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 R of one of the five Platonic polyhedra. The number N coincides with the order vertical bar R vertical bar of R and the particles have all the same mass. Our approach is variational and u(*) is a minimizer of the Lagrangian action A on a suitable subset K of the H-1 T-periodic maps u : R -> R-3N. The set K is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group R. There exist infinitely many such cones K, all with the property that A vertical bar(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.