We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.

Abstract. The aim of the present paper is to study the dynamics of a dumbbell satellite moving in a gravity field generated by an oblate body considering the effect of the zonal harmonic parameter. We prove that the pass trajectory of the mass center of the system is periodic and different from the classical one when the effect of the zonal harmonic parameter is non zero. Moreover, we complete the classical theory showing that the equations of motion in the satellite approximation can be reduced to Beletsky's equation when the zonal harmonic parameter is zero. The main tool for proving these results is the Lindstedt-Poincare's technique.

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Abstract. We provide for a class of Hamiltonian systems in the actionangle variables sufficient conditions for showing the existence of periodic orbits. We expand this result to the study of the existence of periodic orbits of perturbed spatial Keplerian Hamiltonians with axial symmetry. Finally, we apply these general results for finding periodic orbits of the MateseWhitman Hamiltonian, of the spatial anisotropic Hamiltonian and of the spatial generalized van der Waals Hamiltonian.