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	''Mathematical counterpart: [[Noether's theorems/Second theorem/The converse of the second theorem\|The converse of the second theorem]].''		''Mathematical counterpart: [[Noether's theorems/Second theorem/The converse of the second theorem\|The converse of the second theorem]].''
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{{DISPLAYTITLE:Reading the second theorem backwards}}

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

Noether proved both her theorems in both directions. This page is the reverse of
the second one, and it is the direction working physicists actually use.

== The reverse ==

Forward: the theory has a local symmetry, therefore its equations have relations
among them.

Reverse: the equations have relations among them, therefore the theory has a local
symmetry.

== Why the reverse is the working direction ==

You do not usually start by knowing a theory's symmetries. You start with
equations somebody wrote down, often for reasons that had nothing to do with
symmetry, and you want to know what you are holding.

So you count. How many equations, how many variables, and do the equations
determine everything? If you find that some of them follow from the others, the
reverse direction tells you what that means: there is a local symmetry in there,
and it is not an accident of how the equations were written.

This is a diagnosis, not a curiosity. Finding a redundancy tells you that some of
your variables are description rather than physics, that your equations will not
determine everything, and that you will need to make a choice — fix a convention —
before you can compute anything. All of that is forced, and the reverse direction
is what tells you it is forced rather than a mistake.

== A restriction that turns out not to be one ==

Before the reverse direction proper, Noether clears away an assumption she had
been making, and the argument for it is unusually pretty.

She had assumed the symmetry depends on your free shape in the simplest possible
way: draw twice the shape and you get twice the effect.

Suppose it did not. Suppose the effect involved the shape squared, or the shape
multiplied by how fast it varies.

Now do two of these transformations one after the other, and look at what comes
out. Terms appear that cannot be produced by any single free shape — they need
more shapes than you started with. But the family was supposed to be built from a
fixed number of free shapes, and now composing two of its members has produced
something outside it.

So either the family was never closed under composition, in which case it was not
the sort of family we are discussing, or the complicated dependence was not there
after all.

'''The simple dependence is not an assumption. It is forced by the requirement
that doing two symmetries in a row is a symmetry.'''

== Getting the whole symmetry back ==

The reverse direction proper runs the forward proof backwards, and it recovers the
smallest steps of the symmetry — the infinitesimal version from
[[Noether's theorems/Explainers/What a continuous symmetry is|the earlier page]].

Getting from those back to the full-sized transformations is where Noether leans
on Lie, and it is real work rather than a formality: a result of his guarantees
that a family of smallest steps of this kind always belongs to a genuine family of
full-sized transformations, and that every one of those is built up out of the
small ones.

She notes one wrinkle. Sometimes you have to let your free shape change as you
build the transformation up, rather than holding it fixed and repeating. When you
can hold it fixed, everything is simpler.

And, as with
[[Noether's theorems/Explainers/Reading the theorem backwards|the first theorem's
reverse]], there is a case involving symmetries that depend on speeds where extra
care is needed before the same conclusions can be drawn.

== One more assumption removed ==

Throughout, the free shapes were allowed to vary with position but not with the
quantities being described.

Noether closes the section by showing that this too costs nothing. Allow the
shapes to depend on the quantities as well and you get relations of the same kind,
just more of them — and running the reverse direction on those brings you back to
the case already handled.

The same argument settles the mixed case, where a theory has global knobs and
local shapes at once, and with that the classification is complete.

== How a symmetry gets discovered ==

The reverse direction is not only a diagnosis. It is sometimes how the symmetry is
found in the first place.

Somebody writes down equations for a new situation, often by analogy with
something already understood. They count: so many equations, so many quantities to
solve for. The count says the problem should be determined.

It is not. Solutions come in families, differing in ways no measurement
distinguishes.

At that point the reverse direction says something definite. This is not a
mistake in the equations and not a shortage of data. There is a local symmetry in
the theory, it is the reason the solutions come in families, and it can be found
by asking which changes to the description leave every prediction alone.

Several of the symmetries at the centre of modern physics were identified along
exactly this route, from a redundancy noticed first and explained afterwards.

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

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