Newton–Hooke-type symmetry of anisotropic oscillators

Zhang, P.M.; Horváthy, P. A.; Andrzejewski, K.; Gonera, Joanna; Kosiński, Piotr

doi:10.1016/j.aop.2012.11.018

Cited by 25 publications

(24 citation statements)

References 58 publications

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“…(2.14) is well known, in our case it is a linear combinations of trigonometric functions only or both hyperbolic (or linear) and trigonometric ones depending on the values of parameters appearing in (2.14), the form of the coefficients can be quite complicated (for γ = they are presented in [21]). In the general case the transformation to the normal coordinates seems more useful (see also [54,55]). The above results form a setup for our further considerations.…”

Section: )mentioning

confidence: 99%

From polarized gravitational waves to analytically solvable electromagnetic beams

Andrzejewski

Prencel

2019

Phys. Rev. D

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Using the correspondence between solutions of gravitational and gauge theories (the so-called classical double copy conjecture) some electromagnetic fields with vortices are constructed, for which the Lorentz force equations are analytically solvable. The starting point is a certain class of plane gravitational waves exhibiting the conformal symmetry. The notion of the Niederer transformation, crucial for the solvability, is analysed in the case of the Lorentz force equation on the curved spacetimes as well as its derivation by means of integrals of motion (associated with conformal generators preserving these vortices) is presented. Furthermore, some models discussed recently in the context of the intense laser beams are constructed from their gravitational counterparts, with the special emphasis put on the focusing property, and new solvable examples are presented.

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Section: )mentioning

confidence: 99%

From polarized gravitational waves to analytically solvable electromagnetic beams

Andrzejewski

Prencel

2019

Phys. Rev. D

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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. Restricting ourselves to natural orthogonal systems, i.e., such whose Hamiltonian is H = 1 2 n k=1 g k (x 1 , .…”

Section: Classical Separabilitymentioning

confidence: 99%

“…(2.15). Our theory provides us now with D = 2 conserved quantities in involution, namely with the separable 2D Hamiltonian, 4) and with the Runge-Lenz-type conserved quantity…”

Section: Reduction To and Induction From The 2d Problemmentioning

confidence: 99%

“…Here we only show how to use them. Put 4) where the α i s are arbitrary constants, and define the column vector K composed of n functions,…”

Section: )mentioning

confidence: 99%

“…Firstly, the system splits, consistently with Kohn's theorem, into center-of-mass and relative motion and the former system carries a NewtonHooke type symmetry [3,4]. Secondly, for the particular values of the frequency ratios…”

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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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Analytical and numerical analysis of a rotational invariant D = 2 harmonic oscillator in the light of different noncommutative phase-space configurations

Abreu

Mendes

Oliveira

2013

J. High Energ. Phys.

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In this work we have investigated some properties of classical phase-space with symplectic structures consistent, at the classical level, with two noncommutative (NC) algebras: the Doplicher-Fredenhagen-Roberts algebraic relations and the NC approach which uses an extended Hilbert space with rotational symmetry. This extended Hilbert space includes the operators θ ij and their conjugate momentum π ij operators. In this scenario, the equations of motion for all extended phase-space coordinates with their corresponding solutions were determined and a rotational invariant NC Newton's second law was written. As an application, we treated a NC harmonic oscillator constructed in this extended Hilbert space. We have showed precisely that its solution is still periodic if and only if the ratio between the frequencies of oscillation is a rational number. We investigated, analytically and numerically, the solutions of this NC oscillator in a two-dimensional phase-space. The result led us to conclude that noncommutativity induces a stable perturbation into the commutative standard oscillator and that the rotational symmetry is not broken. Besides, we have demonstrated through the equations of motion that a zero momentum π ij originated a constant NC parameter, namely, θ ij = const., which changes the original variable characteristic of θ ij and reduces the phase-space of the system. This result shows that the momentum π ij is relevant and cannot be neglected when we have that θ ij is a coordinate of the system.

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Newton–Hooke-type symmetry of anisotropic oscillators

Cited by 25 publications

References 58 publications

From polarized gravitational waves to analytically solvable electromagnetic beams

From polarized gravitational waves to analytically solvable electromagnetic beams

Separability and dynamical symmetry of Quantum Dots

Analytical and numerical analysis of a rotational invariant D = 2 harmonic oscillator in the light of different noncommutative phase-space configurations

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