Variational formulation of ideal fluid flows according to gauge principle

TL;DR

This paper introduces a gauge principle-based variational formulation for ideal fluid flows, incorporating symmetries of translation and rotation, leading to new insights into vorticity, helicity, and the topology of fluid fields.

Contribution

It develops a novel gauge-theoretic variational framework for ideal fluids that unifies translational and rotational symmetries, extending traditional formulations and revealing topological aspects.

Findings

Derives Euler's equation from a gauge-invariant action principle.

Shows how the Lagrangian $\Lambda_A$ introduces non-vanishing helicity.

Provides a consistent transformation between Lagrangian and Eulerian descriptions.

Abstract

On the basis of the gauge principle of 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. The rotational transformations are regarded as gauge transformations as well as the translational ones. In addition to the Lagrangians representing the translation symmetry, a structure of rotation symmetry is equipped with a Lagrangian $\Lambda_A$ including the vorticity and a vector potential bilinearly. Euler's equation of motion is derived from variations according to the action principle. 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 $\ba$. Without $\Lambda_A$, the action principle results in the Clebsch solution with vanishing helicity. The Lagrangian $\Lambda_A$ yields non-vanishing vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an equation of the gauge field, and the $\Lambda_A$ characterizes topology of the field. The present formulation is comprehensive and provides a consistent basis for a unique transformation between the Lagrangian $\ba$ space and the Eulerian $\bx$ space. In contrast, with translation symmetry alone, there is an arbitrariness in the ransformation between these spaces.