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 for the n-body problem (with potential of homogeneous degree -, with >0) which ensures that the restriction of the Lagrangian action to the space ^G of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. In proving that local minima of ^G are free of collisions we develop an averaging technique based on Marchals 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. Mathematics Subject Classification (2000) 70F10, 70F16, 37C80, 70G75