To be a little specific, here is the question that I am trying to solve:
Determine, with justification, if the following curves give a reparametrization of the curve $γ : (0, ∞) → \mathbb{R^2}, γ(t) = (\cos t^2,\sin t^2) $
$γ_1(t) = (\sin t,\cos t)$, where $γ_1 : (0, ∞) → \mathbb{R^2}$
$γ_2(t) = (\cos^2t,\sin^2t)$, where $γ_2 : (0, ∞) → \mathbb{R^2}$
Now, let me summarize what I have done to make things a little clear.
For question 1, to see that $γ_1$ is a reparametrization of $γ$, we have to find a reparametrization map $φ$ such that $(\cos φ(t)^2,\sin φ(t)^2) = (\sin t, \cos t).$ We observe that $φ(t)=\pm\sqrt{(\pi/2-t)}$ which is clearly not smooth in $[\pi/2,\infty)$. Reparametrization maps have to be smooth. Since $φ$ is not smooth, hence $\gamma_1$ can't be a reparametrization of $\gamma$.
For question 2, we see that the level curve associated with $\gamma_2$ and $\gamma$ are different. $\gamma$ satisfies the level curve $x^2+y^2=1$ while $\gamma_2$ doesn't. So $\gamma_2$ can't be a reparametrization of $\gamma$.
Now my question is whether or not there is a concrete way to determine whether or not a curve is the reparametrization of another curve or is it just intelligent guesswork?