A generalization of vortex lines

TL;DR

This paper generalizes the classical vortex line concepts in fluid dynamics using the theory of integral invariants, extending Helmholtz's theorems to more abstract mathematical frameworks.

Contribution

It introduces a generalized mathematical framework for vortex lines and tubes based on integral invariants, broadening the classical understanding in fluid dynamics.

Findings

Vortex lines can be viewed as integral surfaces of a 1-dimensional integrable distribution.

The generalized framework encompasses properties of vortex lines and tubes in a more abstract setting.

The approach links Helmholtz's theorems with Poincare and Cartan's theory of integral invariants.

Abstract

Helmholtz theorem states that, in ideal fluid, vortex lines move with the fluid. Another Helmholtz theorem adds that strength of a vortex tube is constant along the tube. The lines may be regarded as integral surfaces of a 1-dimensional integrable distribution (given by the vorticity 2-form). In general setting of theory of integral invariants, due to Poincare and Cartan, one can find $d$-dimensional integrable distribution whose integral surfaces show both properties of vortex lines: they move with (abstract) fluid and, for appropriate generalization of vortex tube, strength of the latter is constant along the tube.