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The standard way to write the Maxwell equations (say in vacuum in absence of charges) as Euler-Lagrange (EL) equations is to take the first pair of the Maxwell equaitons and to deduce from it existence of electromagnetic potential. Substituting it into the second pair of Maxwell equations one gets second order equations for the potential. They can be presented as Euler-Lagrange equations for the potential.

I am wondering if there is a way to present the Maxwell equations as an EL-equation in terms of electromagnetic fields only rather than potential.

I think I can prove that this is impossible if one requires in addition that the Lagrangian density is invariant under the Poincare group.

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    $\begingroup$ @Qmechanic: I tihnk the answers there do not answer my question. $\endgroup$
    – MKO
    Dec 25, 2020 at 19:40
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    $\begingroup$ The questions and answers Qmechanic refers to give you an action/Lagrangian in terms of the stress tensor (which contains the fields) and also give you the Maxwell equations solely in terms of the stress tensor. So yes, they do answer your question. You dont need the potential, it's just a corrolary of the EL equation $dF=0$ together with Poincare's lemma. $\endgroup$
    – NDewolf
    Dec 26, 2020 at 10:03
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    $\begingroup$ @NDewolf : I am not agree. While the Lagrangian density does not use the potential, to get EL equations coinciding with Maxwell equations, one has to use that $F_{\mu,\nu}$ is expressed via potential and then use integration by parts. $\endgroup$
    – MKO
    Dec 26, 2020 at 10:45
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    $\begingroup$ I see, you also want to express the intermediate steps without potentials. I'm not sure a gauge theory approach is useful then, since this built around the potiential/gauge connection. However, is there a good reason why you'd want to do this? $\endgroup$
    – NDewolf
    Dec 26, 2020 at 12:14
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    $\begingroup$ The question says "in terms of electromagnetic fields only", which suggests that additional fields (auxiliary fields) are not allowed. If auxiliary fields were allowed, then the question would have already been answered in physics.stackexchange.com/a/343082. The question also seems to require that all of Maxwell's equations must be contained in the Euler-Lagrange equations alone, with no additional constraints. (I tried posting an answer that respects those rules, but the answer was flawed for other reasons, so I deleted it.) $\endgroup$
    – Chiral Anomaly
    Dec 27, 2020 at 1:37

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In the case of time-harmonic fields, it can be formulated in terms of fields without employing the auxiliary potentials. The Lagrangian in this case is $$L=\int_V \frac{1}{2}i\omega\mu H^2-\frac{1}{2}i\omega\epsilon E^2-\mathbf{E}\cdot \nabla \times\mathbf{H}+\frac{1}{2}\sigma E^2+\mathbf{J}\cdot \mathbf{E} \; d^3r.$$ The Maxwell's equations in time-harmonic case are $$\nabla \times \mathbf{E}=i\omega\mu\mathbf{H},$$ $$\nabla\times \mathbf{H}=-i\omega\epsilon\mathbf{E}+\mathbf{J},$$ $$\nabla\cdot\epsilon\mathbf{E}=\rho,$$ $$\nabla\cdot\mu\mathbf{H}=0.$$ The continuity equation then becomes $$\nabla\cdot\mathbf{J}=i\omega\rho.$$ For the general case, as far as I know, no such Lagrangian has been found so far.

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