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Lagrangian for fields AND particles?

  1. Jan 7, 2010 #1
    In general what does a Lagrangian for a system consisting of interacting fields and particles look like?

    It can't be, for example,

    [tex]L=\sum{\frac{1}{2}mv_j^2+A(x_j)\inner v_j}[/tex]

    That would be for a system of particles in a fixed, i.e. "background", field. I'm interested in how we can mix particles and fields in Lagrangian mechanics. I know how to write down, as above, the Lagrangian for particles influenced by a field. And I know how to write down a Lagrangian (density) for a field with fixed (continuum) sources. But what does a Lagrangian (density?) that governs both fields and discrete sources look like?

    No need to lay out the most general case. Just a simple example will suffice.
  2. jcsd
  3. Jan 9, 2010 #2
    Ok. No replies. I can take this now to the next step myself. Then maybe someone else can help from there.

    Supposedly, the full action for both EM fields and (dynamic) sources is

    [tex] -m \int d\tau \sqrt{- g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu} + q \int dx'^4 \int d\tau ~ \delta^4(x'-x(\tau)) \frac{dx^\mu (\tau)}{d\tau} A_\mu - \frac{1}{4} \int d^4 x F^{\alpha \beta} F_{\alpha \beta} [/tex]

    See this thread: https://www.physicsforums.com/showthread.php?t=222066

    Ok. Now - how do we get the equations of motion from this action? How do we apply the Euler-Lagrange equations to an action of mixed particles and fields?

    References to helpful source material would be much appreciated.
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