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Derivation of Lagrangian for Classical Electrodynamics

  1. Sep 16, 2009 #1

    Born2bwire

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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,

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

    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.
     
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  3. Sep 16, 2009 #2

    gabbagabbahey

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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 [itex]dq=\rho dV[/itex] with current density [itex]\textbf{J}=\rho\textbf{v}[/itex].
     
  4. Sep 16, 2009 #3

    Born2bwire

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    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.
     
  5. Sep 18, 2009 #4
    This looks like the relativistic Lagrangian density to me, just not in four vector and tensor form?
     
  6. Sep 19, 2009 #5
    A derivation is given in the book "Quantum Field Theory" by Claude Itzykson and Jean-Bernard Zuber, page 7 and further.
     
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