(i) In Hamiltonian mechanics the action is written

$$S=\int d t\left(p^{a} \dot{q}^{a}-H\left(q^{a}, p^{a}, t\right)\right)$$

Starting from Maupertius' principle $\delta S=0$, derive Hamilton's equations

$$\dot{q}^{a}=\frac{\partial H}{\partial p^{a}}, \quad \dot{p}^{a}=-\frac{\partial H}{\partial q^{a}} .$$

Show that $H$ is a constant of the motion if $\partial H / \partial t=0$. When is $p^{a}$ a constant of the motion?

(ii) Consider the action $S$ given in Part (i), evaluated on a classical path, as a function of the final coordinates $q_{f}^{a}$ and final time $t_{f}$, with the initial coordinates and the initial time held fixed. Show that $S\left(q_{f}^{a}, t_{f}\right)$ obeys

$$\frac{\partial S}{\partial q_{f}^{a}}=p_{f}^{a}, \quad \frac{\partial S}{\partial t_{f}}=-H\left(q_{f}^{a}, p_{f}^{a}, t_{f}\right)$$

Now consider a simple harmonic oscillator with $H=\frac{1}{2}\left(p^{2}+q^{2}\right)$. Setting the initial time and the initial coordinate to zero, find the classical solution for $p$ and $q$ with final coordinate $q=q_{f}$ at time $t=t_{f}$. Hence calculate $S\left(t_{f}, q_{f}\right)$, and explicitly verify (2) in this case.