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

Владислав Витальевич Тишков

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

В. В. Тишков

2026-07-21

https://doi.org/10.24108/preprints-3115944

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.

Ссылка для цитирования:

Тишков В. В. 2026. A Generalisation of Bertrand's Theorem to Anisotropic Harmonic Oscillators: Geometry and Classification of Lissajous Helices. PREPRINTS.RU. https://doi.org/10.24108/preprints-3115944

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