I was reading a paper called Parametrically enhanced hidden photon search by Peter Graham et al. In the paper, a Lagrangian that describes the theory of the hidden photon is $$\mathcal{L}=-\frac{1}{4}(F_{\mu\nu}F^{\mu\nu}+F'_{\mu\nu}F'^{\mu\nu})+\frac{1}{2}m_{\gamma'}^2A'_\mu A'^{\mu}-eJ_{EM}^\mu(A_{\mu}+\varepsilon A'_\mu).$$
The paper says in a following section that this Lagrangian leads to equations of motion:
$$\begin{align}(\partial_t^2+\nabla^2)V&=\rho_{EM}\tag{1}\\ (\partial_t^2+\nabla^2)\vec{A}&=\vec{J}_{EM}\tag{2}\\ \dot{V}+\nabla\cdot \vec{A}&=0\tag{3}\\ (\partial_t^2+\nabla^2+m_{\gamma'}^2)V'&=\varepsilon\rho_{EM}\tag{4}\\ (\partial_t^2+\nabla^2+m_{\gamma'}^2)\vec{A}'&=\varepsilon\vec{J}_{EM}\tag{5}\\ \dot{V}'+\nabla\cdot \vec{A}'&=0\tag{6} \end{align}$$
My question is how to get these equations of motion from that Lagrangian. I tried using the Euler-Lagrange Equation. Using $A^{\nu}$ gives the inhomogeneous Maxwell's equations. I'm to use $A'^{\nu}$ now. I think it gives some modified Maxwell's equations, but I still can't quite get to these equations.
