Weak positive Poisson stability and Hamiltonian vector fields in mechanical systems

We consider mechanical systems whose configuration manifold is Q = G x S, where G is a compact Lie group and S is a smooth manifold. Under an additional assumption of symmetry, we show that the dynamics of the system over the phase space T*Q can be reduced to G x T*S. We then show that the component of the dynamics on G is weakly positively Poisson stable. We apply this result to analyze global attitude controllability of a spacecraft with two rotors.