Hyperbolic Shirts fit a 4-body Problem

TL;DR

This paper investigates the curvature of geodesic flows in a 4-body problem with inverse cube potential, confirming a numerical conjecture and analyzing dynamics on specific invariant surfaces.

Contribution

It provides a detailed analysis of the curvature properties of the reduced space in the 4-body problem with inverse cube potential, confirming a prior numerical conjecture.

Findings

Curvature analysis of geodesic flow on collinear and parallelogram invariant surfaces.

Confirmation of a numerical conjecture regarding the 4-body problem.

Insights into the dynamical behavior of the system based on curvature properties.

Abstract

Consider the equal mass planar $4$-body problem with a potential corresponding to an inverse \textit{cube} force. The Jacobi-Maupertuis principle reparametrizes the dynamics as geodesics of a certain metric. We examine the curvature of this geodesic flow in the reduced space on the collinear and parallelogram invariant surfaces and derive some dynamical consequences. This proves a numerical conjecture of [4].