# About EM Lagrangian: Should electric flux density $D$ be a canonical conjugate variable of electric field $E$?

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.

• I've never seen $D$ described as the canonical conjugate of $E$... what book is this? May 4, 2020 at 18:24
• Possible duplicates: physics.stackexchange.com/q/356856/2451 May 4, 2020 at 18:34
• May 4, 2020 at 22:57