Mathematics

We will describe some methods to get parabolic orbits, also called homoclinics to infinity, as well as quasi-periodic orbits for some singular Hamiltonian systems satisfying the strong force hypothesis. For parabolic orbits, we use some double approximatiom method, applying first classical variational methods to get heteroclinic orbits between two fixed points, then we let the points go to infinity to get parabolic orbits. As for quasi-periodic orbits, we get the solutions by applying variational methods to an associated second order elliptic P.D.E. on a torus using a generalized Poincare inequality, the we show that the obtained weak solutions correspond to regular quasi-periodic solutions of the Hamiltonian system. These are joint works with Pablo Padilla (UNAM, Mexico).