Hamilton's principle/Mathematics/Noether's first theorem

Plain-language version: Symmetry and conservation

Noether's first theorem states that a continuous symmetry of an action integral produces a conserved quantity, one for each parameter of the symmetry group. It subsumes every first integral found piecemeal in this volume so far, and it explains why those integrals existed rather than merely recording that they did.

Provenance note

Symmetry to conservation, four examples.

Noether's 1918 paper is a verified source for this volume, but her mathematics and her prose stand differently. The equations below are her own, from the 1918 original, and are reproduced in her notation. The English wording is a 1971 translation still under copyright, so her prose is paraphrased throughout this page and never transcribed. Where a phrase of hers is characterised, it is characterised, not quoted. See Sources.

Setup and notation

Noether works with independent variables $x_1,\dots ,x_n$ and dependent variables $u_1,\dots ,u_{\mu }$, and with the integral

$I=\int \cdots \int f(x,u,\frac{\partial u}{\partial x},\dots )dx\mathrm{(}\mathrm{1}\mathrm{)}$

She writes ${\psi }_i$ for the Lagrange expressions — the left-hand sides of the Euler-Lagrange equations of Several independent variables — so that the first variation is

$\delta I=\int \cdots \int \sum {\psi }_i\delta u_{i}dx\mathrm{(}\mathrm{2}\mathrm{)}$

when the variations vanish at the boundary. Retaining the boundary terms instead gives an identity free of integrals,

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

where $\mathrm{D}\mathrm{i}\mathrm{v}A=\partial A_1/\partial x_1+\cdots +\partial A_n/\partial x_n$ and $A$ is linear in $\delta u$ and its derivatives. Equation (3) holds for any $f$ and any variation; it is the partial-integration identity of Several independent variables written without discarding anything.

The two kinds of variation

A transformation moves both the independent and dependent variables:

$y_i=x_i+\Delta x_i,v_i(y)=u_i+\Delta u_i\mathrm{(}\mathrm{4}\mathrm{)}$

Noether distinguishes $\Delta u$, the total change, from the variation at fixed argument,

$\delta u_i=v_i(x)-u_i(x)=\Delta u_i-\sum \frac{\partial u_i}{\partial x_{\lambda }}\Delta x_{\lambda }\mathrm{(}\mathrm{5}\mathrm{)}$

Equation (5) is the step most modern accounts compress, and compressing it is where sign errors come from. The correction term accounts for the fact that the field is being compared at a shifted point.

Deriving the divergence relations

Invariance of $I$ under the infinitesimal transformation means the integral is unchanged when computed over the transformed region. Converting that region back to the original gives an extra divergence, and the condition becomes

$\delta f+\mathrm{D}\mathrm{i}\mathrm{v}(f\cdot \Delta x)=0\mathrm{(}\mathrm{6}\mathrm{)}$

Substituting (3) into (6) to eliminate $\delta f$,

$\sum {\psi }_i\delta u_i=\mathrm{D}\mathrm{i}\mathrm{v}B,B=A-f\cdot \Delta x\mathrm{(}\mathrm{7}\mathrm{)}$

For a group with $\rho$ parameters ${\epsilon }_1,\dots ,{\epsilon }_{\rho }$, both $\delta u$ and $B$ are linear in the parameters. Equating coefficients gives $\rho$ separate relations:

$\sum {\psi }_i\delta u_i^{(1)}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(1)},\dots \sum {\psi }_i\delta u_i^{(\rho )}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\rho )}\mathrm{(}\mathrm{8}\mathrm{)}$

Theorem I. If $I$ is invariant under a $\rho$-parameter continuous group, then $\rho$ linearly independent combinations of the Lagrange expressions are divergences. The converse also holds.

Statement paraphrased from Noether 1918, §1, Theorem I. The equations are hers; the English is this volume's.

Passing to conservation laws

Everything above is an identity, true whether or not the field equations hold. Impose them — set ${\psi }_i=0$, which Noether notes is what physicists call the field equations — and (8) collapses to

$\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(1)}=0,\dots \mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\rho )}=0\mathrm{(}\mathrm{9}\mathrm{)}$

These are the conservation laws. Noether observes that such divergence equations had lately come to be called laws of conservation, the phrasing marking how recent the terminology then was.

In one independent variable a divergence is a total derivative, so (9) reads $B^{(k)}=\text{const}$ and gives $\rho$ first integrals. That recovers the results of the Euler-Lagrange equation and the Beltrami identity as the one-dimensional case of a single theorem.

The two-step structure is the theorem's real content. Symmetry alone gives an identity. Symmetry plus the equations of motion gives a conservation law. The identity is the stronger statement and it survives where the field equations do not hold.

The symmetry-to-conservation table

symmetry	generator	conserved quantity
time translation	$\Delta t=\epsilon$	energy
space translation	$\Delta x_k=\epsilon$	momentum, component $k$
rotation	$\Delta \theta =\epsilon$	angular momentum
Galilean boost	$\Delta x=\epsilon t$	centre-of-mass motion
phase rotation of a complex field	$\Delta \psi =i\epsilon \psi$	charge
scaling, when $L$ admits it	$\Delta x=\epsilon x$	virial quantity

The first three rows are the classical integrals, and Hamilton derived exactly them from his characteristic function in 1834, obtaining the law of motion of the centre of gravity and the law of description of areas by considering common translations and rotations of the whole system. That is Noether's theorem for a particular group, eighty-four years early, and it shows how natural the reasoning is once the action is the object of study.

Two cautions

Independent symmetries need not give independent conservation laws. Noether says so explicitly, and gives an example where the first integrals obtained satisfy a nonlinear dependency. The free particle of Several dependent variables is the elementary case: energy conservation there follows from momentum conservation and is not new information.

The symmetry must be a symmetry of the action, not of the equations of motion. These differ. Equations of motion can have symmetries the action lacks, and those produce no conservation law by this theorem. The hypothesis is invariance of $I$.

A third point deserves its own page. Noether's theorem gives nothing useful when the group depends on arbitrary functions rather than on parameters. The "conserved" quantities then turn out to vanish identically or to be mathematically trivial. That failure is not a defect of the theorem; it is the content of the second one, and it is why there are two theorems rather than one.

Next

Noether's second theorem treats the infinite-parameter case and its consequence for general relativity.