The Geodesics for Poincaré’s Half-Plane: a Nonstandard Derivation

Constants of motion in Mechanics are usually inferred from groups of symmetry transformations of the given system, as, for example, a Lagrangian function that is time-invariant implies the conservation of energy. Here we wish to show that useful properties of a mechanical system can sometimes be deduced from a family of Noether-like transformations that are not inspired by any symmetry whatsoever. The sample system we concentrate on is the Lagrangian interpretation of Poincaré’s half plane of hyperbolic geometry, and the properties we will derive in a new way are the shape and the time parameterization of its geodesics.