We consider a spherical pendulum, where a mass $m$ experiencing gravity, is attached to a spherical joint via a massless rod of length $b$. The position of the mass can be completely characterized by specifying the two angular coordinates $\theta$ and $\phi$. We can think of this system as a point mass constraint to move on the surface of the sphere under the influence of gravity. Then we get the Lagrangian: $$L(\theta, \phi, \dot{\theta}, \phi)=\frac{mb^2}{2}\left(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2\right)+mbg\cos(\theta).$$ And we have to calculate the Euler-Lagrange equations associated with $L$ and verify that angular momentum is $$p_{\phi}=mb^2\sin^2(\theta)\dot{\phi}~?$$
I have tried to find the Euler-Lagrange equations, and got:
$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}}\right)=\frac{\partial L}{\partial \theta}$$ $$mb^2 \ddot{\theta}=mb^2\sin(\theta)\cos(\theta)\dot{\phi}^2-mgb \sin(\theta)$$ $$mb^2 \ddot{\theta}=\frac{mb^2}{2}\sin(2\theta)\dot{\phi}^2-mgb \sin(\theta)$$ $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right)=\frac{\partial L}{\partial \phi}$$ $$\frac{d}{dt}(mb^2 \sin^2(\theta) \dot{\phi})=0$$ $$mb^2( \sin^2(\theta) \ddot{\phi}+2\sin(\theta)\cos(\theta) \dot{\phi}\dot{\theta})=0$$ $$mb^2( \sin^2(\theta) \ddot{\phi}+\sin(2\theta) \dot{\phi}\dot{\theta})=0.$$ But how can I use this to show that $p_{\phi}=mb^2\sin^2(\theta)\dot{\phi}$? I can not see a $p$ in my Euler-Lagrange equations. What to do then? And what will the equations of motion be?