An algorithm for coalescence of classical objects and formation of planetary systems

Isaac Newton formulated the central difference algorithm (Eur. Phys. J. Plus (2020) 135:267) when he derived his second law. The algorithm is under various names ("Verlet, leap-frog,..") the most used algorithm in simulations of complex system in Physics and Chemistry, and it is also applied in Astrophysics. His discrete dynamics has the same qualities as his exact analytic dynamics for continuous space and time with time reversibility, symplecticity and conservation of momentum, angular momentum and energy. Here, the algorithm is extended to include the fusion of objects at collisions. The extended algorithm is used to obtain the self-assembly of celestial objects at the emergence of planetary systems. The emergence of twelve planetary systems is obtained. The systems are stable over very long times, even when two "planets" collide or if a planet is engulfed by its sun.