Three-body planetary problem: study of KAM stability for the secular part of the Hamiltonian

The Hamiltonian representing the average over the mean--motion angles (i.e. the secular part) of the three--body planetary problem is considered. An efficient algorithm constructing invariant tori for the trajectories in phase space is provided. To give a possible practical application, we consider a toy--model including the main terms of the secular part of a hypothetic Sun--Jupiter--Saturn system having eccentricities and inclinations equal to $1/20$ of the true ones. The scheme of a KAM proof of the stability of the model is sketched. The proof is ``computer assisted''.