The Motion of a Gyrostat in a Central Gravitational Field: Phase Portraits of an Integrable Case

Abstract In this paper we describe the Hamiltonian dynamics, in some invariant manifolds of the motion of a gyrostat in Newtonian interaction with a spherical rigid body. Considering a first integrable approximation of this roto-translatory problem, by means of Liouville-Arnold theorem and some specifics techniques, we obtained a complete topological classification of the phase flow associated to this system. The action-angle variables regions are obtained. These variables allow us to calculate the modified Keplerian elements of this problem useful to elaborate a perturbation theory. The results of this work have a direct application to the study of two body roto-translatory pro-blems where the rotation of one of them influences strongly in the orbital motion of the system. In particular, we can apply these results to binary asteroids.