Jet space and prolongation

Plain-language counterpart: Why Noether needed a third symbol.

Jet space: a function and its first jet

The prolongation page did the bookkeeping by hand. This page gives the apparatus that makes it systematic, and derives the one fact about it that explains a choice Noether made without explaining.

Jet space

Take independent variables $x$ and dependent variables $u$. Form the space whose points are

$(x,u,u^{(1)},u^{(2)},\dots ,u^{(k)})\mathrm{(}\mathrm{1}\mathrm{)}$

where the $u^{(j)}$ are treated as independent coordinates, not as derivatives of anything. This is the $k$-jet space.

A function of $x$ then determines a point of jet space at each $x$, by supplying its actual derivatives for the $u^{(j)}$ — but the space itself does not know that, and that is the point of the construction.

Differentiation along a function is recovered by the total derivative,

$D_x=\frac{\partial }{\partial x}+u^{(1)}\frac{\partial }{\partial u}+u^{(2)}\frac{\partial }{\partial u^{(1)}}+\dots \mathrm{(}\mathrm{2}\mathrm{)}$

which differentiates with respect to $x$ while carrying the chain rule through the jet coordinates.

What this buys. Noether's integrand depends on $x$, on $u$, and on derivatives to some order. On jet space it is an ordinary function of the coordinates, and a transformation that mixes the variables and their derivatives is an ordinary transformation of that space, with an ordinary generator of the form on the previous page. Prolongation becomes the statement that a generator downstairs determines a unique generator upstairs.

Why the barred variation, and not the total change

Here is a fact Noether uses everywhere and never remarks on, and it is the reason her three-way split exists.

Her barred variation compares the new configuration with the old at the same point:

$\overline{\delta }{u}_i=v_i(x)-u_i(x)\mathrm{(}\mathrm{N}\mathrm{9},\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{)}$

Differentiate that with respect to $x_{\lambda }$. Both terms are functions of the same argument, so differentiation passes straight through:

$\frac{\partial }{\partial x_{\lambda }}\left(\overline{\delta }{u}_i\right)=\frac{\partial v_i}{\partial x_{\lambda }}-\frac{\partial u_i}{\partial x_{\lambda }}=\overline{\delta }\left(\frac{\partial u_i}{\partial x_{\lambda }}\right)\mathrm{(}\mathrm{3}\mathrm{)}$

The barred variation commutes with differentiation.

Now the same question for the total change. From the prolongation page,

$\Delta \left(\frac{\partial u}{\partial x_{\varkappa }}\right)=\frac{\partial \Delta u}{\partial x_{\varkappa }}-\sum _{\lambda }\frac{\partial u}{\partial x_{\lambda }}\frac{\partial \Delta x_{\lambda }}{\partial x_{\varkappa }}\mathrm{(}\mathrm{4}\mathrm{)}$

which is $\partial (\Delta u)/\partial x_{\varkappa }$ plus a correction. The total change does not commute with differentiation.

That is the whole reason for her second symbol. Her proof integrates by parts repeatedly, which means moving derivatives across variations dozens of times. With the barred variation every such move is free. With the total change every one would carry a correction term, and the proof would be unreadable.

A treatment that compresses her three symbols into two has therefore not merely lost a distinction. It has lost the property that makes the calculation possible.

Not derived here

UNVERIFIED. The general prolongation formula to arbitrary order, the statement that prolongation is a Lie algebra homomorphism, and the variational bicomplex — the modern framework in which both theorems are corollaries of a statement about a differential complex — are stated in the literature and are not derived on this page. Neither theorem needs any of them, and Noether proves both without the apparatus.

Provenance: (N9) and (4) are Noether's, from her second and third sections, reproduced in her notation from the public-domain German original. The jet-space definitions (1) and (2) are standard and are stated, not derived. The commutation result (3) is derived above, and the observation that it explains her choice of symbol is this volume's own. See Sources.