Derivation of Lagrangian for Classical Electrodynamics

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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.

gabbagabbahey
Homework Helper
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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 $\textbf{J}=\rho\textbf{v}$.

Gold Member
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.