On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

TL;DR

This paper investigates the stability of symmetric periodic orbits in the Newtonian N-body problem related to Platonic polyhedra, providing an algorithm for their enumeration, computational methods for their analysis, and evidence of their instability.

Contribution

It introduces an algorithm to enumerate symmetric periodic orbits, describes a computational approach to analyze their stability, and offers computer-assisted proofs of their instability.

Findings

All studied orbits appear unstable.

Computational methods effectively analyze orbit stability.

Computer-assisted proofs confirm instability in some cases.

Abstract

In (Fusco et. al., 2011) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T>0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in (Fusco et. al., 2011). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic.