On the solution and Lagrangian representation of Duffing oscillator with damping

Abstract

We construct the equation of Duffing oscillator in a dissipative medium using certain concepts from elementary mechanics. The Duffing equation (DE) without damping can be solved analytically. This is not true for a DE that involves a damping term. We remove the damping term from a linearly damped DE and thus obtain a simple analytical solution x(t) of the damped Duffing equation in the weak damping limit. The constructed solution allows us to examine the effect of damping on the phase path of the oscillator. The phase path is a parametric plot of x(t) and x = dxdt on the plane (x, x). While the phase path of the un-damped Duffing oscillator is an isolated limit cycle, the corresponding phase path for the Duffing oscillator with damping is a distorted one. We confirm our observation on the effect of dissipation by numerical simulation. We point out that it is often of interest to study dissipative systems at the quantum level and construct Lagrangian representations for both un-damped and damped Duffing oscillators. These results are expected to play a role to quantize the systems. We make some additional comments in respect of this.