數(shù)學(xué)物理中的全局分析

內(nèi)容概要

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.

書籍目錄

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

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