I was reading the part that Euler-Lagrange equation holds even on changing the coordinates. In the book by David Morin, the author talks of geometrical picture of the change of coordinates. He makes the following point,
"If you plot a function and then stretch the horizontal axis in an arbitrary manner, a stationery value will be a stationary value after stretching."
My question is, how is change of coordinates equivalent to stretching the horizontal axis?
David Morin's pictorial argument of horizontal stretching is already pretty convincing. Alternatively, covariance follows from the fact that the functional derivative/EL-expression $$ \frac{\delta S[q]}{\delta q^i(t)}~=~\frac{\partial q^{\prime j}}{\partial q^i}\frac{\delta S[q]}{\delta q^{\prime j}(t)}\tag{1}$$ transforms as a co-vector [i.e. a $(0,1)$ tensor] under coordinate transformations $$q^{\prime j}~=~f^j(q,t), \tag{2}$$ cf. e.g. this Phys.SE post.
If one wants to rescale the horizontal axis, then is simply means that one scale the independent variable of a function $$ f(x) \rightarrow f(ax) , $$ where $a$ is an arbitrary constant. For example, if $$ f(x) = \cos(kx) , $$ and $a=2$ then $$ f(2x) = \cos(2kx) . $$ The latter will oscillate twice as fast as the former when plotted as a function of $x$.
