Comparison between the variational equations of the averaged Lagrangian method and a nonlinear Schroedinger equation for waves on the surface of a fluid layer

By the example of waves on the surface of a fluid layer, the variational equations obtained in the averaged Lagrangian method of Whitham are demonstrated to become equivalent to a nonlinear Schroedinger equation, if the construction of the Lagrangian involves a term related to fundamental harmonic in addition to the terms from the second and zero harmonics considered by Whitham to get linear corrections to the field functions.