A Generalisation of Bertrand's Theorem to Anisotropic Harmonic Oscillators: Geometry and Classification of Lissajous Helices

We prove a complete classification theorem for bounded orbits of a three-dimensional anisotropic harmonic oscillator with free motion along one axis. This provides a natural generalisation of Bertrand's classical theorem (1873) to non-central potentials. We show that all bounded trajectories are Lissajous helices of the form r(t) = (a cos(ωₓt + φₓ), b cos(ωᵧt + φᵧ), ct + z₀). A trajectory is closed in R³ if and only if the longitudinal velocity vanishes (c = 0) and the frequency ratio ωₓ/ωᵧ is rational. We derive closed-form expressions for curvature, torsion, and arc length. The results have immediate applications to ion traps, optical lattices, molecular vibrations, and superintegrable systems.