Variational methods and Hamiltonian perturbation theory applied to the N-body problem: a theoretical and computational approach

This thesis is divided in two parts: the first part is dedicated to variational methods applied to the N-body problem, from both the theoretical and the computational point of view, while the second part concerns the Hamiltonian perturbation theory applied to the N-body problem with orbit crossing singularities. In the first part we start reviewing the Calculus of Variation technique and the proof of the existence of periodic orbits sharing the symmetry of Platonic polyhedra. With this basis, we first set up an algorithm to produce a list of periodic orbits, then we modify the system adding a central body of large mass. Here we are still able to find periodic solutions. Moreover, applying the Gamma-convergence theory, we describe the asymptotic behavior of such periodic orbits, as the central mass becomes larger and larger. Non-rigorous and rigorous numerical methods aimed to the computation of periodic orbits are outlined, and they are applied to the above case. In particular, we were able to provide a computer-assisted proof of the instability of the orbits with the symmetry of Platonic Polyhedra. Moreover, we use numerical methods in order to compute periodic orbits in a system of charged particles. In the second part, we study the relation between the solutions of the full circular restricted 3-body problem and the solutions of the averaged equations, even when the averaging principle can not be applied. In particular, taking into account many solutions of the full equations, all with the same initial conditions for the slow variables, numerical simulations show a relation between them and the solution of the equations of motion of the normal form, in the sense that the solutions obtained through the normal form give statistical information on the solutions of the full circular restricted 3-body problem.