A note on the Lagrangian method for nonlinear dispersive waves

A few remarks are made on the method of the averaged Lagrangian developed by Whitham to describe slow variations of nonlinear wave trains. The concepts of multiple scales is incorporated into the variational formalism, and a consistent development to higher approximation is suggested as formal perturbation based on the variational principle. The propagation of weakly dispersive long waves is also reconsidered in relation to the Lagrangian method. It is demonstrated that the non-linear Schroedinger equation and the Korteweg-de Vries equation can be derived from the Euler-Lagrange equations of the perturbed Lagrangian. (author)