Commensurate Harmonic Oscillators: Classical Symmetries

TL;DR

This paper investigates the classical symmetry structures of N-dimensional harmonic oscillators with rational frequency ratios, revealing differences between isotropic and commensurate cases and clarifying the global nature of their symmetries.

Contribution

It provides a global analysis of symmetries in commensurate harmonic oscillators, distinguishing local and global transformations and resolving a longstanding ambiguity.

Findings

Invariant phase-space functions form su(N) algebra.

SU(N) symmetry group exists only on reduced phase space for commensurate oscillators.

Global symmetry transformations are limited by flow singularities.

Abstract

The symmetry properties of a classical N-dimensional harmonic oscillator with rational frequency ratios are studied from a global point of view. A commensurate oscillator possesses the same number of globally defined constants of motion as an isotropic oscillator. In both cases invariant phase-space functions form the algebra su(N) with respect to the Poisson bracket. In the isotropic case, the phase-space flows generated by the invariants can be integrated globally to a set of finite transformations isomorphic to the group SU(N). For a commensurate oscillator, however, the group SU(N) of symmetry transformations is found to exist only on a reduced phase space, due to unavoidable singularities of the flow in the full phase space. It is therefore crucial to distinguish carefully between local and global definitions of symmetry transformations in phase space. This result solves the longstanding problem of which symmetry to associate with a commensurate harmonic oscillator.