<p>Celestial Newtonian systems have regular dynamics, but with classical Keplerian rotation velocities, which disagree with the observed rotation of galaxies in the Universe. However, modifications of the classical accelerations or gravitational attractions can overcome this defect. The large-scale simulations of galaxies are with approximations with “particle–mesh” (PM) substitutions of the attractions from objects far away, which affects the regular dynamics. Here, we investigate the impact of the PM approximation and of the modifications of accelerations or gravitational attractions on the stability of the regular dynamics in a celestial system. The simple three-body system (TBS) is the simplest system to test the stability of the regular dynamics with approximations or with modifications of celestial dynamics, and it is easy to implement on a computer. Simulations of the TBS show that the PM approximation, and the modification of the accelerations (MOND), destabilizes TBS. In contrast, a modification of gravity by replacing Newton’s inverse square attraction with an increased attraction (Yukawa, MOGA) for faraway interactions stabilizes the system. The PM approximation and the MOND modification of classical dynamics do not preserve the momentum and angular momentum of a conservative system exactly, and they do not obey Newton’s third law. Although these errors and shortcomings are small, they eventually cause the instability of the regular dynamics.</p>