Synopses & Reviews
This book gives a common treatment to three areas of application of Global analysis to Mathematical Physics previously considered quite distant from each other. These areas are the geometry of manifolds applied to classical mechanics, stochastic differential geometry used in quantum and statistical mechanics, and infinite-dimensional differential geometry fundamental for hydrodynamics.
Review
"this book proves that mechanics is a field where geometric and stochastic methods have very interesting applications. It should be recommended to researchers in probability theory who want to apply stochastic calculus to physics, and also to researchers in mathematical physics who want to learn how probabilistic tools can be applied to problems in mechanics. ZENTRALBLATT MATH"
Synopsis
The first edition of this book entitled Analysis on Riemannian Manifolds and Some Problems of Mathematical Physics was published by Voronezh Univer sity Press in 1989. For its English edition, the book has been substantially revised and expanded. In particular, new material has been added to Sections 19 and 20. I am grateful to Viktor L. Ginzburg for his hard work on the transla tion and for writing Appendix F, and to Tomasz Zastawniak for his numerous suggestions. My special thanks go to the referee for his valuable remarks on the theory of stochastic processes. Finally, I would like to acknowledge the support of the AMS fSU Aid Fund and the International Science Foundation (Grant NZBOOO), which made possible my work on some of the new results included in the English edition of the book. Voronezh, Russia Yuri Gliklikh September, 1995 Preface to the Russian Edition The present book is apparently the first in monographic literature in which a common treatment is given to three areas of global analysis previously consid ered quite distant from each other, namely, differential geometry and classical mechanics, stochastic differential geometry and statistical and quantum me chanics, and infinite-dimensional differential geometry of groups of diffeomor phisms and hydrodynamics. The unification of these topics under the cover of one book appears, however, quite natural, since the exposition is based on a geometrically invariant form of the Newton equation and its analogs taken as a fundamental law of motion."
Description
Includes bibliographical references (p. [203]-209) and index.
Table of Contents
Contents: Some Geometric Constructions in Calculus on Manifolds.- Geometric Formalism of Newtonian Mechanics.- Accessible Points of Mechanical Systems.- Stocastic Differential Equations on Riemannian Manifolds.- Langevin's Equation.- Mean Derivatives, Nelson's Stochastic Mechanics and Quantization.- Geometry of Manifolds of Diffeomorphisms.- Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid.- Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms.