The action $S$ of a Hamiltonian system may be regarded as a function of the final coordinates $q^{a}, a=1, \ldots, N$, and the final time $t$ by setting

$$S\left(q^{a}, t\right)=\int_{\left(q_{i}^{a}, t_{i}\right)}^{\left(q^{a}, t\right)} d t^{\prime}\left[p^{a}\left(t^{\prime}\right) \dot{q}^{a}\left(t^{\prime}\right)-H\left(p^{a}\left(t^{\prime}\right), q^{a}\left(t^{\prime}\right), t^{\prime}\right)\right]$$

where the initial coordinates $q_{i}^{a}$ and time $t_{i}$ are held fixed, and $p^{a}\left(t^{\prime}\right), q^{a}\left(t^{\prime}\right)$ are the solutions to Hamilton's equations with Hamiltonian $H$, satisfying $q^{a}(t)=q^{a}, q^{a}\left(t_{i}\right)=q_{i}^{a}$.

(a) Show that under an infinitesimal change of the final coordinates $\delta q^{a}$ and time $\delta t$, the change in $S$ is

$$\delta S=p_{a}(t) \delta q_{a}-H\left(p^{a}(t), q^{a}(t), t\right) \delta t$$

(b) Hence derive the Hamilton-Jacobi equation

$$\frac{\partial S}{\partial t}\left(q^{a}, t\right)+H\left(\frac{\partial S}{\partial q^{a}}\left(q^{a}, t\right), q^{a}, t\right)=0$$

(c) If we can find a solution to $(*)$,

$$S=S\left(q^{a}, t ; P^{a}\right),$$

where $P^{a}$ are $N$ integration constants, then we can use $S$ as a generating function of type $I I$, where

$$p^{a}=\frac{\partial S}{\partial q^{a}} \quad, \quad Q^{a}=-\frac{\partial S}{\partial P^{a}}$$

Show that the Hamiltonian $K$ in the new coordinates $Q^{a}, P^{a}$ vanishes.

(d) Write down and solve the Hamilton-Jacobi equation for the one-dimensional simple harmonic oscillator, where $H=\frac{1}{2}\left(p^{2}+q^{2}\right)$. Show the solution takes the form $S(q, t ; E)=W(q, E)-E t$. Using this as a generating function $F_{I I}(q, t, P)$ show that the new coordinates $Q, P$ are constants of the motion and give their physical interpretation.