On the stability of tetrahedral relative equilibria in the positively curved 4-body problem

TL;DR

This paper investigates the spectral stability of tetrahedral orbits in a positively curved 4-body gravitational system, combining numerical experiments with rigorous analysis near specific limit cases.

Contribution

It provides the first detailed stability analysis of tetrahedral configurations in curved space, supported by both numerical and analytical methods.

Findings

Identified regions of stability and instability depending on mass ratios and latitude.

Performed highly precise numerical experiments to map stability zones.

Provided rigorous proofs near limit cases with extreme mass or latitude values.

Abstract

We consider the motion of point masses given by a natural extension of Newtonian gravitation to spaces of constant positive curvature. Our goal is to explore the spectral stability of tetrahedral orbits of the corresponding 4-body problem in the 2-dimensional case, a situation that can be reduced to studying the motion of the bodies on the unit sphere. We first perform some extensive and highly precise numerical experiments to find the likely regions of stability and instability, relative to the values of the masses and to the latitude of the position of three equal masses. Then we support the numerical evidence with rigorous analytic proofs in the vicinity of some limit cases in which certain masses are either very large or negligible, or the latitude is close to zero.