Given a Hamiltonian system with variables $\left(q_{i}, p_{i}\right), i=1, \ldots, n$, state the definition of a canonical transformation

$$\left(q_{i}, p_{i}\right) \rightarrow\left(Q_{i}, P_{i}\right),$$

where $\mathbf{Q}=\mathbf{Q}(\mathbf{q}, \mathbf{p}, t)$ and $\mathbf{P}=\mathbf{P}(\mathbf{q}, \mathbf{p}, t)$. Write down a matrix equation that is equivalent to the condition that the transformation is canonical.

Consider a harmonic oscillator of unit mass, with Hamiltonian

$$H=\frac{1}{2}\left(p^{2}+\omega^{2} q^{2}\right) .$$

Write down the Hamilton-Jacobi equation for Hamilton's principal function $S(q, E, t)$, and deduce the Hamilton-Jacobi equation

$$\frac{1}{2}\left[\left(\frac{\partial W}{\partial q}\right)^{2}+\omega^{2} q^{2}\right]=E$$

for Hamilton's characteristic function $W(q, E)$.

Solve (1) to obtain an integral expression for $W$, and deduce that, at energy $E$,

$$S=\sqrt{2 E} \int d q \sqrt{\left(1-\frac{\omega^{2} q^{2}}{2 E}\right)}-E t$$

Let $\alpha=E$, and define the angular coordinate

$$\beta=\left(\frac{\partial S}{\partial E}\right)_{q, t}$$

You may assume that (2) implies

$$t+\beta=\left(\frac{1}{\omega}\right) \arcsin \left(\frac{\omega q}{\sqrt{2 E}}\right)$$

Deduce that

$$p=\frac{\partial S}{\partial q}=\frac{\partial W}{\partial q}=\sqrt{\left(2 E-\omega^{2} q^{2}\right)}$$

from which

$$p=\sqrt{2 E} \cos [\omega(t+\beta)] .$$

Hence, or otherwise, show that the transformation from variables $(q, p)$ to $(\alpha, \beta)$ is canonical.