The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature

TL;DR

This paper extends the classical harmonic oscillator to Riemannian and Lorentzian spaces of constant curvature, analyzing its properties, solutions, and orbits in a unified geometric framework.

Contribution

It introduces a Cayley-Klein approach to study harmonic oscillators on various curved spaces, deriving explicit solutions and characterizing orbits as conics in these geometries.

Findings

Orbits are conics centered at the potential origin in all CK spaces.

Explicit solutions for the oscillator equations are obtained via three methods.

The properties of conics are extended to spaces of constant curvature.

Abstract

The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (indefinite). In this paper we study the main properties of these `curved' harmonic oscillators simultaneously on any such configuration space, using a Cayley-Klein (CK) type approach, with two free parameters $\ki, \kii$ which altogether correspond to the possible values for curvature and signature type: the generic Riemannian and Lorentzian spaces of constant curvature (sphere ${\bf S}^2$, hyperbolic plane ${\bf H}^2$, AntiDeSitter sphere ${\bf AdS}^{\unomasuno}$ and DeSitter sphere ${\bf dS}^{\unomasuno}$) appear in this family, with the Euclidean and Minkowski spaces as flat limits. We solve the equations of motion for the `curved' harmonic oscillator and obtain explicit expressions for the orbits by using three different methods: first by direct integration, second by obtaining the general CK version of the Binet's equation and third, as a consequence of its superintegrable character. The orbits are conics with centre at the potential origin in any CK space, thereby extending this well known Euclidean property to any constant curvature configuration space. The final part of the article, that has a more geometric character, presents those results of the theory of conics on spaces of constant curvature which are pertinent.