Joint work with F. Gabern

In this talk we will focus on the dynamics near the Lagrangian points of the Sun-Jupiter system. To try to account for the effect of Saturn, we will develop a specific model based on the computation of a true solution of the planar three-body problem for Sun, Jupiter and Saturn, close to the real motion of these three bodies. Then, we will derive the equations of motion of a fourth infinitesimal particle moving under the attraction of these three masses. Using suitable coordinates, the model will be written as a time-dependent perturbation of the well-known spatial Restricted Three-Body Problem.

Next, we will discuss some techniques to study these four body models. They are based on computing, up to high order, suitable normal forms and first integrals. From these expansions, it is not difficult to derive approximations to invariant tori (of dimensions 2, 3 and 4) as well as bounds on the speed of diffusion on suitable domains.