Noether's theorems/First theorem/The action and the variational setup

Plain-language counterpart: What an action is.

A varied path against a symmetry-transformed path

Everything in both theorems rests on one object and one condition on it. This page fixes the object, the notation, and what invariance is going to mean. The notation is Noether's own; the notation page gives the modern equivalents.

The variables

Let $x_1,\dots ,x_n$ be independent variables and $u_1(x),\dots ,u_{\mu }(x)$ functions of them. Noether assumes every function appearing is analytic, or at least continuous and finitely often continuously differentiable, and single-valued on the region considered.

Two abbreviations of hers are used throughout and are easy to misread. She suppresses indices wherever she can, including under summation signs, so ${\partial }^{2}u/\partial x^2$ stands for ${\partial }^2{u}_{\alpha }/\partial x_{\beta }\partial x_{\gamma }$ and similarly at higher order. And she writes $dx$ for $d{x}_1\dots d{x}_n$. Neither is a restriction; both are shorthand.

What a transformation does

Subject the $x$ and $u$ to the transformations of a group. Because the transformations are invertible, exactly $n$ independent variables must again be present among the transformed quantities; call them $y_1,\dots ,y_n$, and call the remaining dependent ones $v_1(y),\dots ,v_{\mu }(y)$. The transformations may also involve the derivatives of the $u$ with respect to the $x$. That possibility is not decoration; it is what prolongation is about, and it is one of the places where her statement is more general than the textbook one.

A function is called an invariant of the group when

$P(x,u,\frac{\partial u}{\partial x},\frac{{\partial }^{2}u}{\partial x^2},\dots )=P(y,v,\frac{\partial v}{\partial y},\frac{{\partial }^{2}v}{\partial y^2},\dots )\mathrm{(}\mathrm{1}\mathrm{)}$

The integral, and what invariance of it means

The object both theorems are about is an integral

$I=\int \dots \int f(x,u,\frac{\partial u}{\partial x},\frac{{\partial }^{2}u}{\partial x^2},\dots )dx\mathrm{(}\mathrm{2}\mathrm{)}$

taken over an arbitrary real region of $x$. Physics calls $f$ the Lagrangian density and $I$ the action. Noether calls them the integrand and the integral, and nothing in either theorem depends on the physical reading.

The integral is an invariant of the group when

$I=\int \dots \int f(x,u,\frac{\partial u}{\partial x},\frac{{\partial }^{2}u}{\partial x^2},\dots )dx=\int \dots \int f(y,v,\frac{\partial v}{\partial y},\frac{{\partial }^{2}v}{\partial y^2},\dots )dy\mathrm{(}\mathrm{N}\mathrm{1}\mathrm{)}$

integrated over an arbitrary real $x$-region and the corresponding $y$-region.

Read the equality carefully. It is not a statement that $f$ is unchanged. It is a statement that the integral of $f$ over a region equals the integral of the same functional form over the transformed region, with the transformed arguments. The integrand may change; the two changes — in the function and in the region — must cancel.

Three remarks that matter later

The condition is on the integral, not on the equations of motion. This is the single most common error in accounts of the theorems, and it has its own page. A transformation that carries solutions to solutions need not satisfy (N1), and if it does not, no conservation law follows from it.

Both theorems take (N1) as given and ask what follows. Neither theorem tells you which groups leave a given integral invariant. Finding those is a separate problem, and Noether treats it only in examples.

Invariance up to a divergence is not (N1). The weakening that admits Galilean boosts is Bessel-Hagen's, from 1921, not hers, and this volume keeps the attribution straight. See the page on the invariance condition.

Arguments and coefficients: Noether takes the arguments $x,u,\epsilon ,p(x)$ real, while allowing the coefficients to be complex. Because the final results are identities in those quantities, they hold for complex values too, provided every function appearing is analytic.

Provenance: (N1) is Noether's equation (1), reproduced in her notation from the German original, which is public domain. Equations (1) and (2) above are this page's own restatements of definitions she gives in prose. No English wording of hers is transcribed. See Sources.