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?
