Montreal ASI: SMS july 2007 4 lectures on the N-body problem

In the first two lectures, hamiltonian techniques are applied to avatars of the N-body problem of interest to astronomers: the first one intro-duces one of the simplest non integrable equations, the planar circular restricted problem in the lunar case, where most degeneracies of the gen-eral (non restricted) problem are not present; the second one is a quick introduction to Arnold’s theorem on the stability of the planetary prob-lem where degeneracies are dealt with thanks to Herman’s normal form theorem. The last two lectures address the general (non perturbative) N-body problem: in the third one, a sketch of proof is given of Marchal’s theorem on the absence of collisions in paths of N-body configurations with given endpoints which are local action minimizers; in the last one, this theorem is used to prove the existence of various families of periodic and quasi-periodic solutions with prescribed symmetries and in particular to extend globally Liapunov families bifurcating from polygonal relative equilibria. Celestial mechanics is famous for demanding extensive compu-tations which hardly appear here: these notes only describe the skeleton on which these computations live