The 1918 paper read in its own order

The paper runs to twenty-three pages in six numbered sections. This page follows its order and its notation, and describes its arguments without reproducing them; the derivations are in the first theorem and second theorem tracks.

Her notation

She writes f for the integrand and I for the integral, which physics now calls the Lagrangian density and the action. The dependent variables are u, the independent ones x. She calls the group with a finite number of parameters a endliche kontinuierliche Gruppe and writes it with a Fraktur G carrying the parameter count; the group depending on arbitrary functions is the unendliche kontinuierliche one, written with a subscript that marks it as infinite.

Two of her symbols need care.

ψ denotes what she calls the Lagrangesche Ausdrücke, the Lagrange expressions: the left-hand sides of the equations of the variational problem. Setting them to zero is what physics calls imposing the field equations. She keeps them as expressions, not equations, throughout — which is the whole reason her identities can hold off-shell.

She distinguishes three variations, not two. Δ is the total change in a quantity under a transformation that moves both the dependent and the independent variables. δ is the ordinary variation of the calculus of variations. And δ̄, a delta with a bar, is the change at fixed argument — what is left of Δ after the motion of the independent variables has been subtracted out. Her equation (9) is the relation between them, and it is the source of most sign errors in later derivations of her theorems. Anyone who writes her result with two symbols where she used three has already lost the distinction the proof turns on.

The notation page tabulates all of this against modern usage.

§1, the preliminaries and the two theorems

The first section defines the two kinds of group, sets up the integral and its first variation, and identifies the Lagrange expressions. It then states both theorems, in six lines each.

The first theorem: if the integral is invariant under a group with a finite number of essential parameters, then that many linearly independent combinations of the Lagrange expressions become divergences; and conversely, that condition implies the invariance. She notes it survives the limiting case of infinitely many parameters.

The second: if the integral is invariant under a group depending on arbitrary functions, in which those functions occur up to some order of derivative, then there exist that many identical relations among the Lagrange expressions and their derivatives up to that order; and here too the converse holds.

She then does something most later accounts omit. She says what happens on passing to the variational problem — that is, on setting the Lagrange expressions to zero. In one dimension the first theorem then yields first integrals; in several dimensions it yields the divergence equations that had recently begun to be called Erhaltungssätze, conservation laws. The second theorem says something different: that some of the Lagrange equations are a consequence of the rest.

Two examples follow immediately, before any proof. The Weierstrass parameter representation is the simplest case of the second theorem. General relativity is the other, and she names it as an instance rather than as the subject.

§2, the identities

The second section derives both results. It is the technical core and it is short.

She writes a general infinitesimal transformation, imposes invariance, converts the integration over the transformed region into an integration over the original region, introduces the barred variation at equation (9), and reaches a single identity, her equation (12), holding for every invariant integral. From that one identity both theorems follow by specialising the group: constant parameters give the divergence relations, arbitrary functions give the dependencies.

Note the order. She does not prove two theorems. She proves one identity and takes two readings of it. The theorems are distinct because their inputs are distinct, not because their derivations are.

§3 and §4, the converses

Two sections, one for each theorem, each running the argument of §2 backwards. The finite case is §3, the infinite case §4.

These sections are why the volume can say that a conservation law is evidence of a symmetry. Most textbook treatments prove one direction and stop. Noether proved both, and the harder direction is the one that gets dropped.

§5, invariance of the parts

A technical section, restricting to the case where the transformations involve no derivatives of the dependent variables and the new independent variables depend only on the old ones. Under that restriction the individual pieces of the identities are themselves invariant, and so are the first integrals.

§6, the Hilbert assertion

The final section, and the reason the paper was written.

She defines a divergence relation as uneigentlich, improper, when the conserved quantity can be assembled from the Lagrange expressions and their derivatives — an identity wearing the costume of a conservation law. All others are eigentlich, proper.

She then proves the criterion. The divergence relations belonging to a finite group are improper if and only if that group is a subgroup of an infinite group under which the integral is invariant. Specialising to the translation group gives Hilbert's claim: if the integral admits translations, its energy relations are improper exactly when it is invariant under an infinite group containing the translations.

She adds the remark that decides how the whole subject should be read. The property is group-theoretic. General relativity is one theory that has it. It is not the reason.

She closes by noting that Hilbert's assertion holds literally only if "general relativity" is understood more broadly than usual, and endorses Klein's suggestion that the physicists' term Relativität would be better replaced by invariance with respect to a group.

Provenance: the German original was read in full and is public domain. Section structure, notation and equation numbers are reported from it directly. Noether's prose is paraphrased and marked as paraphrased; no English wording of hers is transcribed. Where a German term of art appears above it is her own word, given because no English equivalent is standard. See Sources.