Relative Equilibria in curved restricted 4-body problems

TL;DR

This paper investigates the existence and characteristics of special relative equilibrium configurations in curved 4-body problems on spherical and hyperbolic geometries, introducing new criteria and specific solution types.

Contribution

It provides new existence criteria for quadrilateral configurations and identifies specific relative equilibria types in restricted curved 4-body problems.

Findings

Existence of quadrilateral configurations on the sphere's equator.

Proof of square-like relative equilibria in a restricted 4-body problem.

Discussion of kite-shaped relative equilibria in a different restricted scenario.

Abstract

We consider the curved 4-body problems on spheres and hyperbolic spheres. After obtaining a criterion for the existence of quadrilateral configu- rations on the equator of the sphere, we study two restricted 4-body problems, one in which two masses are negligible, and another in which only one mass is negligible. In the former we prove the evidence square-like relative equilibria, whereas in the latter we discuss the existence of kite-shaped relative equilibria.