Noether's second theorem

Remarks

Invariance up to a divergence gives $\sum_\alpha E_\alpha(u) \, \delta u_\alpha = \operatorname{div} S$ for some expression $S$ that is linear in the $g_j$ and their derivatives. Integrating by parts moves the derivatives from $g_j$ onto $E_\alpha$, so $\sum_j g_j \sum_\alpha \mathcal{D}^*_{\alpha j} E_\alpha$ is a divergence too. An expression of the form $\sum_j g_j F_j$ that is a divergence for all compactly supported $g_j$ must have $F_j = 0$.

Example: the length $J(x, y) = \int_a^b \sqrt{\dot x^2 + \dot y^2} \, dt$ of a regular plane curve, one with $(\dot x, \dot y) \ne 0$ everywhere, is invariant under reparametrisations $t \mapsto t + \varepsilon g(t)$. The variations are $\delta x = \dot x \, g$ and $\delta y = \dot y \, g$, up to sign, so here $\mathcal{D}$ is multiplication by $\dot x$ and $\dot y$. The Euler–Lagrange expressions are $E_x = -\frac{d}{dt} T_1$ and $E_y = -\frac{d}{dt} T_2$, with $T = (\dot x, \dot y) / \abs{(\dot x, \dot y)}$. The identity is $\dot x \, E_x + \dot y \, E_y = 0$ for every regular curve; it holds because $T \cdot \dot T = 0$.

Unlike the first Noether's theorem, the result gives identities that hold for all $u$, not conservation laws along solutions. So the Euler–Lagrange equations are not independent, and their solutions are not determined by initial data alone.

Going deeper

For a student

This is the theorem behind gauge theories and general relativity. In electromagnetism, gauge invariance gives the identity that the divergence of the Maxwell equations vanishes identically, which is linked to charge conservation. In general relativity, invariance under diffeomorphisms gives the contracted Bianchi identities. The identities explain why such equations are underdetermined and need gauge fixing before one can solve an initial value problem.