= Semicircular extremals of the hyperbolic length functional
{title2=$I[y]=\int \sqrt{1+y'^2}\,dx/y$}
For positive twice differentiable <functions> $y$, the <Euler-Lagrange equation> for this <functional> reduces to $yy''+1+y'^2=0$. Thus $(yy')'=-1$ and integration gives $(x-h)^2+y^2=R^2$. The upper <circular arc> is a graph wherever its radicand is positive. A fixed horizontal endpoint with free height imposes $F_{y'}=y'/(y\sqrt{1+y'^2})=0$, so that endpoint must lie above the centre. Distinct horizontal endpoints cannot both satisfy this <natural boundary conditions for a free endpoint>. The <Beltrami identity> alone also admits constant functions, which are excluded by the full <Euler-Lagrange equation>.
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