Editing Noether's theorem (section)

=== Generalization of the proof ===
This applies to ''any'' local symmetry derivation ''Q'' satisfying ''QS''&nbsp;≈&nbsp;0, and also to more general local functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields. Let ''ε'' be any arbitrary smooth function of the spacetime (or time) manifold such that the closure of its support is disjoint from the boundary. ''ε''&nbsp;is a [[test function]]. Then, because of the variational principle (which does ''not'' apply to the boundary, by the way), the derivation distribution q generated by ''q''[''ε''][Φ(''x'')] = ''ε''(''x'')''Q''[Φ(''x'')] satisfies ''q''[''ε''][''S'']&nbsp;≈&nbsp;0 for every&nbsp;''ε'', or more compactly, ''q''(''x'')[''S'']&nbsp;≈ 0&nbsp;for all ''x'' not on the boundary (but remember that ''q''(''x'') is a shorthand for a derivation ''distribution'', not a derivation parametrized by ''x'' in general). This is the generalization of Noether's theorem.

To see how the generalization is related to the version given above, assume that the action is the spacetime integral of a Lagrangian that only depends on <math>\varphi</math> and its first derivatives. Also, assume

:<math>Q[\mathcal{L}]\approx\partial_\mu f^\mu</math>

Then,

:<math>
\begin{align}
q[\varepsilon][\mathcal{S}] & = \int q[\varepsilon][\mathcal{L}] d^{n} x  \\[6pt]
& = \int \left\{ \left(\frac{\partial}{\partial \varphi}\mathcal{L}\right) \varepsilon Q[\varphi]+ \left[\frac{\partial}{\partial (\partial_\mu \varphi)}\mathcal{L}\right]\partial_\mu(\varepsilon Q[\varphi]) \right\} d^{n} x \\[6pt]
& = \int \left\{ \varepsilon Q[\mathcal{L}] + \partial_{\mu}\varepsilon \left[\frac{\partial}{\partial \left( \partial_\mu \varphi\right)} \mathcal{L} \right] Q[\varphi] \right\} \, d^{n} x \\[6pt]
& \approx \int \varepsilon \partial_\mu \left\{f^\mu-\left[\frac{\partial}{\partial (\partial_\mu\varphi)}\mathcal{L}\right]Q[\varphi]\right\} \, d^{n} x
\end{align}
</math>

for all <math>\varepsilon</math>.

More generally, if the Lagrangian depends on higher derivatives, then

:<math>
  \partial_\mu\left[
       f^\mu
    -  \left[\frac{\partial}{\partial (\partial_\mu \varphi)} \mathcal{L} \right] Q[\varphi]
    - 2\left[\frac{\partial}{\partial (\partial_\mu \partial_\nu \varphi)} \mathcal{L}\right]\partial_\nu Q[\varphi]
    +  \partial_\nu\left[\left[\frac{\partial}{\partial (\partial_\mu \partial_\nu \varphi)}\mathcal{L}\right] Q[\varphi]\right]
    -  \,\dotsm
  \right] \approx 0.
</math>