Bifurcations of MacLaurin spheroids. A Hamiltonian perspective

Dirichlet’s problem for the dynamics of fluid bodies with ellipsoidal shape can be formulated as a Hamiltonian system invariant under the action of a symmetry Lie group. I apply methods from Hamiltonian bifurcation theory to the study of the branch of solutions known as MacLaurin spheroids. I show that all its bifurcations are into three types named I, $S$ and adjoint $S$ ellipsoids in agreement with previous necessary conditions obtained by Chandrasekhar by linearizing the hydrodynamic equations.