Supersymmetric partner chirping of Newtonian free damping

We connect the classical free damping cases by means of Ros-ner's construction in supersymmetric quantum mechanics. Starting with the critical damping, one can obtain in the underdamping case a chirping of instantaneous physical frequency ω 2 (t) ∝ ω 2 u sech 2 (ω u t), whereas in the overdamped case the " chirping " is of the (unphysical) type ω 2 (t) ∝ ω 2 o sec 2 (ω o t), where ω u and ω o are the underdamped and overdamped frequency parameters, respectively. The techniques of supersymmetric quantum mechanics gave a new impetus to many research areas in the last two decades [1]. In this note, we start with the critical free damping case in classical mechanics and construct the corresponding supersymmetric partners. The free damping Newton equation reads m d 2 x dt 2 + γ dx dt + kx = 0. (1) Using the gauge change of dependent variable x = y exp(− γ 2m t), this equation can be put in the following Schrödinger form in the time domain y ′′ − γ 2m 2 − k m y = 0 , (2) or y ′′ − ω 2 d y = 0, where ω 2 d = (γ/2m) 2 − k/m. Thus, one can discuss separately, the classical cases of underdamping (oscillating relaxation), critical damping (rapid nonoscillating relaxation), and overdamping (slow nonoscillating relaxation), i.e., −ω 2 u = ω 2 d < 0, ω 2 c = ω 2 d = 0 and ω 2 o = ω 2 d > 0, respectively [2]. Notice that the physical ω 2 d frequencies are negative, the positive ones are only convenient mathematical symbols corresponding in fact to nonoscillating regimes. We now proceed with the supersymmetric scheme that we apply in a manner similar to Rosner [3]. Thus, we start with the case of critical damping corresponding in quantum mechanics to a potential which is zero and relate it to a Schrödinger-type equation in the time domain which has a fundamental frequency at −ω 2 u , i.e., a single oscillating relaxation mode that we consider as the equivalent of a bound state in the usual quantum mechanics. In other words, we solve the " fermionic " Riccati equation W 2 1 − W ′ 1 + ω 2 d = 0 ,