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	''Mathematical counterpart: [[Noether's theorems/First theorem/The first theorem stated and proved\|The first theorem stated and proved]].''		''Mathematical counterpart: [[Noether's theorems/First theorem/The first theorem stated and proved\|The first theorem stated and proved]].''
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	''No mathematics is used on this page. The mathematical treatment is on the		''No mathematics is used on this page. The mathematical treatment is on the
	counterpart page named above.''		counterpart page named above.''

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Noether theorems volume, initial import (B-07)

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{{DISPLAYTITLE:Symmetry and conservation: the first theorem}}

''Mathematical counterpart: [[Noether's theorems/First theorem/The first theorem stated and proved|The first theorem stated and proved]].''

[[File:NT-F-01.png|thumb|right|360px|alt=An irregular closed region representing all field configurations, with a curve running through it marked as one symmetry orbit. Points along the curve are marked, one of them highlighted as a solution. The caption states that the action has the same value everywhere along the orbit.|A symmetry orbit through the space of configurations]]

This is the primer. If you read one page of this volume, read this one.

== The statement ==

Take a scoring rule for whole histories — an
[[Noether's theorems/Explainers/What an action is|action]]. Suppose there is a
smoothly adjustable change that leaves every score alone. Count its knobs.

Then for each knob there is a quantity that flows without accumulating anywhere.
And once the system is actually obeying its equations of motion, that quantity is
conserved: it has the same value at the end as at the beginning.

Turn a knob for time and you get energy. Turn one for place and you get momentum.
Turn one for direction and you get angular momentum.

== What is surprising about this ==

Nothing in the setup mentions energy. Nothing mentions conservation. You put in a
statement about the shape of a scoring rule and you get out a quantity that
nature keeps constant.

The three quantities above were known to be conserved for centuries before 1918.
Noether did not discover them. What she supplied was the reason — and the reason
turned out to be the same reason in all three cases, which nobody had known.

She also supplied the count. Three knobs for rotation, three conserved
quantities, and no fourth to look for.

== Two things the theorem does not say ==

'''It does not say every conserved quantity comes from a symmetry.''' Some do
not, and this volume has
[[Noether's theorems/Explainers/A conserved quantity Noether does not explain|a
page on one]].

'''It does not say conservation is the output.''' This is the point most
compressed accounts lose, and it is worth being slow about.

What the theorem gives you first is a statement true of '''every''' possible
history, including impossible ones. It says a certain combination flows without
piling up. That statement holds for a ball on a path it would never take.

Conservation arrives only when you add the extra condition that the system is
doing what physics says. That is a second step, Noether takes it separately and
visibly, and there is a
[[Noether's theorems/Explainers/True everywhere against true for real motions|whole
page]] on why keeping the two apart is worth the trouble.

== It works backwards too ==

Noether proved the reverse as well: find such a flowing quantity, and there must
be a symmetry behind it.

Most textbooks give one direction only. The reverse is the direction that does
real work, because it turns an observation into evidence. Something is conserved
and you do not know why? The theorem says look for the symmetry, and it is there.
See
[[Noether's theorems/Explainers/Reading the theorem backwards|Reading the theorem
backwards]].

== And there is a second theorem ==

The page you are reading covers one of Noether's two theorems.

The other one takes a different kind of symmetry — one that can be dialled
differently from place to place — and it does not give conservation laws at all.
It gives something stranger and, for modern physics, more important.

It is the theorem the paper was written to prove, and it is the one that usually
gets left out. See
[[Noether's theorems/Explainers/The second theorem, and why it is not the first|The
second theorem, and why it is not the first]].

== One slogan to be careful with ==

You will often see the theorem compressed to four words: symmetry implies
conservation.

It is a good slogan and it is not the theorem. It drops the identity that holds
before physics is imposed, it drops the reverse direction, and it leaves no room
at all for the second theorem. Use it to remember the shape. Do not use it to
decide what is true.

== One case, all the way through ==

A pendulum clock in a sealed room.

'''The scoring rule.''' At each instant the pendulum has a position and a speed.
The rule combines them into a number, and adding that number up over the whole
swing gives the score for that history.

'''The symmetry.''' Nothing in the rule mentions what time it is. Start the clock
now or in an hour, and the rule is the same rule. One knob: how far you shift the
start.

'''The count.''' One knob, so exactly one conserved quantity, and no second one to
hunt for.

'''The quantity.''' It turns out to be the sum of the energy of motion and the
energy of position — the total energy. Nobody put energy into the setup. It fell
out of the fact that the rule does not mention the time.

'''The check.''' Now open a window and let the room warm through the afternoon,
so that the air resistance changes as the day goes on. The rule now does mention
the time. The symmetry is gone, and so is the conservation law — the pendulum
really does lose energy, and the theorem correctly predicts that it may.

That last step is the one worth remembering. The theorem is not a guarantee that
energy is conserved. It is a statement about exactly when it is.

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

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