Suppose that you have the Lagrangian density for $\phi(\vec{x}, t)$ and $\vec{A}(\vec{x}, t)$ given as follows:
$$\mathcal{L} = \frac{\epsilon_{0}}{2} |-\nabla\phi - \partial_{t}\vec{A}|^{2} - \frac{1}{2\mu_{0}}|\nabla \times \vec{A}|^{2} - \rho \phi + \vec{j}\cdot \vec{A}.$$
Now in general, the Euler-Lagrangian equations for fields are give by:
$$\partial_{\mu} \left(\frac{\partial \mathcal{L}}{\partial( \partial_{\mu} \phi)}\right) = \frac{\partial \mathcal{L}}{\partial \phi}.$$
where $\mathcal{L} = \mathcal{L}(\phi, \partial_{\mu}\phi)$ and $\phi$ is the field. For the E&M $\mathcal{L}$, can I write the equations of motion for $\phi, \vec{A}$ as follows:-
$$\partial_{t}\left(\frac{\partial \mathcal{L}}{\partial \dot{\phi}}\right) + \nabla \cdot\left(\frac{\partial \mathcal{L}}{\partial(\nabla \phi)}\right) = \frac{\partial \mathcal{L}}{\partial \phi}$$
$$\partial_{t}\left(\frac{\partial \mathcal{L}}{\partial \dot{\vec{A}}}\right) + \nabla \times\left(\frac{\partial \mathcal{L}}{\partial(\nabla \times \vec{A})}\right) = \frac{\partial \mathcal{L}}{\partial \vec{A}}.$$
If $$\mathcal{L} = -\frac{1}{4}F^{\mu\nu}F_{\mu \nu} + \mu_{0}j_{\mu}A^{\mu},$$ then I could have used the definition of $F^{\mu\nu}$ and derived the equations, however, I am not sure that once I break it into spacial and time components, the Euler Lagrangian equations would end up what I wrote.
