Synopses & Reviews
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 Poincaré--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.
Synopsis
Asymptotic methods of nonlinear mechanics developed by N. M. Krylov and N. N. Bogoliubov originated new trend in perturbation theory. They pene- trated deep into various applied branches (theoretical physics, mechanics, ap- plied astronomy, dynamics of space flights, and others) and laid the founda- tion for lrumerous generalizations and for the creation of various modifications of thesem. E f, hods. A great number of approaches and techniques exist and many differen. t classes of mathematical objects have been considered (ordinary differential equations, partial differential equations, delay diffe, 'ential equations and others). The stat. e of studying related problems was described in mono- graphs and original papers of Krylov N. M., Bogoliubov N. N. 1], 2], Bogoli- ubov N. N 1J, Bogoliubov N. N., Mitropolsky Yu. A. 1], Bogoliubov N. N., Mitropol- sky Yu. A., Samoilenko A. M. 1], Akulenko L. D. 1], van den Broek B. 1], van den Broek B., Verhulst F. 1], Chernousko F. L., Akulenko L. D. and Sokolov B. N. 1], Eckhause W. l], Filatov A. N. 2], Filatov A. N., Shershkov V. V. 1], Gi- acaglia G. E. O. 1], Grassman J. 1], Grebennikov E. A. 1], Grebennikov E. A., Mitropolsky Yu. A. 1], Grebennikov E. A., Ryabov Yu. A. 1], Hale J . K. I]' Ha- paev N. N. 1], Landa P. S. 1), Lomov S. A. 1], Lopatin A. K. 22]- 24], Lykova O. B.
Table of Contents
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.