  The inverse square force law admits a conserved vector that lies in the plane
of motion. This vector has been associated with the names of Laplace, Runge,
and Lenz, among others. Many workers have explored aspects of the symmetry and
degeneracy associated with this vector and with analogous dynamical symmetris.
We define a conserved dynamical variable $\alpha$ that characterizes the
orientation of the orbit in two-dimensional configuration space for the Kepler
problem and an analogous variable $\beta$ for the isotropic harmonics
oscillator. This orbit orientation variable is canonically conjugate to the
angular momentum component normal to the plane of motion. We explore the
canoncial one-parameter group of transformations generated by $\alpha (\beta).$
Because we have an obvious pair of conserved canonically conjugate variables,
it is desirable to us them as a coordinate-momentum pair. In terms of these
phase space coordinates, the form of the Hamiltonian is nearly trivial because
neither member of the pair can occur explicitly in the Hamiltonian. From these
considerations we gain a simple picture of the dynamics in phase space. The
procedure we use is in the spirit of the Hamilton-Jacobi method.
