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

Jacques Féjoz

doi:10.3934/dcds.2013.33.3555

Article Contents

2013, Volume 33, Issue 8: 3555-3565. Doi: 10.3934/dcds.2013.33.3555

This issue Previous Article Discretization of dynamical systems with first integrals Next Article Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations

On "Arnold's theorem" on the stability of the solar system

Jacques Féjoz^1,

1.
Université Paris-Dauphine, CEREMADE, Place du Maréchal de Lattre de Tassigny, Paris

Received: September 30, 2012

Revised: October 31, 2012

Published: July 2013

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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.

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Mathematics Subject Classification: Primary: 37J40, 37N05, 70F15, 70H08, 70K20, 70K45; Secondary: 11K60, 37C65, 58C15.

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