 Canonical momenta for a squared coordinate

On either branch $q_1\ne 0$, all local canonical momenta for these coordinates have the form $P_1=(p_1-p_2)/(2q_1)+{\partial }_{Q_1}S$ and $P_2=p_2+{\partial }_{Q_2}S$, with arbitrary smooth $S(Q_1,Q_2)$. The mixed coordinate–momentum brackets fix the linear momentum terms, while the momentum–momentum bracket requires the remaining shift one-form to be closed. It is locally a gradient. The transformation is singular at $q_1=0$ and is not globally one-to-one across both signs of $q_1$.