Abstract
As an extension of the familiar conserved vector of the nonrelativistic Kepler problem, the rotating Laplace-Runge-Lenz vector is introduced to describe the precessing elliptical orbits of the Kepler problem modified by an inverse cube force and the relativistic Kepler problem. This vector, which has a constant length, is always oriented to the moving perihelion points of the orbit. Using a transformation defined by an exponential operator with the vector product, this rotating vector is derived from the conserved vector. The transformations yield the dynamical systems without precessions, so that the energy eigenvalues of the two problems are derived without solving Schrodinger or Klein-Gordon equations in quantum mechanics.
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