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Bifurcation of liquid drops

The reduced energy momentum method of Simo et al. (1991) may be used to locate bifurcations of steady motions of conservative systems from symmetric steady motions. A family of functionals parametrized by the isotropy algebra can be used to characterize a symmetric relative equilibrium. A variation of the Equivariant Branching Lemma is used to construct a scalar equilibrium equation on an appropriate submanifold of the configuration manifold. This equation can be solved for the free parameter associated to the isotropy subalgebra to obtain stability and bifurcation results. Applications to three mechanical systems are considered: the heavy Lagrange top, the Riemann ellipsoids, and planar incompressible free boundary flow. The bifurcation analyses of the heavy top and the Riemann ellipsoids recapture the classical results of Routh and Riemann. The analysis of swirling planar flows establishes the existence of nonrigid steady motions of asymmetric fluid regions bifurcating from a rigidly rotating fluid disc and improves an existing formal stability result for the rigidly rotating disc. A common feature of these three examples is the existence of a range of angular velocities for which the symmetric steady motions are both (formally) orbitally and linearly stable and yet bifurcations to asymmetric steady motions occur.