Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$.
However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (where $\alpha$ is any constant) and $L + \frac{df}{dt}$ (where $f$ is any function) lead to the same equations of motion.
Can we then consider that the Lagrangian is invariant under a transformation if we find $\delta L=\alpha L$ or $\delta L=\frac{df}{dt}$ instead of $\delta L=0$?
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