View source for Noether's theorems/Second theorem/Groups depending on arbitrary functions

{{DISPLAYTITLE:Groups depending on arbitrary functions}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Symmetry that can differ from place to place|Symmetry that can differ from place to place]].''

The second theorem has exactly one input the first does not: the group. This page
fixes what kind of group, and separates it from two things it is regularly
confused with.

== The definition ==

A group is '''infinite continuous''', written
<math>\mathfrak{G}_{\infty\varrho}</math>, when its most general transformations
depend on <math>\varrho</math> essential arbitrary functions <math>p(x)</math>
and their derivatives, analytically or at least continuously and finitely often
continuously differentiably.

Noether takes the underlying definition from Lie: an infinite continuous group is
a transformation group whose transformations are given by the most general
solutions of a system of partial differential equations, provided those solutions
do not depend on only a finite number of parameters.

Write <math>\sigma</math> for the highest order of derivative of the
<math>p(x)</math> that occurs. The theorem's output is counted by
<math>\varrho</math> and its depth by <math>\sigma</math>, and both appear in the
statement.

== What it is not ==

'''It is not a group with infinitely many parameters.''' Noether places that case
between the two as an intermediate: a group depending on infinitely many
parameters but not on arbitrary functions. She notes that the first theorem
survives that limiting case. The second theorem does not apply to it, because
the second theorem's proof turns on being able to choose the arbitrary functions
freely on a region and make them vanish near its boundary, which a parameter
cannot do.

'''It is not a bigger version of the finite group.''' A parameter is one number
for the whole region. An arbitrary function is a different number at every point.
That is a difference in kind, and it is the reason the two theorems return
different objects.

A group depending on both arbitrary functions and parameters Noether calls
'''mixed'''. Both theorems apply to it, and both dependencies and independent
divergence relations occur.

== Linearity in the arbitrary functions ==

As with the finite case, <math>\Delta x</math> and <math>\Delta u</math> are
taken linear in the <math>p(x)</math> and their derivatives. Here the
demonstration that this is no restriction is more direct than in the finite case
and does not need the converse: it follows from the fact that
<math>\mathfrak{G}_{\infty\varrho}</math> depends formally on
<math>\varrho</math> and only <math>\varrho</math> arbitrary functions. The
argument is on
[[Noether's theorems/Second theorem/The converse of the second theorem|the
converse page]].

== Her two examples ==

Noether gives both before any proof, which is worth imitating.

'''Weierstrass parameter representation,''' the simplest case, with
<math>\sigma=0</math>. An integral homogeneous of the first order is invariant
when the independent variable is replaced by an arbitrary function of itself,
leaving the dependent variables unchanged:

<math>y=p(x);\qquad v_{i}(y)=u_{i}(x)\qquad\qquad\mathrm{(1)}</math>

One arbitrary function occurs, and no derivatives of it. The corresponding
relation among the Lagrange expressions is the familiar linear one,

<math>\sum\psi_{i}\frac{du_{i}}{dx}=0\qquad\qquad\mathrm{(2)}</math>

This case has no converse, and Noether says so when she introduces it.

'''General relativity,''' with <math>\sigma=1</math>. The group is all
transformations of the independent variables,

<math>y_{i}=p_{i}(x)\qquad\qquad\mathrm{(3)}</math>

while the dependent variables — which she denotes <math>g_{\mu\nu}</math> and
<math>q</math> — are subjected to the transformations thereby induced for the
coefficients of a quadratic and a linear differential form. Those induced
transformations contain the first derivatives of the arbitrary functions.
Corresponding to this are the known <math>n</math> dependencies between the
Lagrange expressions and their first derivatives.

'''Note the framing.''' She names general relativity as an example of a group of
this kind. She does not name it as the subject, and
[[Noether's theorems/Second theorem/Proper and improper conservation laws|her
remark at the end of the paper]] makes the point explicitly.

''Provenance: the definitions, the intermediate and mixed cases, and both
examples are Noether's, from her first and second sections. Displays (1) to (3)
are hers, set as running text in the original and numbered here. Reproduced in
her notation from the public-domain German original; her prose is paraphrased,
and the Lie citation is hers, not read independently. See
[[Noether's theorems/Sources|Sources]].''