Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem

We consider an n-degrees of freedom Hamiltonian system near an elliptic equilibrium point. The system is put in normal form (up to an arbitrary order and with respect to some resonance module) and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form. A bound to the rate of diffusion is thus found, and by optimizing the order of normalization exponential estimates of Nekhoroshev's type are obtained. This provides explicit estimates for the stability properties of the elliptic point, and leads in some cases to "effective stability," i.e., stability up to finite but long times. An application to the stability of the triangular libration points in the spatial restricted three body is also given.