Two-body problem. Part I: Newtonian theory and Kepler's laws

Abstract

In this paper, which contains the part $1$ of a long work on the classical problem of the motion of two bodies atracting each other mutually through forces acting along the straight-line joining them, we present  a complete text, rich in technical details on the gravitational problem of two bodies interacting via Newton's universal law of gravitation (1687), which certainly will be useful as a didatic material for both teachers and students enrolled in Classical Mechanics courses. After finding the exact analytical solution for the equation of motion and all its corresponding trajectories, we show that the bounded periodic orbits which are non-circular are always elliptic ones without any advance of the perihelion, in full agreement to the three Kepler laws. We also indicate how to find the results corresponding to the electrodynamics analogous, i.e., when two charged corpuscles attract themselves with Coulomb's force. In the part $2$, to be published in this same journal, we shall present the unpublished translation from German to Portuguese of Einstein's original paper (1915), in which he applies his General Theory of Relativity to get an approximate solution to the relativistic gravitational two-body problem, which led to elliptic orbits with a rate for the precession of the perihelion that is in good agreement to that observed for Mercury, but not for the other planets of the solar system. At last, in part $3$ we shall present the exact solution yielded by Assis's Relational Mechanics (1989), which will allow us to compare the theoretical models from Newton, Einstein and Assis, showing that, though the last two models furnish the same rate of precession ($43$''/century), the equations of motion are not identical. We shall also show that the advance of the perihelion predicted by Relational Mechanics takes place with respect to the fixed stars, just as it has been observed and measured by astronomers, contrarily to which is found in relativistic models.