Scattering in a Coulomb potential: a velocity space point of view

The unbounded motion of a particle in a Coulomb field is analyzed from the point of view of velocity space using both the hodograph and the properties of the Hamilton vector. Many features of the motion even the classical deflection function and the differential scattering cross section in velocity space follow from simple geometrical considerations, the standard Rutherford scattering formula in configuration space can then be simply obtained from them. We address the connection between initial conditions and the properties of the scattering orbits with the help of the Hamilton vector. We also discuss an approximate method for calculating the effect of a central perturbation on the properties of the hodograph and on the Rutherford's differential scattering cross section.