Planar rotational symplectic reduction
= Planar rotational symplectic reduction
{title2=$\omega_{\mathrm{red}}=dr\wedge dp_r$}
For a planar <cotangent bundle>, rotation has <moment map> $\ell=xp_y-yp_x$. At $\ell\ne0$ its level is regular and the circle action is free. Polar canonical coordinates give $\omega=dr\wedge dp_r+d\theta\wedge d\ell$. Fixing $\ell$ and quotienting $\theta$ leaves $T^*\mathbb R_{>0}$ with the displayed form. The unit-mass free <Hamiltonian> becomes $p_r^2/2+\ell^2/(2r^2)$.