Lagrangian Formalism in Perturbed Nonlinear Klein-Gordon Equations

We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term we are able to write the Lagrangian density and then, calculate the Lagrangian as a function of collective variables. We use the Lagrangian formalism together with the Rice {\it Ansatz} to derive the equations of motion of the collective coordinates (CCs) for the perturbed sine-Gordon (sG) and $\phi^{4}$ systems. For the $N$ collective coordinates, regardless of the {\it Ansatz} used, we show that, for the nonlinear Klein-Gordon equations, this approach is equivalent to the {\it Generalized Traveling Wave Ansatz} ({\it GTWA})