A particle of mass $m$ is constrained to move on the surface of a sphere of radius $\ell$.

The Lagrangian is given in spherical polar coordinates by

$$L=\frac{1}{2} m \ell^{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+m g \ell \cos \theta,$$

where gravity $g$ is constant. Find the two constants of the motion.

The particle is projected horizontally with velocity $v$ from a point whose depth below the centre is $\ell \cos \theta=D$. Find $v$ such that the particle trajectory

(i) just grazes the horizontal equatorial plane $\theta=\pi / 2$;

(ii) remains at depth $D$ for all time $t$.