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

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. [ABSTRACT FROM AUTHOR]