Abstract
For a gyrostat in a incompressible ideal fluid, by writing Kirchhoff's equations as a Lie-Poisson system and using a non-canonical Hamiltonian formulation, we provide the expressions of the equilibria when the gyrostatic momentum is constant with the form I = (0, 0, l) and present necessary and sufficient conditions for the stability of some of them via the energy-Casimir method and the study of the linearized equations of the motion. Finally, using a recent geometric method introduced by Hanssmann and Van der Meer, we give a sufficient condition for the existence of a non-degenerate Hamiltonian Hopf bifurcation at those equilibria when the gyrostat is symmetric. (C) 2012 Elsevier B.V. All rights reserved.