Newton's laws of motion in form of Riccati equation

Abstract

We discuss two applications of Riccati equation to Newton's laws of motion. The first one is the motion of a particle under the influence of a power law central potential $V(r)=k r^ε$. For zero total energy we show that the equation of motion can be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems