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Part Ⅰ．Finite-Dimensional Differential Geometry and Mechanics 3

Chapter 1 Some Geometric Constructions in Calculus on Manifolds 3

1．Complete Riemannian Metrics and the Completeness of Vector Fields 3

1.A A Necessary and Sufficient Condition for the Completeness of a Vector Field 3

1.B A Way to Construct Complete Riemannian Metrics 5

2．Riemannian Manifolds Possessing a Uniform Riemannian Atlas 7

3．Integral Operators with Parallel Translation 10

3.A The Operator S 10

3.B The Operator Γ 12

3.C Integral Operators 14

Chapter 2 Geometric Formalism of Newtonian Mechanics 17

4．Geometric Mechanics:Introduction and Review of Standard Examples 17

4.A Basic Notions 17

4.B Some Special Classes of Force Fields 19

4.C Mechanical Systems on Groups 20

5．Geometric Mechanics with Linear Constraints 22

5.A Linear Mechanical Constraints 22

5.B Reduced Connections 23

5.C Length Minimizing and Least-Constrained Nonholonomic Geodesics 24

6．Mechanical Systems with Discontinuous Forces and Systems with Control:Differential Inclusions 26

7．Integral Equations of Geometric Mechanics:The Velocity Hodograph 28

7.A General Constructions 29

7.B Integral Formalism of Geometric Mechanics with Constraints 31

8．Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force 32

Chapter 3 Accessible Points of Mechanical Systems 39

9．Examples of Points that Cannot Be Connected by a Trajectory 40

10．The Main Result on Accessible Points 41

11．Generalizations to Systems with Constraints 45

Part Ⅱ．Stochastic Differential Geometry and its Applications to Physics 49

Chapter 4 Stochastic Differential Equations on Riemannian Manifolds 49

12．Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces 49

12.A Wiener Processes 49

12.B The It？ Integral 50

12.C The Backward Integral and the Stratonovich Integral 53

12.D The It？ and Stratonovich Stochastic Differential Equations 54

12.E Solutions of SDEs 56

12.F Approximation by Solutions of Ordinary Differential Equations 57

12.G A Relationship Between SDEs and PDEs 58

13．Stochastic Differential Equations on Manifolds 59

14．Stochastic Parallel Translation and the Integral Formalism for the It？ Equations 67

15．Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations 76

15.A Wiener Processes on Riemannian Manifolds 76

15.B Stochastic Equations 78

15.C Equations with Identity as the Diffusion Coefficient 80

16．Stochastic Differential Equations with Constraints 83

Chapter 5 The Langevin Equation 87

17．The Langevin Equation of Geometric Mechanics 87

18．Strong Solutions of the Langevin Equation,Ornstein-Uhlenbeck Processes 91

Chapter 6 Mean Derivatives,Nelson's Stochastic Mechanics,and Quantization 95

19．More on Stochastic Equations and Stochastic Mechanics in Rn 96

19.A Preliminaries 96

19.B Forward Mean Derivatives 97

19.C Backward Mean Derivatives and Backward Equations 98

19.D Symmetric and Antisymmetric Derivatives 101

19.E The Derivatives of a Vector Field Along ξ(t)and the Acceleration of ξ(t) 106

19.F Stochastic Mechanics 107

20．Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds 109

20.A Mean Derivatives on Manifolds and Related Equations 109

20.B Geometric Stochastic Mechanics 114

20.C The Existence of Solutions in Stochastic Mechanics 115

21．Relativistic Stochastic Mechanics 125

Part Ⅲ．Infinite-Dimensional Differential Geometry and Hydrodynamics 133

Chapter 7 Geometry of Manifolds of Diffeomorphisms 133

22．Manifolds of Mappings and Groups of Diffeomorphisms 133

22.A Manifolds of Mappings 133

22.B The Group of Hs-Diffeomorphisms 134

22.C Diffeomorphisms of a Manifold with Boundary 136

22.D Some Smooth Operators and Vector Bundles over Ds(M) 137

23．Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms 139

23.A The Case of a Closed Manifold 139

23.B The Case of a Manifold with Boundary 141

23.C The Strong Riemannian Metric 141

24．Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid 142

24.A Diffuse Matter 142

24.B A Barotropic Fluid 143

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 147

25.A Volume-Preserving Diffeomorphisms of a Closed Manifold 148

25.B Volume-Preserving Diffeomorphisms of a Manifold with Boundary 151

25.C LHS's of an Ideal Incompressible Fluid 152

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 156

27．The Regularity Theorem and a Review of Results on the Existence of Solutions 164

Chapter 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms 171

28．Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus 172

29．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 180

Covariant Derivatives 181

Connection Coefficients and Christoffel Symbols 183

Second-Order Differential Equations and the Spray 185

The Exponential Map and Normal Charts 186

B．Introduction to the Theory of Set-Valued Maps 186

C．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．The It？ Group and the Principal It？ Bundle 190

E．Sobolev Spaces 191

F．Accessible Points and Closed Trajectories of Mechanical Systems(by Viktor L.Ginzburg) 192

Growth of the Force Field and Accessible Points 193

Accessible Points in Systems with Constraints 197

Closed Trajectories of Mechanical Systems 198

References 203

Index 211