Naive questions on the concept of effective Lagrangian and equations of motion? Let us consider a LC circuit containing an electric dipole moment, the quantum system (electric field $E$ coupled with a dipole moment) can be described by the path integral
$$Z=\int DEDxe^{i\int dtL},$$
where the total Lagrangian 
$$L=\frac{1}{2g}(\dot{E}^2-\omega _{LC}^2E^2)+\frac{m}{2}\dot{x}^2-\frac{m\omega _{0}^2}{2}x^2+exE.$$
After integrating out the dipole $x$, we obtain an effective Lagrangian 
$L_{eff}$ for the electric field 
$$L_{eff}=\frac{1}{2g}(\dot{E}^2-\omega _{LC}^2E^2)+\frac{e^2}{2m}E(\partial_t^2+\omega _{0}^2)^{-1}E.$$
On the other hand, from the classical point of view, by solving the total Lagrangian $L$, we can obtain a 4th order equation of motion for the electric field 
$$[\partial_t^4+(\omega _{0}^2+\omega _{LC}^2)\partial_t^2+\omega _{0}^2\omega _{LC}^2-\frac{e^2g}{m}]E=0.\tag{a}$$
My questions are:


*

*Can the second term in $L_{eff}$ be written as a function of $\dot{E}$ and $E$?

*Can we derive an 'Euler-Lagrangian' equation from the effective Lagrangian $L_{eff}$?
If yes, is this equation the same as the above 4th order equation of motion $(a)$ of the classical system?

*Can we construct another effective Lagrangian from the classical dynamics which gives rise to Eq.$(a)$? Is the concept of effective Lagrangian ONLY meaningful for the quantum system?
 A: 1 : I don't think so
2 : 
Note that $L_{eff}$, may be written, thanks to an integration by parts  $(\partial_t E)^2 = \partial_t(E\partial_t E) - E \partial_t^2E$, and  neglecting the surface term due to the the total derivative : 
$$ L_{eff}=E\quad (\frac{1}{2g}(-\partial_t^2 -\omega _{LC}^2)+\frac{e^2}{2m}(\partial_t^2+\omega _{0}^2)^{-1})\quad E \tag{1}$$
The equation of movement is then :
$$(\frac{1}{2g}(-\partial_t^2 -\omega _{LC}^2)+\frac{e^2}{2m}(\partial_t^2+\omega _{0}^2)^{-1})\quad E = 0 \tag{2}$$
Multiplying $(2)$ by $(\partial_t^2+\omega _{0}^2)$ gives you  the equation $(a)$
3 : Different lagrangians may give the same equation of movement.
A: OP's system is two coupled harmonic oscillators
$$\tag{1} L~=~\frac{1}{2}(m\dot{x}^2 - k x^2) +  \frac{1}{2}(M\dot{y}^2 - K y^2) - \kappa xy. $$
It seems a steep price to pay to create a non-local formulation by integration out one variable by brute force as OP does. Here we instead find the normal modes of the system of two coupled harmonic oscillators.
The equations of motion are 
$$\tag{2} \begin{pmatrix}\ddot{x} \\ \ddot{y}\end{pmatrix} ~=~- \Lambda \begin{pmatrix}x \\ y\end{pmatrix}, \qquad \Lambda ~:=~ \begin{pmatrix}\frac{k}{m}&\frac{\kappa}{m} \\ \frac{\kappa}{M} &\frac{K}{M} \end{pmatrix}. $$
Interestingly, the real $2\times 2$ matrix $\Lambda$ is not symmetric if $m\neq M$. The two eigenvalues of $\Lambda$ are real
$$\tag{3} \lambda_{\pm} ~=~ \frac{{\rm tr}(M)}{2} \pm  \sqrt{\Delta}, $$
$$\tag{4} \Delta~:=~\left(\frac{{\rm tr}(M)}{2}\right)^2-\det(M)~\geq~0. $$ 
If the matrix $T$ diagonalizes the matrix
$$\tag{5}\Lambda~=~TDT^{-1}, \qquad 
D ~:=~ \begin{pmatrix}\lambda_+&0\\ 0 &\lambda_- \end{pmatrix}, $$
then define new variables
$$ \tag{6} \begin{pmatrix}x_+ \\ x_-\end{pmatrix} 
~=~T^{-1} \begin{pmatrix}x \\ y\end{pmatrix}. $$
Then the equations of motion are 
$$  \tag{7} \ddot{x}_{\pm}~=~-\lambda_{\pm} x_{\pm}, $$
with Lagrangian
$$  \tag{8}  \tilde{L}~=~\sum_{\pm} \frac{m_{\pm}}{2}\left(\dot{x}^2_{\pm} - \lambda_{\pm} x^2_{\pm} \right), $$
where the masses $m_{\pm}$ depends on the parameters of the theory.
