Recently I've reviewed classical Electromagnetism to understand D-brane more deeply, and I find I don't understand basic at all.

My book explains what is the action of electromagnetic field. To be general, we start with this (generalized) Maxwell's equations:

\begin{align*} &\mathrm{div}D = \rho \\ &\mathrm{rot}H = j + \frac{\partial D}{\partial t}\\ &\mathrm{div}B = \rho_m \\ &\mathrm{rot}E = -j_m - \frac{\partial B}{\partial t} \end{align*}

First, we assume there are no charges: $\rho, \rho_m, j, j_m=0$.

The book says we should choose 2 independent variables of 4 ($D, H, B, E$) to construct action. However, the Maxwell's equations don't tell us how $D, E (\textrm{or} B)$ or $H, B (\textrm{or} E)$ are related so I think I can't say these 4 is not independent in general.

This is my first question.

Of course, with some assumptions we introduce relations like $D = \epsilon E$, but in general how we conclude that these 4 variables aren't independent?

By the way, the book chooses $E, B$ as 2 independent dynamical variables. That's OK, but it says "Then, $D$ and $H$ become canonical conjugates of $E, B$: $D = \frac{\partial\mathcal{L}}{\partial E}, H = - \frac{\partial\mathcal{L}}{\partial B}$".

How this is derived? This is my main question.

Assuming we only know above Maxwell's equation, then can we say $D$ and $H$ become canonical conjugates of $E, B$ if we would like to construct the action and choose $E, B$ as 2 dynamical variables?

I think this is very basic question, sorry.



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