Stability of Some Equilibrium Solutions of the Problem of Motion of a Gyrostat in an Incompressible Ideal Fluid

Kirchhoff’s equations for a gyrostat in a incompressible ideal fluid, have been written as a Lie-Poisson system, using a non-canonical Hamiltonian formulation. Next, we provide some equilibria of the problem, when the gyrostatic momentum is constant and adopts the form l = (0, 0, l). By means of the energy-Casimir method, we obtain sufficient conditions for the stability of some of these equilibria. Besides, by studying the linearized equations of motion, we offer the necessary conditions for the stability.