Motion on the n-dimensional ellipsoid under the influence of a harmonic force revisited

TL;DR

This paper explicitly finds integrals of motion for an n-dimensional ellipsoid under harmonic force, explores classical and quantum separation of variables, and introduces a generalized parametrization, revealing conditions for free motion equivalence.

Contribution

It provides explicit integrals of motion and a generalized orthogonal parametrization for the harmonic oscillator on an ellipsoid, extending classical and quantum separation methods.

Findings

Explicit integrals of motion for the system

A generalized orthogonal parametrization depending on two parameters

Conditions under which harmonic motion reduces to free motion

Abstract

The $n$ integrals in involution for the motion on the $n$-dimensional ellipsoid under the influence of a harmonic force are explicitly found. The classical separation of variables is given by the inverse momentum map. In the quantum case the Schr\"odinger equation separates into one-dimensional equations that coincide with those obtained from the classical separation of variables. We show that there is a more general orthogonal parametrisation of Jacobi type that depends on two arbitrary real parameters. Also if there is a certain relation between the spring constants and the ellipsoid semiaxes the motion under the influence of such a harmonic potential is equivalent to the free motion on the ellipsoid.