Luca SBANO

Via C.Dossi 45 
00137- Roma
Italy
L.Sbano@mclink.it sbano@atmos.ifa.rm.cnr.it

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

Abstract : We consider the planar three-body problem (3BP) with a Newtonian-like potential of the form $\sum_{i\neq j}m_im_j \|x_i-x_j\|^{-\alpha}$ with $\alpha\geq 2$ and also the Newtonian case \alpha=1$. We study the Least Action principle on the manifold defined by the vanishing of the total angular momentum. Using the topology of the reduced configuration space we are able to find a class of trajectories on which a generalizationof the Poincaré inequality holds, then we can apply the direct method of calculus of variations. We prove the existence of a periodic solution: for the Newtonian potential ($\alpha=1$), this periodic solution has at most a finitenumber of collisions. For $\alpha\geq 2$ 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 $\Z *\Z$.

\vfill {\it To appear in NONLINEARITY}