In an earlier paper we have
shown a method which may be used to construct an SU(n) group as the
symmetry group of the harmonic oscillator in classical mechanics. The method
is applicable to quadratic Hamiltonians, and was applied in a subsequent
paper to the charged harmonic oscillator in a magnetic field. We now apply
the technique to the Kepler problem, which may be made equivalent to a
harmonic oscillator by a suitable transformation. An SU(3) group is found,
generated by constants of the motion which are the angular momentum, the
Runge vector, which points to the perihelion, and a vector along the line of
nodes. Different groups are found by separation in spherical and parabolic
coordinates, while yet another group is found by inspection in parabolic
coordinates. One purpose of our investigation is to find symmetry groups for
quantum mechanical problems. While our results dispel the thought that there
might not be sets of constants of the motion closed with respect to Poisson
Brackets and thus generating a Lie symmetry group, they do show that
the functional relationship involved may make it very difficult to use the
correspondence principle to construct satisfactory quantum mechanical operators.
Our SU(3) symmetry group is not isomorphic to the R(4) symmetry group of the
hydrogen atom found by Fock and Bargmann; the angular momentum and
Runge vector are nonlinear functions of a subset of the generators of our
SU(3) group. It is not possible to find an operator generalization of these
functions which can be satisfied by irreducible representations of SU(3) and
R(4).
