\underline{L}(t)= \underline{L}_{res}(t)+
\underline{L}_{em}(t)+\underline{L}_{int}(t)$$ $$\label{c8}
\underline{L}_{res}(t)=\int_0^\infty d\omega \int'd^3k
\left(|\dot{\underline{\vec{X}}}_\omega|^2-\omega^2|\underline{\vec{X}}_\omega|^2\right)
 +\int_0^\infty d\omega\int'd^3k
\left(|\dot{\underline{\vec{Y}}}_\omega|^2-\omega^2|\underline{\vec{Y}}_\omega|^2\right)$$ $$\label{c9}
\underline{L}_{em}(t)= \int'd^3k
\left(\varepsilon_0|\dot{\underline{\vec{A}}}|^2 +\varepsilon_0
|\vec{k}\
\underline{\varphi}|^2-\frac{|\vec{k}\times\underline{\vec{A}}|^2}{\mu_0}
\right) +\varepsilon_0\int' d^3k\left( -\imath \vec{k}\cdot
\dot{\underline{\vec{A}}}\ \  \underline{\varphi}^*+h.c\right)$$ $$\begin{aligned}
\label{c10}
&&\underline{L}_{int}(t)=\nonumber\\
&-&\int_0^\infty d\omega\ \int' d^3q \int' d^3p\left[ \left(
\underline{\dot{\vec{A}}}(\vec{q},t)+\imath\vec{ q}\ \
\underline{\varphi}(\vec{q},t)\right)\cdot\underline{f}( \omega,
-\vec{q} ,\vec{p})
\cdot\underline{\vec{X}}_\omega(\vec{p},t)+h.c\right]\nonumber\\
&-&\int_0^\infty d\omega\ \int' d^3q \int' d^3p\left[ \left(
\underline{\dot{\vec{A^*}}}(\vec{q},t)-\imath\vec{ q}\
\underline{\varphi}^*(\vec{q},t)\right)\cdot\underline{f}( \omega,
\vec{q} ,\vec{p})\cdot
\underline{\vec{X}}_\omega(\vec{p},t)+h.c\right]\nonumber\\
&+&\int_0^\infty d\omega\ \int' d^3q \int' d^3p\left[ \left(
\imath\vec{q}\times
\underline{\vec{A}}(\vec{q},t)\right)\cdot\underline{g}( \omega,
-\vec{q} ,\vec{p})
\cdot\underline{\vec{Y}}_\omega(\vec{p},t)+h.c\right]\nonumber\\
&+&\int_0^\infty d\omega\ \int' d^3q \int' d^3p\left[\left(
-\imath\vec{q}\times \underline{\vec{A^*}}(\vec{q},t)\right)\cdot
\underline{g}( \omega, \vec{q} ,\vec{p})\cdot
\underline{\vec{Y}}_\omega(\vec{p},t)+h.c\right]\end{aligned}$$ where $\displaystyle \int'd^3k $ implies the integration over the half space $ k_z\geq 0$ ( hereafter , we apply the symbol $\displaystyle \int'd^3k $ for the integration on the half space $
k_z\geq 0 $ and $\displaystyle \int d^3k $ for the integration on the total reciprocal space). In the reciprocal space the total Lagrangian (\[c7\])- (\[c10\]) do not involve the space derivatives of the dynamical variables of the system and the classical equations of the motion of the system can be obtained using the principle of the Hamilton’s least action, $ \displaystyle \delta \int dt \ \underline{L}(t)=0$. These equations are the Euler-Lagrange equations. For the vector potential $ \underline{\vec{A}}(\vec{k},t)$ and the scalar potential $
\underline{\varphi}(\vec{k},t)$ 