# Maxwell Lagrangian where $F$ and its derivatives are the variables (i.e., without replacing $F={\rm d}A$) [duplicate]

The way the electromagnetic Lagrangian is usually constructed is by noticing that the EM fields are always constrained to satisfy $${\rm d}F=0$$ (half of Maxwell's equations). We can immediately solve this constraint equation through $$F={\rm d}A$$. Then the following action,

$$S[A,{\rm d}A] = \int {\rm d}A \wedge \star {\rm d}A,$$

leads to the remaining Maxwell's equations.

That's fine, but I want an action where $$F$$ is the dynamical variable and the constraint $${\rm d}F=0$$ isn't solved a priori, but rather follows by introducing a Lagrange multiplier $$\lambda$$. Concretely, I want an action of the form,

$$S[F,{\rm d}F,\lambda]=?,$$

such that:

• From varying $$F$$, we get the equation of motion $${\rm d}\star F=0$$.
• From varying $$\lambda$$, we get the equation of motion $${\rm d} F=0$$.

Is such a thing possible?

EDIT: The linked answer doesn't satisfy my demands. While it derived $${\rm d}F=0$$ by varying a Langrange multiplier, which I'm allowing, it also derives the equation of motion $${\rm d}\star F=0$$ by varying a Lagrange multiplier. I do not want this. The eom should come by varying $$F$$.

• There is something quite close, the so-called "first order formalism". Nov 15, 2018 at 19:06
• Nov 15, 2018 at 19:07