= Noether second theorem
{c}
{title2=$\sum_a[\mathcal E_aR_a-\partial_\mu(\mathcal E_aR_a^\mu)]\equiv0$}
Suppose an action is invariant under a local transformation $\delta\Phi_a=R_a\alpha+R_a^\mu\partial_\mu\alpha$ for every compactly supported function $\alpha$. Variation and integration by parts give the off-shell identity
$$
\sum_a\left(\mathcal E_aR_a-\partial_\mu(\mathcal E_aR_a^\mu)\right)\equiv0.
$$
Thus arbitrary local symmetry parameters imply dependencies among the field equations. In <gauge theory>, these identities accompany constraints rather than independent physical charges for each function. This complements the conserved-current statement of <Noether theorem>.
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