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

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

2026-09-16T04:58:00Z

Pipe internal links so cross-references read as titles, not paths

← Older revision		Revision as of 04:58, 16 September 2026
Line 17:		Line 17:
	translation still under copyright, so her prose is '''paraphrased''' throughout		translation still under copyright, so her prose is '''paraphrased''' throughout
	this page and never transcribed. Where a phrase of hers is characterised, it is		this page and never transcribed. Where a phrase of hers is characterised, it is
	characterised, not quoted. See [[Hamilton's principle/Sources]].		characterised, not quoted. See [[Hamilton's principle/Sources\|Sources]].

	== Setup and notation ==		== Setup and notation ==
Line 30:		Line 30:
	She writes <math>\psi_i</math> for the '''Lagrange expressions''' — the left-hand		She writes <math>\psi_i</math> for the '''Lagrange expressions''' — the left-hand
	sides of the Euler-Lagrange equations of		sides of the Euler-Lagrange equations of
	[[Hamilton's principle/Mathematics/Several independent variables]] — so that the		[[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]] — so that the
	first variation is		first variation is

Line 46:		Line 46:
	<math>\delta u</math> and its derivatives. Equation (3) holds for any		<math>\delta u</math> and its derivatives. Equation (3) holds for any
	<math>f</math> and any variation; it is the partial-integration identity of		<math>f</math> and any variation; it is the partial-integration identity of
	[[Hamilton's principle/Mathematics/Several independent variables]] written		[[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]] written
	without discarding anything.		without discarding anything.

Line 121:		Line 121:
	<math>B^{(k)} = \text{const}</math> and gives <math>\rho</math> first integrals.		<math>B^{(k)} = \text{const}</math> and gives <math>\rho</math> first integrals.
	That recovers the results of		That recovers the results of
	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]] and		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]] and
	[[Hamilton's principle/Mathematics/The Beltrami identity]] as the		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] as the
	one-dimensional case of a single theorem.		one-dimensional case of a single theorem.

Line 163:		Line 163:
	says so explicitly, and gives an example where the first integrals obtained		says so explicitly, and gives an example where the first integrals obtained
	satisfy a nonlinear dependency. The free particle of		satisfy a nonlinear dependency. The free particle of
	[[Hamilton's principle/Mathematics/Several dependent variables]] is the		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] is the
	elementary case: energy conservation there follows from momentum conservation and		elementary case: energy conservation there follows from momentum conservation and
	is not new information.		is not new information.
Line 180:		Line 180:
	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Noether's second theorem]] treats the		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]] treats the
	infinite-parameter case and its consequence for general relativity.		infinite-parameter case and its consequence for general relativity.

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

2026-09-16T04:46:40Z

Add DISPLAYTITLE: title the page by its section rather than its path

← Older revision		Revision as of 04:46, 16 September 2026
Line 1:		Line 1:
			{{DISPLAYTITLE:Noether's first theorem}}
	{{Plebeian\|Hamilton's principle/Explainers/Symmetry and conservation}}		{{Plebeian\|Hamilton's principle/Explainers/Symmetry and conservation}}

Line 185:		Line 186:

	<references />		<references />

BFP Rauch: 1 revision imported

2026-09-09T05:05:15Z

1 revision imported

← Older revision	Revision as of 05:05, 9 September 2026
(No difference)

hamiltons-principle>HPVBuild: Initial import of M3-05

2026-09-08T12:00:00Z

Initial import of M3-05

New page

{{Plebeian|Hamilton's principle/Explainers/Symmetry and conservation}}

'''Noether's first theorem''' states that a continuous symmetry of an action
integral produces a conserved quantity, one for each parameter of the symmetry
group. It subsumes every first integral found piecemeal in this volume so far,
and it explains why those integrals existed rather than merely recording that
they did.

== Provenance note ==

[[File:OPL-HP-F-09.png|thumb|right|400px|Symmetry to conservation, four examples.]]

Noether's 1918 paper is a verified source for this volume, but her mathematics
and her prose stand differently. The equations below are her own, from the 1918
original, and are reproduced in her notation. The English wording is a 1971
translation still under copyright, so her prose is '''paraphrased''' throughout
this page and never transcribed. Where a phrase of hers is characterised, it is
characterised, not quoted. See [[Hamilton's principle/Sources]].

== Setup and notation ==

Noether works with independent variables <math>x_1, \ldots, x_n</math> and
dependent variables <math>u_1, \ldots, u_\mu</math>, and with the integral

:<math>I = \int \cdots \int f\!\left( x, u, \frac{\partial u}{\partial x},
\ldots \right) dx
\qquad\qquad\mathrm{(1)}</math>

She writes <math>\psi_i</math> for the '''Lagrange expressions''' — the left-hand
sides of the Euler-Lagrange equations of
[[Hamilton's principle/Mathematics/Several independent variables]] — so that the
first variation is

:<math>\delta I = \int \cdots \int \sum \psi_i\, \delta u_i \; dx
\qquad\qquad\mathrm{(2)}</math>

when the variations vanish at the boundary. Retaining the boundary terms instead
gives an identity free of integrals,

:<math>\sum \psi_i\, \delta u_i = \delta f + \mathrm{Div}\, A
\qquad\qquad\mathrm{(3)}</math>

where <math>\mathrm{Div}\, A = \partial A_1/\partial x_1 + \cdots +
\partial A_n/\partial x_n</math> and <math>A</math> is linear in
<math>\delta u</math> and its derivatives. Equation (3) holds for any
<math>f</math> and any variation; it is the partial-integration identity of
[[Hamilton's principle/Mathematics/Several independent variables]] written
without discarding anything.

== The two kinds of variation ==

A transformation moves both the independent and dependent variables:

:<math>y_i = x_i + \Delta x_i,
\qquad
v_i(y) = u_i + \Delta u_i
\qquad\qquad\mathrm{(4)}</math>

Noether distinguishes <math>\Delta u</math>, the total change, from the variation
at fixed argument,

:<math>\delta u_i = v_i(x) - u_i(x)
= \Delta u_i - \sum \frac{\partial u_i}{\partial x_\lambda}\, \Delta x_\lambda
\qquad\qquad\mathrm{(5)}</math>

'''Equation (5) is the step most modern accounts compress, and compressing it is
where sign errors come from.''' The correction term accounts for the fact that
the field is being compared at a shifted point.

== Deriving the divergence relations ==

Invariance of <math>I</math> under the infinitesimal transformation means the
integral is unchanged when computed over the transformed region. Converting that
region back to the original gives an extra divergence, and the condition becomes

:<math>\delta f + \mathrm{Div}(f \cdot \Delta x) = 0
\qquad\qquad\mathrm{(6)}</math>

Substituting (3) into (6) to eliminate <math>\delta f</math>,

:<math>\sum \psi_i\, \delta u_i = \mathrm{Div}\, B,
\qquad B = A - f\cdot\Delta x
\qquad\qquad\mathrm{(7)}</math>

For a group with <math>\rho</math> parameters
<math>\varepsilon_1, \ldots, \varepsilon_\rho</math>, both <math>\delta u</math>
and <math>B</math> are linear in the parameters. Equating coefficients gives
<math>\rho</math> separate relations:

:<math>\sum \psi_i\, \delta u_i^{(1)} = \mathrm{Div}\, B^{(1)},
\quad \ldots \quad
\sum \psi_i\, \delta u_i^{(\rho)} = \mathrm{Div}\, B^{(\rho)}
\qquad\qquad\mathrm{(8)}</math>

'''Theorem I.''' If <math>I</math> is invariant under a <math>\rho</math>-parameter
continuous group, then <math>\rho</math> linearly independent combinations of the
Lagrange expressions are divergences. The converse also holds.

<blockquote>
Statement paraphrased from Noether 1918, §1, Theorem I. The equations are hers;
the English is this volume's.
</blockquote>

== Passing to conservation laws ==

Everything above is an '''identity''', true whether or not the field equations
hold. Impose them — set <math>\psi_i = 0</math>, which Noether notes is what
physicists call the field equations — and (8) collapses to

:<math>\mathrm{Div}\, B^{(1)} = 0,
\quad \ldots \quad
\mathrm{Div}\, B^{(\rho)} = 0
\qquad\qquad\mathrm{(9)}</math>

These are the conservation laws. Noether observes that such divergence equations
had lately come to be called laws of conservation, the phrasing marking how
recent the terminology then was.

In one independent variable a divergence is a total derivative, so (9) reads
<math>B^{(k)} = \text{const}</math> and gives <math>\rho</math> first integrals.
That recovers the results of
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]] and
[[Hamilton's principle/Mathematics/The Beltrami identity]] as the
one-dimensional case of a single theorem.

'''The two-step structure is the theorem's real content.''' Symmetry alone gives
an identity. Symmetry plus the equations of motion gives a conservation law. The
identity is the stronger statement and it survives where the field equations do
not hold.

== The symmetry-to-conservation table ==

{| class="wikitable"
! symmetry !! generator !! conserved quantity
|-
| time translation || <math>\Delta t = \varepsilon</math> || energy
|-
| space translation || <math>\Delta x_k = \varepsilon</math> || momentum, component <math>k</math>
|-
| rotation || <math>\Delta\theta = \varepsilon</math> || angular momentum
|-
| Galilean boost || <math>\Delta x = \varepsilon t</math> || centre-of-mass motion
|-
| phase rotation of a complex field || <math>\Delta\psi = i\varepsilon\psi</math> || charge
|-
| scaling, when <math>L</math> admits it || <math>\Delta x = \varepsilon x</math> || virial quantity
|}

The first three rows are the classical integrals, and Hamilton derived exactly
them from his characteristic function in 1834, obtaining the law of motion of the
centre of gravity and the law of description of areas by considering common
translations and rotations of the whole system.<ref>Hamilton, ''On a General
Method in Dynamics'' (1834), §6, equations (O.), (P.), (15.) and (16.). His
argument is that <math>V</math> cannot change under a motion that alters neither
configuration nor their mutual relation. ESTABLISHED source.</ref> That is
Noether's theorem for a particular group, eighty-four years early, and it shows
how natural the reasoning is once the action is the object of study.

== Two cautions ==

'''Independent symmetries need not give independent conservation laws.''' Noether
says so explicitly, and gives an example where the first integrals obtained
satisfy a nonlinear dependency. The free particle of
[[Hamilton's principle/Mathematics/Several dependent variables]] is the
elementary case: energy conservation there follows from momentum conservation and
is not new information.

'''The symmetry must be a symmetry of the action, not of the equations of
motion.''' These differ. Equations of motion can have symmetries the action lacks,
and those produce no conservation law by this theorem. The hypothesis is
invariance of <math>I</math>.

A third point deserves its own page. Noether's theorem gives nothing useful when
the group depends on '''arbitrary functions''' rather than on parameters. The
"conserved" quantities then turn out to vanish identically or to be
mathematically trivial. That failure is not a defect of the theorem; it is the
content of the second one, and it is why there are two theorems rather than one.

== Next ==

[[Hamilton's principle/Mathematics/Noether's second theorem]] treats the
infinite-parameter case and its consequence for general relativity.

== References ==

<references />