Let $\mathbf{r}:[a,b]\to\mathbb{R}^2,\ a,b\in\mathbb{R}, a<b$, be a $C^{\infty}([a,b])$ application (smooth). Is it true that we can always find two functions $r,\varphi :[a,b]\to\mathbb{R}$, $r\in C^{0}([a,b])$ (continuous) and $\varphi\in C^{\infty}([a,b])$ such that:
$$\mathbf{r}(t)=r(t)(\cos\varphi(t),\sin\varphi (t)),\ \forall\ t\in [a,b]$$
? What can be said about the uniqueness of $r$ and $\varphi$, given $\varphi (t_0)=\varphi_0$?