The gyrostat with a fixed point in a Newtonian force field: Relative equilibria and stability

In this paper we consider the non-canonical Hamiltonian dynamics of a gyrostat with a fixed point in a Newtonian force field. By means of geometric-mechanics methods we study the relative equilibria of these systems for different approximations of the potential function. In particular, we obtain all the equilibria of a generalized Lagrange–Poisson problem under different potentials. we use the Energy-Casimir method to obtain sufficient conditions of stability of equilibria in complex problems of gyrostat dynamics. By means of this method and spectral stability analysis we have obtained necessary and sufficient conditions of stability for equilibria for any potential with axial symmetry U ( k 3 ) and a Newtonian potential U ( 3 ) . The advantages of the Energy-Casimir method, in opposition with the classical Lyapunov–Chetaev method in stability problems of gyrostat dynamics is clear and we illustrate it with several interesting examples.