Noether's theorems/Notation

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

This volume reproduces Noether's equations in her own symbols. Her notation is not modern notation, and three of her letters collide with standard modern uses. Both tables below are checked against every mathematical expression in the volume.

Her symbols against modern usage

Noether	meaning	modern equivalent
$f$	the integrand	Lagrangian, or Lagrangian density
$I$	the integral	the action
$x_1,\dots ,x_n$	independent variables	time, or spacetime coordinates
$u_1,\dots ,u_{\mu }$	dependent variables	the fields, or the generalised coordinates
${\psi }_i$	the Lagrange expressions	left side of the Euler-Lagrange equation
$\mathrm{D}\mathrm{i}\mathrm{v}$	divergence	${\partial }_a{A}^a$
${𝔊}_{\varrho }$	finite continuous group, $\varrho$ parameters	global symmetry group
${𝔊}_{\mathrm{\infty }\varrho }$	infinite continuous group, $\varrho$ arbitrary functions	gauge, or local symmetry group
$\epsilon$	the group parameters	same
$p(x)$	the arbitrary functions	the gauge functions
$\varkappa$	highest derivative order in $f$	same
$\sigma$	highest derivative order of the $p(x)$	same
$A$, $B$	the divergence and the current, $B=A-f\Delta x$	the Noether current
$\Gamma$	divergence generated by partial integration	no standard name

The three variations

Noether uses three, and most later accounts use two. The distinction is load-bearing and is the subject of the boundary term page.

symbol	meaning	commutes with differentiation?
$\Delta u$	total change: new configuration at the new point against old at the old	no
$\overline{\delta }u$	change at fixed argument, the drift subtracted out	yes
$\delta u$	ordinary variation of the calculus of variations; no transformation of the $x$	not applicable

The last column is derived on the jet space page and is the reason the third symbol exists.

Collisions, and how this volume resolves them

symbol	Noether's use	competing use	resolution here
$\mu$	count of the dependent variables	spacetime index, as in the metric	Noether's own collision. She writes both, using the metric form in her general-relativity example. Reproduced as she wrote it; this volume adds no third use
$\varrho$ vs $\rho$	$\varrho$ is her count of parameters or arbitrary functions	$\rho$ is a density	both occur, and they are near-identical in some fonts. Density appears only on the vibrating-string page
$\tau$	derivative order in her partial-integration identity	affine parameter along a geodesic	both occur, on different tracks, and neither page uses the other's meaning
$\psi$	the Lagrange expressions	a fermion field	reserved to Noether volume-wide. The competing use does not appear
moment map	—	conventionally $\mu$	written $J$
spacetime indices	—	conventionally $\mu ,\nu$	written $a,b,c$

Equation numbering

Equations reproduced from the 1918 paper carry her numbers with an N prefix, as (N9), and those numbers are unique across the volume. Each page's own equations number from (1) independently.

Seventeen of her twenty numbered equations appear in this volume: (N1) through (N17). Her (18), (19) and (20) belong to a construction the volume does not reproduce.

texvc constraints

MediaWiki checks mathematics with texvc, which is stricter than LaTeX. Two constructions are rejected and are avoided throughout: \tag, for which equation numbers are set inside the mathematics as \qquad\qquad\mathrm{(n)}; and \lvert and \rvert, for which \vert is used.

Every one of the volume's expressions was passed through texvc's own checker before delivery. Her notation survives it intact, including the Fraktur group symbols, the barred variation and the binomial coefficients set with \atop.