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I've been unable to find any references for the lagrangian density in presence of matter. It's a lot easier to find this expressions in vacuum, but, when it comes to matter, I couldn't find any clear information. I mean, when we're referring to this equations: $$\begin{cases} \nabla \cdot \mathbf{E}=\large{\frac{\rho}{\epsilon_0}} \\ \\ \nabla \cdot \mathbf{B}=0 \\ \\ \nabla \times \mathbf{E}=-\large{\frac{\partial \mathbf{B}}{\partial t}} \\ \\ \nabla \times \mathbf{B}=\mu_0 \mathbf{J}+\large{\frac{1}{c^2}\frac{\partial\mathbf{E}}{\partial t}} \end{cases}$$

I know the associated lagrangian density is:

$$\mathcal{L}=-\frac{1}{4\mu_0} F_{\mu \nu} F^{\mu \nu}-J_{\mu}A^{\mu}$$

Nevertheless, I was seeking how these other equations would be written in terms of the lagrangian formulation:

$$\begin{cases} \nabla \cdot \mathbf{D}=\rho \\ \\ \nabla \cdot \mathbf{B}=0 \\ \\ \nabla \times \mathbf{E}=-\large{\frac{\partial \mathbf{B}}{\partial t}} \\ \\ \nabla \times \mathbf{H}=\mathbf{J}+\large{\frac{\partial\mathbf{D}}{\partial t}} \end{cases}$$

And the only thing I got, is the link I leave below:

https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism#Matter

It says, " Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows":

$$\mathcal{L}=-\frac{1}{4\mu_0} F_{\mu \nu} F^{\mu \nu}-J_{\mu}A^{\mu}+\frac{1}{2} F_{\mu \nu}\mathcal{M^{\mu \nu}}$$

Where $\mathcal{M^{\mu \nu}}$ is the magnetization-polarization tensor.

However, in this link, it's not explained why the term corresponding to the bound currents is the way it is, and I couldn't find anywhere information that clarifies my doubts.

So, my question is: Why the term corresponding to the bound currents is the way it is?

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  1. The classical electrodynamics equations in matter is nonsense because the matter itself is a subject to the same equations. So the vacuum is the only option.
  2. The Maxwell's equations in vacuum contain 10 unknown functions (3E, 3H, 4J). We have 6 independent equations in Maxwell's equations. In order for theory to be definite we need to add 4 more equations. They called dynamics equations. It could've been Newton's dynamics but the problem is: Newton's dynamics originally formulated for a trajectory of a particle - it is not a field dynamics. It is not easy to close the system. After watching all attempts I've chosen to develop the simplest one. I introduced the invariant charge density term in Lagrange Density and got rid of interaction term (the magnetization polarization term equuals to zero in vacuum). See my article http://gsjournal.net/Science-Journals/Research%20Papers-Mathematical%20Physics/Download/1741
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