= Solution
For an affinely parametrized <geodesic> $s\mapsto(x(s),y(s))$, the energy <Lagrangian> is $L=(\dot x^2+\dot y^2)/(2y^2)$. The coordinate $x$ is cyclic, so the <Euler-Lagrange equation> gives $\dot x/y^2=c$, a constant. Parametrize a nonconstant <geodesic> by hyperbolic <arc length>; then
$$
\dot x=cy^2,\qquad\dot x^2+\dot y^2=y^2.
$$
If $c=0$, $x$ is constant: these are vertical lines. If $c\ne0$, on intervals with $\dot y\ne0$,
$$
\frac{dx}{dy}=\frac{\pm cy}{\sqrt{1-c^2y^2}}.
$$
Integration gives $(x-a)^2+y^2=c^{-2}$. The two branches join smoothly at the highest point. Conversely these curves satisfy the <geodesic equations>, or are images of vertical lines under half-plane <Möbius transformations> that are <isometries>. Thus the complete nonconstant <geodesics> are
$$
\boxed{x=\text{constant},\qquad(x-a)^2+y^2=R^2\ (y>0),}
$$
namely vertical lines and upper semicircles orthogonal to the real axis. Their subarcs give <geodesic segments>.
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