The Kepler Problem

Cordani, Bruno

doi:10.1007/978-3-0348-8051-0

Cited by 89 publications

(53 citation statements)

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“…According to the Robertson Theorem ( [7] (Sec. 8.1.3., p. 169), see also [8]), classical separability does imply, in our case, that of the Schrödinger equation 4.…”

Section: Classical Separabilitymentioning

confidence: 99%

Separability and dynamical symmetry of Quantum Dots

et al. 2014

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The separability and Runge-Lenz-type dynamical symmetry of the internal dynamics of certain two-electron Quantum Dots, found by Simonović et al. [1], is traced back to that of the perturbed Kepler problem. A large class of axially symmetric perturbing potentials which allow for separation in parabolic coordinates can easily be found. Apart of the 2:1 anisotropic harmonic trapping potential considered in [1], they include a constant electric field parallel to the magnetic field (Stark effect), the ring-shaped Hartmann potential, etc. The harmonic case is studied in detail.

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“…According to the Robertson Theorem ( [7] (Sec. 8.1.3., p. 169), see also [8]), classical separability does imply, in our case, that of the Schrödinger equation 4.…”

Section: Classical Separabilitymentioning

confidence: 99%

Separability and dynamical symmetry of Quantum Dots

et al. 2014

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“…As we will see in the next section, this variant generalizes Hamilton's eccentricity vector [13] which is known to be a first integral for inverse-square central forces.…”

Section: First Integrals In N Dimensionsmentioning

confidence: 97%

“…This means that the expression (7.6) provides a general Laplace-Runge-Lenz vector that points toward the periapsis point for all three types of trajectories. In n = 3 dimensions, this vector is exactly the usual Laplace-Runge-Lenz vector [3,13] A * = v × L − k| r| −1 r (7.11)…”

Section: Examples Of N-dimensional General Laplace-runge-lenz Vectormentioning

confidence: 99%

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

Anco

Meadows

Pascuzzi

2016

Journal of Mathematical Physics

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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).

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“…Due to the singular motions, it is non-Hausdorff. Other modern treatments of Kepler's dynamical system may be found in the books by V. Guillemin and S. Sternberg [5] and by B. Cordani [3].…”

Section: The Manifold Of Motions Of a Planet In Kepler's Approximationmentioning

confidence: 99%