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The inverse square force law admits a conserved vector that lies in the planeof motion. This vector has been associated with the names of Laplace, Runge,and Lenz, among others. Many workers have explored aspects of the symmetry anddegeneracy associated with this vector and with analogous dynamical symmetris.We define a conserved dynamical variable $\alpha$ that characterizes theorientation of the orbit in two-dimensional configuration space for the Keplerproblem and an analogous variable $\beta$ for the isotropic harmonicsoscillator. This orbit orientation variable is canonically conjugate to theangular momentum component normal to the plane of motion. We explore thecanoncial 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 thesephase space coordinates, the form of the Hamiltonian is nearly trivial becauseneither member of the pair can occur explicitly in the Hamiltonian. From theseconsiderations we gain a simple picture of the dynamics in phase space. Theprocedure we use is in the spirit of the Hamilton-Jacobi method.Comment: 15 pages, 1 figure, to be published in American Journal of Physic

Illustrating dynamical symmetries in classical mechanics: The Laplace–Runge–Lenz vector revisited