What "Noether's theorem" now means
The name is used by two communities and does not denote the same thing in both. The divergence is not a matter of emphasis. It is a difference in which theorem is meant, at what generality, and in which direction.
What a physicist means
Singular: Noether's theorem. It means the first theorem, in the field-theory special case, and it is usually stated in one line — every continuous symmetry of the action yields a conserved current.
Around that line, the standard presentation:
- proves the forward direction only;
- states it for first derivatives and for transformations that do not mix dependent and independent variables;
- includes divergence symmetries, which Bessel-Hagen added;
- often states it on-shell from the start, so the identity that holds before the field equations are imposed never appears;
- treats the second theorem, where it appears at all, under other names and usually without attribution to the same paper.
The theorem in this sense is a tool. It is applied several times per course, it works, and the presentation is adequate to what it is used for.
What a mathematician means
Plural: Noether's theorems. Two results about variational problems admitting continuous groups, at the generality of the 1918 paper — any number of independent variables, derivatives of arbitrary order, transformations mixing dependent and independent variables — with both converses proved.
In this sense the second theorem is the more interesting of the two. It is the origin of the identities among the field equations of a gauge system, of the constraint structure of such systems, and of the modern treatment of variational problems on jet bundles. The first theorem, from here, is the easier corollary.
Why they differ
The two readings descend from different routes.
The physicist's descends from the recovery of the 1950s and 1960s, which took from the paper what field theory needed at the time and left the rest. Once that version was in the textbooks it reproduced itself, because the next textbook was written from the last one and not from the paper.
The mathematician's descends through the calculus of variations and Lie theory, where the paper had always been read as what it is, and where the generality was never the obstacle it was for physics.
Neither community was wrong about its own subject. They stopped reading the same document.
One slogan to distrust
"Symmetry implies conservation" is not her statement of the first theorem. Her statement is that invariance of the integral makes a certain number of linearly independent combinations of the Lagrange expressions into divergences — an identity, holding whether or not anything satisfies the field equations. Conservation appears only on passing to the variational problem, which she does explicitly and separately. The slogan compresses an identity and its specialisation into one clause, drops the converse, and leaves no room at all for the second theorem.
It is a good slogan. It is not the theorem.
What has changed recently
The historical record has been repaired, and the repair has not yet reached the teaching.
Kosmann-Schwarzbach's book-length study of the theorems, their precursors and their reception appeared in French in 2004 and in English in 2011. The centenary in 2018 produced a body of work returning to the paper itself. The German original is now freely readable online, in a transcription proofread against the source, with its mathematics intact — which was not true for most of the period during which the textbook version formed.
A reader today can therefore do what almost no physicist between 1930 and 2000 could conveniently do: read the paper. This volume exists to make the difference between the paper and the slogan visible, and the shortest route to it is the paper read in its own order followed by the two theorems set side by side.
Provenance: the characterisation of Noether's own statement is read from the German original, which is public domain; her prose is paraphrased and no English wording of hers is transcribed. The account of current usage in the two communities is this volume's own assessment and is editorial, not sourced. See Sources.