Riccati parameter modes from Newtonian free damping motion by supersymmetry

Abstract

We determine the class of damped modes \tilde{y} which are related to the common free damping modes y by supersymmetry. They are obtained by employing the factorization of Newton's differential equation of motion for the free damped oscillator by means of the general solution of the corresponding Riccati equation together with Witten's method of constructing the supersymmetric partner operator. This procedure leads to one-parameter families of (transient) modes for each of the three types of free damping, corresponding to a particular type of %time-dependent angular frequency. %time-dependent, antirestoring acceleration (adding up to the usual Hooke restoring acceleration) of the form a(t)=\frac{2γ^2}{(γt+1)^{2}}\tilde{y}, where γis the family parameter that has been chosen as the inverse of the Riccati integration constant. In supersymmetric terms, they represent all those one Riccati parameter damping modes having the same Newtonian free damping partner mode