(a) In the Hamiltonian framework, the action is defined as

$$S=\int\left(p_{a} \dot{q}_{a}-H\left(q_{a}, p_{a}, t\right)\right) d t$$

Derive Hamilton's equations from the principle of least action. Briefly explain how the functional variations in this derivation differ from those in the derivation of Lagrange's equations from the principle of least action. Show that $H$ is a constant of the motion whenever $\partial H / \partial t=0$.

(b) What is the invariant quantity arising in Liouville's theorem? Does the theorem depend on assuming $\partial H / \partial t=0$ ? State and prove Liouville's theorem for a system with a single degree of freedom.

(c) A particle of mass $m$ bounces elastically along a perpendicular between two parallel walls a distance $b$ apart. Sketch the path of a single cycle in phase space, assuming that the velocity changes discontinuously at the wall. Compute the action $I=\oint p d q$ as a function of the energy $E$ and the constants $m, b$. Verify that the period of oscillation $T$ is given by $T=d I / d E$. Suppose now that the distance $b$ changes slowly. What is the relevant adiabatic invariant? How does $E$ change as a function of $b$ ?