SPONTANEOUS SYMMETRY BREAKING AND BIFURCATIONS FROM THE MACLAURIN AND JACOBI ELLIPSOIDS

The equilibrium of a rotating self-gravitating fluid is governed by nonlinear 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 a 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.