Classical and quantum dynamics of over-damped non linear systems

Overdamping is a regime in which friction is sufficiently large that the motion either decays to its equilibrium position or it crosses the equilibrium position exactly once before returning monotonically towards the equilibrium position. The phenomena of overdamping has been studied classically and quantum mechanically only for the case of the linear damped harmonic oscillator. Here we study the classical and quantum dynamics of a family of over-damped non linear systems. The main objective of this paper is to find a Lagrangian and Hamiltonian framework to study over-damped non linear systems and to show that a quantum mechanical description can be developed in the momentum representation. Our results reduce to the well known solution of the linear damped harmonic oscillator when the non linear part is set to zero.