Finiteness of polygonal relative equilibria for generalised quasi-homogeneous $n$-body problems and $n$-body problems in spaces of constant curvature

TL;DR

This paper proves the finiteness of polygonal relative equilibria in generalized quasi-homogeneous and curved space $n$-body problems, showing at most one class exists under fixed masses and rotation.

Contribution

It establishes the uniqueness of certain polygonal relative equilibria in generalized and curved space $n$-body problems, extending previous results to broader settings.

Findings

At most one equivalence class of polygonal relative equilibria with all points on a circle.

At most one equivalence class with all but one point on a circle rotating around the remaining point.

Results apply to problems with fixed masses and rotation in spaces of negative constant curvature.

Abstract

We prove for generalisations of quasi-homogeneous $n$-body problems with center of mass zero and $n$-body problems in spaces of negative constant Gaussian curvature that if the masses and rotation are fixed, there exists, for every order of the masses, at most one equivalence class of relative equilibria for which the point masses lie on a circle, as well as that there exists, for every order of the masses, at most one equivalence class of relative equilibria for which all but one of the point masses lie on a circle and rotate around the remaining point mass. The method of proof is a generalised version of a proof by J.M. Cors, G.R. Hall and G.E. Roberts on the uniqueness of co-circular central configurations for power-law potentials.