Lagrangian for E and B fields, not vector potential?

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SUMMARY

The discussion centers on the search for a Lagrangian that directly incorporates the electric field (E) and magnetic field (B) to derive Maxwell's equations. A proposed Lagrangian is L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}, which successfully yields the second-order Maxwell equations but does not produce the standard first-order equations. The relationship between these two forms of Maxwell's equations is questioned, particularly regarding their equivalence.

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pellman
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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?
 
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[tex]L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/tex]

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

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