The post-post-Newtonian problem in classical electromagnetic theory

We find the Lagrangian to order c^?4 for two charged bodies (with $\text{e}{}_1\text{m}{}_1\text{=}\text{e}{}_2\text{m}{}_2$) 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 equationsof 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 c^?2). 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.