BFP Rauch: 1 revision imported

2026-09-10T13:29:56Z

1 revision imported

← Older revision	Revision as of 13:29, 10 September 2026
(No difference)

NT>NoetherVolumeBuild: Noether theorems volume, initial import (M-02)

2026-09-10T00:00:00Z

Noether theorems volume, initial import (M-02)

New page

{{DISPLAYTITLE:Jet space and prolongation}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Why Noether needed a third symbol|Why Noether needed a third symbol]].''

[[File:NT-F-20.png|thumb|right|360px|alt=Axes showing a curve and, above it, a dashed curve giving its slope. Vertical dotted lines join three points on the curve to the corresponding points on the slope curve. A side panel shows a jet coordinate tuple and notes that the derivatives are treated as independent coordinates.|Jet space: a function and its first jet]]

[[Noether's theorems/First theorem/Prolongation to derivatives|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 <math>x</math> and dependent variables
<math>u</math>. Form the space whose points are

<math>\left(x,\ u,\ u^{(1)},\ u^{(2)},\dots,u^{(k)}\right)\qquad\qquad\mathrm{(1)}</math>

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

A function of <math>x</math> then determines a point of jet space at each
<math>x</math>, by supplying its actual derivatives for the
<math>u^{(j)}</math> — 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''',

<math>D_{x}=\frac{\partial}{\partial x}+u^{(1)}\frac{\partial}{\partial u}+u^{(2)}\frac{\partial}{\partial u^{(1)}}+\dots\qquad\qquad\mathrm{(2)}</math>

which differentiates with respect to <math>x</math> while carrying the chain rule
through the jet coordinates.

'''What this buys.''' Noether's integrand depends on <math>x</math>, on
<math>u</math>, 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
[[Noether's theorems/Mathematics/Lie groups and Lie algebras|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''':

<math>\bar{\delta}u_{i}=v_{i}(x)-u_{i}(x)\qquad\qquad\mathrm{(N9, first part)}</math>

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

<math>\frac{\partial}{\partial x_{\lambda}}\left(\bar{\delta}u_{i}\right)=\frac{\partial v_{i}}{\partial x_{\lambda}}-\frac{\partial u_{i}}{\partial x_{\lambda}}=\bar{\delta}\left(\frac{\partial u_{i}}{\partial x_{\lambda}}\right)\qquad\qquad\mathrm{(3)}</math>

'''The barred variation commutes with differentiation.'''

Now the same question for the total change. From
[[Noether's theorems/First theorem/Prolongation to derivatives|the prolongation
page]],

<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{(4)}</math>

which is <math>\partial(\Delta u)/\partial x_{\varkappa}</math> 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 [[Noether's theorems/Sources|Sources]].''

Noether's theorems/Mathematics/Jet space and prolongation - Revision history

BFP Rauch: 1 revision imported

NT>NoetherVolumeBuild: Noether theorems volume, initial import (M-02)