Polygonal homographic orbits in spaces of constant curvature

TL;DR

This paper proves that in spaces of constant curvature, irregular polygonal homographic orbits cannot exist for the 2D n-body problem when n is at least 3.

Contribution

It establishes a non-existence result for irregular polygonal homographic solutions in curved spaces, extending classical n-body problem insights.

Findings

Irregular polygonal homographic orbits are impossible in curved spaces for n ≥ 3.

The proof applies to spaces with non-zero constant curvature.

Results generalize classical Euclidean n-body problem to curved geometries.

Abstract

We prove that the geometry of the 2-dimensional $n$-body problem for spaces of constant curvature $\kappa\neq 0$, $n\geq 3$, does not allow for polygonal homographic solutions, provided that the corresponding orbits are irregular polygons of non-constant size.