Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem

Abstract

We prove the existence of a number of smooth periodic motions u ^sub ^ 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 ∣ ℛ ∣ of ℛ and the particles have all the same mass. Our approach is variational and u ^sub ^ is a minimizer of the Lagrangian action 𝒜 on a suitable subset 𝒦 of the H ^sup 1^ T-periodic maps u:[arrow right]^sup 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 ∣ 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.[PUBLICATION ABSTRACT]