Existence of global symmetries of divergence-free fields with first integrals

TL;DR

This paper investigates the relationship between symmetries and first integrals of divergence-free vector fields in three dimensions, with implications for plasma physics and magnetic confinement, including conditions for the existence of flux coordinates.

Contribution

It establishes conditions under which the converse of a Noether-type theorem holds for divergence-free fields, and provides new proofs for the existence of flux coordinates without symmetry assumptions.

Findings

Converse of Noether-type theorem holds on the toroidal region.

Quick proofs of flux coordinate existence in high generality.

No need for symmetry assumptions like MHS or quasi-symmetry.

Abstract

The relationship between symmetry fields and first integrals of divergence-free vector fields is explored in three dimensions in light of its relevance to plasma physics and magnetic confinement fusion. A Noether-type Theorem is known: for each such symmetry, there corresponds a first integral. The extent to which the converse is true is investigated. In doing so, a reformulation of this Noether-type Theorem is found for which the converse holds on what is called the toroidal region. Some consequences of the methods presented are quick proofs of the existence of flux coordinates for magnetic fields in high generality; without needing to assume a symmetry such as in the cases of magneto-hydrostatics (MHS) or quasi-symmetry.