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			{{Noether's theorems map}}

	''Mathematical counterpart: [[Noether's theorems/First theorem/On-shell and off-shell\|On-shell and off-shell]].''		''Mathematical counterpart: [[Noether's theorems/First theorem/On-shell and off-shell\|On-shell and off-shell]].''
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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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{{DISPLAYTITLE:True everywhere against true for real motions}}

''Mathematical counterpart: [[Noether's theorems/First theorem/On-shell and off-shell|On-shell and off-shell]].''

[[File:NT-F-05.png|thumb|right|360px|alt=A large outlined region labelled all configurations, containing a smaller shaded region labelled solutions. The Noether identity is annotated as holding over the whole large region; the conservation law only over the inner one.|On-shell and off-shell as nested regions]]

Two statements have been in play in this volume and they are not the same
statement. This page separates them. It is the least glamorous page here and
probably the most useful.

== Two kinds of true ==

A company's books satisfy a rule: what the company owns equals what it owes plus
what the owners hold. That is true of every set of books ever written, including
fraudulent ones, badly kept ones, and ones for a company that does not exist. It
is true because of what the words mean.

The same company made a profit last year. That is also true. But it is true only
of this company, in this year. Change the facts and it stops being true.

The first is an '''identity'''. The second is a '''claim about the actual
case'''.

Noether's theorems produce the first kind. Conservation laws are the second kind.

== What the theorem actually delivers ==

The first theorem's output is an identity. It says a certain combination of
sensitivities — the leftovers from
[[Noether's theorems/Explainers/Nudging a path and seeing what changes|the nudging
page]] — flows without accumulating.

This is true of any configuration you care to name. A chain hanging in a shape no
chain would ever hang in. A planet on an orbit that violates gravity. The
statement holds because of how the scoring rule is built, not because of what the
system is doing.

Then you add one condition: the system is obeying its equations of motion. The
sensitivities go to zero, the left-hand side of the identity vanishes, and what is
left says the flow balances.

'''That is the conservation law, and it arrived at the second step, not the
first.'''

== The trade ==

The identity is true of everything and says less. The conservation law is true of
real motions only and says something you can measure.

Neither contains the other. You get the second from the first by adding physics.
You cannot get the first back from the second, because the second says nothing
about the configurations you threw away.

Physicists have names for the two situations. On the actual solutions is
'''on-shell'''. True regardless is '''off-shell'''. Noether has no such words;
she just says what she is doing, which is passing to the variational problem.

== Three reasons to keep them apart ==

'''The reverse direction needs the identity.''' Running the theorem backwards
starts from the identity and reconstructs the symmetry. Start from the
conservation law alone and there is no route back.

'''The non-isolated case needs the identity.''' When something outside is feeding
the system, the sensitivities are not zero — they equal whatever is being fed in.
That case is available only because the underlying statement never assumed
otherwise.

'''The second theorem never makes the crossing at all.''' Its output is an
identity too, and adding physics does not turn it into a conservation law. It
turns it into a statement about the equations themselves.

That third reason is the important one. A reader who has only ever met the first
theorem in its on-shell form has nowhere to put the second theorem, and will
reach for the nearest available shape — a conservation law — which is the wrong
one. Almost every mis-statement of the second theorem starts here.

== The habit worth forming ==

When you meet a claim in this subject, ask which kind it is. True of every
configuration, or true of the real one?

Noether was careful about this on every page. Most accounts of her are not, and
the carelessness is not visible in the formulas. It is visible only in what people
then believe the theorems say.

== How to tell which one you are being told ==

A practical test, for reading anyone's account of this subject.

Find the statement. Then ask: is this true of a configuration that breaks the laws
of physics?

If yes, it is an identity. If no, it is a claim about actual solutions.

Now check whether the author told you which. Most do not, because in the first
theorem's case the two look almost identical on the page — one has some terms on
the left, the other has zero there, and everything else is the same.

'''If an account never distinguishes them, it has silently chosen the second, and
its reader has no way to reach the second theorem from there.'''

== A note on the words ==

Physicists say "on-shell" for true of actual solutions and "off-shell" for true
regardless. The words come from an unrelated corner of particle physics and are
not intuitive.

Noether has no such words. She simply says what she is doing: she passes to the
variational problem, meaning she now assumes the system is doing what the scoring
rule says it should. Her way of putting it is clearer, and it is a small loss that
the modern terms replaced it.

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

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