[QUE/CM-10008]

1. Show that \begin{eqnarray} Q_1=\frac{1}{1}{\sqrt{2}}\Big( q_1+ \frac{p_2}{m\omega}\Big) &\qquad& P_1=(p_1-m\omega q_2)\\ Q_2=\frac{1}{\sqrt{2}}\Big( q_1- \frac{p_2}{m\omega}\Big) &\qquad& P_2=(p_1+m\omega q_2) \end{eqnarray} (where \(m\omega\) is a constant) is a canonical transformation by Poisson bracket test. This requires six simple Poisson brackets.
2. Find the generating function\(F(q_,q_2,Q_1,P_2)\) for this transformation, type 1 in the first degree of freedom and type 2 in the second degree of freedom.

Source:Calkin