Multiphase averaging for classical systems. With applications to adiabatic theorems. Transl. from the French by H. S. Dumas
type: Item
instance of: scholarly article
title: Multiphase averaging for classical systems. With applications to adiabatic theorems. Transl. from the French by H. S. Dumas
zbMATH Open document ID: 0668.34044
published in: Applied Mathematical Sciences
publication date: 1992-09-17T00:00:00Z
review text: Three significant books in the development of the theory of oscillations -- all by Soviet authors -- are ``Theory of oscillations'' by \textit{A. A. Andronov} and \textit{C. E. Chaikin (S. E. Khaĭkin)} [Princeton, N.J.: Princeton University Press (1949; Zbl 0032.36501)], ``Introduction to nonlinear mechanics'' by \textit{N. M. Krylov} and \textit{N. N. Bogolyubov} [Princeton, N.J.: Princeton University Press (1943; Zbl 0063.03382)] and ``Asymptotic methods in the theory of nonlinear oscillations'' by \textit{N. N. Bogolyubov} and \textit{Yu. A. Mitropol'skiĭ} [New York: Gordon and Breach (1961; Zbl 0151.12201), Russian original Moscow: FizMatLit (1958; Zbl 0083.08101)]. The notion and development of averaging is associated particularly with the latter two publications. Since 1960 general multiphase averaging theorems have been developed, again mostly by Soviet mathematicians, including D. Anosov, V. Arnold and A. Neĭshtadt. The intended purpose of the book under review is, according to the authors, to ``clarify and render accessible the main results in averaging for ODE's since the appearance of Bogolyubov and Mitropol'skiĭ's book'' (Foreword, p. v). This is a worthy aim for non-Russian readers since much source material remains in untranslated Soviet sources, although this book is itself a translation of a text originally published in French. Chapter 1 is a short introduction which outlines the general problem and the notation used. The book is mainly concerned with the standard system \[ dI/dt=\varepsilon f(I,\phi,\varepsilon),\quad d\phi /dt=\omega (I,\phi)+\varepsilon g(I,\phi,\varepsilon), \] where \(I\) and \(\phi\) denote the slow and fast variables respectively and \(\varepsilon\) is a small parameter. The averaged system for the slow motion associated with these differential equations is given by \(dI/dt=\varepsilon <f>(I)\), \(<f>(I)=\int f(I,\phi,0)\,d\mu_{\phi}\) where \(d\mu_\phi\) denotes the invariant measure in the unperturbed motion. Chapter 2 is co…
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34-01
Mathematics Subject Classification ID: 70H08
Mathematics Subject Classification ID: 70H09
zbMATH DE Number: 41889
zbMATH Keywords: Kolmogorov-Arnold-Moser theory
zbMATH Keywords: multiphase averaging theorems
zbMATH Keywords: optimality
zbMATH Keywords: perturbation of integrable Hamiltonian systems
author: Pierre Lochak
author: Claude Meunier
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 41889