Noether's theorems/Second theorem/The Noether identities

Plain-language counterpart: Equations that were never independent.

Identities among the field equations from a local symmetry

The relations (N16) are what later literature calls the Noether identities. This page says what kind of object they are and works the two cases Noether gives.

What they assert

$\varrho$ relations, one for each arbitrary function, holding among the Lagrange expressions and their derivatives up to order $\sigma$. They are identities in all the arguments — the independent variables, the dependent variables, and every derivative appearing.

Compare the output of the first theorem, (N13). Both are identities. But (N13) equates a combination of Lagrange expressions to a divergence, which is a statement that something flows. (N16) equates a combination of Lagrange expressions to zero, which is a statement that the Lagrange expressions are not free of one another.

Nothing flows. There is no current here and no charge. A reader who arrives expecting the second theorem to produce a conserved quantity has been mis-prepared by the presentation of the first.

The Weierstrass case, worked

The simplest instance, from the previous page: one arbitrary function, no derivatives of it, so $\varrho =1$ and $\sigma =0$. The theorem promises one identity involving the Lagrange expressions and no derivatives of them, and what comes out is

$\sum {\psi }_i\frac{d{u}_i}{dx}=0\mathrm{(}\mathrm{1}\mathrm{)}$

a single linear relation among the ${\psi }_i$ themselves. The count matches: one function in, one identity out, at order zero.

The practical reading: a first-order homogeneous integral cannot determine its own parametrisation, so one of its Lagrange equations is redundant. That is not a defect of the formulation. It is the price of writing a curve without committing to how it is traced.

The general-relativity case

Here $\varrho =n$, one arbitrary function for each independent variable, and $\sigma =1$, since the induced transformations of the metric coefficients contain the first derivatives of the arbitrary functions. The theorem therefore promises $n$ identities among the Lagrange expressions and their first derivatives, and those are the dependencies already known in the literature by 1918.

The route by which they were known matters for the history. Hilbert had asserted them for a particular invariant, and Klein had derived them in general form by the method of infinitesimal transformations, both before Noether's paper. Hilbert's own 1924 reprint records that division of credit, and the history track gives the detail. What Noether supplied was the theorem of which they are one instance.

Mixed groups

For a group depending on both arbitrary functions and parameters, set the $p(x)$ to zero and the divergence relations (N13) appear; set the $\epsilon$ to zero and the dependencies (N16) appear. Both hold, and they are independent of one another. Noether states this in one sentence at the end of her second section, and it is the cleanest statement anywhere that the two theorems are not competing readings of one result.

The name

Noether calls them dependencies, or identical relations. The term Noether identities is later. It is used in this volume because it is standard, and is recorded here as an anachronism, as "on-shell" is on its page.

Provenance: (1) and the mixed-group statement are Noether's, from her first and second sections. The account of Hilbert's and Klein's priority for the general-relativity dependencies rests on Hilbert's 1924 reprint, which was not reached in full text and is summarised from the secondary literature. Her prose is paraphrased. See Sources.