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