A tutorial on KAM theory. (Q2778778)

Initiated by A. N. Kolmogorov in 1954, and then carried through by V. I. Arnold and J. Moser in the 1960s, the classical KAM theory resolved fundamental stability problems in celestial mechanics and ergodic theory, by establishing the persistence of quasiperiodic motions under small perturbations of completely integrable Hamiltonian systems. The subsequent developments have expanded the scope of the theory greatly, and the theory now consists of a very rich collection of results and techniques for dealing with a variety of problems in perturbation theory, connected with small divisors. NEWLINENEWLINEThe purpose of the present paper is to provide an introduction into some of the main ideas involved in the KAM theory, and to describe and compare some of the main methods of proof. In the first section of the paper, the author presents an extensive list of recent developments in the KAM theory, always providing complete references and explaining different interrelations. After the following section devoted to some motivating examples, such as the classical Lindstedt series for the twist map, there comes an extensive section on the preliminaries, presenting the relevant background in analysis (interpolation inequalities for classical function spaces), number theory, and symplectic geometry. In the fourth section, the author illustrates the most basic principle of the KAM method, namely that quadratic convergence can overcome small divisors, by discussing the Siegel center theorem. There follows a section on the Nash-Moser implicit function theorems, and the next section is devoted to an extensive discussion of the classical KAM theorem on the existence of invariant Diophantine tori for small perturbations of integrable systems. Here, the author first presents an original proof by Kolmogorov, which produces an invariant torus with a fixed given frequency, and then proceeds to discuss Arnold's approach, as well as a proof based on the Lagrangian formalism. When discsussing the latter, the author follows some lecture notes of J. Moser. A proof not involving changes of variables is also described. The paper concludes with a section on computer assisted proofs. NEWLINENEWLINEAltogether, in the opinion of this reviewer, the present paper provides a broad and comprehensive introduction to the KAM theory. In particular, it should prove an ideal source of information for those knowing the basics of the theory and wishing to learn more.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].