= Solution
A <canonical transformation> is a differentiable invertible change of phase-space coordinates preserving the <symplectic form>, equivalently the canonical <Poisson brackets>. At each fixed time, let $D$ be its Jacobian in the ordering $(q,p)\mapsto(Q,P)$ and put $J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}$. The condition is
$$
\boxed{DJD^T=J,}
$$
which is equivalent to $D^TJD=J$. Time dependence changes the <Hamiltonian> by a generating-function time derivative, but not this fixed-time symplectic condition.
The <Hamilton-Jacobi equation> is $S_t+\tfrac12[(S_q)^2+\omega^2q^2]=0$. Separating $S=W(q,E)-Et$ gives $(W_q)^2=2E-\omega^2q^2$. On the positive-momentum branch, therefore,
$$
\boxed{W(q,E)=\int^q\sqrt{2E-\omega^2s^2}\,ds,\qquad S=W-Et.}
$$
Choose the integration constant so that the given angle relation is obtained. For $E>0$, $\beta=S_E$ yields $q=\sqrt{2E}\sin[\omega(t+\beta)]/\omega$, while $p=S_q=\sqrt{2E-\omega^2q^2}$ on this branch. Continuing the angle across the turning points gives the full orbit
$$
\boxed{q=\frac{\sqrt{2\alpha}}\omega\sin[\omega(t+\beta)],\qquad p=\sqrt{2\alpha}\cos[\omega(t+\beta)],\qquad\alpha=E.}
$$
The square-root expression is locally the positive branch; the cosine formula correctly changes sign on the rest of the orbit.
The generating function has differential $dS=p\,dq+\beta\,d\alpha-H\,dt$. At fixed time, taking an exterior derivative gives $dq\wedge dp=d\beta\wedge d\alpha$. Thus \b[the canonical new coordinate is $Q=\beta$ and its conjugate momentum is $P=\alpha$]. Directly, $\alpha=(p^2+\omega^2q^2)/2$ and $\beta=\operatorname{atan2}(\omega q,p)/\omega-t$ locally imply
$$
\beta_q=\frac p{p^2+\omega^2q^2},\qquad\beta_p=-\frac q{p^2+\omega^2q^2},\qquad\boxed{\{\beta,\alpha\}=1.}
$$
The angle is a local chart or a periodic variable, and is undefined at the zero-energy point. The source's phrase “variables $(\alpha,\beta)$” lists the new momentum before its coordinate: if interpreted instead as the conventional coordinate-then-momentum ordering, that ordered pair would have $\{\alpha,\beta\}=-1$ and be anti-canonical.
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