Noether's theorems/Index of symmetries

Noether's theorems: map
Reading the volume How to read this volume Notation Glossary Index of symmetries Figure gallery
Reading by level Start here: the plain-language primer The first theorem, stated and proved The second theorem, stated and proved The two theorems side by side The mathematics: Lie groups and Lie algebras The applications: General relativity and the energy question The history: Göttingen and the energy problem
Sources Sources Bibliography Chronology Further reading
Related Hamilton's principle On Physical Lines of Force

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

symmetry	what it returns	page
translation of the independent variable	energy	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 $N$-particle system	the total momentum, and Newton's third law as an off-shell identity	Point mechanics
rotation of two dependent variables, internal	electric charge	Centre-of-mass motion and electric charge
time translation of a string	energy density and energy flux	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	Killing vectors
spatial Killing vector of an expanding universe	comoving momentum; hence photon energy falling as the inverse stretch factor	Photon energy
the flow generated by any quantity	the generator identity; conservation is the antisymmetry of the bracket	Poisson brackets
rigid shift of a discrete action	a discrete momentum, conserved exactly at every step	Variational integrators

First theorem, with Bessel-Hagen's weakened condition

symmetry	what it returns	page
Galilean boost	uniform motion of the centre of mass	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

symmetry	what it returns	page
Weierstrass reparametrisation, one free function, order zero	one linear relation among the Lagrange expressions	The Noether identities
coordinate relabelling in general relativity, $n$ functions, order one	$n$ identities; the count is a theorem, their identification with the Bianchi identities is not derived here	Bianchi identity
electromagnetic gauge freedom, one function	one identity, which forces charge conservation rather than permitting it	Electromagnetism
Yang-Mills gauge freedom	identities, one per free function; the count only, nothing derived	Yang-Mills
Noether's logarithmic group	improper energy relations, with no physics attached	Proper and improper
translation group inside any infinite group	energy relations improper, if and only if	same

Unresolved, and outside both theorems

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	Kelvin's circulation theorem
the Runge-Lenz vector	the conserved quantity is exhibited; the symmetry behind it is not derived	Runge-Lenz
topological charge	neither theorem. Conserved without the field equations being used, so no symmetry is responsible	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?