While one are using Hamilton Principle on density Lagrangian, he may find this
$$\delta (\partial_{\mu} \phi_{i}) = \partial_{\mu} (\delta \phi_{i})$$
Why this statement are true?
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Related: Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $.The argument is essentially the same in field theory. – Qmechanic Jun 05 '20 at 14:39