Four 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 introduces one of the simplest non integrable equations, the planar circular restricted problem in the lunar case, where most degeneracies of the general (non-restricted) problem are not present; the second one is a quick introduction to Arnold's theorem on the stability of the planetary problem 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 Lyapunov families bifurcating from polygonal relative equilibria. Celestial mechanics is famous for demanding extensive computations which hardly appear here: these notes only describe the skeleton on which these computations live.