Consider the action of a free particle in the space $$ s=\int_{t_1}^{t_2} \frac{m v^2}{2} d t.\tag{*} $$ The change of time coordinates $t'=\alpha t$, where $\alpha\in(0,1]$, preserves the form of the Euler-Lagrange equations. In fact, the action under the new coordinates is $$ s' = \alpha\int_{t_1'}^{t_2'} \frac{m}{2} {v'}^2 d t'. $$ Such an integral has the following Euler lagrange equations $$ \alpha m\frac{d}{dt'} v' = 0 $$

However, the Noether's theorem for equation (*) states that such an integral is only invariat with respect to time translations.

Is this an special kind of invariance? which is the different between Noether's type invariances and this?


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