Polygonal rotopulsators of the curved $n$-body problem

TL;DR

This paper investigates specific polygonal solutions called rotopulsators in curved 3D spaces, proving their existence, properties, and finiteness, and clarifying relationships among different solution classes in the curved $n$-body problem.

Contribution

It establishes the existence of polygonal rotopulsator solutions in curved spaces and characterizes their mass conditions and finiteness properties, expanding understanding of the curved $n$-body problem.

Findings

Existence of polygonal positive and negative elliptic rotopulsators in $ extbf{H}^3$ and $ extbf{S}^3$

Masses must be equal for nonconstant size rotopulsators

Finiteness of negative and positive elliptic relative equilibria under size bounds

Abstract

We revisit polygonal positive elliptic rotopulsator solutions and polygonal negative elliptic rotopulsator solutions of the $n$-body problem in $\mathbb{H}^{3}$ and $\mathbb{S}^{3}$ and prove existence of these solutions, prove that the masses of these rotopulsators have to be equal if the rotopulsators are of nonconstant size and show that the number of negative elliptic relative equilibria of this type is finite, as is the number of positive elliptic relative equilibria if an upper bound on the size of the relative equilibrium is imposed. Additionally, we prove that a class of negative hyperbolic rotopulsators is in fact a subclass of the class of polygonal negative elliptic rotopulsators.