Comparison of the Newtonian and relativistic predicted trajectories for a low-speed periodically delta-kicked system.

The dynamics of a periodically delta-kicked Hamiltonian system moving at low speed (i.e., at speed much less than the speed of light) is studied numerically. In particular, the trajectory of the system predicted by Newtonian mechanics is compared with the trajectory predicted by special relativistic mechanics for the same parameters and initial conditions. We find that the Newtonian trajectory, although close to the relativistic trajectory for some time, eventually disagrees completely with the relativistic trajectory, regardless of the nature (chaotic, nonchaotic) of each trajectory. However, the agreement breaks down very fast if either the Newtonian or relativistic trajectory is chaotic, but very much slower if both the Newtonian and relativistic trajectories are nonchaotic. In the former chaotic case, the difference between the Newtonian and relativistic values for both position and momentum grows, on average, exponentially. In the latter nonchaotic case, the difference grows much slower, for example, linearly on average.