# Invariance with respect to time dilations of the free particle

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?