Physical origin of the Runge-Lenz vector

The dynamical symmetry of the non-relativistic Kepler problem has attracted much attention in the scientific literature. In the present paper, we show that the Runge-Lenz vector, which accounts for the presence of this symmetry, has its physical origin in the generator of Lorentz transformations for the relativistic two-body problem. We reach this conclusion by considering the relativistic two-body problem, for the electromagnetic as well as for the gravitational interaction, in the 1/c2 approximation.