  We numerically solve the functional differential equations (FDE's) of
2-particle electrodynamics, using the full electrodynamic force obtained from
the retarded Lienard-Wiechert potentials and the Lorentz force law. In
contrast, the usual formulation uses only the Coulomb force (scalar potential),
reducing the electrodynamic 2-body problem to a system of ordinary differential
equations (ODE's). The ODE formulation is mathematically suspect since FDE's
and ODE's are known to be incompatible; however, the Coulomb approximation to
the full electrodynamic force has been believed to be adequate for physics. We
can now test this long-standing belief by comparing the FDE solution with the
ODE solution, in the historically interesting case of the classical hydrogen
atom. The solutions differ.
  A key qualitative difference is that the full force involves a `delay'
torque. Our existing code is inadequate to calculate the detailed interaction
of the delay torque with radiative damping. However, a symbolic calculation
provides conditions under which the delay torque approximately balances (3rd
order) radiative damping. Thus, further investigations are required, and it was
prematurely concluded that radiative damping makes the classical hydrogen atom
unstable. Solutions of FDE's naturally exhibit an_infinite_ spectrum of
_discrete_ frequencies. The conclusion is that (a) the Coulomb force is_not_ a
valid approximation to the full electrodynamic force, so that (b) the n-body
interaction needs to be reformulated in various current contexts such as
molecular dynamics.
