= Delaunay variables
{c}
{title2=$(\mathcal L,\mathcal G,\mathcal H;\ell,\omega,\Omega)$}
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, $J_r=\mathcal L-\mathcal G$, $J_\theta=\mathcal G-\mathcal H$, $J_\phi=\mathcal H$; the type-two <canonical transformation> is generated by $F_2=(\mathcal L-\mathcal G)w_r+(\mathcal G-\mathcal H)w_\theta+\mathcal H w_\phi$. 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.
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