The Schubart orbits on the circle

TL;DR

This paper investigates the three-body problem on a circle with cotangent potential, constructing singular and symmetric periodic orbits, including Schubart orbits, using topological methods in phase space.

Contribution

It introduces the existence of Schubart orbits on the circle and constructs homothetic orbits ending in various singularities, expanding understanding of three-body dynamics on curved spaces.

Findings

Existence of homothetic orbits ending in singularities

Construction of Schubart orbits with two equal masses

Use of Wazewski set in phase space for proofs

Abstract

We consider the three body problem on $S^1$ under the cotangent potential. We first construct homothetic orbits ending in singularities, including total collision singularity and collision-antipodal singularity. Then certain symmetrical periodic orbits with two equal masses, called Schubart orbits, are shown to exist. The proof is based on the construction of a Wazewski set in the phase space.