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.
In this work we study a generalized integrable biparametric family of 4-D isotropic oscillators. This family allows to treat, in a unified way, oscillators defined by the potentials given by Hartmann and Quesne and other ring-shaped systems. Using the Liouville–Arnold theorem and the analysis of the momentum map in its critical points, we obtain a complete topological classification of the different invariant sets of the phase flow of this problem. By this topological study and the calculation of the action-angle variables we obtain the full classification of periodic and quasiperiodic orbits for this system.
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