 Semicircular extremals of the hyperbolic length functional

For positive twice differentiable functions $y$, the Euler-Lagrange equation for this functional reduces to $y{y}^{\prime \prime }+1+y^{\prime 2}=0$. Thus $(y{y}^{\prime }{)}^{\prime }=-1$ 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 $F_{y^{\prime }}=y^{\prime }/(y\sqrt{1+y^{\prime 2}})=0$, 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.