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Published from the Principia Machina source

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	{{DISPLAYTITLE:Index of symmetries and their conserved quantities}}		{{DISPLAYTITLE:Index of symmetries and their conserved quantities}}
			{{Noether's theorems map}}

	[[File:NT-F-07.png\|thumb\|right\|360px\|alt=Five symmetries in a left column, each joined by an arrow to the quantity it conserves in a right column: clock shift to energy, spatial motion to momentum, rotation to angular momentum, field rotation to electric charge, and steady-speed change to uniform centre-of-mass motion. The last arrow is dashed to mark that it needs the weaker invariance condition.\|The symmetry-to-conservation map]]		[[File:NT-F-07.png\|thumb\|right\|360px\|alt=Five symmetries in a left column, each joined by an arrow to the quantity it conserves in a right column: clock shift to energy, spatial motion to momentum, rotation to angular momentum, field rotation to electric charge, and steady-speed change to uniform centre-of-mass motion. The last arrow is dashed to mark that it needs the weaker invariance condition.\|The symmetry-to-conservation map]]
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	The last row is the volume's boundary marker. The test that produces it is a		The last row is the volume's boundary marker. The test that produces it is a
	single question: were the field equations used?		single question: were the field equations used?

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NT>NoetherVolumeBuild: Noether theorems volume, initial import (R-02)

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Noether theorems volume, initial import (R-02)

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{{DISPLAYTITLE:Index of symmetries and their conserved quantities}}

[[File:NT-F-07.png|thumb|right|360px|alt=Five symmetries in a left column, each joined by an arrow to the quantity it conserves in a right column: clock shift to energy, spatial motion to momentum, rotation to angular momentum, field rotation to electric charge, and steady-speed change to uniform centre-of-mass motion. The last arrow is dashed to mark that it needs the weaker invariance condition.|The symmetry-to-conservation map]]

Every symmetry treated anywhere in this volume, with what it returns and '''which
theorem applies'''. The last column is the distinction the volume exists to keep,
and it is the reason this index is worth more than a list of conserved
quantities.

== First theorem: global symmetries, conserved quantities ==

{| class="wikitable"
! symmetry !! what it returns !! page
|-
| translation of the independent variable || energy || [[Noether's theorems/First theorem/Energy, momentum and angular momentum|Energy, momentum and angular momentum]]
|-
| translation of a dependent variable || the conjugate momentum; the coordinate is cyclic || same
|-
| rotation of two dependent variables, in space || angular momentum || same
|-
| rigid translation of an <math>N</math>-particle system || the total momentum, and Newton's third law as an off-shell identity || [[Noether's theorems/Applications/Conserved quantities in point mechanics|Point mechanics]]
|-
| rotation of two dependent variables, internal || electric charge || [[Noether's theorems/First theorem/Centre-of-mass motion and electric charge|Centre-of-mass motion and electric charge]]
|-
| time translation of a string || energy density and energy flux || [[Noether's theorems/Applications/Continuum mechanics|Continuum mechanics]]
|-
| space translation of a string || the wave's momentum density, which is not the material's momentum || same
|-
| a Killing vector of a curved space || a quantity constant along every geodesic || [[Noether's theorems/Mathematics/Killing vectors and conserved quantities along geodesics|Killing vectors]]
|-
| spatial Killing vector of an expanding universe || comoving momentum; hence photon energy falling as the inverse stretch factor || [[Noether's theorems/Applications/Photon energy in an expanding universe|Photon energy]]
|-
| the flow generated by any quantity || the generator identity; conservation is the antisymmetry of the bracket || [[Noether's theorems/Mathematics/Poisson brackets and the generator identity|Poisson brackets]]
|-
| rigid shift of a '''discrete''' action || a discrete momentum, conserved exactly at every step || [[Noether's theorems/Applications/Variational integrators and discrete symmetry|Variational integrators]]
|}

== First theorem, with Bessel-Hagen's weakened condition ==

{| class="wikitable"
! symmetry !! what it returns !! page
|-
| Galilean boost || uniform motion of the centre of mass || [[Noether's theorems/First theorem/Centre-of-mass motion and electric charge|Centre-of-mass motion and electric charge]]
|}

Strict invariance gives nothing here. The weakened condition is Bessel-Hagen's,
from 1921, and is attributed on every page where it is used.

== Second theorem: local symmetries, identities ==

{| class="wikitable"
! symmetry !! what it returns !! page
|-
| Weierstrass reparametrisation, one free function, order zero || one linear relation among the Lagrange expressions || [[Noether's theorems/Second theorem/The Noether identities|The Noether identities]]
|-
| coordinate relabelling in general relativity, <math>n</math> functions, order one || <math>n</math> identities; the count is a theorem, their identification with the Bianchi identities is not derived here || [[Noether's theorems/Applications/General relativity and the Bianchi identity|Bianchi identity]]
|-
| electromagnetic gauge freedom, one function || one identity, which '''forces''' charge conservation rather than permitting it || [[Noether's theorems/Applications/Electromagnetism and charge conservation|Electromagnetism]]
|-
| Yang-Mills gauge freedom || identities, one per free function; the count only, nothing derived || [[Noether's theorems/Applications/Yang-Mills theory|Yang-Mills]]
|-
| Noether's logarithmic group || improper energy relations, with no physics attached || [[Noether's theorems/Second theorem/Proper and improper conservation laws|Proper and improper]]
|-
| translation group inside any infinite group || energy relations improper, if and only if || same
|}

== Unresolved, and outside both theorems ==

{| class="wikitable"
! case !! status !! page
|-
| particle relabelling in an ideal fluid → circulation || the symmetry is a free shape, so this sits between the theorems. Which one applies is '''not resolved''' in this volume || [[Noether's theorems/Applications/Kelvin's circulation theorem|Kelvin's circulation theorem]]
|-
| the Runge-Lenz vector || the conserved quantity is exhibited; the symmetry behind it is '''not derived''' || [[Noether's theorems/Applications/The Runge-Lenz vector|Runge-Lenz]]
|-
| topological charge || '''neither theorem.''' Conserved without the field equations being used, so no symmetry is responsible || [[Noether's theorems/Applications/Topological charge, which is not a Noether charge|Topological charge]]
|}

The last row is the volume's boundary marker. The test that produces it is a
single question: were the field equations used?

Noether's theorems/Index of symmetries - Revision history

PMA Publisher: Published from the Principia Machina source

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