Special Issue: 70 Years of KAM Theory (Issue Editors: Alessandra Celletti, Luigi Chierchia, and Dmitry Treschev)

Nineteen Fifty-Four: Kolmogorov’s New “Metrical Approach” to Hamiltonian Dynamics

Luigi Chierchia^a, Isabella Fascitiello<sup>b

a</sup> Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy
^b Dipartimento of Education, Università Roma Tre, 00185 Roma, Italy

Abstract: We review Kolmogorov’s 1954 fundamental paper On the persistence of conditionally periodic motions under a small change in the Hamilton function (Dokl. akad. nauk SSSR, 1954, vol. $\bf 98$, pp. 527–530), both from the historical and the mathematical point of view. In particular, we discuss Theorem 2 (which deals with the measure in phase space of persistent tori), the proof of which is not discussed at all by Kolmogorov, notwithstanding its centrality in his program in classical mechanics.
In Appendix, an interview (May 28, 2021) to Ya. Sinai on Kolmogorov's legacy in classical mechanics is reported.

Keywords: Kolmogorov’s theorem on invariant tori, KAM theory, history of dynamical systems, small divisors, Hamiltonian systems, perturbation theory, symplectic transformations, nearlyintegrable systems, measure of invariant tori

MSC: 01A60, 37J40, 37J05, 37J25, 70H08

Received: 31.01.2024
Accepted: 27.05.2024

Language: English