(a) Write the Hamiltonian for a spherical pendulum of length \(\ell\) and mass \(m\), using the polar angle \(\theta\) and azimuthal angle \(\varphi\) as generalized coordinates.

(b) Then write out Hamilton's equations of motion, and identify two first-integrals of motion.

(c) Find a first-order differential equation of motion involving \(\theta\) alone and its first time derivative.

(d) Sketch contours of constant \(H\) in the \(\theta, p_{\theta}\) phase plane, and use it to identify the types of motion one expects.