Hamilton's Eccentricity Vector Generalised to Newton Wonders

Abstract

The vectorial velocity is given as a function of the position of a particle in orbit when a Newtonian central force is supplemented by an inverse cubic force as in Newton's theorem on revolving orbits. Such expressions are useful in fitting orbits to radial velocities of orbital streams. The Hamilton-Laplace-Runge-Lenz eccentricity vector is generalised to give a constant of the motion for these systems and an approximate constant for orbits in general central potentials. A related vector is found for Hooke's centred ellipse.