Stability of Hamiltonian relative equilibria in mechanical systems and the problem of Riemann ellipsoids

We will shortly review the theory of stability for Hamiltonian relative equilibria and present a method for testing the stability for mechanical systems of the form ``kinetic plus potential energy that allows singularities of the moment map associated to point symmetry transformations. This method relies on the particular form of the Hamiltonian and the fibered geometry of the phase space