Variational formulation of the motion of an ideal fluid on the basis of gauge principle

TL;DR

This paper introduces a new variational formulation for ideal fluid flow based on gauge principles, deriving fundamental equations and conserved quantities, and highlighting the role of vorticity and topology in fluid dynamics.

Contribution

It presents a gauge-theoretic variational framework for ideal fluids, incorporating vorticity and topology, and clarifies the transformation between Lagrangian and Eulerian descriptions.

Findings

Derivation of Euler's equation from gauge-invariant action

Identification of conserved currents via Noether's theorem

Vorticity equation as a gauge field equation

Abstract

On the basis of gauge principle in the field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. A structure of rotation symmetry is equipped with a Lagrangian $\Lambda_A$ including vorticity, in addition to Lagrangians of translation symmetry. From the action principle, Euler's equation of motion is derived. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate $\ba$. It is shown that, with the translation symmetry alone, there is freedom in the transformation between the Lagrangian $\ba$-space and Eulerian $\bx$-space. The Lagrangian $\Lambda_A$ provides non-trivial topology of vorticity field and yields a source term of the helicity. The vorticity equation is derived as an equation of the gauge field. Present formulation provides a basis on which the transformation between the $\ba$ space and the $\bx$ space is determined uniquely.