We find a two-degree-of-freedom Hamiltonian for the time-symmetric problem of
straight line motion of two electrons in direct relativistic interaction. This
time-symmetric dynamical system appeared 100 years ago and it was popularized
in the 1940s by the work of Wheeler and Feynman in electrodynamics, which was
left incomplete due to the lack of a Hamiltonian description. The form of our
Hamiltonian is such that the action of a Lorentz transformation is explicitly
described by a canonical transformation (with rescaling of the evolution
parameter). The method is closed and defines the Hamiltonian in implicit form
without power expansions. We outline the method with an emphasis on the physics
of this complex conservative dynamical system. The Hamiltonian orbits are
calculated numerically at low energies using a self-consistent steepest-descent
method (a stable numerical method that chooses only the nonrunaway solution).
The two-degree-of-freedom Hamiltonian suggests a simple prescription for the
canonical quantization of the relativistic two-body problem.