Hamilton symmetry in relativistic Coulomb systems

Abstract: Hamilton's hodograph method geometrizes, in a simple and very elegant way, in velocity space, the full dynamics of classical particles in $1/r$ potentials. States of given energy and angular momentum are represented by circular hodographs whose radii depend only on the angular momentum, and hodographs differing only in the energy are related by uniform translations. This feature indicates the existence of an internal symmetry, named here after Hamilton. The hodograph method and the Hamilton symmetry are extended here for relativistic charged particles in a Coulomb field, on the relativistic velocity space which is a 3D hyperboloid $H^3$ embedded in a 3+1 pseudo-Euclidean space. The key for the simplicity and elegance of the velocity-space method is the linearity of the velocity equation, a unique feature of $1/r$ interactions for Newtonian and relativistic systems alike. Although with hodographs much more complicated than for Newtonian systems, the main features of the Hamilton symmetry persist in the full relativistic picture : (1) general hodographs may be represented as linearly displaced base energy-independent circles, (2) hodographs corresponding to same angular momentum but with different energies are connected via translations along geodesics of the velocity space. As an internal symmetry over and beyond central symmetry, the Hamilton symmetry is equivalent to the Laplace-Runge-Lenz symmetry and complements it.