A "Backwards" Two-Body Problem of Classical Relativistic Electrodynamics

The two-body problem of classical electrodynamics can be formulated in terms of action at a distance by using the retarded Liénard-Wiechert potentials (or a combination of retarded and advanced potentials). The resulting equations of motion in the retarded case, for example, form a complicated functional delay-differential system. For such equations in the case of one-dimensional motion, as shown in an earlier paper, one can specify rather arbitrary past histories of the particles, and then solve for the future trajectories. Yet it is often assumed or asserted that unique trajectories would be determined by the specification of Newtonian initial data (the positions and velocities of the particles at some instant). Simple examples of delay-differential equations should lead one to doubt this. For example, even if the values of $x$ and all its derivatives are given at $t_0$, the equation $x^{\prime }\mathrm{}\left(t\right)=ax\left(t\right)+bx(t-\tau )$, with $\tau >0\mathrm{}$ and $b\ne 0$, still has infinitely many solutions valid for all $t$. Nevertheless, under certain special conditions for the electrodynamic equations it is found that instantaneous values of positions and velocities do indeed determine the solution uniquely. The case treated in this paper involves two charges of like sign moving on the $x$ axis, assumed to have been subject only to their mutual retarded electrodynamic interaction for all time in the past. The similar questions of existence and uniqueness for a more general model, e.g., three-dimensional motion or half-retarded and half-advanced interactions, or even for charges with opposite signs remain open.