Choreographic 4-Body Orbits with Solely Attractive Linear Forces

Abstract The choreographic plane motion of four or more bodies with equal masses has been analytically described with a combination of attractive and repulsive linear forces. In this work, we describe families of choreographic solutions to the equations of motion in a plane with only attractive linear forces along the lines joining the bodies. The conditions that produce collisions between any of the four bodies can be isolated and thus avoided. The expressions of many relevant properties of the system can be obtained in closed form. The angular momentum kernel is evaluated in order to establish the set of choreographic solutions that satisfy the zero angular momentum condition. Depending on the initial parameters, there can be other symmetries and conserved quantities in addition to the energy, momentum and centre of mass. The masses and restoring force coefficients necessary conditions that allow choreographic behaviour are established. The curves admit a larger number of bodies. The analytic choreographic trajectories are similar to those obtained numerically under a Newtonian force. This all attractive forces scheme, albeit linear, moves us closer to the Newtonian gravitational paradigm.