Multiphase averaging for classical systems. With applications to adiabatic theorems. Transl. from the French by H. S. Dumas. (English) Zbl 0668.34044

Applied Mathematical Sciences, 72, New York etc.: Springer-Verlag. xi, 360 p. DM 78.00 (1988).
Three significant books in the development of the theory of oscillations – all by Soviet authors – are “Theory of oscillations” by A. A. Andronov and C. E. Chaikin (S. E. Khaĭkin) [Princeton, N.J.: Princeton University Press (1949; Zbl 0032.36501)], “Introduction to nonlinear mechanics” by N. M. Krylov and N. N. Bogolyubov [Princeton, N.J.: Princeton University Press (1943; Zbl 0063.03382)] and “Asymptotic methods in the theory of nonlinear oscillations” by N. N. Bogolyubov and 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 concerned with ergodicity, and a general averaging theorem due to Anosov is stated and proved.
The present state of averaging for one frequency systems is reviewed in Chapter 3, together with an introduction to two-frequency systems. In Chapters 4, 5 and 6 the authors discuss Neĭshtadt’s results. In Chapter 4 the optimal averaging theorem is proved in detail for two frequency systems. Anosov’s method is used in Chapter 5 to develop an averaging theorem for n frequency systems although is not necessarily optimal. Optimality for a class of systems is proved in Chapter 6. Chapter 7 is concerned with the important subject of perturbation of integrable Hamiltonian systems. This has now become an extensive subject: here the authors mainly discuss the Kolmogorov-Arnold-Moser (KAM) theory of Hamiltonian systems.
In the remaining three chapters the author look at adiabatic invariance. This is a careful treatment in which adiabatic theorems are developed rigorously from averaging. The problem is examined in the context of the isochronous pendulum which is the classical problem of a pendulum whose length is increased at a rate much more slowly than its period.
There are nine appendices totalling 68 pages which fill in some of the required analytical background. The bibliography contains about 150 items.
This monograph is a good mathematical introduction to the subject matter of the title. Whilst there are passing references to applications, the reader will need to look elsewhere for details of the model problems which give rise to the differential equations.
Reviewer: Peter Smith (Keele


34C29 Averaging method for ordinary differential equations
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
70H08 Nearly integrable Hamiltonian systems, KAM theory
70H09 Perturbation theories for problems in Hamiltonian and Lagrangian mechanics