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	''Mathematical counterpart: [[Noether's theorems/First theorem/The converse of the first theorem\|The converse of the first theorem]].''		''Mathematical counterpart: [[Noether's theorems/First theorem/The converse of the first theorem\|The converse of the first theorem]].''
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NT>NoetherVolumeBuild: Noether theorems volume, initial import (B-10)

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Noether theorems volume, initial import (B-10)

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{{DISPLAYTITLE:Reading the theorem backwards}}

''Mathematical counterpart: [[Noether's theorems/First theorem/The converse of the first theorem|The converse of the first theorem]].''

Noether proved the theorem in both directions. Textbooks give one. This page is
about the other one.

== The forward and the reverse ==

Forward: there is a symmetry, therefore there is a conserved quantity.

Reverse: there is a conserved quantity, therefore there is a symmetry.

Both are in the 1918 paper. The forward direction gets a few lines; the reverse
gets a whole section, because it is harder.

== Why the reverse is the useful one ==

Footprints in snow.

The forward statement is: a person walked here, so there are footprints. True,
and not much use — you already knew about the person.

The reverse is: there are footprints, so a person walked here. That is the one you
use, because footprints are what you actually find.

Physics is in the same position. Conserved quantities are what turns up. Something
holds steady when you expected it to drift, and you want to know why.

The reverse direction says: there is a symmetry, it is not a coincidence, and here
is how to find it. Without the reverse, a conserved quantity is consistent with a
symmetry and evidence of nothing.

== How the proof goes ==

Backwards, and that is not a joke — it is the actual structure.

Noether takes the forward argument and runs each step in reverse. The conserved
flows come first, they are combined back into the single relation the forward
proof had produced, and from that relation the transformation is reconstructed
explicitly. Not shown to exist. Written down.

Then one more thing has to be checked, and it is not automatic: that the
transformations you have recovered form a proper family — that doing two of them
in a row lands you inside the family rather than outside it. Noether checks it.

== Where she stops ==

There is a case she cannot settle, and she says so.

When the transformation depends on speeds as well as positions — the situation on
[[Noether's theorems/Explainers/Why speed has to come along for the ride|the
prolongation page]] — one possibility remains that her argument does not
eliminate. Her text says the matter must remain undecided.

'''This volume reports that rather than smoothing it over.''' The reverse
direction is not unconditional, its author knew it, and a modern account that
states it flatly has added a certainty she declined to claim.

== One thing the reverse direction settles for free ==

Early in the paper Noether makes an assumption to keep things manageable: that
turning a knob twice as far does twice as much, at least for the smallest steps.
It looks like a real restriction.

The reverse direction shows it is not. If the knobs behaved otherwise, you would
get relations of the same kind, just more of them — and feeding those back through
the reverse direction returns you to a symmetry whose knobs behave simply after
all. Nothing was lost by assuming it.

This is a small point and a good example of what the reverse direction is for.
It answers a question about the setup that the forward direction cannot reach.

== A conserved quantity nobody had a reason for ==

The best illustration of why the reverse direction earns its keep is a planet.

A planet orbiting the sun conserves energy, because time is uniform. It conserves
angular momentum, because space has no preferred direction. Both of those were
understood.

It also conserves a third thing — a quantity that points along the long axis of the
ellipse and keeps the orbit from slowly rotating. This was noticed in the
eighteenth century and had no explanation for a very long time. There was no
obvious symmetry to blame. Rotations were already used up on angular momentum.

The reverse direction says there must be one anyway. And there is: a symmetry that
does not move the planet around in space at all, but mixes together the different
ways of describing where it is and how fast it is going. It is invisible if you
only look for symmetries that move things about a room.

'''That is the reverse direction doing exactly what it is for.''' Somebody found a
conserved quantity, could not account for it, and the theorem said: there is an
account, keep looking. There was.

The full story is on
[[Noether's theorems/Applications/The Runge-Lenz vector|the Runge-Lenz page]].

''No mathematics is used on this page. The mathematical treatment is on the
counterpart page named above.''

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