We fmd the Lagrangian to order cm4 for two charged bodies (with cl/ml = ee/mz) in electromagnetic theory. This Lagrangian contains acceleration terms in its final form and we show why it is incorrect to eliminate these terms by using the equations of motion in the Lagrangian as was done by Golubenkov and Smorodinskii, and by Landau and Lifshitz. We find the center of inertia and show that the potential energy term does not split equally between particles 1 and 2 as it does in the Darwin Lagrangian (Lagrangian to order CC"). In addition to the infinite self-energy terms in the electromagnetic energy-momentum tensor, which are eliminated using Gupta's method, some new type of divergent terms are found in the moment of electromagnetic field energy and in the electromagnetic field momentum which cancel in the final conservation law for the center of inertia.
I. INTRODUCTION
For certain n-body Lagrangians (to order c? and in standard coordinates [I]), such as the Darwin (pure electromagnetism) or Einstein-Infeld-Hoffmann (pure gravitation) or BaZaliski (gravitation and electromagnetism) Lagrangians, it is well known [2, 3,4, 51 that in finding the center of inertia the potential energy terms -Grnirnj/rii and eiei/rij must be split equally between the particles i and j. We have recently shown [6] that this +, 4 split, as we shall call it, also holds for the case of the gravitational n-body Lagrangian (to order c-~ and in stqndard coordinates [I]) with parameterized post-Newtonian (PPN) parameters y and /3. We have also shown [6] that the 4, + split holds only for certain coordinate systems. We have found [6], for the case of the Bazanski Lagrangian, coordinate systems where something other than the &, 4 split occurs.
In this paper we shall turn our attention to the two-body post-post-Newtonian (i.e., to order c-3 Lagrangian in electromagnetism. In order to postpone dipole radiation from the c-~ to the c-~ order we must require [7] that cl/ml = e2/m2 . If we were doing the n-body problem we would have to require that the charge to mass ratio for all the particles was the same. As we are dealing with pure electromagnetism 358