Abstract

We describe the coadjoint orbits of the group of volume preserving diffeomorphisms of R3 associated to the motion of closed vortex sheets in ideal 3D fluids. We show that these coadjoint orbits can be identified with nonlinear Grassmannians of compact surfaces enclosing a given volume and endowed with a closed 1-form describing the vorticity density. Goldin, Menikoff, and Sharp present in [1] the case of open vortex sheets (tubes/ribbons). If the vorticity density has a discrete period group and is nonvanishing, the vortex sheet is given by a surface of genus one fibered by its vortex lines over a circle. We determine the Hamilton equations for such vortex sheets relative to the Hamiltonian function suggested by Khesin in [2] and we prove that there are no stationary solutions having rotational symmetries. These coadjoint orbits are shown to be prequantizable if the period group of the 1-form and the volume enclosed by the surface satisfy an Onsager-Feynman relation. Joint work with Francois Gay-Balmaz, ENS Paris. [1] Goldin, G.A., Menikoff, R., Sharp, D.H., Quantum vortex configurations in three dimensions, Phys. Rev. Lett., 67 (1991), 3499-3502. [2] Khesin, B., Symplectic structures and dynamics on vortex membranes, Moscow Math. Journal 12 (2012), 413-434.