The Lagrange expressions and the first variation

Plain-language counterpart: Nudging a path and seeing what changes.

This page does the work of the ordinary calculus of variations and stops one step short of where textbooks stop. That step is the whole point.

The first variation

Take any integral of the form (2) on the previous page — invariant or not. Form its first variation $\delta I$ and reduce it by integration by parts. Assuming $\delta u$ and all its derivatives that appear vanish on the boundary, and are otherwise arbitrary, this gives

$\delta I=\int \dots \int \delta fdx=\int \dots \int (\sum {\psi }_i(x,u,\frac{\partial u}{\partial x},\dots )\delta u_i)dx\mathrm{(}\mathrm{N}\mathrm{2}\mathrm{)}$

The ${\psi }_i$ are what Noether calls the Lagrange expressions. They are the left-hand sides of the Lagrange equations of the associated variational problem $\delta I=0$.

They are expressions, not equations. Setting ${\psi }_i=0$ is what physics calls imposing the field equations. Noether does not do it here and does not do it in either proof. Everything up to and including both theorems is carried with the ${\psi }_i$ left standing. That is why her results are identities, and it is the reason the second theorem says anything at all.

The integral-free identity

Relation (N2) holds only under the boundary assumption. Drop the assumption and write the boundary terms out, and what results is an identity in $\delta u$ and its derivatives, with no integral sign anywhere.

Integration by parts shows those boundary terms to be integrals over divergences, that is, over expressions of the form

$\mathrm{D}\mathrm{i}\mathrm{v}A=\frac{\partial A_1}{\partial x_1}+\dots +\frac{\partial A_n}{\partial x_n}\mathrm{(}\mathrm{1}\mathrm{)}$

with $A$ linear in $\delta u$ and its derivatives. Therefore

$\sum {\psi }_i\delta u_i=\delta f+\mathrm{D}\mathrm{i}\mathrm{v}A\mathrm{(}\mathrm{N}\mathrm{3}\mathrm{)}$

This is the workhorse. It holds identically, for every integral, invariant or not, on-shell or off. Both theorems are obtained by combining it with an invariance condition and nothing else.

The three special cases

Noether writes (N3) out in three cases, and the third is the one that fixes the generality of her result.

For a simple integral with $f$ containing only first derivatives, (N3) becomes the relation Heun called the Lagrangian central equation:

$\sum {\psi }_i\delta u_i=\delta f-\frac{d}{dx}(\sum \frac{\partial f}{\partial u{\text{'}}_i}\delta u_i),(u_{i^{\prime }}=\frac{d{u}_i}{dx})\mathrm{(}\mathrm{N}\mathrm{4}\mathrm{)}$

For the $n$-fold integral:

$\sum {\psi }_i\delta u_i=\delta f-\frac{d}{d{x}_1}(\sum \frac{\partial f}{\partial \frac{\partial u_i}{\partial x_1}}\delta u_i)-\dots -\frac{d}{d{x}_n}(\sum \frac{\partial f}{\partial \frac{\partial u_i}{\partial x_n}}\delta u_i)\mathrm{(}\mathrm{N}\mathrm{5}\mathrm{)}$

For a simple integral with $\varkappa$ derivatives of the $u$:

$\sum {\psi }_i\delta u_i=\delta f-\frac{d}{dx}\{\sum (\left(\genfrac{}{}{0ex}{}{1}{1}\right)\frac{\partial f}{\partial u_i^{(1)}}\delta u_i+\left(\genfrac{}{}{0ex}{}{2}{1}\right)\frac{\partial f}{\partial u_i^{(2)}}\delta u_i^{(1)}+\dots +\left(\genfrac{}{}{0ex}{}{\varkappa }{1}\right)\frac{\partial f}{\partial u_i^{(\varkappa )}}\delta u_i^{(\varkappa -1)})\}+\dots +(-1{)}^{\varkappa }\frac{d^{\varkappa }}{d{x}^{\varkappa }}\{\sum \left(\genfrac{}{}{0ex}{}{\varkappa }{\varkappa }\right)\frac{\partial f}{\partial u_i^{(\varkappa )}}\delta u_i\}\mathrm{(}\mathrm{N}\mathrm{6}\mathrm{)}$

A corresponding identity holds for the $n$-fold integral, and there $A$ contains $\delta u$ up to the $(\varkappa -1)$th derivative.

A transcription note. In the electronic text of the German original the first bracket of (N5) reads with $x_i$ where the pattern $d/d{x}_1,\dots ,d/d{x}_n$ requires $x_1$. It is written $x_1$ above. Whether the slip is the 1918 printing's or the transcription's has not been checked against the page images, and Sources records it as open.

Why these define the Lagrange expressions

Noether points out that (N4), (N5) and (N6) actually define the ${\psi }_i$, and gives the reason in one sentence: the combinations on the right eliminate all higher derivatives of the $\delta u$, while relation (N2), to which integration by parts leads uniquely, is satisfied.

That is worth restating because it is the licence for everything after. The ${\psi }_i$ are not postulated and are not borrowed from physics. They are whatever the reduction produces, and the reduction is unique.

Provenance: (N2) to (N6) are Noether's equations (2) to (6), reproduced in her notation from the public-domain German original, with the one deviation noted above. Equation (1) is her definition of a divergence, set as a numbered display by this volume. Her prose is paraphrased. See Sources.