Reduction, Relative Equilibria and Stability for a Gyrostat in the N-body Problem

We consider the non-canonical Hamiltonian dynamics of a gyrostat in the nbody problem. Using the symmetries of the system we carry out a reduction process in two steps, giving explicitly at each step the Poisson structure of the reduced system. Next, we obtain general properties of the relative equilibria of the problem and if we restrict to different approximations of the gravitational potential function, some particular cases are studied and, by means of energy-Casimir and spectral methods, sufficient and necessary conditions of stability can be obtained. We extend some results by Fanny and Badaoui (1998) and by Mondéjar, Vigueras and Ferrer (2001). In [3] the case of a rigid body in the three-body problem, in terms of the global variables in the unreduced problem, was considered, and in [12] the case of a gyrostat in the three-body problem was studied, but working now in the reduced system; in this way natural simplifications in the conditions of the equilibria appear. As a particular case, the problem of three bodies is considered for an arbitrary approximation of the potential function and the conditions for existence of Eulerian and Lagrangian equilibria are given in different cases. Here, as in [12], we use geometric methods, developed in part by Marsden, and others (see [6], [7], [8], [9] and [10]).