Spontaneous symmetry breaking and bifurcations from the Maclaurin and Jacobi ellipsoids

The equilibrium of a rotating self-gravitating fluid is governed by non-linear equations. The equilibrium solutions, parametrized in terms of the angular momentum squared, exhibit the phenomenon of bifurcation, accompanied by spontaneous symmetry breaking. Under very general assumptions, a set of selection rules can be derived, which drastically restrict the patterns of symmetry breaking that are allowed to appear. Bifurcations of this kind are similar to second-order phase transitions à la Landau. The method is illustrated by the simple example of an incompressible fluid in rigid rotation. However, the selection rules are more general; they apply also to models which approximate a rotating star more realistically.