Regolarizzazione delle collisioni binarie nel problema degli N-corpi

We consider the dynamic of N point masses that interact through Newton's Law of Gravitation, known as N-body problem. We prove the existence of collision-free periodic or with fixed-end solutions, basing our proof upon variational arguments. Then we investigate the collisional trajectories, which correspond to a singularity of the equation of motion. In the framework of the Kepler problem and the circular, restricted, three-body problem, we present the regularization techniques known as Levi-Civita and Kustaanheimo-Stiefel transformation. The Levi-Civita transformation allows us to study the planar case, while the KS-transformation the spatial one. Both these methods are adapted to desingularize binary collisions. Finally we consider the spatial perturbed Kepler problem. We prove that, in the in the neighborhood of a binary collision singularity, the perturbed motion has the same basic properties as in the unperturbed case. Then we proceed with the regularization of the perturbed equation of motion through KS-transformation.