Improved numerical plotting of elliptical orbits using radial action coordinates -- has the symmetry of Leibniz radial theory based on inertia versus gravity been ignored?

Abstract

We that show two body gravitational orbits may be plotted using a radial reference frame rather than the customary Newtonian rectilinear inertial frame. Infinitesimal calculus cofounder and continental contemporary of Newton, Leibniz claimed that the second radial derivative could be found by taking the difference between an inertial force varying inversely, with the cubed radius and the gravitational force varying with an inversed squared radius. His radial method was severely criticised by Newton and supporters, who preferred the rectilinear inertial frame, also claiming Leibniz failed to satisfy Newton's third law of gravity. We show that these two approaches are equivalent if the central body is much larger than that in orbit. Furthermore we justify the Leibniz least action approach using the semi-latus rectum of the orbit, characteristic of the eccentricity to calculate the Newtonian centripetal acceleration opposed by the inertial acceleration, with the net radial acceleration controlling radius also allowing estimation of angular displacement and the action for each interval. Numerically, our Leibniz model can be more accurate since it requires no assumption that linear vectors adequately simulate a curvilinear trajectory. Our review of the Newton gravitational equation using centre of mass coordinates concludes that the third law is satisfied for symmetry if the gravitational and inertial masses are taken separately as equating force expressions for gravity and radial inertia, directed to the centre of mass for radial inertia. This symmetry validates centre of mass radial coordinates for more accurate plotting of orbits including dual stars, possibly applicable for better understanding of Einstein relativity. An advantage of radial coordinates is that orbiting couples can also accept perturbations from other nearby bodies acting centripetally, but not inertially.