= Spherical-geodesic first integral
{title2=$h=\sin^2\theta\,d\phi/ds$}
For a <sphere> with metric $a^2(d\theta^2+\sin^2\theta\,d\phi^2)$, physical <arc length> $s$ gives $h=\sin^2\theta\,d\phi/ds$. The <cyclic coordinate> $\phi$ proves conservation of $a^2h$, while unit-speed normalization gives
$$
\left(\frac{d\theta}{d\phi}\right)^2=\frac{\sin^4\theta}{a^2h^2}-\sin^2\theta.
$$
This azimuthal form requires $h\ne0$. The solutions are <great circles>; meridians have $h=0$ and require another parameter.
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