Analytic Five-Body Gerono Lemniscata Choreography

The trajectories and interaction forces of five bodies with equal masses are analytically described. The bodies follow a choreographic motion in a figure of eight, Gerono lemniscata periodic orbit without collisions. The forces are a linear combination of pairwise distances with either attractive or repulsive coefficients. The center of mass is fixed at the origin. The moment of inertia, as well as the kinetic and potential energies of the system, are constant. Notably, the angular momentum is zero just as in the Bernoulli Lemniscata parametrized by elliptic functions. The number of bodies in the choreography can be increased in a straightforward way. The orbit can be readily generalized to symmetric chains with an arbitrary number of loops.