The $n$-body problem -- an alternative scheme for determining solutions for planetary systems

Abstract

This study presents a general alternative scheme of the procedure and necessary conditions for solving the $n$-body problem. The presented solution is not a solution of the classical problem, where the initial conditions of positions and initial velocities/momenta of the bodies are known. Starting from the standard initial condition the procedure treats contributions to momentum from particular pair-wise interactions as independent variables from which the total momentum and velocity of a given mass is then reproduced. Initial values of those contributions are arbitrary as long as the resulting velocities match the initial condition. The obtained solutions take into account the gravitational interactions between each pair of bodies, as a result of which they are characterized by higher stability than the solutions of the classical problem. The presented procedure was used to calculate the positions and mutual velocities of three bodies: the Sun, the Earth and the Moon. It has also been tested for our entire planetary system (8 planets) with the Sun and the Moon. For these types of systems, the method allows obtaining solutions that are stable. The procedure was also tested on the example of the Pythagorean system of bodies.