= Solution
The <chain rule> and
$$
\theta_x=\Theta_X=k,
\qquad
\theta_t=\Theta_T=-\omega,
\qquad
X_x=T_t=\varepsilon
$$
give
$$
\phi_x=k\Phi_\theta+\varepsilon\Phi_X,
\qquad
\phi_t=-\omega\Phi_\theta+\varepsilon\Phi_T.
$$
Applying the same rule to $L_1$ and $L_2$ in the <Euler-Lagrange field equation> yields the exact modulated equation
$$
\boxed{
k(L_1)_\theta+\varepsilon(L_1)_X
-\omega(L_2)_\theta+\varepsilon(L_2)_T-L_3=0.}
$$
Multiply this equation by $\Phi_\theta$. Since $k$ and $\omega$ are independent of the fast phase,
$$
L_\theta
=L_1(k\Phi_{\theta\theta}+\varepsilon\Phi_{X\theta})
+L_2(-\omega\Phi_{\theta\theta}+\varepsilon\Phi_{T\theta})
+L_3\Phi_\theta.
$$
The <product rule> then rearranges the field equation into the exact <modulated-wave first integral>
$$
\boxed{
\partial_\theta\left[(kL_1-\omega L_2)\Phi_\theta-L\right]
+\varepsilon\partial_X(\Phi_\theta L_1)
+\varepsilon\partial_T(\Phi_\theta L_2)=0.}
$$
Average this identity over one $2\pi$ period in $\theta$. <Periodic function>[Periodicity] kills the first term, while differentiation of the <averaged Lagrangian> gives
$$
\frac{\partial\overline L}{\partial k}
=\left\langle\Phi_\theta L_1\right\rangle,
\qquad
\frac{\partial\overline L}{\partial\omega}
=-\left\langle\Phi_\theta L_2\right\rangle.
$$
Consequently
$$
\boxed{
\partial_X\frac{\partial\overline L}{\partial k}
-\partial_T\frac{\partial\overline L}{\partial\omega}=0.}
$$
The same two equations follow directly from the modulated <principle of stationary action>[variational principle]. Varying $\Phi$ gives
$$
\partial_\theta(kL_1-\omega L_2)
+\varepsilon\partial_XL_1
+\varepsilon\partial_TL_2-L_3=0,
$$
which is the exact field equation above. For a variation of $Θ$, use $\delta k=\partial_X\delta\Theta$ and $\delta\omega=-\partial_T\delta\Theta$. <Integration by parts> in $X$ and $T$ gives the averaged equation. Thus variation with respect to the periodic profile reproduces the local wave equation, whereas variation with respect to its slow phase gives the <Whitham modulation equation> for wave action.
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