# Derivation of Lagrangian for Classical Electrodynamics

• Born2bwire

#### Born2bwire

Gold Member
Is there a derivation for the classical electrodynamic Lagrangian? I have taken a look at a few textbooks that I have on hand but all of them just state the Lagrangian (in the voodoo four-vector talk, \glares) without explaining the reasoning behind it. I know that the Lagrangian for a charged particle can be found by working it out but I am interesed in the Lagrangian from current and charge sources. What I want to do is apply the non-relativistic Lagrangian density,

$$\mathcal{L} = \frac{1}{2}\left(\epsilon E^2-\frac{1}{\mu}B^2\right) - \phi\rho + \mathbf{J}\cdot\mathbf{A}$$

and add in the contribution due to fictious magnetic charges and currents. We often use magnetic currents in our work to simplify the solution process and increase robustness and though I am tempted to just add in the analogue terms from the dual I do not want to just haphazardly cram in terms that look like they are correct without knowing that the principles are sound.

If you know how to work out the Lagrangian for a charged particle, you can just use that! Treat each infinitesimal piece of your extended charge/current distribution as a point charge $dq=\rho dV$ with current density $\textbf{J}=\rho\textbf{v}$.

If you know how to work out the Lagrangian for a charged particle, you can just use that! Treat each infinitesimal piece of your extended charge/current distribution as a point charge $dq=\rho dV$ with current density $\textbf{J}=\rho\textbf{v}$.

Ok, I was figuring that was going to be it, I got as far as that in Jackson before I saw something shiny and then before I knew it's bedtime. I'll give this a go when I reboot in the morning. Unfortunately this isn't looking like it's going to set itself up the way I would like it to unless I can play around with the gauges... Eh screw it, I'm going to bed.

...What I want to do is apply the non-relativistic Lagrangian density,

$$\mathcal{L} = \frac{1}{2}\left(\epsilon E^2-\frac{1}{\mu}B^2\right) - \phi\rho + \mathbf{J}\cdot\mathbf{A}$$

and add in the contribution due to fictious magnetic charges and currents.

This looks like the relativistic Lagrangian density to me, just not in four vector and tensor form?

A derivation is given in the book "Quantum Field Theory" by Claude Itzykson and Jean-Bernard Zuber, page 7 and further.