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Noether theorems volume, initial import (F-03)

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{{DISPLAYTITLE:One-parameter groups and their generators}}

''Plain-language counterpart: [[Noether's theorems/Explainers/What a continuous symmetry is|What a continuous symmetry is]].''

[[File:NT-F-15.png|thumb|right|360px|alt=A family of parallel flow curves. One point carries a tangent arrow labelled the generator, and a series of dots along the curve shows the finite transformation built by repeating the small step. A caption states that both proofs run on the generator because the group is recoverable from it.|A one-parameter group: flow lines and the generator]]

The two theorems differ in exactly one input: the kind of group. This page
defines both kinds, because the distinction has to be in place before either
theorem can be stated.

== Group, finite, infinite, mixed ==

A transformation group is a system of transformations such that every
transformation has an inverse in the system, and the composition of any two
belongs to the system.

The group is called '''finite continuous''', written
<math>\mathfrak{G}_{\varrho}</math>, when its transformations are contained in a
most general one depending analytically on <math>\varrho</math> essential
parameters <math>\varepsilon</math>. Essential means the <math>\varrho</math>
parameters cannot be represented as <math>\varrho</math> functions of fewer
parameters.

The group is called '''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.

Between the two sits the group depending on infinitely many parameters but not
on arbitrary functions. A group depending on both arbitrary functions and
parameters is called '''mixed'''.

Noether takes her definition of the infinite continuous group from Lie, who
defines it as 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.

'''The first theorem takes <math>\mathfrak{G}_{\varrho}</math>. The second takes
<math>\mathfrak{G}_{\infty\varrho}</math>. That is the whole of the
difference in input,''' and
[[Noether's theorems/Second theorem/The two theorems side by side|everything that
follows from it]] is downstream of this one distinction.

== Normalising the identity ==

For any continuous group, finite or infinite, the parameters can always be
arranged so that the identity transformation corresponds to zero values of the
<math>\varepsilon</math>, or of the arbitrary functions <math>p(x)</math>.
Noether cites Lie for this. Where arbitrary functions are involved, the special
parameter values are replaced by fixed functions and their derivatives.

== The general transformation and its lowest terms ==

With the identity at zero, the most general transformation has the form

<math>\begin{array}{lll}
y_{i} & =A_{i}\left(x,\ u,\ \dfrac{\partial u}{\partial x},\dots\right) & =x_{i}+\Delta x_{i}+\dots\\
v_{i}(y) & =B_{i}\left(x,\ u,\ \dfrac{\partial u}{\partial x},\dots\right) & =u_{i}+\Delta u_{i}+\dots
\end{array}\qquad\qquad\mathrm{(1)}</math>

where <math>\Delta x_{i}</math> and <math>\Delta u_{i}</math> are the terms of
lowest dimension in <math>\varepsilon</math>, or in <math>p(x)</math> and its
derivatives. Noether assumes them linear in those quantities and remarks that
this will be shown to be no restriction of generality — the demonstration is in
[[Noether's theorems/First theorem/The converse of the first theorem|the converse]].

The '''infinitesimal transformation''' is what is left when the higher terms are
dropped:

<math>y_{i}=x_{i}+\Delta x_{i};\qquad v_{i}(y)=u_{i}+\Delta u_{i}\qquad\qquad\mathrm{(2)}</math>

== Recovering the finite transformations ==

The infinitesimal transformations are not a weaker object. The finite ones are
obtained back from them by integrating the simultaneous system

<math>\frac{dx_{i}}{dt}=\Delta x_{i};\quad\frac{du_{i}}{dt}=\Delta u_{i};\quad\left(\begin{array}{c}x_{i}=y_{i}\\u_{i}=v_{i}\end{array}\ \text{for}\ t=0\right)\qquad\qquad\mathrm{(N17)}</math>

and the resulting finite transformations contain <math>\varrho</math> parameters,
namely the products <math>t\varepsilon_{1},\dots,t\varepsilon_{\varrho}</math>.

This is why the whole of both proofs can be run on infinitesimal transformations
without loss: the group is recoverable from its generators. Noether uses the
recovery explicitly in both converses.

'''One caution.''' When <math>\Delta x</math> and <math>\Delta u</math> contain
derivatives of the <math>u</math>, the finite transformations obtained this way
can depend on infinitely many derivatives, because the count grows at each step
of the integration. That is treated on
[[Noether's theorems/First theorem/Prolongation to derivatives|the prolongation
page]].

''Provenance: (N17) is Noether's equation (17), reproduced in her notation from
the public-domain German original. Equations (1) and (2) are displays this volume
has numbered; the array in (1) is hers, set unnumbered in the original. Her prose
is paraphrased and the Lie citations are hers, not read independently. See
[[Noether's theorems/Sources|Sources]].''

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