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Hello everyone,

I tried to describe my question but it seems getting too complicated and confusing to write down my thoughts in detail, so I am trying to start with the following question...

Are invariance of the Lagrangian under a transformation and invariance of the action equivalent to each other? or even the same?

Physiana

P.S.: I know it sounds stupid. But I was wondering if there is a difference in physics by demanding the Lagrangian being invariant or just transforming in a certain way. As for example for time dilatation [itex]

t \rightarrow t' = \lambda t

[/itex] the Lagrangian transforms as[itex]

L \rightarrow L'= \frac{L}{\lambda}

[/itex] which is not "exactly" invariant, although "a" is constant and probably does not really matter. So I wondered if there are more transformations that "change" the Lagrangian but leave the action invariant.

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# Invariant Lagrangian or action

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