Inequivalent Lagrangians for the damped harmonic oscillator

Abstract

A constant of the motion, in addition to what exists in the literature, is presented for the damped harmonic oscillator and its dynamical origin is investigated. These two constants of motion are used to construct expressions for a hierarchy of inequivalent Lagrangians. It is shown that each inequivalent Lagrangian may be related to a higher order degenerate Lagrangian. The hierarchical Lagrangians tend to pose some characteristic problems for discussing the corresponding phase-space structure. PACS Nos.: 47.20.Ky, 42.81.Dp