Loose ends in a strong force 3-body problem

TL;DR

This paper explores the geometric structure of a specific three-body problem with inverse cube force, analyzing the visibility of orbits near binary collisions through the lens of negatively curved surfaces.

Contribution

It introduces a geometric framework for understanding the 3-body problem's orbits as geodesics on a negatively curved surface with boundary, providing new insights into collision behavior.

Findings

Orbits correspond to geodesics on a complete, negatively curved surface.

Binary collisions are represented as the ends of the surface.

Visibility properties of the surface relate to the dynamics of the orbits.

Abstract

Up to symmetries, the orbits of three equal masses under an inverse cube force with zero angular momentum and constant moment of inertia can be reparametrized as the geodesics of a complete, negatively curved metric on a pair of pants. The ends of the pants represent binary collisions. Here we will examine the visibility properties of such negatively curved surfaces, allowing a description of orbits beginning or ending in binary collisions of this 3-body problem.