Symplectic cotangent lift with a closed momentum shift ($x=\chi (x^{\prime }),y=\eta \circ \chi +D{\chi }^{-T}{y}^{\prime }$)

If $x=\chi (x^{\prime })$ is an angular diffeomorphism and $\eta \cdot dx$ is a closed differential one-form, the map $y=\eta (\chi (x^{\prime }))+D\chi (x^{\prime }{)}^{-T}{y}^{\prime }$ is a symplectomorphism. It combines a cotangent lift of a diffeomorphism with translation by a closed differential one-form. In particular $\eta =du+a$ is allowed on a torus even when the constant one-form $a\cdot dx$ is not exact.

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 $q_1$.