Noether's theorems/First theorem/The first theorem stated and proved

Plain-language counterpart: Symmetry and conservation: the first theorem.

A symmetry orbit through the space of configurations

The three levels of generality

Everything needed is now in place. The proof is short.

The statement

Theorem I. If the integral $I$ is invariant under a ${𝔊}_{\varrho }$, then $\varrho$ linearly independent combinations of the Lagrange expressions become divergences — and conversely, the invariance of $I$ under a ${𝔊}_{\varrho }$ follows from that. The theorem also holds in the limiting case of infinitely many parameters.

Three things to note before the proof.

It is a statement about divergences, not about conservation. The word conservation does not appear. What appears is that certain combinations become divergences, which is an identity.

The converse is part of the statement, not an afterthought. It is proved on its own page.

The count is exact. $\varrho$ parameters give $\varrho$ relations, and they are linearly independent.

The proof

Take $𝔊$ to be a finite continuous ${𝔊}_{\varrho }$, and suppose (N1) holds.

Step 1. Eliminate the barred variation of the integrand. In (N11), express $\overline{\delta }f$ through the Lagrange expressions using the identity (N3), which holds for any variation whatever and therefore for $\overline{\delta }$. This gives

$\sum {\psi }_i\overline{\delta }{u}_i=\mathrm{D}\mathrm{i}\mathrm{v}B,(B=A-f\cdot \Delta x)\mathrm{(}\mathrm{N}\mathrm{12}\mathrm{)}$

Relation (N12) is an identity in all the arguments occurring, for every invariant integral $I$. It is the form Noether was looking for of Lie's differential equations for $I$. The theorem is now essentially proved; what remains is to separate it by parameter.

Step 2. Split by parameter. By assumption $\Delta u$ and $\Delta x$ are linear in the parameters ${\epsilon }_1,\dots ,{\epsilon }_{\varrho }$. By (N9) the same holds for $\overline{\delta }u$ and its derivatives, and therefore $A$ and $B$ are linear in the $\epsilon$ too. Write

$B=B^{(1)}{\epsilon }_1+\dots +B^{(\varrho )}{\epsilon }_{\varrho };\overline{\delta }u=\overline{\delta }{u}^{(1)}{\epsilon }_1+\dots +\overline{\delta }{u}^{(\varrho )}{\epsilon }_{\varrho }\mathrm{(}\mathrm{1}\mathrm{)}$

where the $\overline{\delta }{u}^{(1)},\dots$ are functions of $x,u,\partial u/\partial x,\dots$. Since the $\epsilon$ are independent, (N12) holds coefficient by coefficient, and the $\varrho$ divergence relations follow:

$\sum {\psi }_i\overline{\delta }{u}_i^{(1)}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(1)};\dots \sum {\psi }_i\overline{\delta }{u}_i^{(\varrho )}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\varrho )}\mathrm{(}\mathrm{N}\mathrm{13}\mathrm{)}$

Step 3. Linear independence. Suppose the $\varrho$ combinations were dependent. By (N9), $\overline{\delta }u=0$ together with $\Delta x=0$ would force $\Delta u=0$ and $\Delta x=0$, that is, a dependence among the infinitesimal transformations. No such dependence holds for any parameter value: if one did, the group recovered from the infinitesimal transformations by integrating (N17) would depend on fewer than $\varrho$ essential parameters, contrary to the definition of ${𝔊}_{\varrho }$. The remaining possibility, $\overline{\delta }u=0$ with $\mathrm{D}\mathrm{i}\mathrm{v}(f\Delta x)=0$, is excluded by the same construction. These conclusions survive the limiting case of infinitely many parameters.

The theorem is proved.

What has and has not been obtained

Obtained: $\varrho$ identities, (N13), holding at every point, for all values of the arguments, whether or not anything satisfies a field equation.

Not obtained: any conservation law. Conservation appears only on passing to the variational problem, and that step is separate and explicit. The quantities $B^{(\lambda )}$ that become the conserved currents are identified on the current and charge page.

Nothing in this proof used anything not established on the six preceding pages: (N3) from the Lagrange-expressions page, (N9) and (N11) from the boundary-term page, the linearity in $\epsilon$ and the recovery (N17) from the groups page, and the definition of ${𝔊}_{\varrho }$ from the same. The invariance condition (N1) is from the setup page and enters only through (N11).

Provenance: the theorem statement is a paraphrase of Noether's Satz I; no English wording of hers is transcribed. (N12), (N13) and (N17) are her equations (12), (13) and (17), and the display (1) above is hers, set unnumbered in the original. All reproduced in her notation from the public-domain German original. See Sources.