Démonstration du " théorème d'Arnold " sur la stabilité du système planétaire (d'après Michael Herman)
Résumé
V.I. Arnold~[Small denominators and problems of stability of motion in classical and celestial mechanics (in Russian), \textit{Uspehi Mat. Nauk.}~\textbf{18} (1963), 91--192] stated and partly proved the following theorem~: in the Newtonian model of the Solar system with $n\geq 2$ planets in space, if the masses of the planets are small enough compared to the mass of the Sun, there is a subset of the phase space of positive measure, in the neighborhood of circular and coplanar Keplerian motions, leading to quasiperiodic motions with $3n-1$ frequencies. This article details the proof of this theorem, following M.R. Herman's lectures~[Proof of a theorem of V.I. Arnold, \textit{S{é}minaire de Sys\-t{é}mes Dynamiques} and manuscripts, 1998].