Lagrangian description, symplectization, and Eulerian dynamics of incompressible fluids

Authors: HASAN GÜMRAL

Abstract: Eulerian dynamical equations in a three-dimensional domain are used to construct a formal symplectic structure on time-extended space. Symmetries, invariants, and conservation laws are related to this geometric structure. The symplectic structure incorporates dynamics of helicities as identities. The generator of the infinitesimal dilation for symplectic two-form can be interpreted as a current vector for helicity. Symplectic dilation implies the existence of contact hypersurfaces. In particular, these include contact structures on the space of streamlines and on the Bernoulli surfaces.

Keywords: Incompressible fluid, symplectic and contact structures, symplectic dilation, helicity conservation, Lagrangian description

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