The topology of the planar three-body problem with zero total angular momentum and the existence of periodic orbits

We consider the planar three-body problem (3BP) with a Newtonian-like potential of the form Sigma(i not equal j)m(i)m(j)\\x(i)-x(j)\\(-alpha) with alpha greater than or equal to 2 and also the Newtonian case alpha = 1. We study the least action principle on the manifold defined by the vanishing of the total angular momentum. Using the topology of the reduced configuration space we are able to find a class of trajectories on which a generalization of the Poincare inequality holds, this then enables us to apply the direct method of calculus of variations. We prove the existence of a periodic solution: for the Newtonian potential (alpha = 1), this periodic solution has at most a finite number of collisions. For alpha greater than or equal to 2 the solution is in the homotopic class of the trajectories that go through at least three different collinear configurations. This result is a direct consequence of the first homotopy group of the reduced configuration space without coincidence set, this group is isomorphic to the free-generated group Z * Z.