Paper detail

Planetary systems with forces other than gravitational forces

A discrete and exact algorithm for obtaining planetary systems is derived in a recent article (Eur. Phys. J. Plus 2022, 137:99). Here the algorithm is used to obtain planetary systems with forces different from the Newtonian inverse square gravitational forces. A Newtonian planetary system exhibits regular elliptical orbits, and here it is demonstrated that a planetary system with pure inverse forces also is stable and with regular orbits, whereas a planetary system with inverse cubic forces is unstable and without regular orbits. The regular orbits in a planetary system with inverse forces deviate, however, from the usual elliptical orbits by having revolving orbits with tendency to orbits with three or eight loops. Newton's Proposition 45 in $\textit{Principia}$ for the Moon's revolving orbits caused by an additional attraction to the gravitational attraction is confirmed, but whereas the additional inverse forces stabilize the planetary system, the additional inverse cubic forces can destabilize the planetary system at a sufficient strength.