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Global Analysis in Mathematical Physics
Band 122

Global Analysis in Mathematical Physics

Geometric and Stochastic Methods

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Fr. 182.00

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Global Analysis in Mathematical Physics

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eBook

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Beschreibung

Details

Einband

Gebundene Ausgabe

Erscheinungsdatum

13.12.1996

Verlag

Springer Us

Seitenzahl

216

Maße (L/B/H)

24.1/16/1.8 cm

Beschreibung

Details

Einband

Gebundene Ausgabe

Erscheinungsdatum

13.12.1996

Verlag

Springer Us

Seitenzahl

216

Maße (L/B/H)

24.1/16/1.8 cm

Gewicht

524 g

Auflage

1997

Übersetzt von

V.L. Ginzburg

Sprache

Englisch

ISBN

978-0-387-94867-6

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  • Global Analysis in Mathematical Physics
  • I. Finite-Dimensional Differential Geometry and Mechanics.- 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 S.- 3.B The Operator ?.- 3.C Integral Operators.- 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.CLength 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.- 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.- II. Stochastic Differential Geometry and its Applications to Physics.- 4 Stochastic Differential Equations on Riemannian Manifolds.- 12. Review of the Theory of Stochastic Equations and Integrals on Finite-Dimensional Linear Spaces.- 12.A Wiener Processes.- 12.B The Itô Integral.- 12.C The Backward Integral and the Stratonovich Integral.- 12.D The Itô 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 Itô 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.- 5 The Langevin Equation.- 17. The Langevin Equation of Geometric Mechanics.- 18. Strong Solutions of the Langevin Equation, Ornstein—Uhlenbeck Processes.- 6 Mean Derivatives, Nelson’s Stochastic Mechanics, and Quantization.- 19. More on Stochastic Equations and Stochastic Mechanics in ?n.- 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.- III. Infinite-Dimensional Differential Geometry and Hydrodynamics.- 7 Geometry of Manifolds of Diffeomorphisms.- 22. Manifolds of Mappings and Groups of Diffeomorphisms.- 22.A Manifolds of Mappings.- 22.B The Group of H8-Diffeomorphisms.- 22.C Diffeomorphisms of a Manifold with Boundary.- 22.D Some Smooth Operators and Vector Bundles over Ds(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.- 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.- 25.A Volume-Preserving Diffeomorphisms of a Closed Manifold.- 25.B Volume-Preserving Diffeomorphisms of a Manifold with Boundary.- 25.C 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.- 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic 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 Derivatives.- Connection 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 Itô Group and the Principal Itô 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.