On Periodic orbits of the Planar N-body Problem

By introducing a new coordinate system, we prove that there are abundant new periodic orbits near relative equilibrium solutions of the N-body problem. We consider only Lagrange relative equilibrium of the three-body problem and Euler-Moulton relative equilibrium of the N-body problem, although we believe that there are similar results for general relative equilibrium solutions. All of these periodic orbits lie on a 2d-dimensional central manifold of the planar N-body problem. Besides d one parameter family of periodic orbits which are well known as Lyapunov's orbits or Weinstein's orbits, we further prove that periodic orbits are unexpectedly abundant: generically the relative measure of the closure of the set of periodic orbits near relative equilibrium solutions on the 2d-dimensional central manifold is close to 1. These abundant periodic orbits are named Conley-Zender's orbits, since to find them is based on an extended result of Conley and Zender on the local existence result for periodic orbits near an elliptic equilibrium point of a Hamiltonian. In particular, the results provide some evidences to support the well known claim of Poincar\'{e} on the conjecture of periodic orbits of the N-body problem.