= Brachistochrone on a horizontal half-cylinder
{title2=$z'(\theta)=RK\sqrt{\cos\theta}/\sqrt{1-K^2\cos\theta}$}
Measure $\theta\in[-\pi/2,\pi/2]$ from the vertically downward direction on the lower half of a cylinder of radius $R$, and let $z$ be axial distance. From rest at an edge, <conservation of energy> gives speed $\sqrt{2gR\cos\theta}$. The <travel time> is $T=(2gR)^{-1/2}\int\sqrt{R^2+z'^2}/\sqrt{\cos\theta}\,d\theta$. Since $z$ is a <cyclic coordinate>, its conjugate quantity $z'/[\sqrt{\cos\theta}\sqrt{R^2+z'^2}]=K$ is constant, giving the displayed formula. For finite positive axial separation, $0<K<1$ is fixed uniquely by integrating $z'$ between the edges: the separation grows strictly from zero to infinity as $K$ increases from zero to one. The time integrand is strictly convex in $z'$, so its supporting-line inequality proves global minimality for the fixed endpoints. As the starting edge is approached, $z'\to0$: the initial direction is directly down the curved surface, with no axial component. This differs from the constant-pitch shortest <geodesic> on the unrolled cylinder.
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