Conserved and piecewise-conserved Runge vectors for the isotropic harmonic oscillator

Abstract

A procedure for constructing a Runge vector for a particle moving (classically or relativistically) under any central force has been given by Fradkin. Serebrennikov and Shabad have pointed out that such vectors may be piecewise conserved, rather than constants for the entire motion. It is shown herein that, for the classical isotropic harmonic oscillator, the above procedure can lead to a piecewise-conserved Runge vector as well as to a conserved one (of the same magnitude). We relate this ambiguity to the symmetries of the orbit.