Helicity current as a symplectic dilation

TL;DR

This paper introduces a symplectic framework for analyzing the evolution of helicity in incompressible fluid flows, linking geometric structures to fluid dynamics and symmetries.

Contribution

It constructs a formal symplectic structure on the flow domain and relates helicity evolution to symplectic dilation, providing new geometric insights into fluid dynamics.

Findings

Helicity density evolution is expressed via divergence of Liouville vector field.

Inviscid flow leads to a helicity conservation law.

Symplectic dilation generates Hamiltonian automorphisms related to flow symmetries.

Abstract

A formal symplectic structure on RxM is constructed for the unsteady flow of an incompressible viscous fluid on a three dimensional domain M. The evolution equation for the helicity density is expressed via the divergence of the Liouville vector field that generates symplectic dilation. For an inviscid fluid this equation reduces to a conservation law. As an application the symplectic dilation is used to generate Hamiltonian automorphisms of the symplectic structure which are then related to the symmetries of the velocity field.