On the symmetries and invariants of the harmonic oscillator

Abstract

A general formulation of Noether's theorem is applied to the equation of a harmonic oscillator. The definition of symmetry includes the usual Lie invariance as a special case and (unlike standard formulations) generates the full set of invariants (i.e. gives closure under functional composition). The analysis for a time-dependent oscillator casts doubt on the importance of a known class of invariants. The existence of a Lagrangian function is shown to be inessential to the analysis.