数学物理中的全局分析 几何及随机方法2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Part Ⅰ．Finite-Dimensional Differential Geometry and Mechanics3

Chapter 1 Some Geometric Constructions in Calculus on Manifolds3

1．Complete Riemannian Metrics and the Completeness of Vector Fields3

1.A A Necessary and Sufficient Condition for the Completeness of a Vector Field3

1.B A Way to Construct Complete Riemannian Metrics5

2．Riemannian Manifolds Possessing a Uniform Riemannian Atlas7

3．Integral Operators with Parallel Translation10

3.A The Operator S10

3.B The Operator Γ12

3.C Integral Operators14

Chapter 2 Geometric Formalism of Newtonian Mechanics17

4．Geometric Mechanics:Introduction and Review of Standard Examples17

4.A Basic Notions17

4.B Some Special Classes of Force Fields19

4.C Mechanical Systems on Groups20

5．Geometric Mechanics with Linear Constraints22

5.A Linear Mechanical Constraints22

5.B Reduced Connections23

5.C Length Minimizing and Least-Constrained Nonholonomic Geodesics24

6．Mechanical Systems with Discontinuous Forces and Systems with Control:Differential Inclusions26

7．Integral Equations of Geometric Mechanics:The Velocity Hodograph28

7.A General Constructions29

7.B Integral Formalism of Geometric Mechanics with Constraints31

8．Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force32

Chapter 3 Accessible Points of Mechanical Systems39

9．Examples of Points that Cannot Be Connected by a Trajectory40

10．The Main Result on Accessible Points41

11．Generalizations to Systems with Constraints45

Part Ⅱ．Stochastic Differential Geometry and its Applications to Physics49

Chapter 4 Stochastic Differential Equations on Riemannian Manifolds49

12．Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces49

12.A Wiener Processes49

12.B The It？ Integral50

12.C The Backward Integral and the Stratonovich Integral53

12.D The It？ and Stratonovich Stochastic Differential Equations54

12.E Solutions of SDEs56

12.F Approximation by Solutions of Ordinary Differential Equations57

12.G A Relationship Between SDEs and PDEs58

13．Stochastic Differential Equations on Manifolds59

14．Stochastic Parallel Translation and the Integral Formalism for the It？ Equations67

15．Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations76

15.A Wiener Processes on Riemannian Manifolds76

15.B Stochastic Equations78

15.C Equations with Identity as the Diffusion Coefficient80

16．Stochastic Differential Equations with Constraints83

Chapter 5 The Langevin Equation87

17．The Langevin Equation of Geometric Mechanics87

18．Strong Solutions of the Langevin Equation,Ornstein-Uhlenbeck Processes91

Chapter 6 Mean Derivatives,Nelson's Stochastic Mechanics,and Quantization95

19．More on Stochastic Equations and Stochastic Mechanics in Rn96

19.A Preliminaries96

19.B Forward Mean Derivatives97

19.C Backward Mean Derivatives and Backward Equations98

19.D Symmetric and Antisymmetric Derivatives101

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

19.F Stochastic Mechanics107

20．Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds109

20.A Mean Derivatives on Manifolds and Related Equations109

20.B Geometric Stochastic Mechanics114

20.C The Existence of Solutions in Stochastic Mechanics115

21．Relativistic Stochastic Mechanics125

Part Ⅲ．Infinite-Dimensional Differential Geometry and Hydrodynamics133

Chapter 7 Geometry of Manifolds of Diffeomorphisms133

22．Manifolds of Mappings and Groups of Diffeomorphisms133

22.A Manifolds of Mappings133

22.B The Group of Hs-Diffeomorphisms134

22.C Diffeomorphisms of a Manifold with Boundary136

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

23．Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms139

23.A The Case of a Closed Manifold139

23.B The Case of a Manifold with Boundary141

23.C The Strong Riemannian Metric141

24．Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid142

24.A Diffuse Matter142

24.B A Barotropic Fluid143

Chapter 8 Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid147

25．Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid147

25.A Volume-Preserving Diffeomorphisms of a Closed Manifold148

25.B Volume-Preserving Diffeomorphisms of a Manifold with Boundary151

25.C LHS's of an Ideal Incompressible Fluid152

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 Manifold156

27．The Regularity Theorem and a Review of Results on the Existence of Solutions164

Chapter 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms171

28．Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus172

29．A Viscous Incompressible Fluid175

Appendices179

A．Introduction to the Theory of Connections179

Connections on Principal Bundles179

Connections on the Tangent Bundle180

Covariant Derivatives181

Connection Coefficients and Christoffel Symbols183

Second-Order Differential Equations and the Spray185

The Exponential Map and Normal Charts186

B．Introduction to the Theory of Set-Valued Maps186

C．Basic Definitions of Probability Theory and the Theory of Stochastic Processes188

Stochastic Processes and Cylinder Sets188

The Conditional Expectation188

Markovian Processes189

Martingales and Semimartingales190

D．The It？ Group and the Principal It？ Bundle190

E．Sobolev Spaces191

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

Growth of the Force Field and Accessible Points193

Accessible Points in Systems with Constraints197

Closed Trajectories of Mechanical Systems198

References203

Index211