Five-body choreography on the algebraic lemniscate is a potential motion

TL;DR

This paper extends the concept of choreographies on the algebraic lemniscate from three to five bodies, discovering new solutions with explicit potentials and demonstrating superintegrability.

Contribution

It introduces two new five-body choreographies on the algebraic lemniscate and derives their explicit pairwise interaction potentials, expanding the understanding of superintegrable systems.

Findings

Two new five-body choreographies on the algebraic lemniscate are found.

Explicit form of the interaction potentials for these choreographies is derived.

The systems are shown to be superintegrable with ten constants of motion.

Abstract

In a remarkable paper of 2003 by Fujiwara et al. \cite{Fujiwara2003}, a figure-eight three-body choreography on the algebraic lemniscate by Bernoulli was discovered. Such a choreography was found to be driven by the action of a pairwise potential $ V(r_{ij}) $, depending only on the mutual relative distances $r_{ij}, i,j=1,2,3$. In the present Letter we show that two different choreographies of five bodies on the same algebraic lemniscate exist and correspond to solutions of ten coupled Newton equations of motion with a pairwise interaction potential. For each choreography the explicit form of the potential is found and ten constants of motion are presented, thus, it is superintegrable.