Lagrangian for E and B fields, not vector potential?

[pellman]

Anyone know of a Lagrangian given in terms of E and B (or equivalently the tensor F) that yields Maxwell equations? A link or reference would be appreciated.

I can write down such a Lagrangian which yields the two second-order Maxwell equations, but not the usual four 1st order equations. And I'm not sure: are the second order equations equivalent to the first order form?

 

[madness]

$$L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

But then F is defined in terms of (the derivatives of) the potential, not E and B.