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Noether theorems volume, initial import (A-10)

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{{DISPLAYTITLE: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
[[Noether's theorems/Second theorem/Groups depending on arbitrary functions|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 [[Noether's theorems/Sources|Sources]].''

Noether's theorems/Applications/Kelvin's circulation theorem - Revision history

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