(i) Derive Hamilton's equations from Lagrange's equations. Show that the Hamiltonian $H$ is constant if the Lagrangian $L$ does not depend explicitly on time.

(ii) A particle of mass $m$ is constrained to move under gravity, which acts in the negative $z$-direction, on the spheroidal surface $\epsilon^{-2}\left(x^{2}+y^{2}\right)+z^{2}=l^{2}$, with $0<\epsilon \leqslant 1$. If $\theta, \phi$ parametrize the surface so that

$$x=\epsilon l \sin \theta \cos \phi, y=\epsilon l \sin \theta \sin \phi, z=l \cos \theta,$$

find the Hamiltonian $H\left(\theta, \phi, p_{\theta}, p_{\phi}\right)$.

Show that the energy

$$E=\frac{p_{\theta}^{2}}{2 m l^{2}\left(\epsilon^{2} \cos ^{2} \theta+\sin ^{2} \theta\right)}+\frac{\alpha}{\sin ^{2} \theta}+m g l \cos \theta$$

is a constant of the motion, where $\alpha$ is a non-negative constant.

Rewrite this equation as

$$\frac{1}{2} \dot{\theta}^{2}+V_{\mathrm{eff}}(\theta)=0$$

and sketch $V_{\mathrm{eff}}(\theta)$ for $\epsilon=1$ and $\alpha>0$, identifying the maximal and minimal values of $\theta(t)$ for fixed $\alpha$ and $E$. If $\epsilon$ is now taken not to be unity, how do these values depend on $\epsilon$ ?