Some new aspects of first integrals and symmetries for central force dynamics

Abstract

1 department of mathematics and statistics, brock university st. catharines, on l2s3a1, canada 2 department of mathematics and statistics, mcmaster university hamilton, on l8s 4k1, canada 3 department of physics, university of toronto toronto, on m5s 1a7, canadaAbstract. For the general central force equations of motion in n > 1 dimensions, a complete set of 2n first integrals is derived in an explicit algorithmic way without the use of dynamical symmetries or Noether's theorem. The derivation uses the polar formulation of the equations of motion and yields energy, angular momentum, a generalized LaplaceRunge-Lenz vector, and a temporal quantity involving the time variable explicitly. A variant of the general Laplace-Runge-Lenz vector, which generalizes Hamilton's eccentricity vector, is also obtained. The physical meaning of the general Laplace-Runge-Lenz vector, its variant, and the temporal quantity are discussed for general central forces. Their properties are compared for precessing bounded trajectories versus non-precessing bounded trajectories, as well as unbounded trajectories, by considering an inverse-square force (Kepler problem) and a cubically perturbed inverse-square force (Newtonian revolving orbit problem).

Global versus local superintegrability of nonlinear oscillators

“…For any solution (r(t), θ(t)) of the equations of motion, L is the planar angular momentum; E is the energy (Hamiltonian); and Θ is the angle reached at some point r = r 0 . As shown in [6], a natural intrinsic choice of r 0 is any turning point r * or any inertial point r * , which are given by U eff (r * ) = E or U ′ eff (r * ) = 0 in terms of the effective potential U eff (r) = U (r) + 1 2 L 2 /r 2 − U (r equil ). In addition to the three functionally-independent c.o.m.…”

“…Thus, Θ is single-valued and nonsingular in these two cases. In particular, if r 0 = r * is a turning point at which r(t) reaches a local maximum, then as shown in [6], Θ is the angle of the LRL vector, which is a c.o.m. for the Coulomb potential and the isotropic oscillator potential.…”

“…Point symmetries: The point symmetries of the central force equations of motion are generated by X L = −∂/∂θ and X E = ∂/∂t for a general potential U (r). Additional point symmetries are admitted only for [6] two special potentials: U (r) = kr p , which admits X 1 = t∂/∂t − 2 p r∂/∂r; U (r) = kr + K/r 3 , which admits X 2 = e 2 √ kt (∂/∂t + √ kr∂/∂r).…”

“…Next, and most importantly, we show how to use the symmetry method outlined in Ref. [6] to do the reverse process: Hamiltonian system ⇒ local symmetries ⇒ first integrals. This method is systematic and explicit, and no ansatzes are needed.…”

“…The main results in the paper will be to show how to go systematically from the Hamiltonian system to local symmetries and then to first integrals, and reciprocally, from the Hamiltonian system directly to first integrals and then to local symmetries. This will be accomplished without the need for ansatzes or guess-work by adapting the extended symmetry method from [6]. As will be seen, superintegrability of the system is not related in any straightforward or universal way to either its point symmetries or its variational symmetry algebra.…”