THE N-BODY PROBLEM, THE BRAID GROUP, AND ACTION-MINIMIZING PERIODIC-SOLUTIONS

A reduced periodic orbit is one which is periodic module a rigid motio n. If such an orbit for the planar N-body problem is collision free th en it represents a conjugacy class in the projective coloured braid gr oup. Under a 'strong force' assumption which excludes the original 1/r Newtonian potential we prove that in most conjugacy classes there is a collision-free reduced periodic solution to Newton's N-body equation s. These are the classes that are 'tied' in the sense of Gordon. We gi ve 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 regardi ng the shape of our solutions which leads to a partial symbolic dynami cs.