ROTATING LIQUID DROPS: HAMILTONIAN STRUCTURE, STABILITY AND BIFURCATION

ncompressible, inviscid, free boundary fluid flows are Hamiltonian systems, i.e., the evolution of the flow is determined by a Poisson bracket and a Hamiltonian function. The Poisson structure for such flows generalizes a previous structure of Zakharov for irrotational free boundary flow and a structure of Arnold for incompressible flow in a region with fixed boundary. The Poisson bracket is determined via reduction from canonical variables in the Lagrangian (material) description. The configuration space for the flow may be regarded as a principal bundle and the terms of the bracket identified with various bundle constructions. A generalized Poisson bracket is computed; for two dimensional flows, the generalized enstrophy is a Casimir function with respect to this bracket.;A rigidly rotating plane circular liquid drop of radius R, with surface tension coefficient ;A functional appropriate to the stability analysis is shown to possess a strong minimum among the class of maps satisfying certain integral constraints. In general, given an integral functional...