Trifolium Rose as an Analytic Four-Body Choreography

Source abstract

Abstract The trajectories and interaction forces of four bodies with equal masses are analytically described. The bodies follow a choreographic motion in a trifolium rose periodic orbit without collisions. The center of mass is fixed at the origin. The nonzero angular momentum and the energy of the system are constant. The interaction forces are not Newtonian, but linear attractive and repulsive along the lines joining the bodies. Due to the linearity of the forces, the curve can be arbitrarily scaled in orthogonal directions. For equal scaling, several other quantities, such as the moment of inertia, as well as the kinetic and potential energies, are constant. This choreography is closely related to Pascal’s limaçon four-body choreography.