Illustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited

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.