View source for Noether's theorems/First theorem/Prolongation to derivatives

{{DISPLAYTITLE:Prolongation to derivatives}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Why speed has to come along for the ride|Why speed has to come along for the ride]].''

[[File:NT-F-03.png|thumb|right|360px|alt=Two axes side by side. The left shows a curve with tangent arrows marking its slope at three points. The right shows those slopes plotted as a curve of their own against the same horizontal axis, joined by an arrow labelled lift. A note says choosing how the points move fixes how the slopes move.|Prolongation: a curve and its lifted derivative data]]

A transformation that moves the variables also moves their derivatives, and it
does not move them freely: how the derivatives transform is determined by how
the variables do. Extending a transformation to the derivatives in the way the
chain rule forces is called prolongation.

Noether needs this twice. Her integrand contains derivatives, so the invariance
condition (N1) already involves transformed derivatives. And she permits the
transformations themselves to contain derivatives of the <math>u</math>, which
is the more general case and the one most later treatments drop.

== How a first derivative moves ==

Under the infinitesimal transformation (2) of
[[Noether's theorems/First theorem/One-parameter groups and their generators|the
previous page]], the change induced in a first derivative is

<math>\Delta\left(\frac{\partial u}{\partial x_{\varkappa}}\right)=\frac{\partial\Delta u}{\partial x_{\varkappa}}-\sum_{\lambda}\frac{\partial u}{\partial x_{\lambda}}\frac{\partial\Delta x_{\lambda}}{\partial x_{\varkappa}}\qquad\qquad\mathrm{(1)}</math>

Two terms, and both are needed. The first is what one would guess: the
derivative of the change in <math>u</math>. The second is the correction for the
fact that the independent variable has moved as well, so the derivative is being
taken with respect to a shifted coordinate.

'''If <math>\Delta x=0</math> the second term vanishes and prolongation is
trivial.''' That is the case in most textbook treatments, which is why
prolongation rarely appears in them by name. It is not the case here.

== Why the derivative count can grow without bound ==

Noether makes a remark in her third section that is easy to pass over and
governs how far the machinery reaches.

Suppose <math>\Delta x</math> and <math>\Delta u</math> themselves contain
derivatives of the <math>u</math>. Integrating the system (N17) to recover the
finite transformations requires
<math>d^{2}x_{i}/dt^{2}</math> and <math>d^{2}u_{i}/dt^{2}</math>, and computing
those requires (1). But (1) introduces one more derivative of <math>u</math> than
it consumed. At the next step the same thing happens again. The number of
derivatives of <math>u</math> appearing therefore increases, in general, at every
step, and the finite transformations can depend on infinitely many of them.

This is not a pathology to be excluded. It is the price of the generality that
lets the theorems apply to transformations mixing the dependent and independent
variables, and Noether pays it explicitly rather than assuming it away.

== Her example ==

Take

<math>f=\tfrac{1}{2}u'^{2};\qquad\psi=-u'';\qquad\psi\cdot x=\frac{d}{dx}\left(u-u'x\right);\qquad\bar{\delta}u=x\cdot\varepsilon\qquad\qquad\mathrm{(2)}</math>

for which

<math>\Delta x=\frac{-2a}{u'^{2}}\varepsilon;\qquad\Delta u=\left(x-\frac{2u}{u'}\right)\cdot\varepsilon\qquad\qquad\mathrm{(3)}</math>

The infinitesimal transformation contains <math>u'</math> in its denominators.
This is the simplest case in which the generated finite transformations depend
on derivatives, and it is her own.

The <math>\bar{\delta}u</math> appearing here is the barred variation, defined on
[[Noether's theorems/First theorem/The boundary term|the next page]]. It is
introduced there and used here only as a label.

== Where this goes ==

Prolongation is what
[[Noether's theorems/Mathematics/Jet space and prolongation|jet space]] was
invented to make systematic. A jet is the bookkeeping device that records a
function together with its derivatives up to some order as a single point, so
that a prolonged transformation is an ordinary transformation of that space.
Noether does the bookkeeping by hand. The modern apparatus is not needed to state
or prove either theorem, and this volume introduces it as a separate subject
rather than as a prerequisite.

''Provenance: equation (1) is Noether's, given in her third section as running
text rather than as a numbered display; this volume has numbered it. Equations
(2) and (3) are her worked example, likewise unnumbered in the original. All are
reproduced in her notation from the public-domain German original. Her prose is
paraphrased. See [[Noether's theorems/Sources|Sources]].''