Nonlinear Mechanics, Groups and Symmetry

ISBN-13: 9789048145171 / Angielski / Miękka / 2010 / 382 str.

Yuri A. Mitropolsky; A. K. Lopatin

Opis

This unique monograph deals with the development of asymptotic methods of perturbation theory, making wide use of group- theoretical techniques. Various assumptions about specific group properties are investigated, and are shown to lead to modifications of existing methods, such as the Bogoliubov averaging method and the Poincare--Birkhoff normal form, as well as to the formulation of new ones. The development of normalization techniques of Lie groups is also treated. The wealth of examples demonstrates how these new group theoretical techniques can be applied to analyze specific problems. This book will be of interest to researchers and graduate students in the field of pure and applied mathematics, mechanics, physics, engineering, and biosciences.

Spis treści

Introduction. 1. Vector Fields, Algebras and Groups Generated by a System of Ordinary Differential Equations and their Properties. 2. Decomposition of Systems of Ordinary Differential Equations. 3. Asymptotic Decomposition of Ordinary Differential Equations with a Small Parameter. 4. Asymptotic Decomposition of Almost Linear Systems of Differential Equations with Constant Coefficients and Perturbations in the Form of Polynomials. 5. Asymptotic Decomposition of Differential Systems with Small Parameter in the Representation Space of Finite-Dimensional Lie Group. 6. Asymptotic Decomposition of Differential Systems where Zero Approximation has Special Properties. 7. Asymptotic Decomposition of Pfaffian Systems with a Small Parameter. Appendix: A: Lie Series and Lie Transformation. B: The Direct Product of Matrices. C: Conditions for the Solvability of Systems of Linear Equations. D: Elements of Lie Group Analysis of Differential Equations on the Basis of the Theory of Extended Operators. Bibliographical Comments. References. Index.