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That is, either (1) solve the Lorentz force equation or (2) Maxwell's equations. But what does the complete problem (1) + (2) look like? Oh, I know, I know, its not solvable. I'd just like to see the differential equations once.

The problem is that all my references give Maxwell's equations in terms of a continuous charge distribution ([tex]\rho[/tex] and [tex]J[/tex]) while the Lorentz force is given for point particles. If we stick to a continuous charge distribution, what I need then, I guess, is the Maxwell equations (got 'em) + the continuity equation (got it) + plus the continuous version of the Lorentz force in terms of [tex]\rho[/tex] and [tex]J[/tex] and depending on E and B (need it).

If someone could provide that last piece, I'd appreciate it.

Interestingly, in quantum theory this is no big deal. It falls right out of SU(1) gauge invariance, replacing the derivatives of the fields with the "covariant derivative", i.e. the derivative plus a term proportional to the EM vector potential (which really just amounts to the classical p --> p - eA). I am wondering how the classical version compares with the quantum case.

Todd