Lagrangian
In a vacuum, the Lagrangian for classical electrodynamics (in joules/meter3) is a scalar density
where
The four-current should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables.
If we separate free currents from bound currents, the Lagrangian becomes
Read more about this topic: Maxwell's Equations In Curved Spacetime