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

TL;DR

This paper proves the existence of symmetric, collision-free periodic solutions to the classical Newtonian N-body problem, invariant under the rotation groups of Platonic polyhedra, using variational methods with topological and symmetry constraints.

Contribution

It introduces a variational approach to find symmetric, collision-free periodic solutions of the N-body problem constrained by Platonic symmetries and topological conditions.

Findings

Existence of symmetric periodic solutions invariant under Platonic rotation groups.

Construction of variational cones where the action functional is coercive.

Identification of collision-free minimizers with complex geometric structures.

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 ${\cal R}$ 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.