Delaunay variables ($(\mathcal{L},\mathcal{G},\mathcal{H};\ell ,\omega ,\Omega )$)

For a bound noncollision Kepler orbit with gravitational parameter $\mu$, the canonical actions per unit mass are $\mathcal{L}=\sqrt{\mu a}$, $\mathcal{G}=\mathcal{L}\sqrt{1-e^2}$ and $\mathcal{H}=\mathcal{G}\cos i$. Their angles are mean anomaly $\ell$, argument of periapsis $\omega$ and longitude of ascending node $\Omega$. The Hamiltonian is $-{\mu }^2/(2{\mathcal{L}}^2)$, so only $\ell$ advances in the unperturbed problem. In the prograde spherical-action chart, $J_r=\mathcal{L}-\mathcal{G}$, $J_{\theta }=\mathcal{G}-\mathcal{H}$, $J_{\varphi }=\mathcal{H}$; the type-two canonical transformation is generated by $F_2=(\mathcal{L}-\mathcal{G})w_r+(\mathcal{G}-\mathcal{H})w_{\theta }+\mathcal{H}{w}_{\varphi }$. Retrograde spherical actions have $J_{\theta }=\mathcal{G}-|\mathcal{H}|$, so the absolute value cannot be dropped globally. Circular and coplanar limits are singular angle charts, not singular orbits.