The Classical KAM Theory at the Dawn of the Twenty-First Century

Abstract

We survey several recent achievements in KAM theory. The achievements chosen pertain to Hamiltonian systems only and are closely connected with the content of Kolmogorov's original theorem of 1954. They include weak non-degeneracy conditions, Gevrey smoothness of families of perturbed invariant tori, "exponential condensation" of perturbed tori, destruction mechanisms of resonant unperturbed tori, excitation of the elliptic normal modes of the unperturbed tori, and "atropic" invariant tori (i. e., tori that are neither isotropic nor coisotropic). The exposition is informal and nontechnical, and, as a rule, the methods of proofs are not discussed.

Theorem on perturbation of coisotropic invariant tori of locally Hamiltonian systems and its applications

“…the system with Lagrangian L (11) has a three-dimensional invariant torus all motions on which are quasiperiodic with frequencies (12).…”

“…Later, numerous works were devoted to the improvement of the procedure used in the proof of KAM-theorems and to the development of new directions and approaches, among which we mention the results of Pöschel on the imbedding of isotropic invariant tori into a family smooth in the Whitney sense [8], the modified Sevryuk-Herman method of artificial parameters [9], and the determination of new conditions for the nondegeneracy of Hamiltonian systems by Rüssmann [10]. For fairly complete information about the contemporary state of the KAM theory, see the surveys [11][12][13].…”

Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmann's non-degeneracy condition

“…The well-known KAM theorems assert that majority of invariant tori of integrable Hamiltonian systems persist under small perturbation if some non-degeneracy conditions hold ( [2,3,5,[9][10][11][12][13]16,17], etc.). These KAM tori usually form a parameterized family with some parameters varying on a Cantor set.…”

Hamiltonian Systems: Stable or Unstable?

“…Later, my colleagues in Nanjing had a work on this topic also ([68]). For more information one can refer to the review [62]. Vol In this direction, Fejoz proved recently that KAM tori exist in a planetary n-body problem which looks like a solar system, a remarkable progress since the work started by Arnold (cf.…”