The geodesics for Poincaré’s half-plane: a nonstandard derivation

The paper proves that nonconstant geodesics in the Poincaré half-plane with L = (ẋ^2 + ẏ^2)/(2y^2) are exactly vertical half-lines (p=0) or Euclidean circles orthogonal to the x-axis (p≠0), using two standard first integrals E and p (equations (2.3)) and a nonlocal constant of motion derived from q2-translations (Theorem 2.1), which yields K(t) and the linear ODE for z = 1/y: z̈ − 2Ez = 0; from there, they reconstruct the geodesics and obtain (x − c)^2 + y^2 = 2E/p^2 (equation (3.7)) . The candidate solution uses the same nonlocal constant-of-motion to get the same ODE for 1/y, then classifies geodesics by introducing c = x + ẏ/(py) to show directly that (x − c)^2 + y^2 is constant when p≠0, and that x is constant when p=0. The approaches share the crucial nonlocal-constant step and E,p integrals; they differ only in the final algebraic reconstruction (paper via eliminating exponentials to an ellipse then using 8c1c2E = p^2, model via the conserved c). Both are correct and essentially the same method at a high level.