Modulated-wave first integral

For a periodic travelling profile $U(\theta )$, an autonomous fast-phase Euler-Lagrange equation has a first integral. For example, the field Lagrangian $\mathcal{L}=\frac{1}{2}{u}_t^2-\frac{c^2}{2}{u}_x^2-V(u)$ and phase $\theta =kx-\omega t$ giveMultiplication by $U_{\theta }$ yieldsThus the bracketed profile energy is constant during one fast cycle. In a wave described by Whitham modulation theory, its parameters also vary on slow scales. The fast identity alone does not determine that slow evolution: one must retain the next-order terms and impose their averaged solvability condition, or vary the averaged Lagrangian. A periodic total phase derivative averages to zero, which is the useful simplification in either route.