On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

In Fusco et al (2011 Inventiones Math. 185 283-332) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T > 0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in Fusco et al (2011 Inventiones Math. 185 283-332). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic.