Could anyone tell me what equations can I obtain from the Lagrangian density
$${\cal L}(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i,\,\,A_{i,j})~=~\frac{1}{2}|\dot A+\nabla\phi|^2-\frac{1}{2}|\nabla \times A|^2-\rho\phi+J\cdot A$$ by the Euler-Lagrange equations?
If your $A$ is a 3D vector potential $\mathbf{A}$, then your Lagrangian is a Lagrangian for electromagnetic field potentials $(\phi,\mathbf{A})$ created with a charge $\propto c$ and the 3D charge velocity $\propto d$. The exact proportionality coefficients depend on units.
