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 $A_H=A_0+\epsilon A_1$, in which $\epsilon$ is the inverse central mass and $A_0$ is the Kepler action. At $\epsilon =0$ 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.

30 pages, 2 figures