Symplectic structures and dynamics on vortex membranes

Details Symplectic structures and dynamics on vortex membranes Symplectic structures and dynamics on vortex membranes

2012-01-27

arxiv.org/abs/1201.5914v1

Mathematics

Symplectic Geometry

20 pages, to appear in Moscow Mathematical Journal

Scientific paper

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes of codimension 2 in any dimension, which generalizes the classical binormal, or vortex filament, equation in 3D. This framework also allows one to define the symplectic structures on the spaces of vortex sheets, which interpolate between the corresponding structures on vortex filaments and smooth vorticities.

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