Can you write and solve the EM field Lagrangian density without reference to the EM potential?

Question

Is it possible to write the Lagrangian density for the EM field and charges $$L=-\frac{1}{4\mu_0}F^{\mu \nu}F_{\mu \nu}+j^{\mu}A_{\mu}$$ only in terms of the Electromagnetic Tensor and current vector? Can you also solve the Euler-Lagrange equation without reference to the potential? For example, witht the Lagrangian density for empty space: $$L=-\frac{1}{4\mu_0}F^{\mu \nu}F_{\mu \nu}$$ There are no derivatives of $F^{\mu \nu}$, so I don't see how you could derive the field equations from it.

Answer 1

The field strength (Faraday Tensor) is defined as:

$$F_{μν}= \partial_{μ}Α_{ν} - \partial_{ν} Α_{μ}$$

You have to vary with respect to the electromagnetic $U(1)$ field $A_{μ}$ everywhere it appears.