Polygonal Homographic Orbits of the Curved n-Body Problem

TL;DR

This paper investigates polygonal homographic orbits in the curved n-body problem, establishing conditions for their existence, especially for regular polygons, and explores relative equilibria with non-equal masses on curved surfaces.

Contribution

It provides necessary and sufficient conditions for polygonal homographic orbits in curved spaces and characterizes when regular polygons occur, including cases with non-equal masses.

Findings

Regular n-gon is a homographic orbit iff all masses are equal.

Existence of scalene triangular relative equilibria on the sphere for n=3.

Not all sets of three masses can form a triangular relative equilibrium.

Abstract

In the $2$-dimensional $n$-body problem, $n\ge 3$, in spaces of constant curvature, $\kappa\ne 0$, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then consider the case of regular polygons. We further use this criterion to show that, for any $n\ge 3$, the regular $n$-gon is a polygonal homographic orbit if and only if all masses are equal. Then we prove the existence of relative equilibria of non-equal masses on the sphere of curvature $\kappa>0$ for $n=3$ in the case of scalene triangles. Such triangular relative equilibria occur only along fixed geodesics and are generated from fixed points of the sphere. Finally, through a classification of the isosceles case, we prove that not any three masses can form a triangular relative equilibrium.