that the metric is generic). There is, however, a gap between the large and small values of c where neither of these theorems can be applied. Moreover, there exists a metric and a magnetic field such that a given energy level El contains no closed trajectories [58]. The example is the horocycle flow (on a compact surface) well known in hyperbolic geometry. We conclude this appendix by mentioning that here we analyzed only a few

aspects of the problem. The reader interested in more results on accessible points and the existence of closed trajectories should consult [58] and [96] and the Bibliography therein.

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Index

acceleration of a stochastic process anticipative stochastic integral antisymmetric derivative 101

53

backward covariant mean derivative 113 bac kward equati on

17

geodesic spray of a connection 185 geodesic spray S 154 geodesic spray Z 140 group exponential on diffeomorphism groups 140 gyroscopic force 20

horizontal bundle 179 horizontal lift 179 179 180

19

constraint 22 - holonomic 23 - non-holonomic 23 covariant derivative 181 current velocity 101 curvature tensor 182 differential inclusion 187 diffusion coefficient 56 diffusion process 56 59

diffusion-type process drift 56

essential extension

96

future a-algebra

Cartan's development 13 Christoffel symbol 184 conditional expectation 188 configuration space 17 connection on a principal bundle connection on a tangent bundle connector 181 conservative force

113

forward mean derivative 97 forward stochastic differential 53 fundamental vector field 180

100

backward mean derivative 98 backward stochastic integral 53 backward Wiener process 100 180 basic vector field

- generator

force field

forward covariant mean derivative

106

56

26

inertia operator (tensor) 18 integral operators with parallel translation 9 Ito equation 56 - on a manifold 60 Ito equation and vector field, canonically corresponding 63 Ito integral 51 - with parallel translation 69 Ito line integral with parallel translation 72 Ito process

55

- on a manifold

71

Ito vector field 62 Ito's bundle 60 Ito's development 71 Ito's formula 55

- backward 101 Ito's group 190

Euclidean connection over a chart 185

kinetic energy

exponential map of a connection 186 - on diffeomorphism groups 140

Lagrangian

17 19

212

Index

Langevin's equation 87 Laplace-Beltrami operator 182 Laplace-de Rham operator 182 left translation 20 Levi-Civita connection 182 linear constraint 22 local coefficient of a connection 184 local connector 183

tetrad Christoffel symbol torsion tensor 182

martingale 189 mechanical system 17 metric tensor 184 momentum 21 multiple integral 52

vector force field 17 velocity 17 velocity hodograph 30

operator S

11

operator r

12

osmotic velocity "past" a-algebra phase space 17

projection P projection P projection R

a-lemma

30

179

E(x)

180

Eb(x)

180

dw(t)

51 53 53

d.w(t) dsw(t)

180 137 137 157

D

D. D

97 98 113

D.

Ricci curvature (tensor) 182 Riemannian connection 182 right-invariant vector field 135 right translation 20

113 101 DA 114 Ds 101 Ds 114 D 176 D 115 D 176 D. 176 D, 115 D. 176 D/dt 181 D/dt 22 D/dt 152 DB(M) 134 Ds(T?) 172 Dµ(M) 150 174

DA

semimartingale 190 set-valued map 186 - upper semicontinuous 186 - lower semicontinuous 186 set-valued vector field 26 spray 185 stochastically complete Riemannian manifold 77 Stratonovich equation 55 53

strong solution of Ito equation symmetric derivative 101

135 135

w-lemma

96

Stratonovich integral

uniform Riemannian atlas 7 uniformly complete Riemannian manifold 68

180

weak solution of Ito equation white noise 52 Wiener process 49 - on a manifold 77 Weitzenbock's formula 183

present a-algebra 96 principal Ito bundle 190 potential 19 potential energy 19 projection A

totally non-holonomic constraint

vertical (sub)bundle vertical lift 181

101

parallel translation

19

total energy

- equation

Newton's equation 18 Newton-Nelson equation 107 - on Riemannian manifold 114

184

56

D' (-)

56

23

Index

S 11 S(a(T), A(-))(t) 69 S (a(r) dr + A(T) dw(T)) (t)

trA'(A)

K

K K K

18 181 139 153

L9

20

O(M)

P P Pe

Pt R9 RI

Vs-1

r r; 180

22 137 137? 96

20 71

56

tr fl' (A, A) tr r,,, (A, A) 141

12

184

1;j

184

rr,a

113

r,,,( , )

183

55 64

80

213

Applied Mathematical Sciences (continued from page ii)

61. SattingerlWeaver: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics.

62. LaSalle: The Stability and Control of Discrete Processes.

63. Grasman: Asymptotic Methods of Relaxation Oscillations and Applications. 64. Hsu: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. 65. Rand/Armbruster: Perturbation Methods, Bifurcation Theory and Computer Algebra. 66. HlavdceklHaslinger/Necasl/Lovisek: Solution of Variational Inequalities in Mechanics. 67. Cercignani: The Boltzmann Equation and Its Applications. 68. Temam: Infinite Dimensional Dynamical Systems in Mechanics and Physics. 69. GolubitskylStewart/Schaeffer: Singularities and Groups in Bifurcation Theory, Vol. 11. 70. Constantin/Foias/Nicoluenko/Temam: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. 71. Catlin: Estimation, Control, and the Discrete Kalman Filter. 72. Lochak/Meunier: Multiphase Averaging for Classical Systems. 73. Wiggins: Global Bifurcations and Chaos. 74. Mawhin/Willem: Critical Point Theory and Hamiltonian Systems. 75. Abraham/Marsden/Ratiu: Manifolds, Tensor Analysis, and Applications, 2nd ed. 76. Lagerstrom: Matched Asymptotic Expansions: Ideas and Techniques.

Aldous: Probability Approximations via the Poisson Clumping Heuristic. 78. Dacorogna: Direct Methods in the Calculus of Variations. 79. Herndndez-Lerma: Adaptive Markov Processes. 80. Lawden: Elliptic Functions and Applications. 81. Bluman/Kumei: Symmetries and Differential Equations. 82. Kress: Linear Integral Equations. 83. Bebernes/Eberly: Mathematical Problems from Combustion Theory. 84. Joseph: Fluid Dynamics of Viscoelastic Fluids. 85. Yang: Wave Packets and Their Bifurcations in Geophysical Fluid Dynamics. 86. Dendrinos/Sonic: Chaos and Socio-Spatial Dynamics. 87. Weder: Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media. 77.

88. Bogaevski/Povzner: Algebraic Methods in

89. O'Malley: Singular Perturbation Methods for Ordinary Differential Equations. 90. Meyer/Hall: Introduction to Hamiltonian Dynamical Systems and the N-body Problem. 91. Straughan: The Energy Method, Stability, and Nonlinear Convection. 92. Naber: The Geometry of Minkowski Spacetime. 93. Colton/Kress: Inverse Acoustic and Electromagnetic Scattering Theory. 94. Hoppensteadt: Analysis and Simulation of Chaotic Systems. 95. Hackbusch: Iterative Solution of Large Sparse Systems of Equations. 96. Marchioro/Pulvirenti: Mathematical Theory of Incompressible Nonviscous Fluids. 97. Lasota/Mackey: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd ed. 98. de Boor/Hollig/Riemenschneider: Box Splines. 99. Hale/Lune& Introduction to Functional Differential Equations. 100. Sirovich (ed): Trends and Perspectives in Applied Mathematics. 101. Nusse/Yorke: Dynamics: Numerical Explorations. 102. Chossatl/ooss: The Couette-Taylor Problem. 103. Chorin: Vorticity and Turbulence. 104. Farkas: Periodic Motions. 105. Wiggins: Normally Hyperbolic Invariant Manifolds in Dynamical Systems. 106. Cercignani/Illner/Pulvirenti: The Mathematical Theory of Dilute Gases. 107. Antman: Nonlinear Problems of Elasticity. 108. Zeidler: Applied Functional Analysis: Applications to Mathematical Physics. 109. Zeidler: Applied Functional Analysis: Main Principles and Their Applications. 110. Diekmann/van Gils/Verduyn Lunel/Walther: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. 111. Visintin: Differential Models of Hysteresis. 112. Kuznetsov: Elements of Applied Bifurcation Theory.

113. Hislop/Sigal: Introduction to Spectral Theory: With Applications to Schrodinger Operators. 114. Kevorkian/Cole: Multiple Scale and Singular Perturbation Methods. 115. Taylor: Partial Differential Equations I, Basic Theory. 116. Taylor: Partial Differential Equations It, Qualitative Studies of Linear Equations. 117. Taylor: Partial Differential Equations III, Nonlinear Equations.

Nonlinear Perturbation Theory.

(continued on next page)

Applied Mathematical Sciences (continued from previous page) 118. GodlewskilRaviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws. 119. Wu: Theory and Applications of Partial Functional Differential Equations. 120. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems. 121. Brokate/Sprekels: Hysteresis and Phase Transitions. 122. Gliklikh: Global Analysis in Mathematical Physics: Geometric and Stochastic Methods.