On arnold's theorem on the stability of the solar system

Abstract

Arnold's theorem on the planetary problem states that, assuming that the masses of n planets are small enough, there exists in the phase space a set of initial conditions of positive Lebesgue measure, leading to quasiperiodic motions with 3n 1 frequencies. Arnold's initial proof is complete only for the plane 2-planet problem. Arnold had missed a resonance later discovered by Herman. The first complete proof, by Herman-Féjoz, relies on the weak non-degeneracy condition of Arnold-Pyartli. A second proof, by Chierchia- Pinzari, is closer to Arnold's initial idea and shows the strong non-degeneracy of the problem after suitable reduction by (part of) the symmetry of rotation. We review and compare these proofs. In an appendix, we define the Poincaré coordinates and prove their symplectic nature through the shortest possible computation.