数学物理中的全局分析：几何及随机方法

This book is the first in monographic literature giving a common treatment to three areas of applications of Global Analysis in Mathematical Physics previously considered quite distant from each other, namely, differential geometry applied to classical mechanics, stochastic differential geometry used in quantum and statistical mechanics, and infinite-dimensional differential geometry fundamental for hydrodynamics. The unification of these topics is made possible by considering the Newton equation or its natural generalizations and analogues as a fundamental equation of motion. New general geometric and stochastic methods of investigation are developed, and new results on existence, uniqueness, and qualitative behavior of solutions are obtained.
The first edition of this book, entitled Analysis on Riemannian Manifolds and Some Problems of mathematical Physics, was published in Russian by Voronezh University Press in 1989. For its English edition, the book has been substantially revised and expanded.

Part I. Finite-Dimensional Differential Geometry and Mechanics
Chapter 1 Some Geometric Constructions in Calculus on Manifolds
1. Complete Riemannian Metrics and the Completeness of Vector Fields
1 .A A Necessary and Sufficient Condition for the Completeness of a Vector Field
1.B A Way to Construct Complete Riemannian Metrics
2. Riemannian Manifolds Possessing a Uniform Riemannian Atlas
3. Integral Operators with Parallel Translation .
3.A The Operator $ .
3.B The Operator F .
3.C Integral Operators .
Chapter 2
Geometric Formalism of Newtonian Mechanics
4. Geometric Mechanics: Introduction and Review of Standard Examples
4.A Basic Notions .
4.B Some Special Classes of Force Fields
4.C Mechanical Systems on Groups
5. Geometric Mechanics with Linear Constraints
5.A Linear Mechanical Constraints
5.B Reduced Connections
5.C Length Minimizing and Least-Constrained Nonholonomic Geodesics .
6. Mechanical Systems with Discontinuous Forces and Systems with Control: Differential Inclusions
7. Integral Equations of Geometric Mechanics The Velocity Hodograph
7.A General Constructions
7.B Integral Formalism of Geometric Mechanics with Constraints .
8. Mechanical Interpretation of Parallel Translation and Systems with Delayed Control Force .
Chapter 3 Accessible Points of Mechanical Systems .
9. Examples of Points that Cannot Be Connected by a Trajectory
10. The Main Result on Accessible Points
11. Generalizations to Systems with Constraints
Part II. Stochastic Differential Geometry and its Applications to Physics
Chapter 4 Stochastic Differential Equationson Riemannian Manifolds
12. Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces
12.A Wiener Processes.
12.B The It8 Integral
12.C The Backward Integral and the Stratonovich Integral
12.D The It8 and Stratonovich Stochastic Differential Equations .
12.E Solutions of SDEs
12.F Approximation by Solutions of Ordinary Differential Equations .
12.G A Relationship Between SDEs and PDEs
13. Stochastic Differential Equations on Manifolds
14. Stochastic Parallel Translation and the Integral Formalism for the It8 Equations
15. Wiener Processes on Riemannian Manifolds and Related Stochastic Differential Equations
15.A Wiener Processes on Riemannian Manifolds
15.B Stochastic Equations
15.C Equations with Identity as the Diffusion Coefficient
16. Stochastic Differential Equations with Constraints
Chapter 5 The Langevin Equation .
17. The Langevin Equation of Geometric Mechanics
18. Strong Solutions of the Langevin Equation, Ornstein-Uhlenbeck Processes
Chapter 6 Mean Derivatives, Nelson's Stochastic Mechanics, andQuantization
19. More on Stochastic Equations and Stochastic Mechanics in 1Rn
19.A Preliminaries .
19.B Forward Mean Derivatives .
19.C Backward Mean Derivatives and Backward Equations .
19.D Symmetric and Antisymmetric Derivatives
19.E The Derivatives of a Vector Field Along (t) and the Acceleration of (t) .
19.F Stochastic Mechanics .
20. Mean Derivatives and Stochastic Mechanics on Riemannian Manifolds .
20.A Mean Derivatives on Manifolds and Related Equations .
20.B Geometric Stochastic Mechanics .
20.C The Existence of Solutions in Stochastic Mechanics
21. Relativistic Stochastic Mechanics
Part III. Infinite-Dimensional Differential Geometry and Hydrodynamics
Chapter 7 Geometry of Manifolds of Diffeomorphisms
22. Manifolds of Mappings and Groups of Diffeomorphisms
　22.A Manifolds of Mappings .
　 22.B The Group of HS-Diffeomorphisms .
　22.C Diffeomorphisms of a Manifold with Boundary
　 22.D Some Smooth Operators and Vector Bundles over D*(M) .
23. Weak Riemannian Metrics and Connections on Manifolds of Diffeomorphisms
　23.A The Case of a Closed Manifold
　23.B The Case of a Manifold with Boundary .
　23.C The Strong Riemannian Metric
24. Lagrangian Formalism of Hydrodynamics of an Ideal Barotropic Fluid .
　 24.A Diffuse Matter
　 24.B A Barotropic Fluid
Chapter 8 Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid
25. Geometry of the Manifold of Volume-Preserving Diffeomorphisms and LHSs of an Ideal Incompressible Fluid 25A Volume-Preserving Diffeomorphisms of a Closed Manifold
　 25.B Volume-Preserving Diffeomorphisms of a Manifold with Boundary .
　 25C LHS's of an Ideal Incompressible Fluid
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
27. The Regularity Theorem and a Review of Results on the Existence of Solutions .
Chapter 9 Hydrodynamics of a Viscous Incompressible Fluid andStochastic Differential Geometry
of Groups of Diffeomorphisms
28. Stochastic Differential Geometry on the Groups of Diffeomorphisms of the n-Dimensional Torus .
29 A Viscous Incompressible Fluid
Appendices
A Introduction to the Theory of Connections
Connections on Principal Bundles
　　 Connections on the Tangent Bundle
　　 Covariant DerivativesConnection Coefficients and Christoffel Symbols
　　 Second-Order Differential Equations and the Spray
　　 The Exponential Map and Normal Charts
B. Introduction to the Theory of Set-Valued Maps
C Basic Definitions of Probability Theory and
　　 the Theory of Stochastic Processes
　　 Stochastic Processes and Cylinder Sets
　　 The Conditional Expectation
　　 Markovian Processes
　　 Martingales and Semimartingales
D The It8 Group and the Principal It8 Bundle
E Sobolev Spaces
F Accessible Points and Closed Trajectories of
　　 Mechanical Systems (by Viktor L. Ginzburg)
　　 Growth of the Force Field and Accessible Points .
　　 Accessible Points in Systems with Constraints
　　 Closed Trajectories of Mechanical Systems
References
Index

The present book is apparently the first in monographic literature in which a common treatment is given to three areas of global analysis previousty 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 ageo..