Synopses & Reviews
Noting that their research is not yet completed, Prykarpatsky (mathematics, U. of Mining and Metallurgy, Cracow, Poland and mechanics and mathematics, NAS, Lviv, Ukraine) and Mykytiuk (mechanics and mathematics, NAS and Lviv Polytechnic State U., Ukraine) describe some of the ideas of Lie algebra that underlie many of the comprehensive integrability theories of nonlinear dynamical systems on manifolds. For each case they analyze, they separate the basic algebraic essence responsible for the complete integrability and explore how it is also in some sense characteristic for all of them. They cover systems with homogeneous configuration spaces, geometric quantization, structures on manifolds, algebraic methods of quantum statistical mechanics and their applications, and algebraic and differential geometric aspects related to infinite-dimensional functional manifolds. They have not indexed their work.
Synopsis
In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e., characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation)."
Synopsis
This book is unique in providing a detailed exposition of modern Lie-algebraic theory of integrable nonlinear dynamic systems on manifolds and its applications to mathematical physics, classical mechanics and hydrodynamics. The authors have developed a canonical geometric approach based on differential geometric considerations and spectral theory, which offers solutions to many quantization procedure problems. Much of the material is devoted to treating integrable systems via the gradient-holonomic approach devised by the authors, which can be very effectively applied. Audience: This volume is recommended for graduate-level students, researchers and mathematical physicists whose work involves differential geometry, ordinary differential equations, manifolds and cell complexes, topological groups and Lie groups.
Table of Contents
Preface. Background Notations. 1. Dynamical Systems with Homogeneous Configuration Spaces. 2. Geometric Quantization and Integrable Dynamical Systems. 3. Structures on Manifolds and Algebraic Integrability of Dynamical Systems. 4. Algebraic Methods of Quantum Statistical Mechanics and Their Applications. 5. Algebraic and Differential Geometric Aspects of the Integrability of Nonlinear Dynamical Systems on Infinite-Dimensional Functional Manifolds. References.