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Noether theorems volume, initial import (F-01)

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{{DISPLAYTITLE:The action and the variational setup}}

''Plain-language counterpart: [[Noether's theorems/Explainers/What an action is|What an action is]].''

[[File:NT-F-02.png|thumb|right|360px|alt=Three curves between endpoints. A solid curve is the true path; a dashed curve with the same fixed endpoints is a variation; a third curve offset from both, with its own moved endpoints, is a symmetry-transformed path. A note distinguishes testing the action from mapping the whole problem.|A varied path against a symmetry-transformed path]]

Everything in both theorems rests on one object and one condition on it. This
page fixes the object, the notation, and what invariance is going to mean. The
notation is Noether's own; the
[[Noether's theorems/Notation|notation page]] gives the modern equivalents.

== The variables ==

Let <math>x_{1},\dots,x_{n}</math> be independent variables and
<math>u_{1}(x),\dots,u_{\mu}(x)</math> functions of them. Noether assumes every
function appearing is analytic, or at least continuous and finitely often
continuously differentiable, and single-valued on the region considered.

Two abbreviations of hers are used throughout and are easy to misread. She
suppresses indices wherever she can, including under summation signs, so
<math>\partial^{2}u/\partial x^{2}</math> stands for
<math>\partial^{2}u_{\alpha}/\partial x_{\beta}\partial x_{\gamma}</math> and
similarly at higher order. And she writes <math>dx</math> for
<math>dx_{1}\dots dx_{n}</math>. Neither is a restriction; both are shorthand.

== What a transformation does ==

Subject the <math>x</math> and <math>u</math> to the transformations of a group.
Because the transformations are invertible, exactly <math>n</math> independent
variables must again be present among the transformed quantities; call them
<math>y_{1},\dots,y_{n}</math>, and call the remaining dependent ones
<math>v_{1}(y),\dots,v_{\mu}(y)</math>. The transformations may also involve the
derivatives of the <math>u</math> with respect to the <math>x</math>. That
possibility is not decoration; it is what
[[Noether's theorems/First theorem/Prolongation to derivatives|prolongation]] is
about, and it is one of the places where her statement is more general than the
textbook one.

A function is called an invariant of the group when

<math>P\left(x,\ u,\ \frac{\partial u}{\partial x},\ \frac{\partial^{2}u}{\partial x^{2}},\dots\right)=P\left(y,\ v,\ \frac{\partial v}{\partial y},\ \frac{\partial^{2}v}{\partial y^{2}},\dots\right)\qquad\qquad\mathrm{(1)}</math>

== The integral, and what invariance of it means ==

The object both theorems are about is an integral

<math>I=\int\dots\int f\left(x,\ u,\ \frac{\partial u}{\partial x},\ \frac{\partial^{2}u}{\partial x^{2}},\dots\right)dx\qquad\qquad\mathrm{(2)}</math>

taken over an arbitrary real region of <math>x</math>. Physics calls
<math>f</math> the Lagrangian density and <math>I</math> the action. Noether
calls them the integrand and the integral, and nothing in either theorem depends
on the physical reading.

The integral is an invariant of the group when

<math>I=\int\dots\int f\left(x,\ u,\ \frac{\partial u}{\partial x},\ \frac{\partial^{2}u}{\partial x^{2}},\dots\right)dx=\int\dots\int f\left(y,\ v,\ \frac{\partial v}{\partial y},\ \frac{\partial^{2}v}{\partial y^{2}},\dots\right)dy\qquad\qquad\mathrm{(N1)}</math>

integrated over an arbitrary real <math>x</math>-region and the corresponding
<math>y</math>-region.

'''Read the equality carefully.''' It is not a statement that <math>f</math> is
unchanged. It is a statement that the integral of <math>f</math> over a region
equals the integral of the same functional form over the transformed region,
with the transformed arguments. The integrand may change; the two changes — in
the function and in the region — must cancel.

== Three remarks that matter later ==

'''The condition is on the integral, not on the equations of motion.''' This is
the single most common error in accounts of the theorems, and it has
[[Noether's theorems/First theorem/On-shell and off-shell|its own page]]. A
transformation that carries solutions to solutions need not satisfy (N1), and if
it does not, no conservation law follows from it.

'''Both theorems take (N1) as given and ask what follows.''' Neither theorem
tells you which groups leave a given integral invariant. Finding those is a
separate problem, and Noether treats it only in examples.

'''Invariance up to a divergence is not (N1).''' The weakening that admits
Galilean boosts is
[[Noether's theorems/History/Bessel-Hagen and the first uses|Bessel-Hagen's,
from 1921]], not hers, and this volume keeps the attribution straight. See
[[Noether's theorems/First theorem/Invariance of the action, strict and up to a divergence|the page on the invariance condition]].

Arguments and coefficients: Noether takes the arguments
<math>x,\ u,\ \varepsilon,\ p(x)</math> real, while allowing the coefficients to
be complex. Because the final results are identities in those quantities, they
hold for complex values too, provided every function appearing is analytic.

''Provenance: (N1) is Noether's equation (1), reproduced in her notation from the
German original, which is public domain. Equations (1) and (2) above are this
page's own restatements of definitions she gives in prose. No English wording of
hers is transcribed. See [[Noether's theorems/Sources|Sources]].''

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