View source for Hamilton's principle/History/Noether 1918

Emmy Noether's ''Invariante Variationsprobleme'' was presented to the Göttingen
Scientific Society on 26 July 1918 and published in that year's ''Nachrichten'',
pages 235 to 257. It contains two theorems relating the symmetries of a
variational problem to the structure of its equations, and it settled a question
about energy conservation in general relativity that had occupied Göttingen for
two years.

== Provenance note ==

The paper is ESTABLISHED on [[Hamilton's principle/Sources]]: the full text was
obtained with its mathematics intact. But the text obtained is Tavel's 1971
English translation, which is under copyright, while the 1918 German original is
public domain.

So this page follows the volume's translation rule. Noether's '''equations''' are
reproduced elsewhere in her own notation; her '''prose is paraphrased''' here and
never transcribed. Nothing on this page is a quotation.

== The question ==

General relativity, published in 1915, made trouble for energy conservation. In
the older physics, energy conservation is a dynamical law with a definite local
content. In the new theory the quantity that plays the role of gravitational
energy turned out to be coordinate-dependent, and no satisfactory local
expression could be found.

Hilbert asserted that this failure was characteristic of general relativity
rather than a defect of anyone's construction. The claim was debated in Göttingen
through 1917 and 1918, with Klein and Einstein both involved. It was a claim
without a proof.

Noether had been brought to Göttingen by Hilbert and Klein in 1915, in the face of
faculty opposition to a woman lecturer, and was working on invariant theory. She
was well placed to answer a question about what invariance implies.

== What she proved ==

The paper contains two theorems, stated together in §1 and proved in the sections
following.

'''Theorem I''' treats invariance under a group with finitely many parameters. If
the action integral is invariant under a group with <math>\rho</math> parameters,
then <math>\rho</math> independent combinations of the Lagrange expressions are
divergences. Imposing the field equations turns those divergences into
conservation laws. This is the result universally known as Noether's theorem, and
it is developed on
[[Hamilton's principle/Mathematics/Noether's first theorem]].

'''Theorem II''' treats invariance under a group depending on arbitrary
functions. The conclusion is different in kind: instead of conservation laws,
such invariance produces '''identities''' among the field equations, holding
before any equation of motion is imposed. This is developed on
[[Hamilton's principle/Mathematics/Noether's second theorem]].

Both theorems carry converses, which Noether proves and which are usually
omitted when the first theorem is quoted.

== The answer to Hilbert ==

§6 is the section the paper was written for, and it is the least quoted.

Noether classifies divergence relations as '''proper''' or '''improper''',
according to whether the conserved quantity is genuinely dynamical or is built out
of the field equations and their derivatives — an identity in the costume of a
conservation law. She then proves that the energy relations of a theory are
improper exactly when the theory is invariant under an infinite group containing
the displacement group as a subgroup.

General relativity is such a theory. Its energy conservation is therefore
improper, and no local gravitational stress-energy tensor of the classical kind
exists. Hilbert's assertion is correct.

But Noether's result is stronger than his claim, and in a direction that
diminishes the role of relativity in it. The property belongs to '''any''' theory
invariant under a group of arbitrary functions. She remarks that for the assertion
to hold literally, the term general relativity would have to be read more broadly
than usual, extending to every such group. The content is group-theoretic; gravity
is one instance.

She closes by endorsing an observation of Klein's, that the word relativity as
physicists use it is replaceable by invariance with respect to a group.

== Why the second theorem was forgotten ==

Theorem I gives physicists a tool of enormous generality and is quoted constantly.
Theorem II tells them that a class of conservation laws they might have wanted does
not exist in the expected form, and is quoted rarely.

The imbalance misrepresents the paper. The two theorems are stated together,
proved by the same method, and the second is the one that answers the question
Noether was asked. A reader who knows only "symmetry implies conservation" has the
corollary and not the argument.

The transmission was slow in any case. Tavel's English translation appeared in
1971, fifty-three years after publication, and the results reached physics
gradually through intermediaries rather than through the paper itself.

== Relation to the rest of this volume ==

Noether's theorems explain results this volume derived piecemeal before reaching
them.

{| class="wikitable"
! earlier result !! which theorem explains it
|-
| conserved momentum for a cyclic coordinate, at [[Hamilton's principle/Mathematics/The Euler-Lagrange equation]] || Theorem I, translation group
|-
| the Beltrami identity, at [[Hamilton's principle/Mathematics/The Beltrami identity]] || Theorem I, time translation
|-
| Hamilton's seven classical integrals, at [[Hamilton's principle/History/Hamilton 1834 and 1835]] || Theorem I, translations and rotations
|-
| the string's energy density and flux, at [[Hamilton's principle/Applications/Continuum systems: the vibrating string]] || Theorem I, in one space dimension
|-
| charge conservation, at [[Hamilton's principle/Applications/Classical field theory]] || Theorem I, internal phase group
|-
| the singular Hessian of the electromagnetic Lagrangian, at [[Hamilton's principle/Applications/The electromagnetic Lagrangian]] || Theorem II, gauge group
|-
| the contracted Bianchi identities, at [[Hamilton's principle/Applications/The Einstein-Hilbert action]] || Theorem II, coordinate group
|-
| the reparametrisation identity for arc length, at [[Hamilton's principle/Applications/Geodesics]] || Theorem II, her own Weierstrass example
|}

Eight results, one paper. That concentration is the reason this volume treats the
1918 paper as a structural pivot rather than as a historical episode.

== Biographical note ==

Noether was not permitted a salaried position at Göttingen at the time she wrote
this paper, and lectured under Hilbert's name. She was dismissed from the
university in 1933 and emigrated to the United States, where she taught at Bryn
Mawr until her death in 1935.

These facts are recorded because they bear on the paper's transmission: work
published from an unsalaried position, by an author outside the ordinary academic
channels, in a journal read mainly by mathematicians, took decades to reach the
physicists it concerned.

<blockquote>
Provenance: nothing on this page is quoted. Noether's theorems and her §6 result
are paraphrased from the ESTABLISHED translation under the rule stated on
[[Hamilton's principle/Sources]]. The Göttingen debate, the dates, and the
biographical facts are editorial summary; Klein's and Hilbert's 1918 notes are not
sources of this volume and are not cited.
</blockquote>

== References ==

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