On the existence of collisionless equivariant minimizers for the classical n-body problem

We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space Lambda for the n-body problem (with potential of homogeneous degree -alpha, with alpha>0) which ensures that the restriction of the Lagrangian action A to the space Lambda(G) of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. In proving that local minima of Lambda(G) are free of collisions we develop an averaging technique based on Marchal's idea of replacing some of the point masses with suitable shapes (see [10]). As an application, several new orbits can be found with some appropriate choice of G. Furthermore, the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits.