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

Plain-language version: When symmetry is local

Noether's second theorem treats invariance under a group depending on arbitrary functions rather than on finitely many parameters. The conclusion is different in kind from the first theorem: instead of conservation laws, such invariance produces identities among the field equations themselves. The theorem is less often taught and it is the one Noether wrote the paper to prove.

The provenance rule of Hamilton's principle/Mathematics/Noether's first theorem applies here too: Noether's equations are reproduced, her English is paraphrased.

Finite against infinite groups

Identities among field equations from a local symmetry.

Noether calls a group finite continuous, $G_{\rho }$, when its transformations depend analytically on $\rho$ parameters, and infinite continuous, $G_{\mathrm{\infty }\rho }$, when they depend on $\rho$ arbitrary functions $p(x)$ and their derivatives.

The distinction is the difference between a rotation, the same everywhere, and a coordinate change that may be chosen independently at each point. Physics calls the second a local or gauge symmetry.

Statement

Theorem II. If $I$ is invariant under a $G_{\mathrm{\infty }\rho }$ in which the arbitrary functions occur up to the $\sigma$-th derivative, then there are $\rho$ identity relationships among the Lagrange expressions and their derivatives up to order $\sigma$. The converse holds.

Statement paraphrased from Noether 1918, §1, Theorem II.

Derivation

Begin from the same relation (7) of Hamilton's principle/Mathematics/Noether's first theorem, $\sum {\psi }_i\delta u_i=\mathrm{D}\mathrm{i}\mathrm{v}B$. Now $\delta u$ is linear in the arbitrary functions and their derivatives:

$\sum {\psi }_i\delta u_i=\sum _{\lambda ,i}{\psi }_i(a_i^{(\lambda )}{p}^{(\lambda )}+b_i^{(\lambda )}\frac{\partial p^{(\lambda )}}{\partial x}+\cdots +c_i^{(\lambda )}\frac{{\partial }^{\sigma }{p}^{(\lambda )}}{\partial x^{\sigma }})\mathrm{(}\mathrm{1}\mathrm{)}$

Move every derivative off $p$ by repeated partial integration, using

$\phi \frac{{\partial }^{\tau }p}{\partial x^{\tau }}=(-1{)}^{\tau }\frac{{\partial }^{\tau }\phi }{\partial x^{\tau }}p\text{mod divergences}\mathrm{(}\mathrm{2}\mathrm{)}$

which gives

$\sum {\psi }_i\delta u_i=\sum _{\lambda }[(a_i^{(\lambda )}{\psi }_i)-\frac{\partial }{\partial x}(b_i^{(\lambda )}{\psi }_i)+\cdots +(-1{)}^{\sigma }\frac{{\partial }^{\sigma }}{\partial x^{\sigma }}(c_i^{(\lambda )}{\psi }_i)]p^{(\lambda )}+\mathrm{D}\mathrm{i}\mathrm{v}\Gamma \mathrm{(}\mathrm{3}\mathrm{)}$

Combining with $\sum {\psi }_i\delta u_i=\mathrm{D}\mathrm{i}\mathrm{v}B$, the bracket multiplied by $p$ equals $\mathrm{D}\mathrm{i}\mathrm{v}(B-\Gamma )$.

The closing step is the arbitrariness of $p$. Integrate over any region and choose the $p$ so that they and enough of their derivatives vanish on the boundary. The divergence integrates to zero. The remaining integral vanishes for arbitrary $p$, so by the fundamental lemma of Hamilton's principle/Mathematics/The fundamental lemma the bracket itself vanishes:

$\sum [(a_i^{(\lambda )}{\psi }_i)-\frac{\partial }{\partial x}(b_i^{(\lambda )}{\psi }_i)+\cdots +(-1{)}^{\sigma }\frac{{\partial }^{\sigma }}{\partial x^{\sigma }}(c_i^{(\lambda )}{\psi }_i)]=0,\lambda =1,\dots ,\rho \mathrm{(}\mathrm{4}\mathrm{)}$

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Equation (4) holds identically — before any field equation is imposed. That is the whole difference from the first theorem, where the corresponding relation became useful only after setting ${\psi }_i=0$.

What an identity means

(4) says the Lagrange expressions are not independent. $\rho$ of the field equations follow from the rest. Three consequences:

The field equations do not determine the evolution. With $\rho$ redundancies, the initial data leave $\rho$ functions free for all time. That freedom is the gauge freedom, and it is not ignorance about the system — it is a statement that distinct solutions describe the same physics.

The Hessian is singular. If $\rho$ combinations of the field equations are identities, the velocities cannot all be solved for in terms of the momenta. This is exactly the degeneracy that blocks the Legendre transform on Hamilton's principle/Mathematics/The Legendre transform. Gauge invariance and Legendre degeneracy are the same fact seen from two sides.

Counting must be done carefully. The naive count of degrees of freedom overcounts by $\rho$, and again by $\rho$ for the constraints among the momenta.

Noether's own examples

She gives the Weierstrass parametric representation as the simplest case: an integral homogeneous of first order in the derivatives is invariant under reparametrisation $y=p(x)$, one arbitrary function without derivatives, and the corresponding identity is

$\sum {\psi }_i\frac{d{u}_i}{dx}=0\mathrm{(}\mathrm{5}\mathrm{)}$

a single linear relation among the Lagrange expressions. Any action written in parametrised form, including the relativistic point particle, carries (5).

Her second example is general relativity: the group of all coordinate transformations $y_i=p_i(x)$, with the metric transforming as induced, and the arbitrary functions entering to first derivative. The identities are then the $n$ relations among the Lagrange expressions and their first derivatives — in modern terms the contracted Bianchi identities, which make the Einstein tensor identically divergence free. This is worked on Hamilton's principle/Applications/The Einstein-Hilbert action.

Proper and improper conservation laws

Noether's §6 is the sharpest part of the paper and the least known.

Suppose $I$ is invariant under a $G_{\mathrm{\infty }\rho }$, and take a finite subgroup $G_{\sigma }$ obtained by specialising the arbitrary functions. The first theorem gives conservation laws for $G_{\sigma }$. But those laws must be consequences of the identities (4), since the identities already imply invariance under any such specialisation.

Noether therefore classifies divergence relations as improper when the $B^{(\lambda )}$ are built from the Lagrange expressions and their derivatives, and proper otherwise. An improper conservation law holds for a reason that has nothing to do with dynamics: it is an identity in disguise.

Specialise to the displacement group $\Delta x_i={\epsilon }_i$, whose invariance means $f$ contains no explicit $x$. The resulting relations,

$\sum {\psi }_i\frac{\partial u_i}{\partial x_{\lambda }}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}\mathrm{(}\mathrm{6}\mathrm{)}$

are the energy relationships, and the $B^{(\lambda )}$ are the energy components. Noether's conclusion:

The energy relationships are improper if and only if $I$ is invariant under an infinite group containing the displacement group as a subgroup.

Statement paraphrased from Noether 1918, §6.

Classical mechanics and special relativity carry no infinite group, so their energy conservation is proper. General relativity does, so its energy conservation is improper — an identity rather than a dynamical law.

The Hilbert assertion

That result was the point of the paper. Hilbert had asserted that the failure of proper energy conservation is characteristic of general relativity, and the claim was contested in the Göttingen debates of 1916 to 1918. Noether's second theorem settles it, and generalises it: the property is not special to relativity but belongs to any theory invariant under a group of arbitrary functions.

She adds the observation that for the assertion to hold literally, the term general relativity would have to be read more broadly than usual, extending to every such group. The substance is group-theoretic, and the physical name is incidental to it.

Provenance: this section paraphrases Noether 1918, §6, and its framing of the Hilbert assertion. Klein's two 1918 notes, which she cites and which carry the other side of the exchange, are not sources of this volume and are not cited. The Noether row of the survey on Hamilton's principle/Sources is ESTABLISHED.

Why the second theorem is the neglected one

The first theorem gives physicists a tool. The second tells them a class of conservation laws they might have hoped for does not exist in the form expected. The first is quoted constantly and the second rarely, and the imbalance misrepresents the paper: the two theorems are stated together, proved together, and the second is the one that answers the question Noether was asked.

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Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton closes the structure track by separating two principles that are routinely conflated.