Noether's theorems/Applications/Kelvin's circulation theorem

This page describes. It does not work its case, because doing so needs the Lagrangian description of a fluid, which this volume does not set up.

The statement

Take an ideal fluid — no viscosity, constant density — and mark out a closed loop of fluid particles. Let it move with the flow, so the loop deforms and drifts but always consists of the same particles.

The circulation around that loop, the integral of the velocity along it, does not change. Kelvin proved this in 1869.

Why it belongs here

UNVERIFIED, the identification that follows.

The symmetry responsible is stated in the literature to be relabelling of the fluid particles: which parcel you call the first, which the second, and so on, is a matter of bookkeeping, and the physics does not depend on it.

Note what kind of symmetry that is. You are not choosing a number. You are choosing a relabelling of every parcel — a whole pattern, one choice at every point. That is a free shape in the sense of the second theorem's page, not a knob.

So circulation conservation sits in the region between the two theorems, and it is the only case in this volume that does. This volume does not resolve which theorem it belongs to, because doing so requires the derivation it does not have.

What can be said without the derivation

The count is unusual, and that is the point. Ordinary conserved quantities come one per knob, and there are finitely many. Circulation is conserved for every loop you can draw — infinitely many conserved quantities. A finite symmetry group cannot produce that. A symmetry with a free shape can.

It marks a boundary of the standard picture. Anyone whose account of Noether's theorem is one conserved quantity per parameter has no way to accommodate this case, and will treat it as a separate result of fluid mechanics. It probably is not.

What would close the gap

Set up the fluid action in the particle-labelling variables, identify the relabelling group and the order to which its arbitrary functions enter, and run it through the machinery. That is a page of work this volume has not done, and the honest state of this page is a flag rather than a result.

Provenance: no source was reached and nothing is derived. The attribution of circulation conservation to relabelling symmetry is tagged UNVERIFIED. Kelvin's date is standard. See Sources.