# Homework central force Lagrangian formulation

For central force $$T = \frac{m}{2}(\dot{r^2} + r^2\dot{\theta^2} + r^2\sin^2{\theta}\dot{\phi^2}$$

Now applying the Lagrangian equation of motion, we get $$\frac{\partial{L}}{\partial{\phi}}=0\implies p_{\phi}=mr^2\sin^2(\theta)\dot{\phi}$$ which means $p_{\phi}$ is conserved and is the usual angular momentum $\vec{l}$

My textbook then uses the argument (which I didn't understand) that since angle at which $\theta$ is measured is not specified, we choose $\theta=\frac{\pi}{2}$ at $\dot{\theta}=0$ and then reduce this problem to 2-D. So can anyone explain what the textbook meant or in general how I use the $\vec{l}$ to show that motion is constrained to a plane? I don't want to use the argument that $\vec{l}$ is constant and hence motion is constrained to a plane because we don't know beforehand that the $p_\phi$ is indeed $\vec{l}$ and hence can't use the property of cross product. Also, how do I find the direction of $p_\phi$ in general from above formulation?

P.S (The other two Lagrangian equation are obivious and hence I have ommited it but my focus still remain to solve that, also I am aware of newtonian formulation of this problem)