Topologically Distinct Collision-Free Periodic Solutions for the \({N}\)-Center Problem

This work concerns the planar \({N}\)-center problem with homogeneous potential of degree \({-\alpha}\) (\({\alpha\in[1,2)}\)). The existence of infinitely many, topologically distinct, non-collision periodic solutions with a prescribed energy is proved. A notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive \({h}\) the existence of a classical periodic solution with energy \({h}\) for the \({N}\)-center problem with \({\alpha\in (1,2)}\) is proven. In case \({\alpha=1}\) a slightly different result is shown: it is the case that there is either a non-collision periodic solution or a collision-reflection solution. The results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions. The proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments.