On weak KAM theory of N-body problems

Abstract. We consider N-body problems with homogeneous potentials, including the Newtonian case. We find upper bounds for the minimal action of paths binding two fixed configurations in a fixed time, from which we deduce regularity of the action potential. We then establish a weak KAM theorem, that is to say, we prove the existence of fixed points of the Lax-Oleinik semigroup, or weak global solutions of the Hamilton-Jacobi equation. We prove in addition, that there are invariant solutions for the action of isometries on the configuration space. AMS classification scheme numbers: 70F10, 70H20, 37J50 1