Are there any relation between two parametric equations of a regular curve? For example, let $\alpha : [a,b] \to \mathbb{R}^2$ and $\beta : [c,d] \to \mathbb{R}^2$ be the parametric equations of a regular curve $C.$ Is there a $C^1$ function $\phi : [a,b] \to [c,d]$ such that $\alpha =\beta \circ \phi?$ [Or $\phi$ is differentiable and $\phi'$ is integrable on $[a,b]?$]
It seems that the existence of $C^1$ function $\phi$ is not guaranteed, but I would be thankful for any comments about this question.
Except at stationary or multiple points of the curve, $\beta$ is invertible and you can write
$$v=\beta^{-1}(\alpha(u))=\phi(u)$$
Then
$$\alpha(u)=\beta(\phi(u)).$$