Quasi periodic Hamiltonian Motions, Scale Invariance, Harmonic Oscillators

TL;DR

This paper reviews the proof of the KAM theorem, framing it as a classical renormalization problem with harmonic oscillators as fixed points, linking Hamiltonian motions to scale invariance concepts.

Contribution

It presents the KAM theorem proof through the lens of renormalization, connecting Hamiltonian dynamics with multiscale analysis and scale invariance.

Findings

KAM theorem can be viewed as a renormalization problem

Harmonic oscillators serve as fixed points in this framework

Links between Hamiltonian motions and scale invariance are established

Abstract

The work of Kolmogorov, Arnold and Moser appeared just before the renormalization group approach to statistical mechanics was proposed by Wilson: it can be classified as a multiscale approach which also appeared in works on the convergence of Fourier's series, or construction of Euclidean quantum fields, or the scaling analysis of the short scale behaviour of Navier-Stokes fluids to name a few which originated a great variety of further problems. In this review the proof of the KAM theorem will be presented as a classical renormalization problem with the harmonic oscillator as a `trivial' fixed point.