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NT>NoetherVolumeBuild: Noether theorems volume, initial import (F-07)

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Noether theorems volume, initial import (F-07)

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{{DISPLAYTITLE:The first theorem stated and proved}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Symmetry and conservation, the first theorem|Symmetry and conservation: the first theorem]].''

[[File:NT-F-01.png|thumb|right|360px|alt=An irregular closed region representing all field configurations, with a curve running through it marked as one symmetry orbit. Points along the curve are marked, one of them highlighted as a solution. The caption states that the action has the same value everywhere along the orbit.|A symmetry orbit through the space of configurations]]
[[File:NT-F-18.png|thumb|right|360px|alt=Three nested-width boxes. The narrowest is point mechanics with one independent variable; the middle is classical field theory with several; the widest is Noether's 1918 statement allowing any number of variables, derivatives of any order, and transformations mixing dependent and independent variables. A note says most textbooks give the middle level and call it the theorem.|The three levels of generality]]

Everything needed is now in place. The proof is short.

== The statement ==

'''Theorem I.''' If the integral <math>I</math> is invariant under a
<math>\mathfrak{G}_{\varrho}</math>, then <math>\varrho</math> linearly
independent combinations of the Lagrange expressions become divergences —
and conversely, the invariance of <math>I</math> under a
<math>\mathfrak{G}_{\varrho}</math> follows from that. The theorem also holds in
the limiting case of infinitely many parameters.

Three things to note before the proof.

'''It is a statement about divergences, not about conservation.''' The word
conservation does not appear. What appears is that certain combinations become
divergences, which is an identity.

'''The converse is part of the statement,''' not an afterthought. It is proved on
[[Noether's theorems/First theorem/The converse of the first theorem|its own page]].

'''The count is exact.''' <math>\varrho</math> parameters give
<math>\varrho</math> relations, and they are linearly independent.

== The proof ==

Take <math>\mathfrak{G}</math> to be a finite continuous
<math>\mathfrak{G}_{\varrho}</math>, and suppose (N1) holds.

'''Step 1. Eliminate the barred variation of the integrand.''' In (N11), express
<math>\bar{\delta}f</math> through the Lagrange expressions using the identity
(N3), which holds for any variation whatever and therefore for
<math>\bar{\delta}</math>. This gives

<math>\sum\psi_{i}\bar{\delta}u_{i}=\mathrm{Div}\,B,\qquad\left(B=A-f\cdot\Delta x\right)\qquad\qquad\mathrm{(N12)}</math>

Relation (N12) is an identity in all the arguments occurring, for every
invariant integral <math>I</math>. It is the form Noether was looking for of
Lie's differential equations for <math>I</math>. '''The theorem is now essentially
proved; what remains is to separate it by parameter.'''

'''Step 2. Split by parameter.''' By assumption <math>\Delta u</math> and
<math>\Delta x</math> are linear in the parameters
<math>\varepsilon_{1},\dots,\varepsilon_{\varrho}</math>. By (N9) the same holds
for <math>\bar{\delta}u</math> and its derivatives, and therefore
<math>A</math> and <math>B</math> are linear in the <math>\varepsilon</math> too.
Write

<math>B=B^{(1)}\varepsilon_{1}+\dots+B^{(\varrho)}\varepsilon_{\varrho};\qquad\bar{\delta}u=\bar{\delta}u^{(1)}\varepsilon_{1}+\dots+\bar{\delta}u^{(\varrho)}\varepsilon_{\varrho}\qquad\qquad\mathrm{(1)}</math>

where the <math>\bar{\delta}u^{(1)},\dots</math> are functions of
<math>x,\ u,\ \partial u/\partial x,\dots</math>. Since the
<math>\varepsilon</math> are independent, (N12) holds coefficient by
coefficient, and the <math>\varrho</math> divergence relations follow:

<math>\sum\psi_{i}\bar{\delta}u_{i}^{(1)}=\mathrm{Div}\,B^{(1)};\quad\dots\quad\sum\psi_{i}\bar{\delta}u_{i}^{(\varrho)}=\mathrm{Div}\,B^{(\varrho)}\qquad\qquad\mathrm{(N13)}</math>

'''Step 3. Linear independence.''' Suppose the <math>\varrho</math> combinations
were dependent. By (N9), <math>\bar{\delta}u=0</math> together with
<math>\Delta x=0</math> would force <math>\Delta u=0</math> and
<math>\Delta x=0</math>, that is, a dependence among the infinitesimal
transformations. No such dependence holds for any parameter value: if one did,
the group recovered from the infinitesimal transformations by integrating (N17)
would depend on fewer than <math>\varrho</math> essential parameters, contrary to
the definition of <math>\mathfrak{G}_{\varrho}</math>. The remaining possibility,
<math>\bar{\delta}u=0</math> with <math>\mathrm{Div}(f\Delta x)=0</math>, is
excluded by the same construction. These conclusions survive the limiting case of
infinitely many parameters.

The theorem is proved.

== What has and has not been obtained ==

'''Obtained:''' <math>\varrho</math> identities, (N13), holding at every point,
for all values of the arguments, whether or not anything satisfies a field
equation.

'''Not obtained:''' any conservation law. Conservation appears only on passing to
the variational problem, and that step is
[[Noether's theorems/First theorem/On-shell and off-shell|separate and explicit]].
The quantities <math>B^{(\lambda)}</math> that become the conserved currents are
identified on
[[Noether's theorems/First theorem/The Noether current and the Noether charge|the
current and charge page]].

Nothing in this proof used anything not established on the six preceding pages:
(N3) from the Lagrange-expressions page, (N9) and (N11) from the boundary-term
page, the linearity in <math>\varepsilon</math> and the recovery (N17) from the
groups page, and the definition of <math>\mathfrak{G}_{\varrho}</math> from the
same. The invariance condition (N1) is from the setup page and enters only
through (N11).

''Provenance: the theorem statement is a paraphrase of Noether's Satz I; no
English wording of hers is transcribed. (N12), (N13) and (N17) are her equations
(12), (13) and (17), and the display (1) above is hers, set unnumbered in the
original. All reproduced in her notation from the public-domain German original.
See [[Noether's theorems/Sources|Sources]].''

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