For a particle of unit mass moving freely on a unit sphere, the Lagrangian in polar coordinates is

$$L=\frac{1}{2} \dot{\theta}^{2}+\frac{1}{2} \sin ^{2} \theta \dot{\phi}^{2}$$

Determine the equations of motion. Show that $l=\sin ^{2} \theta \dot{\phi}$ is a conserved quantity, and use this result to simplify the equation of motion for $\theta$. Deduce that

$$h=\dot{\theta}^{2}+\frac{l^{2}}{\sin ^{2} \theta}$$

is a conserved quantity. What is the interpretation of $h$ ?