A canonical transformation from the phase space coordinates (q, p) to (Q, P) is generated by the function
\(\psi(p, Q)=\frac{p^2}{2 \omega} \tan 2 \pi Q\),
where ω is a positive constant. The function \(\psi\)(p, Q) is related to F(q, Q) by the Legendre transform \(\psi\) = pq - F, where F is defined by df = pdq - PdQ. If the solution for (P, Q) is
\(P(t)=\frac{\omega}{4 \pi} t^2, Q(t)=Q_0\) = constant,
where t is time, then the solution for (p, q) variables can be written as
1
\(p=\frac{\omega t}{2 \pi} \cos 2 \pi Q_0, q=\frac{t}{2 \pi} \sin 2 \pi Q_0\)
2
\(p=-\frac{\omega t}{2 \pi} \cos 2 \pi Q_0, q=\frac{t}{2 \pi} \sin 2 \pi Q_0\)
3
\(p=\frac{\omega t}{2 \pi} \sin 2 \pi Q_0, q=\frac{t}{2 \pi} \cos 2 \pi Q_0\)
4
\(p=-\frac{\omega t}{2 \pi} \sin 2 \pi Q_0, q=\frac{t}{2 \pi} \cos 2 \pi Q_0\)