Noether's theorems/First theorem/The converse of the first theorem

Plain-language counterpart: 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 $\varrho$ linearly independent divergence relations of the form (N13) hold, then $I$ is invariant under a ${𝔊}_{\varrho }$.

The proof, which is the forward proof reversed

Step 1. Multiply the $\varrho$ relations (N13) by the parameters ${\epsilon }_{\lambda }$ and add. This gives (N12).

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

$\overline{\delta }f+\mathrm{D}\mathrm{i}\mathrm{v}(A-B)=0\mathrm{(}\mathrm{1}\mathrm{)}$

Step 3. Define

$\Delta x=\frac{1}{f}\cdot (A-B)\mathrm{(}\mathrm{2}\mathrm{)}$

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

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

Step 5. $\Delta I=0$ entails, in the usual way, invariance under the finite transformations produced by integrating the simultaneous system (N17). Those finite transformations contain $\varrho$ parameters, namely the products $t{\epsilon }_1,\dots ,t{\epsilon }_{\varrho }$.

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 $\partial u/\partial x$. 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 $\varrho$. Since $I$ admits that transformation too, there would be more than $\varrho$ linearly independent divergence relations, contrary to assumption. The only remaining possibility is that the transformation has the special form $\overline{\delta }u=0$, $\mathrm{D}\mathrm{i}\mathrm{v}(f\cdot \Delta x)=0$ — but then $\Delta x$ or $\Delta u$ would depend on derivatives, against the assumption.

Whether that case can occur when derivatives do appear in $\Delta x$ or $\Delta u$, Noether says must remain undecided. When it does, all functions $\Delta x$ with $\mathrm{D}\mathrm{i}\mathrm{v}(f\cdot \Delta x)=0$ must be added to the $\Delta x$ 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

The groups page assumed $\Delta x$ and $\Delta u$ linear in the $\epsilon$ and promised a demonstration. It is a corollary of the converse.

Suppose they were forms of higher degree. Because the power products of the $\epsilon$ are linearly independent, entirely corresponding relations (N13) would follow, only more of them. From those, by the converse, $I$ is invariant under a group whose infinitesimal transformations contain the parameters linearly. If that group is to contain exactly $\varrho$ 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 Sources.