Platonic Polyhedra, Topological Constraints and Periodic Solutions of the Classical $N$-Body Problem

We prove the existence of a number of smooth periodic motions $u_*$ of the classical Newtonian $N$-body problem which, up to a relabeling of the $N$ particles, are invariant under the rotation group ${\cal R}$ of one of the five Platonic polyhedra. The number $N$ coincides with the order of ${\cal R}$ and the particles have all the same mass. Our approach is variational and $u_*$ is a minimizer of the Lagrangean action ${\cal A}$ on a suitable subset ${\cal K}$ of the $H^1$ $T$-periodic maps $u:{\bf R}\to {\bf R}^{3N}$. The set ${\cal K}$ is a cone and is determined by imposing to $u$ both topological and symmetry constraints which are defined in terms of the rotation group ${\cal R}$. There exist infinitely many such cones ${\cal K}$, all with the property that ${\cal A}|_{\cal K}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the $N$-body problem with a rich geometric-kinematic structure.