Lie Poisson structures and dual-spin spacecraft

Although Lie-Poisson structures appear as far back as in Lie\´s work on "function groups" it is only recently that these structures have played a useful role in the investigation of the behavior of solutions of concrete dynamical systems. In this paper we determine the Lie-Poisson structure of rigid spacecraft carrying (motor driven or free-spinning) rotors. This class of spacecraft includes the dualspin spacecraft of interest in attitude acquisition problems. In the latter case a study of the critical point structure of the Hamiltonian restricted to a coadjoint orbit leads to well-known design conditions. We then introduce damping in the free-spinning rotors and investigate asymptotic behavior of solutions. The damped equations are clearly seen to be deformations of the corresponding Lie-Poisson equations. The problem of rigid spacecraft carrying both driven rotors and free-spinning rotors with damping is of much interest for the purposes of attitude acquisition. This problem is dealt with in a sequel to this paper [20].