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Nineteen Fifty-four: Kolmogorov's new "metrical approach" to Hamiltonian Dynamics

Luigi Chierchia, Isabella Fascitiello

Abstract

We review Kolmogorov's 1954 fundamental paper {\sl On the Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian} (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 the author, notwithstanding its centrality in Kolmogorov's program in classical mechanics. \\ In Appendix, a recent interview to Ya. Sinai on KAM Theory is reported.

Nineteen Fifty-four: Kolmogorov's new "metrical approach" to Hamiltonian Dynamics

Abstract

We review Kolmogorov's 1954 fundamental paper {\sl On the Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian} (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 the author, notwithstanding its centrality in Kolmogorov's program in classical mechanics. \\ In Appendix, a recent interview to Ya. Sinai on KAM Theory is reported.
Paper Structure (5 sections, 3 theorems, 51 equations)

This paper contains 5 sections, 3 theorems, 51 equations.

Table of Contents

  1. Historical remarks on Kolmogorov's influence on Classical Mechanics in the 20th Century
  2. The theorems in Kolmogorov's 1954 paper
  3. Theorem 1
  4. Theorem 2
  5. An interview to Ya. Sinai

Key Result

Lemma 1

Let $f$ as above, let ${\lambda} = {\,\rm Lip}_{ {\xi} , {\varepsilon} _0}(f)$, and let $0< {\delta} < {\xi}$. Then, the following holds. (i) Let${ N} _0=\{0,1,2,3,...\}$${\alpha} , {\beta} \in { N} _0^d$ be multi--indices. Then, for a suitable constant depending only on $d$ and $| {\alpha} |+| {\beta} |$. (ii) $\forall n\in { Z} ^d$, the Fourier coefficients $f_n(y, {\omeg

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Proposition 1