On geometry of the Lagrangian description of ideal fluids

TL;DR

This paper explores the geometric and symplectic structures underlying the Euler equations for ideal fluids, revealing connections between vorticity, helicity, and contact geometry in three-dimensional flows.

Contribution

It introduces a symplectic framework for ideal fluid dynamics, linking vorticity, helicity, and contact structures within a unified geometric perspective.

Findings

Symplectic structure on RxM encodes vorticity and velocity fields.

Helicity conservation appears as a symplectic identity.

Contact hypersurfaces include streamlines and Bernoulli surfaces.

Abstract

The Euler equation for an inviscid, incompressible fluid in a three-dimensional domain M implies that the vorticity is a frozen-in field. This can be used to construct a symplectic structure on RxM. The normalized vorticity and the suspended velocity fields are Hamiltonian with the function t and the Bernoulli function, respectively. The symplectic structure incorporates the helicity conservation law as an identity. The infinitesimal dilation for the symplectic two-form can be interpreted as a current vector for the helicity. The symplectic dilation implies the existence of contact hypersurfaces. In particular, these include contact structures on the space of streamlines and the Bernoulli surfaces. The symplectic structure on RxM can be realized as symplectisations of these through the Euler equation.