Let S be a functional $$S=\int{dt\left(\frac{1}{2} m\dot{q}^2-\frac{\alpha}{q^2}\right)} \;.$$ I am trying to apply the following transformations and find a conserved quantity using Noether's theorem, but having much trouble! $$t\to t^\prime=\lambda t$$ and $$q(t)\to q^\prime(t^\prime)=\sqrt{\lambda}q(t) \;.$$ I don't really understand how to apply the transformations, but I do think that once that is done I can find $S^\prime - S$ and with that find the conserved quantity using Noehter's theorem. Is this analysis ok?
Any help is much appreciated!
