  We study the orbits of two interacting particles described by a fully
relativistic classical mechanical Hamiltonian. We use two sets of initial
conditions. In the first set (dynamics 1) the system's center of mass is at
rest. In the second set (dynamics 2) the center of mass evolves with velocity
V. If dynamics 1 is observed from a reference frame moving with velocity -V,
the principle of relativity requires that all observables must be identical to
those of dynamics 2 seen from the lab frame. Our numerical simulations
demonstrate that kinematic Lorentz space-time transformations fail to transform
particle observables between the two frames. This is explained as a result of
the inevitable interaction-dependence of the boost generator in the instant
form of relativistic dynamics. In spite of general inaccuracies of Lorentz
formulas, the orbital periods are correctly predicted by the Einstein's time
dilation factor for all interaction strengths.
