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Published from the Principia Machina source

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	{{DISPLAYTITLE:Notation}}		{{DISPLAYTITLE:Notation}}
			{{Noether's theorems map}}

	This volume reproduces Noether's equations in her own symbols. Her notation is		This volume reproduces Noether's equations in her own symbols. Her notation is
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	symbols, the barred variation and the binomial coefficients set with		symbols, the barred variation and the binomial coefficients set with
	<code>\atop</code>.		<code>\atop</code>.

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NT>NoetherVolumeBuild: Noether theorems volume, initial import (R-01)

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Noether theorems volume, initial import (R-01)

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{{DISPLAYTITLE:Notation}}

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 ==

{| class="wikitable"
! Noether !! meaning !! modern equivalent
|-
| <math>f</math> || the integrand || Lagrangian, or Lagrangian density
|-
| <math>I</math> || the integral || the action
|-
| <math>x_{1},\dots,x_{n}</math> || independent variables || time, or spacetime coordinates
|-
| <math>u_{1},\dots,u_{\mu}</math> || dependent variables || the fields, or the generalised coordinates
|-
| <math>\psi_{i}</math> || the Lagrange expressions || left side of the Euler-Lagrange equation
|-
| <math>\mathrm{Div}</math> || divergence || <math>\partial_{a}A^{a}</math>
|-
| <math>\mathfrak{G}_{\varrho}</math> || finite continuous group, <math>\varrho</math> parameters || global symmetry group
|-
| <math>\mathfrak{G}_{\infty\varrho}</math> || infinite continuous group, <math>\varrho</math> arbitrary functions || gauge, or local symmetry group
|-
| <math>\varepsilon</math> || the group parameters || same
|-
| <math>p(x)</math> || the arbitrary functions || the gauge functions
|-
| <math>\varkappa</math> || highest derivative order in <math>f</math> || same
|-
| <math>\sigma</math> || highest derivative order of the <math>p(x)</math> || same
|-
| <math>A</math>, <math>B</math> || the divergence and the current, <math>B=A-f\Delta x</math> || the Noether current
|-
| <math>\Gamma</math> || 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
[[Noether's theorems/First theorem/The boundary term|the boundary term page]].

{| class="wikitable"
! symbol !! meaning !! commutes with differentiation?
|-
| <math>\Delta u</math> || total change: new configuration at the new point against old at the old || '''no'''
|-
| <math>\bar{\delta}u</math> || change at fixed argument, the drift subtracted out || '''yes'''
|-
| <math>\delta u</math> || ordinary variation of the calculus of variations; no transformation of the <math>x</math> || not applicable
|}

The last column is derived on
[[Noether's theorems/Mathematics/Jet space and prolongation|the jet space page]]
and is the reason the third symbol exists.

== Collisions, and how this volume resolves them ==

{| class="wikitable"
! symbol !! Noether's use !! competing use !! resolution here
|-
| <math>\mu</math> || 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
|-
| <math>\varrho</math> vs <math>\rho</math> || <math>\varrho</math> is her count of parameters or arbitrary functions || <math>\rho</math> is a density || both occur, and they are near-identical in some fonts. Density appears only on the vibrating-string page
|-
| <math>\tau</math> || 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
|-
| <math>\psi</math> || the Lagrange expressions || a fermion field || reserved to Noether volume-wide. The competing use does not appear
|-
| moment map || — || conventionally <math>\mu</math> || written <math>J</math>
|-
| spacetime indices || — || conventionally <math>\mu,\nu</math> || written <math>a,b,c</math>
|}

== 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:
<code>\tag</code>, for which equation numbers are set inside the mathematics as
<code>\qquad\qquad\mathrm{(n)}</code>; and <code>\lvert</code> and
<code>\rvert</code>, for which <code>\vert</code> 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
<code>\atop</code>.

Noether's theorems/Notation - Revision history

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