Rectangular orbits of the curved 4-body problem

TL;DR

This paper investigates special rectangular solutions in the curved 4-body problem, proving that such orbits are necessarily squares on spheres or hyperbolic spaces, expanding understanding of relative equilibria in curved spaces.

Contribution

It establishes that rectangular orbits in the curved 4-body problem are necessarily squares, providing a classification of these special solutions in spherical and hyperbolic geometries.

Findings

Rectangular orbits are necessarily squares in curved spaces.

Such solutions exist as relative equilibria and rotopulsators.

The results apply to both spherical and hyperbolic geometries.

Abstract

We consider the 4-body problem in spaces of constant curvature and study the existence of spherical and hyperbolic rectangular solutions, i.e. equiangular quadrilateral motions on spheres and hyperbolic spheres. We focus on relative equilibria (orbits that maintain constant mutual distances) and rotopulsators (configurations that rotate and change size, but preserve equiangularity). We prove that when such orbits exist, they are necessarily spherical or hyperbolic squares, i.e. equiangular equilateral quadrilaterals.