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

Abstract

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 symmetries. We define a conserved dynamical variable α that characterizes the orientation of the orbit in two-dimensional configuration space for the Kepler problem and an analogous variable β for the isotropic harmonic oscillator. This orbit orientation variable is canonically conjugate to the angular momentum component normal to the plane of motion. We explore the canonical one-parameter group of transformations generated by α (β).Because we have an obvious pair of conserved canonically conjugate variables, it is desirable to use them as a coordinate-mometum 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 dynamics in phase space.The procedure we use is in the spirit of the Hamilton-Jacobi method. 2 I. The Kepler ProblemThere are hundreds of papers 1-10 on the power, applications, and extensions of the

Forced obliquities and moments of inertia of Ceres and Vesta

“…The parameter h represents the eccentricity vector, or Laplace-Runge-Lenz vector, of the orbit (Goldstein, 1976;Kaplan, 1986;O'Connell and Jagannathan, 2003). It has length equal to the eccentricity of the orbit and is oriented along the long axis.…”

Variational contact symmetries of constrained Lagrangians

“…Such contact Noether symmetries of regular Lagrangian systems with a potential have been studied in [28] and it was shown that the corresponding conservation laws follow from the Killing tensors of the kinetic metric and the potential. Some important conservation laws of that kind in physics are: the Runge-Lenz vector field of the Kepler problem, the Ray-Reid invariant, and the Carter constant in Kerr spacetime [29,30,31,32]. As it will be shown, in the case of constrained Lagrangian systems the results are different from that of [28] in that the Noether contact symmetries are related to the Conformal Killing tensors of the underlying space.…”

Dynamical symmetries and observational constraints in scalar field cosmology

“…Indeed the dynamical Noether symmetries provide conserved quantities both in Newtonian physics and in General Relativity which point symmetries cannot. For instance, the well known Runge-Lenz vector field of the Kepler potential [68], the Ermakov integral [69,70], and the Carter constant in the Kerr spacetime [71] all follow from dynamical symmetries and not from point symmetries. These integrals are not linear in the momentum; that is, dynamical Noether symmetries provide new conservation laws in contrast to Noether point symmetries which give integrals linear in the momentum [66,67].…”