Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

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.