Lorentz-invariant Newtonian mechanics for three or more particles

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Abstract
Lorentz-invariant Newtonian equations of motion for three or more interacting particles can be constructed just as for two by assuming that the center-of-mass acceleration is zero when the center-of-mass velocity is zero. But two-particle systems are not obtained from a larger system of this kind when the other particles are widely separated in space. The same is true if the total relativistic kinematic particle momentum and its time derivative are used in place of the center-of-mass velocity and acceleration. The assumption that the center-of-mass acceleration is zero when the center-of-mass velocity is zero does not hold, in particular, for party-conserving forces for three or more identical particles as it does for two.