Averaged Lagrangian

The averaged Lagrangian is the mean over one fast phase of a Lagrangian evaluated on a locally periodic wave profile. Holding its slow parameters fixed during this average yields a function of those parameters, the wavenumber and the angular frequency. A simple explicit example starts fromSince the average of ${\sin }^2$ is $1/2$, one obtainsFor $A\ne 0$, variation with respect to $A$ gives the dispersion relation ${\omega }^2=c^2{k}^2$. Variation of the slow phase instead gives the wave-action conservation law. The distinction matters: substituting the dispersion relation into $\overline{\mathcal{L}}$ before differentiating would set it to zero and erase the nonzero derivatives encoding the conserved wave quantity. General nonlinear waves require averaging their full periodic profile and retaining its shape parameters.