数学物理中的全局分析：几何及随机方法

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

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 Geometryof 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

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.