65 v 1 2 4 Fe b 20 04 Classical Harmonic Oscillator : Supersymmetry and a Dirac - like Coupling Parameter

We obtain a class of oscillation modes with damping and absorption that are connected to the classical harmonic oscillator modes through the supersymmetric one-dimensional matrix procedure similar to relationships of the same type between Dirac and Schrödinger equations in particle physics. A single coupling parameter characterizes both the damping and absorption features of these modes, which in the zero limit of the parameter turn into the known modes of the problem. 1. Factorizations of differential operators describing simple mechanical motion have been only occasionally used in the past, although in quantum mechanics the procedure led to a vast literature under the name of supersymmetric quantum mechanics initiated by a paper of Witten [1]. However, as shown by Rosu and Reyes [2], for the damped Newtonian free oscillator the factorization method could generate interesting results even in an area settled more than three centuries ago. In the following, we apply some of the supersymmetric schemes to the basic classical harmonic oscillator. In particular, we show how a known connection in particle physics between Dirac and Schrödinger equations could lead in the case of harmonic motion to a class of modes depending on one more parameter, denoted by K in this work, besides the natural circular frequency ω 0. The parameter characterizes both the damping and the absorption of the partner modes. 2. The harmonic oscillator can be described by one of the simplest Riccati equation u ′ + u 2 + κω 2 0 = 0, κ = ±1 , (1) where the plus sign is for the normal case whereas the minus sign is for the up side down case. Indeed, employing u = w ′ w one gets the harmonic oscillator differential equation w ′′ + κω 2 0 w = 0 , (2) with the solutions w b = W + cos(ω 0 t + ϕ +) if κ = 1 W − sinh(ω 0 t + ϕ −) if κ = −1 , where W ± and ϕ ± are amplitude and phase parameters, respectively, which can be ignored in the following.