= Canonical momenta for a squared coordinate
{title2=$Q_1=q_1^2,\quad Q_2=q_1+q_2$}
On either branch $q_1\ne0$, 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 $q_1$.
Back to article page
