The problem states as follow:
Given a parametrization $\alpha:I\rightarrow\mathbb{R}^3$ of class $C^k$ $k\geq1$, $\alpha=(\alpha_1,\alpha_2,\alpha_3)$ verifying $$\alpha_1(t)^2+\alpha_2(t)^2$$ prove that there are three functions $r,\theta,\phi \in C^k(I)$ with $r>0$ such
$$\alpha(t)=r(t)(cos\theta(t)cos\varphi(t), sin\theta(t)cos\varphi(t),sin\varphi(t))$$
My attempt:
$r(t)$ can be defined as $||\alpha(t)||$, so is not restrictive to find the functions $\theta, \varphi$ such
$$\alpha(t)=(cos\theta(t)cos\varphi(t), sin\theta(t)cos\varphi(t),sin\varphi(t))$$
for $\alpha:I\rightarrow\mathbb{S}^2$. Then what I did was to match both expressions coordinate to coordinate \begin{align} \alpha_1 &= cos\theta(t)cos\varphi(t)\\ \alpha_2 &= sin\theta(t)cos\varphi(t)\\ \alpha_3 &= sin\varphi(t)\\ \end{align}
and derivate with respect to $t$
\begin{align} \dot{\alpha_1} &= -\dot{\theta}sin\theta(t)cos\varphi(t) - \dot{\varphi}cos\theta(t)sin\varphi(t)\\ \dot\alpha_2 &= \dot{\theta}cos\theta(t)cos\varphi(t) - \dot{\varphi}sin\theta(t)sin\varphi(t)\\ \dot\alpha_3 &= \dot\varphi(t)cos\varphi(t)\\ \end{align}
Now I have a system of differential equation and I tried to apply some result that gave me the existence and uniqueness of solution. In this line I tried the Picard-Lindelof theorem, as with an expression involving $cos$ and $sin$ I wouldn't have problem to prove the expression are lipchitz. The problem is that I can't obtain any function as the two variables as strongly related and for the version of the theorem I've seen you need an expression of the form $x'=f(t,x)$.
My questions would be:
- Is there any other result on system of differential equations that I can use?
- Is there any better approach to take on the problem?