Periodic solutions of the N-body problem with Lennard-Jones-type potentials

Abstract

In this article we consider a system of N identical particles interacting through a potential of Lennard-Jones (LJ) type. We consider subsets of loop space that satisfy certain basic conditions including the notion of ‘tiedness’ introduced by Gordon in [W. Gordon, Conservative dynamical systems involving strong force, Trans. Amer. Math. Soc. 204 (1975), pp. 113–135]. We then show, by means of critical point theory, that this system admits periodic solutions in every homotopy class of these subsets of loop space. More precisely we show that every homotopy class contains at least two periodic solutions for sufficiently large periods. One of these solutions is a local minimum and the other is a mountain-pass critical point of the action functional. We also prove that, given a homotopy class of one of these subsets, there do not exist any periodic solutions in it for sufficiently small periods. Our results have wide applicability. For example, one can consider the space of choreographies and prove existence results for choreographic solutions. Our existence proof relies upon an assumption that global minimizers of standard strong force potentials on suitable spaces are non-degenerate up to some symmetries.