KAM Theory and the Stability of the Solar System

4:00 pm Thursday, March 11, 2010
Jacques Fejoz (Institut de Mathématiques de Jussieu)


In the Newtonian model of the Solar system with n>=2 planets in space, each planet mainly undergoes the attraction of the Sun. If we neglect their mutual attraction, the system becomes the integrable direct product of n uncoupled two-body problems; each planet describes a Keplerian ellipse and thus, solutions of Newton's equations are contained in n-dimensional, invariant, Keplerian tori. The purpose of KAM theory, so-named after Kolmogorov, Arnold and Moser, is to study the preservation of invariant tori. It has led to a partial answer to the question of the stability of the Solar system when, in 1963, Vladimir Arnold stated and partly proved the following theorem~: if the masses of the planets are small, there is a subset of the phase space of positive measure, in the neighborhood of circular and coplanar prograde Keplerian motions, leading to quasi-periodic motions with (3n-1) frequencies. A proof of this theorem will be sketched, relying in particular on some geometric normal form of Hamiltonians introduced by Michael Herman.

 

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