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Is this the same idea as treating [itex]\phi[/itex] and [itex]\partial \phi[/itex] as independent variables in the Lagrangian? Formally, we view the Lagrangian as a function of two variables, and we can differentiate with respect to the two slots as we please, but at the end of the day, we will plug [itex]\phi (t)[/itex] and [itex] \partial \phi (t) [/itex] into the slots. Physicists abbreviate this procedure as treating [itex]\phi[/itex] and [itex]\partial \phi[/itex] as independent variables.

Is this correct?