Mathematical Physics Iii Integrable Systems Of Classical Mechanics
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Mathematical Physics III - Integrable Systems of Classical Mechanics
These Lecture Notes provide an introduction to the modern theory of classical finite-dimensional integrable systems. The first chapter focuses on some classical topics of differential geometry. This should help the reader to get acquainted with the required language of smooth manifolds, Lie groups and Lie algebras. The second chapter is devoted to Poisson and symplectic geometry with special emphasis on the construction of finite-dimensional Hamiltonian systems. Multi-Hamiltonian systems are also considered. In the third chapter the classical theory of Arnold-Liouville integrability is presented, while chapter four is devoted to a general overview of the modern theory of integrability. Among the topics covered are: Lie-Poisson structures, Lax formalism, double Lie algebras, R-brackets, Adler-Kostant-Symes scheme, Lie bialgebras, r-brackets. Some examples (Toda system, Garnier system, Gaudin system, Lagrange top) are presented in chapter five. They provide a concrete illustration of the theoretical part. Finally, the last chapter is devoted to a short overview of the problem of integrable discretization.
Mathematical Physics: Classical Mechanics
As a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior. This book presents the KAM (Kolmogorov-Arnold-Moser) theory and asymptotic completeness in classical scattering. Including a wealth of fascinating examples in physics, it offers not only an excellent selection of basic topics, but also an introduction to a number of current areas of research in the field of classical mechanics. Thanks to the didactic structure and concise appendices, the presentation is self-contained and requires only knowledge of the basic courses in mathematics. The book addresses the needs of graduate and senior undergraduate students in mathematics and physics, and of researchers interested in approaching classical mechanics from a modern point of view.
Topology, Geometry, Integrable Systems, and Mathematical Physics
Author: V. M. Buchstaber
language: en
Publisher: American Mathematical Soc.
Release Date: 2014-11-18
Articles in this collection are devoted to modern problems of topology, geometry, mathematical physics, and integrable systems, and they are based on talks given at the famous Novikov's seminar at the Steklov Institute of Mathematics in Moscow in 2012-2014. The articles cover many aspects of seemingly unrelated areas of modern mathematics and mathematical physics; they reflect the main scientific interests of the organizer of the seminar, Sergey Petrovich Novikov. The volume is suitable for graduate students and researchers interested in the corresponding areas of mathematics and physics.