Helicity invariants in 3D: kinematical aspects

Abstract

Exact, degenerate two-forms Θ k = dθ k on time-extended space R × M which are invariant under the unsteady, incompressible fluid motion on threedimensional region M are introduced. The equivalence class up to exact one-forms of θ k is splitted by the velocity field. The components of this splitting corresponds to Lagrangian and Eulerian conservation laws for helicity densities. These are expressed as the closure of three-forms θ k ∧ Θ l which depend on two discrete and a continuous parameter. Each Θ k is extended to a symplectic form on R × M. The subclasses of θ k 's giving rise to Eulerian helicity conservations is shown to result in conformally symplectic structures on R × M. The connection between Lagrangian and Eulerian conservation laws for helicity is shown to be the same as the conformal equivalence of a Poisson bracket algebra to infinitely many local Lie algebra of functions on R × M.

Lagrangian description, symplectization, and Eulerian dynamics of incompressible fluids

“…The first part has already been studied in our previous works [7] and [8], which we will summarize more compactly in sections two to four. The main contribution of the present work is to analyze the symplectic structure for the existence of contact hypersurfaces and to construct contact structures therein.…”

“…Our aim is, first, to show that a construction of symplectic structure on the time extended space R × M of trajectories that makes use of the (time-dependent) Eulerian equations, as in the case of contact structure for steady flows, is possible and, second, to reduce the dynamics of the Lagrangian description to various threedimensional contact hypersurfaces of symplectic space at our disposal. The first part has already been studied in our previous works [7] and [8], which we will summarize more compactly in sections two to four. The main contribution of the present work is to analyze the symplectic structure for the existence of contact hypersurfaces and to construct contact structures therein.…”

“…With this remark, the following discussions can be extended to the Eulerian equations such as the equations describing the Boussinesq approximation to inhomogeneous Euler equations, equations of barotropic fluids, equations of nonrelativistic superconductivity, equations of ideal magnetohydrodynamics, and dynamo theory (see [8] and the references therein).…”