Correct way for symmetry transform

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Homework Statement



Hi
I'm trying to understand how symmetry transform works.
Suppose a lagrangian [itex]L = q^{-2}[/itex]
(actually it had another kinetic member, but I dont need it for my question here)

The matching action [itex]S = \int dt q^{-2}[/itex]

We were told that it has the next symmetry
[itex]t \rightarrow at[/itex]
[itex]q \rightarrow a^{-1/2}q[/itex]
I tried to figure how, but I cant get it.

The Attempt at a Solution



I thought the correct way is this
[itex]T = at[/itex]
[itex]Q(T) = a^{-1/2}q(T)[/itex]

and now I put it in the action
[itex]S = \int dt q^{-2} = \int (dT/a) (q(T))^{-2} = \int (dT/a) (a^{+1/2}*Q(T))^{-2} = \int (dT/a) (a^{-1})*(Q(T))^{-2} = a^{-2}*S ≠ S[/itex]

Is that the correct way to do it? Obviously I could change the power of a in t/q transform and it would fix it, is that it?

Thanks a lot for reading
Tamir
 

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