Averaged Lagrangian
= Averaged Lagrangian
{title2=$\overline L$}
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 from
$$
\mathcal L=\tfrac12u_t^2-\tfrac{c^2}{2}u_x^2,\qquad u=A\cos(kx-\omega t).
$$
Since the average of $\sin^2$ is $1/2$, one obtains
$$
\overline{\mathcal L}(A,k,\omega)=\frac{A^2}{4}(\omega^2-c^2k^2).
$$
For $A\ne0$, variation with respect to $A$ gives the <dispersion relation> $\omega^2=c^2k^2$. 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.