Polygonal negative hyperbolic rotopulsators of the curved $n$-body problem

TL;DR

This paper investigates the configurations of negative hyperbolic rotopulsators in curved space, proving they form regular polygons unless they are relative equilibria, and showing at most one such equilibrium exists.

Contribution

It establishes the geometric structure of negative hyperbolic rotopulsators and uniqueness of relative equilibria in the curved $n$-body problem.

Findings

Configurations are regular polygons if not relative equilibria.

At most one relative equilibrium exists for these rotopulsators.

Provides conditions for existence of specific solutions in curved space.

Abstract

For the $n$-body problem in spaces of negative constant Gaussian curvature, we prove for a class of negative hyperbolic rotopulsators that if that class exists, the configurations of the point masses of these rotopulsators have to be regular polygons if the rotopulsators are not relative equilibria. Additionally, we prove that if the rotopulsators are relative equilibria, there exists at most one such solution.