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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?

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marked as duplicate by Qmechanic lagrangian-formalism Oct 12 at 19:02

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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.

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