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Noether theorems volume, initial import (S-03)

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{{DISPLAYTITLE:The Noether identities}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Equations that were never independent|Equations that were never independent]].''

[[File:NT-F-13.png|thumb|right|360px|alt=Two groups of five bars representing field equations. On the left, without a local symmetry, all five are independent. On the right, one free function makes the fifth follow from the other four, shown by a curved arrow. A caption states the counting: one arbitrary function gives one identity and one equation that was 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 ==

<math>\varrho</math> relations, one for each arbitrary function, holding among
the Lagrange expressions and their derivatives up to order <math>\sigma</math>.
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
[[Noether's theorems/Second theorem/Groups depending on arbitrary functions|the
previous page]]: one arbitrary function, no derivatives of it, so
<math>\varrho=1</math> and <math>\sigma=0</math>. The theorem promises one
identity involving the Lagrange expressions and no derivatives of them, and what
comes out is

<math>\sum\psi_{i}\frac{du_{i}}{dx}=0\qquad\qquad\mathrm{(1)}</math>

a single linear relation among the <math>\psi_{i}</math> 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 <math>\varrho=n</math>, one arbitrary function for each independent
variable, and <math>\sigma=1</math>, since the induced transformations of the
metric coefficients contain the first derivatives of the arbitrary functions. The
theorem therefore promises <math>n</math> 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
[[Noether's theorems/History/Hilbert and Klein|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
<math>p(x)</math> to zero and the divergence relations (N13) appear; set the
<math>\varepsilon</math> 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
[[Noether's theorems/First theorem/On-shell and off-shell|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 [[Noether's theorems/Sources|Sources]].''

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