One widely accepted model of classical electrodynamics assumes that a moving charged particle produces both retarded and advanced fields. This formulation first appeared at least 75 years ago. It was popularized in the 1940's by work of Wheeler and Feynman. But the most fundamental question associated with the model has remained unanswered: When (if ever) does the two-body problem have a unique solution? The present paper gives an answer in one special case. Imagine two identical charged particles alone in the universe moving symmetrically along the $x$ axis. One is at $x\left(t\right)\mathrm{}$ and the other is at $-x\left(t\right)\mathrm{}$. Their motion is then governed by a system of functional differential equations involving both retarded and advanced arguments. This system together with the Newtonian "initial" data $x\left(0\right)=x_0>0\mathrm{}$ and $x^{\prime }\mathrm{}\left(0\right)=0\mathrm{}$ has a unique solution for all time provided $x_0$ is sufficiently large. Perhaps the existence and uniqueness proof given for this special case will pave the way for more general results on this curious two-body problem.

• Received 10 April 1978

DOI:https://doi.org/10.1103/PhysRevD.19.1098