Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits and subsequently bifurcating families. The Lyapunov families arise from the polygonal equilibrium of n bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases , 4, and 6-9. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link below.