Azimuthal second-order equation for a spherical geodesic
= Azimuthal second-order equation for a spherical geodesic
Differentiating the <spherical-geodesic first integral> gives
$$
\theta''+\sin\theta\cos\theta-
\frac{2\sin^3\theta\cos\theta}{a^2h^2}=0,
\qquad{}'=d/d\phi.
$$
The same equation follows from the affine <geodesic equation> after eliminating the arc-length derivatives. The squared first-derivative equation also fixes the normalization constant; a second-order solution must retain that first integral.