Lorentz-invariant Newtonian mechanics for two particles

Many ordinary Newtonian equations of motion in the center-of-mass frame can be made into Lorentz-invariant equations valid in all inertial frames. This is shown by outlining a construction of Poincar\'e-invariant Newtonian equations of motion, using global Lorentz transformations, starting with the assumption that at zero center-of-mass velocity the center-of-mass acceleration is zero, and the relative acceleration is specified as a function of the relative position and relative velocity, subject to some weak conditions. The only remarkable condition is a limit on repulsive forces, needed to keep the transformed center-of-mass velocity from being zero. This is generally satisfied up to energies comparable with the rest-mass energy. For Coulomb forces, it fails for higher energies, when the particles get closer than the classical electron radius and violate the uncertainty principle.