Math Seminars
J. Fejoz
(Paris 6)
On M. Herman's proof of Arnold's theorem in celestial mechanics
In 1963, Vladimir Arnold [A] stated and partly proved the following
theorem~: in the Newtonian model of the Solar system with n>1
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
direct Keplerian motions, leading to quasiperiodic motions with 3n-1
frequencies. In 1998, in a series of lectures Michael Herman sketched
a complete and more conceptual proof of this theorem [F]. In reviewing
this proof, I will focus on a couple of ideas which make it so
powerful and, I believe, elegant.
[A] V.I. Arnold. Small denominators and problems of the stability of
motion in classical and celestial mechanics (in Russian). Usp.
Mat. Nauk. 18 (1963), 91--192 (English transl., Russ.
Math. Surv. 18 (1963), 85--193).
[F] J. Fejoz, Demonstration du `theoreme d'Arnold' sur la stabilite du
systeme planetaire (d'apres Herman), Michael Herman Memorial Issue,
Ergodic Theory and Dynamical Systems 24:5 (2004)