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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 with and also the Newtonian case . 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 Poincaré 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 , this periodic solution has at most a finite number of collisions. For 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 .