Platonic constellations of periodic motions in the $(n + 1)$-body problem

Abstract

We study the spatial $(n+1)$-body problem formed by one heavy central mass together with $n$ equal masses placed on a single orbit of a polyhedral rotation group $H\in\{T,O,I\}$, so that $n=|H|\in\{12,24,60\}$. Imposing the symmetry $q_{L}=L\,q_{I}$ for $L\in H$ reduces the problem to a single $2\pi$-periodic reference curve, with reduced action $A_{H}=A_{0}+\varepsilon A_{1}$, in which $\varepsilon$ is the inverse central mass and $A_{0}$ is the Kepler action. At $\varepsilon=0$ the critical set contains, as one connected component, the five-dimensional manifold of Kepler ellipses of minimal period $2\pi$, which we prove to be a nondegenerate critical manifold. A Lyapunov--Schmidt reduction along this manifold turns the continuation problem into the search for nondegenerate critical points of an explicit function $\Phi(e,\psi)$ of the eccentricity $e$ and the spatial orientation $\psi$, a nondegeneracy we verify by a computer-assisted proof. We thereby obtain, for each of the three groups, families of periodic solutions of the $(n+1)$-body problem with $n+1\in\{13,25,61\}$ bodies, bifurcating from Kepler ellipses and carrying the full tetrahedral, octahedral, or icosahedral symmetry.