Noether's theorems/Second theorem/The two theorems side by side

Plain-language counterpart: The two theorems compared.

The first and second theorems side by side

Every page in this volume that needs the distinction links here. The comparison below is read off the two finished proofs, not asserted.

The table

	Theorem I	Theorem II
Group	finite continuous, $\varrho$ constant essential parameters	infinite continuous, $\varrho$ arbitrary functions, entering to order $\sigma$
Output	$\varrho$ linearly independent combinations of the Lagrange expressions become divergences	$\varrho$ identical relations among the Lagrange expressions and their derivatives to order $\sigma$
Form of the output	$\sum {\psi }_i\overline{\delta }{u}_i^{(\lambda )}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}$	left side of (N16) $=0$
Where it holds	identically, in all arguments	identically, in all arguments
On setting ${\psi }_i=0$	becomes $\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}=0$, a conservation law	becomes the statement that $\varrho$ field equations follow from the rest
Conserved quantity	yes: the current $B^{(\lambda )}$, the charge its cross-section	none. Nothing flows
Converse	proved, section 3, with one case left undecided	proved, section 4
Where the proofs part	(N12) splits by equating coefficients of the $\epsilon$	(N12) cannot split; derivatives of $p$ are removed by (2) of S-02, then the boundary choice kills the divergence

The one place they diverge

Both proofs run identically from (N1) through (N9), (N11), (N3) and into

$\sum {\psi }_i\overline{\delta }{u}_i=\mathrm{D}\mathrm{i}\mathrm{v}B\mathrm{(}\mathrm{N}\mathrm{12}\mathrm{)}$

That is roughly nine tenths of the work, and it is shared. What follows depends on one property of $\overline{\delta }u$.

If $\overline{\delta }u$ is linear in constant parameters, the parameters are independent, (N12) holds coefficient by coefficient, and $\varrho$ divergence relations drop out.

If $\overline{\delta }u$ is linear in arbitrary functions and their derivatives, no such split exists, because $\partial p/\partial x$ is not independent of $p$. The derivatives must first be traded away against the coefficients, which generates a divergence; then the freedom to choose $p$ vanishing on the boundary kills the whole right-hand side, and what is left is zero rather than a divergence.

The right-hand side is the difference. One theorem ends with a divergence on the right, the other with nothing. Everything else follows from that.

Three errors this page exists to prevent

"The second theorem is the local version of the first." It is not a version of the first. Different input, different output, different consequence of imposing the field equations. The table's third and sixth rows are the check.

"The second theorem gives conservation laws for local symmetries." It gives identities. When a local symmetry does produce something that looks like a conservation law, that law is improper in the sense of the criterion page — its conserved quantity is assembled from the field equations and carries no information they did not.

"General relativity is why energy conservation is degenerate." The criterion is group-theoretic and biconditional. General relativity satisfies its condition. So does the logarithmic group Noether constructs with nothing physical attached. Her closing remark says this in as many words.

Which one is the paper about

Both, but not equally. Theorem I is stated first and is the one that entered physics. Theorem II is the one the paper was written to prove: it is the answer to the Göttingen question, it occupies the final section, and it is what Hilbert's assertion needed.

This volume's editorial position, stated as such: quoting only Theorem I gives the corollary without the argument. Readers are free to disagree, and the history track records how the other reading came to dominate.

Provenance: every row of the table is read off the two proofs in this volume, which are in turn read from the public-domain German original. The three errors and the closing position are this volume's own and are marked as editorial. See Sources.