Spherical-geodesic first integral ($h={\sin }^2\theta d\varphi /ds$)

For a sphere with metric $a^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$, physical arc length $s$ gives $h={\sin }^2\theta d\varphi /ds$. The cyclic coordinate $\varphi$ proves conservation of $a^{2}h$, while unit-speed normalization givesThis azimuthal form requires $h\ne 0$. The solutions are great circles; meridians have $h=0$ and require another parameter.

Differentiating the spherical-geodesic first integral givesThe 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.

With polar angle $0\le \theta \le \pi$ and $0<|ah|\le 1$, the spherical-geodesic first integral gives ${\theta }_{\min }=\arcsin |ah|$ and ${\theta }_{\max }=\pi -\arcsin |ah|$. The maximum absolute latitude is the different angle $\arccos |ah|$. At $|ah|=1$ the great circle is the equator; at $h=0$ it is a meridian reaching both poles.