The Ermanno-Bernoulli constants and representations of the complete symmetry group of the Kepler Problem

The expression of the components of the equation of motion of the classical Kepler problem in terms of the natural variables associated with the Ermanno-Bernoulli constants leads naturally to the same equations as are obtained by the technique of reduction of order developed by Nucci [J. Math. Phys. 37, 1772 (1996)], reported by Nucci and Leach [J. Math. Phys. 42, 746 (2001)]. Three representations of the complete symmetry group of the Kepler problem are obtained from the three standard representations of the complete symmetry group of the simple harmonic oscillator. The algebra of the complete symmetry group of the two-dimensional Kepler problem is identified to be A(1)circle plus{A(3,3)}. The applicability of the results to other classes of problem, such as the Kepler problem with drag, which possess a conserved vector of Laplace-Runge-Lenz type, is indicated. The three-dimensional Kepler problem is shown to be completely specified by six symmetries rather than the eight previously reported by Krause [J. Math. Phys. 35, 5734 (1994)].