Lagrange's equations for a system with generalized coordinates $q_{i}(t)$ are given by

$$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0$$

where $L$ is the Lagrangian. The Hamiltonian is given by

$$H=\sum_{j} p_{j} \dot{q}_{j}-L,$$

where the momentum conjugate to $q_{j}$ is

$$p_{j}=\frac{\partial L}{\partial \dot{q}_{j}}$$

Derive Hamilton's equations in the form

$$\dot{q}_{i}=\frac{\partial H}{\partial p_{i}}, \quad \dot{p}_{i}=-\frac{\partial H}{\partial q_{i}}$$

Explain what is meant by the statement that $q_{k}$ is an ignorable coordinate and give an associated constant of the motion in this case.

The Hamiltonian for a particle of mass $m$ moving on the surface of a sphere of radius $a$ under a potential $V(\theta)$ is given by

$$H=\frac{1}{2 m a^{2}}\left(p_{\theta}^{2}+\frac{p_{\phi}^{2}}{\sin ^{2} \theta}\right)+V(\theta)$$

where the generalized coordinates are the spherical polar angles $(\theta, \phi)$. Write down two constants of the motion and show that it is possible for the particle to move with constant $\theta$ provided that

$$p_{\phi}^{2}=\left(\frac{m a^{2} \sin ^{3} \theta}{\cos \theta}\right) \frac{d V}{d \theta} .$$