Markus Kunze, Herbert Spohn
Slow Motion of Charges Interacting Through the Maxwell Field
(91K, LATEX)

ABSTRACT.  We study the Abraham model for $N$ charges interacting with the Maxwell 
field. On the scale of the charge diameter, $R_{\varphi}$, the charges 
are a distance $\eps^{-1}R_{\varphi}$ apart and have a velocity 
$\sqrt{\eps} c$ with $\eps$ a small dimensionless parameter. We follow 
the motion of the charges over times of the order 
$\eps^{-3/2}R_{\varphi}/c$ and prove that on this time scale their 
motion is well approximated by the Darwin Lagrangian. The mass is 
renormalized. The interaction is dominated by the instantaneous Coulomb 
forces, which are of the order $\eps^{2}$. The magnetic fields and 
first order retardation generate the Darwin correction of the order 
$\eps^{3}$. Radiation damping would be of the order $\eps^{7/2}$.