Platonic Polyhedra, Periodic Orbits and Chaotic Motions in the $$N$$ N -body Problem with Non-Newtonian Forces

Abstract

We consider the \(N\)-body problem with interaction potential \(U_\alpha =\frac{1}{\vert x_i-x_j\vert ^\alpha }\) for \(\alpha >1\). We assume that the particles have all the same mass and that \(N\) is the order \(\vert \mathcal {R}\vert \) of the rotation group \(\mathcal {R}\) of one of the five Platonic polyhedra. We study motions that, up to a relabeling of the \(N\) particles, are invariant under \(\mathcal {R}\). By variational techniques we prove the existence of periodic and chaotic motions.