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∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $|\mathc...
Ausführliche Beschreibung

Abstract 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 $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $|\mathcal{R}|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}|_{{\mathcal {K}}}$ 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. Ausführliche Beschreibung