Figure-eight choreographies of the equal mass three-body problem with Lennard-Jones-type potentials

TL;DR

This paper discovers numerous figure-eight choreographic solutions in a three-body system with Lennard-Jones-type potentials, revealing complex symmetric orbits that include near-homogeneous and highly curved trajectories.

Contribution

It introduces new figure-eight solutions for the three-body problem with Lennard-Jones-type interactions, expanding understanding of choreographies in such systems.

Findings

Multiple figure-eight solutions found via numerical search

Solutions include near-homogeneous and highly curved orbits

Orbits exhibit symmetry and complex shapes

Abstract

We report on figure-eight choreographic solutions to a system of three identical particles interacting through a potential of Lennard-Jones type, $1/r^{12}-1/r^6$ where $r$ is a distance between the particles. By numerical search, we found there are a multitude of such solutions. A series of them are close to a figure-eight solutions to a homogeneous system with no $1/r^{12}$ term in the potential. The rest are very different from them and have several points with large curvatures in their figure-eight orbits, at which particles are repelled. Here figure-eight choreographies are the periodic motion whose shape is symmetric in both horizontal and vertical axis, starting with an isosceles triangle configuration and going back to an isosceles triangle configuration with opposite direction through Euler configuration. Thus the lobe of this figure-eight may be complex shape and needs not to be convex.