An update on the classical and quantum harmonic oscillators on the sphere and the hyperbolic plane in polar coordinates

•Systematic derivation of classical solutions in polar coordinates.•Restriction on angular momentum values for bounded motions on the hyperbolic plane highlighted.•Relation between solutions in polar and cartesian coordinates established.•Jacobi polynomials shown to appear in bound-state radial wavefunctions of quantum model.

A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian coordinates, whose form was previously guessed. In addition, the nature of the classical orthogonal polynomials entering the bound-state radial wavefunctions of the corresponding quantum model is identified.