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{{DISPLAYTITLE:The converse of the first theorem}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Reading the theorem backwards|Reading the theorem backwards]].''

Noether's third section proves the converse for the finite group. Most
treatments omit it. It is the direction that makes a conservation law evidence of
anything.

== The statement ==

If <math>\varrho</math> linearly independent divergence relations of the form
(N13) hold, then <math>I</math> is invariant under a
<math>\mathfrak{G}_{\varrho}</math>.

== The proof, which is the forward proof reversed ==

'''Step 1.''' Multiply the <math>\varrho</math> relations (N13) by the parameters
<math>\varepsilon_{\lambda}</math> and add. This gives (N12).

'''Step 2.''' Apply the identity (N3), which holds for any variation. From (N12)
and (N3) together,

<math>\bar{\delta}f+\mathrm{Div}\,(A-B)=0\qquad\qquad\mathrm{(1)}</math>

'''Step 3.''' Define

<math>\Delta x=\frac{1}{f}\cdot(A-B)\qquad\qquad\mathrm{(2)}</math>

and (1) becomes (N11), the local invariance condition.

'''Step 4.''' Integrate (N11) back up: this gives (N7), that is
<math>\Delta I=0</math>. So <math>I</math> is invariant under the infinitesimal
transformation determined by <math>\Delta x</math> and <math>\Delta u</math>,
where <math>\Delta u</math> is fixed from <math>\Delta x</math> and
<math>\bar{\delta}u</math> by (N9), and both are linear in the parameters.

'''Step 5.''' <math>\Delta I=0</math> entails, in the usual way, invariance under
the finite transformations produced by integrating the simultaneous system
(N17). Those finite transformations contain <math>\varrho</math> parameters,
namely the products <math>t\varepsilon_{1},\dots,t\varepsilon_{\varrho}</math>.

== The group property, and the gap Noether leaves open ==

Step 5 gives transformations. It does not yet give a group, and the argument that
they form one is worth following because of where it stops.

Suppose the finite transformations contain no derivatives
<math>\partial u/\partial x</math>. Then they always form a group. If they did
not, at least one infinitesimal transformation arising from the Lie bracket
process would fail to be a linear combination of the other <math>\varrho</math>.
Since <math>I</math> admits that transformation too, there would be more than
<math>\varrho</math> linearly independent divergence relations, contrary to
assumption. The only remaining possibility is that the transformation has the
special form <math>\bar{\delta}u=0</math>,
<math>\mathrm{Div}\,(f\cdot\Delta x)=0</math> — but then <math>\Delta x</math> or
<math>\Delta u</math> would depend on derivatives, against the assumption.

'''Whether that case can occur when derivatives do appear in
<math>\Delta x</math> or <math>\Delta u</math>, Noether says must remain
undecided.''' When it does, all functions <math>\Delta x</math> with
<math>\mathrm{Div}\,(f\cdot\Delta x)=0</math> must be added to the
<math>\Delta x</math> determined above in order to recover the group property,
and by her convention the parameters so added are not counted.

This is an open point in the paper, marked as such by its author. This volume
does not close it and does not present the converse as unconditional.

== Linearity in the parameters was no restriction ==

[[Noether's theorems/First theorem/One-parameter groups and their generators|The
groups page]] assumed <math>\Delta x</math> and <math>\Delta u</math> linear in
the <math>\varepsilon</math> and promised a demonstration. It is a corollary of
the converse.

Suppose they were forms of higher degree. Because the power products of the
<math>\varepsilon</math> are linearly independent, entirely corresponding
relations (N13) would follow, only more of them. From those, by the converse,
<math>I</math> is invariant under a group whose infinitesimal transformations
contain the parameters linearly. If that group is to contain exactly
<math>\varrho</math> parameters, linear dependencies must hold among the
divergence relations originally obtained. Nothing is lost by assuming linearity
from the start.

== Why the converse matters ==

Without it, a conservation law is a consequence of a symmetry and nothing more.
With it, a conservation law is '''evidence''' of one: observe a conserved
quantity, and the theorem says the action has a corresponding invariance,
recoverable by (2).

That is the direction that does work in practice, and it is the direction
textbooks drop. The gap Noether marks above is the honest limit on how far it can
be pushed.

''Provenance: the whole argument is Noether's third section. Displays (1) and (2)
are hers, given as running text; this volume has numbered them. Her statement
that the derivative case must remain undecided is reported as her own reservation
and is not this volume's assessment. Reproduced in her notation from the
public-domain German original; her prose is paraphrased. See
[[Noether's theorems/Sources|Sources]].''

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