Hamilton's principle/Mathematics/Noether's second theorem - Revision history

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	region and choose the <math>p</math> so that they and enough of their derivatives		region and choose the <math>p</math> so that they and enough of their derivatives
	vanish on the boundary. The divergence integrates to zero. The remaining integral		vanish on the boundary. The divergence integrates to zero. The remaining integral
	vanishes for arbitrary <math>p</math>, so by ~~the fundamental lemma of~~		vanishes for arbitrary <math>p</math>, so by [[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] the bracket itself
	[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] the bracket itself
	vanishes:		vanishes:

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	'''The Hessian is singular.''' If <math>\rho</math> combinations of the field		'''The Hessian is singular.''' If <math>\rho</math> combinations of the field
	equations are identities, the velocities cannot all be solved for in terms of the		equations are identities, the velocities cannot all be solved for in terms of the
	momenta. This is exactly the degeneracy that blocks ~~the Legendre transform on~~		momenta. This is exactly the degeneracy that blocks [[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]. Gauge invariance and
	[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]. Gauge invariance and
	Legendre degeneracy are the same fact seen from two sides.		Legendre degeneracy are the same fact seen from two sides.

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			{{DISPLAYTITLE:Noether's second theorem}}
	{{Plebeian\|Hamilton's principle/Explainers/When symmetry is local}}		{{Plebeian\|Hamilton's principle/Explainers/When symmetry is local}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/When symmetry is local}}

'''Noether's second theorem''' treats invariance under a group depending on
arbitrary '''functions''' rather than on finitely many parameters. The conclusion
is different in kind from the first theorem: instead of conservation laws, such
invariance produces '''identities''' among the field equations themselves. The
theorem is less often taught and it is the one Noether wrote the paper to prove.

The provenance rule of
[[Hamilton's principle/Mathematics/Noether's first theorem]] applies here too:
Noether's equations are reproduced, her English is paraphrased.

== Finite against infinite groups ==

[[File:OPL-HP-F-37.png|thumb|right|400px|Identities among field equations from a local symmetry.]]

Noether calls a group '''finite continuous''', <math>G_\rho</math>, when its
transformations depend analytically on <math>\rho</math> parameters, and
'''infinite continuous''', <math>G_{\infty\rho}</math>, when they depend on
<math>\rho</math> arbitrary functions <math>p(x)</math> and their derivatives.

The distinction is the difference between a rotation, the same everywhere, and a
coordinate change that may be chosen independently at each point. Physics calls
the second a '''local''' or '''gauge''' symmetry.

== Statement ==

'''Theorem II.''' If <math>I</math> is invariant under a
<math>G_{\infty\rho}</math> in which the arbitrary functions occur up to the
<math>\sigma</math>-th derivative, then there are <math>\rho</math> identity
relationships among the Lagrange expressions and their derivatives up to order
<math>\sigma</math>. The converse holds.

<blockquote>
Statement paraphrased from Noether 1918, §1, Theorem II.
</blockquote>

== Derivation ==

Begin from the same relation (7) of
[[Hamilton's principle/Mathematics/Noether's first theorem]],
<math>\sum\psi_i\,\delta u_i = \mathrm{Div}\,B</math>. Now
<math>\delta u</math> is linear in the arbitrary functions and their derivatives:

:<math>\sum \psi_i\, \delta u_i =
\sum_{\lambda,i} \psi_i \left( a_i^{(\lambda)} p^{(\lambda)}
+ b_i^{(\lambda)} \frac{\partial p^{(\lambda)}}{\partial x}
+ \cdots
+ c_i^{(\lambda)} \frac{\partial^\sigma p^{(\lambda)}}{\partial x^\sigma} \right)
\qquad\qquad\mathrm{(1)}</math>

Move every derivative off <math>p</math> by repeated partial integration, using

:<math>\varphi\, \frac{\partial^\tau p}{\partial x^\tau}
= (-1)^\tau\, \frac{\partial^\tau \varphi}{\partial x^\tau}\, p
\quad \text{mod divergences}
\qquad\qquad\mathrm{(2)}</math>

which gives

:<math>\sum \psi_i\, \delta u_i =
\sum_\lambda \left[ (a_i^{(\lambda)}\psi_i)
- \frac{\partial}{\partial x}(b_i^{(\lambda)}\psi_i)
+ \cdots
+ (-1)^\sigma \frac{\partial^\sigma}{\partial x^\sigma}
(c_i^{(\lambda)}\psi_i) \right] p^{(\lambda)}
+ \mathrm{Div}\,\Gamma
\qquad\qquad\mathrm{(3)}</math>

Combining with <math>\sum\psi_i\delta u_i = \mathrm{Div}\, B</math>, the bracket
multiplied by <math>p</math> equals <math>\mathrm{Div}(B - \Gamma)</math>.

'''The closing step is the arbitrariness of <math>p</math>.''' Integrate over any
region and choose the <math>p</math> so that they and enough of their derivatives
vanish on the boundary. The divergence integrates to zero. The remaining integral
vanishes for arbitrary <math>p</math>, so by the fundamental lemma of
[[Hamilton's principle/Mathematics/The fundamental lemma]] the bracket itself
vanishes:

:<math>\sum \left[ (a_i^{(\lambda)}\psi_i)
- \frac{\partial}{\partial x}(b_i^{(\lambda)}\psi_i)
+ \cdots
+ (-1)^\sigma \frac{\partial^\sigma}{\partial x^\sigma}
(c_i^{(\lambda)}\psi_i) \right] = 0,
\qquad \lambda = 1, \ldots, \rho
\qquad\qquad\mathrm{(4)}</math>

<math>\blacksquare</math>

'''Equation (4) holds identically''' — before any field equation is imposed. That
is the whole difference from the first theorem, where the corresponding relation
became useful only after setting <math>\psi_i = 0</math>.

== What an identity means ==

(4) says the Lagrange expressions are not independent. <math>\rho</math> of the
field equations follow from the rest. Three consequences:

'''The field equations do not determine the evolution.''' With
<math>\rho</math> redundancies, the initial data leave <math>\rho</math>
functions free for all time. That freedom is the gauge freedom, and it is not
ignorance about the system — it is a statement that distinct solutions describe
the same physics.

'''The Hessian is singular.''' If <math>\rho</math> combinations of the field
equations are identities, the velocities cannot all be solved for in terms of the
momenta. This is exactly the degeneracy that blocks the Legendre transform on
[[Hamilton's principle/Mathematics/The Legendre transform]]. Gauge invariance and
Legendre degeneracy are the same fact seen from two sides.

'''Counting must be done carefully.''' The naive count of degrees of freedom
overcounts by <math>\rho</math>, and again by <math>\rho</math> for the
constraints among the momenta.

== Noether's own examples ==

She gives the '''Weierstrass parametric representation''' as the simplest case:
an integral homogeneous of first order in the derivatives is invariant under
reparametrisation <math>y = p(x)</math>, one arbitrary function without
derivatives, and the corresponding identity is

:<math>\sum \psi_i\, \frac{du_i}{dx} = 0
\qquad\qquad\mathrm{(5)}</math>

a single linear relation among the Lagrange expressions. Any action written in
parametrised form, including the relativistic point particle, carries (5).

Her second example is '''general relativity''': the group of all coordinate
transformations <math>y_i = p_i(x)</math>, with the metric transforming as
induced, and the arbitrary functions entering to first derivative. The identities
are then the <math>n</math> relations among the Lagrange expressions and their
first derivatives — in modern terms the contracted Bianchi identities, which make
the Einstein tensor identically divergence free. This is worked on
[[Hamilton's principle/Applications/The Einstein-Hilbert action]].

== Proper and improper conservation laws ==

Noether's §6 is the sharpest part of the paper and the least known.

Suppose <math>I</math> is invariant under a <math>G_{\infty\rho}</math>, and take
a finite subgroup <math>G_\sigma</math> obtained by specialising the arbitrary
functions. The first theorem gives conservation laws for
<math>G_\sigma</math>. But those laws must be consequences of the identities (4),
since the identities already imply invariance under any such specialisation.

Noether therefore classifies divergence relations as '''improper''' when the
<math>B^{(\lambda)}</math> are built from the Lagrange expressions and their
derivatives, and '''proper''' otherwise. An improper conservation law holds for a
reason that has nothing to do with dynamics: it is an identity in disguise.

Specialise to the '''displacement group''' <math>\Delta x_i = \varepsilon_i</math>,
whose invariance means <math>f</math> contains no explicit <math>x</math>. The
resulting relations,

:<math>\sum \psi_i\, \frac{\partial u_i}{\partial x_\lambda}
= \mathrm{Div}\, B^{(\lambda)}
\qquad\qquad\mathrm{(6)}</math>

are the '''energy relationships''', and the <math>B^{(\lambda)}</math> are the
energy components. Noether's conclusion:

'''The energy relationships are improper if and only if <math>I</math> is
invariant under an infinite group containing the displacement group as a
subgroup.'''

<blockquote>
Statement paraphrased from Noether 1918, §6.
</blockquote>

Classical mechanics and special relativity carry no infinite group, so their
energy conservation is proper. General relativity does, so its energy
conservation is improper — an identity rather than a dynamical law.

== The Hilbert assertion ==

That result was the point of the paper. Hilbert had asserted that the failure of
proper energy conservation is characteristic of general relativity, and the claim
was contested in the Göttingen debates of 1916 to 1918. Noether's second theorem
settles it, and generalises it: the property is not special to relativity but
belongs to '''any''' theory invariant under a group of arbitrary functions.

She adds the observation 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 substance is group-theoretic, and the physical name is
incidental to it.

<blockquote>
Provenance: this section paraphrases Noether 1918, §6, and its framing of the
Hilbert assertion. Klein's two 1918 notes, which she cites and which carry the
other side of the exchange, are not sources of this volume and are not cited. The
Noether row of the survey on [[Hamilton's principle/Sources]] is ESTABLISHED.
</blockquote>

== Why the second theorem is the neglected one ==

The first theorem gives physicists a tool. The second tells them a class of
conservation laws they might have hoped for does not exist in the form expected.
The first is quoted constantly and the second rarely, and the imbalance
misrepresents the paper: the two theorems are stated together, proved together,
and the second is the one that answers the question Noether was asked.

== Next ==

[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]
closes the structure track by separating two principles that are routinely
conflated.

== References ==

<references />

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	The provenance rule of		The provenance rule of
	[[Hamilton's principle/Mathematics/Noether's first theorem]] applies here too:		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]] applies here too:
	Noether's equations are reproduced, her English is paraphrased.		Noether's equations are reproduced, her English is paraphrased.

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	Begin from the same relation (7) of		Begin from the same relation (7) of
	[[Hamilton's principle/Mathematics/Noether's first theorem]],		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]],
	<math>\sum\psi_i\,\delta u_i = \mathrm{Div}\,B</math>. Now		<math>\sum\psi_i\,\delta u_i = \mathrm{Div}\,B</math>. Now
	<math>\delta u</math> is linear in the arbitrary functions and their derivatives:		<math>\delta u</math> is linear in the arbitrary functions and their derivatives:
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	vanish on the boundary. The divergence integrates to zero. The remaining integral		vanish on the boundary. The divergence integrates to zero. The remaining integral
	vanishes for arbitrary <math>p</math>, so by the fundamental lemma of		vanishes for arbitrary <math>p</math>, so by the fundamental lemma of
	[[Hamilton's principle/Mathematics/The fundamental lemma]] the bracket itself		[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] the bracket itself
	vanishes:		vanishes:

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	equations are identities, the velocities cannot all be solved for in terms of the		equations are identities, the velocities cannot all be solved for in terms of the
	momenta. This is exactly the degeneracy that blocks the Legendre transform on		momenta. This is exactly the degeneracy that blocks the Legendre transform on
	[[Hamilton's principle/Mathematics/The Legendre transform]]. Gauge invariance and		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]. Gauge invariance and
	Legendre degeneracy are the same fact seen from two sides.		Legendre degeneracy are the same fact seen from two sides.

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	first derivatives — in modern terms the contracted Bianchi identities, which make		first derivatives — in modern terms the contracted Bianchi identities, which make
	the Einstein tensor identically divergence free. This is worked on		the Einstein tensor identically divergence free. This is worked on
	[[Hamilton's principle/Applications/The Einstein-Hilbert action]].		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]].

	== Proper and improper conservation laws ==		== Proper and improper conservation laws ==
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	Hilbert assertion. Klein's two 1918 notes, which she cites and which carry the		Hilbert assertion. Klein's two 1918 notes, which she cites and which carry the
	other side of the exchange, are not sources of this volume and are not cited. The		other side of the exchange, are not sources of this volume and are not cited. The
	Noether row of the survey on [[Hamilton's principle/Sources]] is ESTABLISHED.		Noether row of the survey on [[Hamilton's principle/Sources\|Sources]] is ESTABLISHED.
	</blockquote>		</blockquote>

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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]		[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]]
	closes the structure track by separating two principles that are routinely		closes the structure track by separating two principles that are routinely
	conflated.		conflated.