The [url=http://en.wikipedia.org/wiki/Lagrangian_mechanics]lagrange equation[/url] for the system is L=T-V = \tfrac{1}{2}m (L\dot\theta)^2+\tfrac{1}{2}m (L\sin\theta \dot{\phi})^2- (-mgL\cos\theta)

from \frac{d}{dt}(\frac{\partial L}{\partial \dot{\phi}})-\frac{\partial L}{\partial \phi}=0
you will find m L^2 \sin\theta^2 \dot{\phi}=const


The reason is:
since there is no \phi in lagrange equation so \frac{\partial L}{\partial \phi}=0
which mean \frac{d}{dt}(\frac{\partial L}{\partial \dot{\phi}})=0 ,
so (\frac{\partial L}{\partial \dot{\phi}}) must be a constant. i.e. m L^2 \sin\theta^2 \dot{\phi}=const