Lagrangian for fields AND particles?

[pellman]

In general what does a Lagrangian for a system consisting of interacting fields and particles look like?

It can't be, for example,

$$L=\sum{\frac{1}{2}mv_j^2+A(x_j)\inner v_j}$$

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.

 

[pellman]

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

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

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.