The Coulomb problem and Rutherford scattering using the Hamilton vector

Abstract

The motion of a particle in a Coulomb field is analyzed with the help of the conserved Hamilton vector. This affords a simple way of obtaining both the orbit in configuration space and the hodograph in velocity space. We show how to obtain the Hamilton vector, then, with its help, we get the equations of both trajectories. We next show that the trajectories of the Coulomb problem in velocity space are all circular. We also exhibit a geometric method for calculating the deflection angle in the case of scattering trajectories and then we derive the Rutherford scattering formula. We also discuss an approximate method which takes advantage of the Hamilton vector for studying scattering in a centrally perturbed Coulomb field. As an example of the use of this approach the case of an inverse cubic perturbation is discussed in some detail.