Prolongation to derivatives

Plain-language counterpart: Why speed has to come along for the ride.

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 $u$, which is the more general case and the one most later treatments drop.

How a first derivative moves

Under the infinitesimal transformation (2) of the previous page, the change induced in a first derivative is

$\Delta \left(\frac{\partial u}{\partial x_{\varkappa }}\right)=\frac{\partial \Delta u}{\partial x_{\varkappa }}-\sum _{\lambda }\frac{\partial u}{\partial x_{\lambda }}\frac{\partial \Delta x_{\lambda }}{\partial x_{\varkappa }}\mathrm{(}\mathrm{1}\mathrm{)}$

Two terms, and both are needed. The first is what one would guess: the derivative of the change in $u$. 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 $\Delta x=0$ 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 $\Delta x$ and $\Delta u$ themselves contain derivatives of the $u$. Integrating the system (N17) to recover the finite transformations requires $d^2{x}_i/d{t}^2$ and $d^2{u}_i/d{t}^2$, and computing those requires (1). But (1) introduces one more derivative of $u$ than it consumed. At the next step the same thing happens again. The number of derivatives of $u$ 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

$f=\frac{1}{2}u{\text{'}}^2;\psi =-u^{″};\psi \cdot x=\frac{d}{dx}(u-u^{\prime }x);\overline{\delta }u=x\cdot \epsilon \mathrm{(}\mathrm{2}\mathrm{)}$

for which

$\Delta x=\frac{-2a}{u{\text{'}}^2}\epsilon ;\Delta u=(x-\frac{2u}{u^{\prime }})\cdot \epsilon \mathrm{(}\mathrm{3}\mathrm{)}$

The infinitesimal transformation contains $u^{\prime }$ in its denominators. This is the simplest case in which the generated finite transformations depend on derivatives, and it is her own.

The $\overline{\delta }u$ appearing here is the barred variation, defined on the next page. It is introduced there and used here only as a label.

Where this goes

Prolongation is what 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 Sources.