Removing collision singularities from the action minimizers for the N– body problem whith free boundaries

For the planar and spatial N-body problems, it has been proved by Marchal and Chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions. This important result has been extended by Ferrario & Terracini to Newtonian-type problems and equivariant problems. It has also been used to construct many symmetric solutions for the N-body problem. In this paper we are interested in action minimizing solutions in function spaces with free boundaries. The function spaces are imposed with boundary conditions, such that every mass point starts and ends on two transversal proper subspaces of R d, d ≥ 2. We will prove that solutions for this minimizing problem with free boundaries are always free from collisions, including boundary collisions. This result can be used to construct certain classes of relative periodic solutions of the N-body problem. 1