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

Constants of motion are usually derived from groups of symmetry transformation of the system. Here we show that useful properties of the system can be deduced from a family of Noether-like transformations that are not inspired by any symmetry whatsoever. The system here is the Lagrangian interpretation of Poincaré’s half plane, and the property is the shape of the geodesics.