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

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. 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.