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].
Publié le : 2004-07-05
Classification:
Solar System,
stability,
Newton,
Arnold,
Herman,
invariant torus,
Nash-Moser,
small denominator,
resonance,
secular system,
[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]
@article{hal-00633820,
author = {F\'ejoz, Jacques},
title = {D\'emonstration du " th\'eor\`eme d'Arnold " sur la stabilit\'e du syst\`eme plan\'etaire (d'apr\`es Michael Herman)},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {fr},
url = {http://dml.mathdoc.fr/item/hal-00633820}
}
Féjoz, Jacques. Démonstration du " théorème d'Arnold " sur la stabilité du système planétaire (d'après Michael Herman). HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00633820/