Past exam of the theoretical physics I course of the University of Cambridge / 2013 / Paper 1 / 3 / b / Solution

A locally invertible time-independent canonical transformation is canonical whenThe proposed coordinates are locally independent only for $q_1\ne 0$, since their configuration-space Jacobian has determinant $2q_1$. Work on one sign branch of $q_1$.

The mixed brackets first requireThe last bracket condition isExpressing $A,B$ in the $Q$ coordinates turns this into ${\partial }_{Q_2}A={\partial }_{Q_1}B$. Thus their shift one-form is closed and locally is the differential of an arbitrary smooth function $S(Q_1,Q_2)$. The canonical momenta for a squared coordinate therefore have the general local formConversely, any such $S$ satisfies all bracket conditions. This does not give one nonsingular transformation through $q_1=0$, or identify the two sign branches globally.

Choose $S=Q_2^3/3$. Then