Helicity invariants in 3D : kinematical aspects

TL;DR

This paper explores the mathematical structures underlying helicity invariants in 3D fluid dynamics, revealing connections between Lagrangian and Eulerian conservation laws through symplectic and conformally symplectic forms.

Contribution

It introduces a novel framework linking helicity invariants to symplectic geometry and conformal structures, providing new insights into fluid invariants and their algebraic properties.

Findings

Helicity invariants correspond to closed three-forms depending on parameters.

Potential one-forms induce conformally symplectic structures on space-time.

Lagrangian and Eulerian conservation laws are connected via conformal Poisson algebra.

Abstract

Exact, degenerate two-forms on time-extended space R X M which are invariant under the unsteady, incompressible fluid motion on 3D region M are introduced. The equivalence class up to exact one-forms of each potential one-form 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 which depend on two discrete and a continuous parameter. Each two-form is extended to a symplectic form on R X M. The subclasses of potential one-forms giving rise to Eulerian helicity conservations is shown to result in conformally symplectic structures on R X 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 X M.