TheN-body problem, the braid group, and action-minimizing periodic solutions

A reduced periodic orbit is one which is periodic modulo a rigid motion. If such an orbit for the planar N-body problem is collision free then it represents a conjugacy class in the projective coloured braid group. Under a `strong force' assumption which excludes the original Newtonian potential we prove that in most conjugacy classes there is a collision-free reduced periodic solution to Newton's N-body equations. These are the classes that are `tied' in the sense of Gordon. We give explicit homological conditions which ensure that a class is tied. The method of proof is the direct method of the calculus of variations. For the three-body problem we obtain qualitative information regarding the shape of our solutions which leads to a partial symbolic dynamics.