Arnold Diffusion and the D'Alаmbert Precession Problem

A planet can be described by an homogeneous rigid ellipsoid with flatness $\eta$, moving on a Keplerian orbit around a star and subject only to Newtonian forces. It was proposed in 1994 in [2] that, for suitable initial data, the precession cone can change $O(1)$ in a finite time, no matter how small $\eta$ is, as a consequence of Arnold diffusion mechanism. One can start introducing some simplifications in the original model, neglecting a term in its Hamiltonian so that the problem is reduced to a priori unstable three time scale system; for such systems a general theory of Arnold diffusion can indeed be developed (mainly in [2], [8], [10], [11]). In this paper we will review the main results about Arnold diffusion in three time scale a priori unstable systems and we discuss their relevance for a complete understanding of the precession problem.