Esistenza di soluzioni espansive nel problema degli N corpi

The aim of this thesis is to study some recent analytical methods to prove the existence of expansive solutions in the Newtonian model of gravitation, which is also known as the classical N-body problem. The first chapter of the thesis deals with a method obtained by Ezequiel Maderna and Andrea Venturelli to prove the existence of hyperbolic motions in the N-body problem, given any initial configuration of the N bodies, any collisionless asymptotic velocity and any positive value of the energy. In particular, their approach consists in using a variational setting for the problem to build a special class of weak solutions, called viscosity solutions, of the Hamilton-Jacobi equation associated to Newton’s equations. The existence of partially hyperbolic motions is also proved in this chapter by analyzing an article published by Juan Manuel Burgos. The second chapter is based on a paper published by Alberto Boscaggin, Walter Dambrosio, Guglielmo Feltrin and Susanna Terracini and it treats a functional method to prove the existence of parabolic arcs in the perturbed Kepler problem, whose equation represents many significant problems in celestial mechanics. This approach consists in using a perturbative argument, after reformulating the problem in a proper functional space X. In the last chapter of the thesis, a new proof for the existence of hyperbolic solutions in the N-body problem is presented. In this case, the proof consists in applying the generalized Weierstrass Theorem on the space X, in order to obtain the existence of minimizers for a renormalized Lagrangian action.