Motion On the Sphere: Integrability and Families of Orbits

Voyatzi, Despina; Ichtiaroglou, S.

doi:10.1007/s10569-005-1783-3

Cited by 9 publications

(6 citation statements)

References 13 publications

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“…These results have been recently obtained by Voyatzi and Ichtiaroglou [17]. The straight lines v = c are geodesics and the circles u = c are orthogonal to them, see Fig.…”

Section: Examplessupporting

confidence: 83%

“…the angular momentum), while a family of concentric circles guarantees that the potential is separable in polar coordinates and thus possesses an integral, quadratic in the velocities. On the other hand, Voyatzi and Ichtiaroglou [17] studied motion on the two-dimensional sphere and showed that the permissibility of a family of meridians or a family of parallels guarantees integrability, with linear or quadratic integrals respectively.…”

Section: Introductionmentioning

confidence: 99%

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Motion on a given surface: Monoparametric families of orbits sufficient for separability

Kotoulas

Ichtiaroglou

2006

Journal of Geometry and Physics

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“…These results have been recently obtained by Voyatzi and Ichtiaroglou [17]. The straight lines v = c are geodesics and the circles u = c are orthogonal to them, see Fig.…”

Section: Examplessupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Motion on a given surface: Monoparametric families of orbits sufficient for separability

Kotoulas

Ichtiaroglou

2006

Journal of Geometry and Physics

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“…We also note that our results contain those of [61,64] We emphasize that the potentials listed in Table 4.12 concern dynamical systems with Lagrangian (4.42) which are integrable via Noether point symmetries.…”

Section: Noether Symmetriessupporting

confidence: 59%

Symmetries of Differential equations with applications in Relativistic Physics

Παλιαθανάσης¹

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Σε αυτή την εργασία μελετάμε τους μονοπαραμετρικούς μετασχηματισμούς κάτω από τους οποίους οι διαφορικές εξισώσεις είναι αναλλοίωτες. Ειδικότερα μελετάμε τις σημειακές συμμετρίες Lie και Noether διαφορικών εξισώσεων τάξεως. Αναπτύσσουμε μια γεωμετρική μέθοδο για τον υπολογισμό των συμμετριών η οποία συνδέει τις σημειακές συμμετρίες των διαφορικών εξισώσεων με τις συμμετρίες του χώρου που πραγματοποιείται η κίνηση. Η γεωμετρική μέθοδος εφαρμόζεται σε διάφορα προβλήματα όπως: η κατηγοριοποίηση των συμμετριών Νευτώνειων συστημάτων δύο και τριών διαστάσεων, η γενίκευση του συστήματος Kepler-Ermakov σε καμπύλους χώρους, η σύνδεση των συμμετριών ανάμεσα σε κλασσικά και κβαντικά συστήματα και η αναζήτηση Τύπου ΙΙ κρυφών συμμετριών στην κυματική εξίσωση και στην εξίσωση διάδοσης θερμότητας σε καμπύλους χώρους. Τέλος, η γεωμετρική μέθοδος εφαρμόστηκε σαν γεωμετρικό κριτήριο για την επιλογή διάφορων μοντέλων στις εναλλακτικές θεωρίες βαρύτητας.

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“…In fact, it has been studied by many authors (see e.g. [48][49][50][51][52][53][54][55][56][57][58][59] and references therein) mainly in relation of the theory of dynamical symmetries.…”

Section: On the Relation Of This Nonlinear Model With The Harmonic Osmentioning

confidence: 99%

The quantum harmonic oscillator on the sphere and the hyperbolic plane

2007

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A nonlinear model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature is exactly solved. This model depends of a parameter $\la$ that is related with the curvature of the space. Firstly the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a $\la$-dependent invariant measure $d\mu_\la$ and expressing the Hamiltonian as a function of the Noether momenta. In the second part the quantum superintegrability of the Hamiltonian and the multiple separability of the Schr\"odinger equation is studied. Two $\la$-dependent Sturm-Liouville problems, related with two different $\la$-deformations of the Hermite equation, are obtained. This leads to the study of two $\la$-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions $\Psi_{m,n}$ and the energies $E_{m,n}$ of the bound states are exactly obtained in both the sphere $S^2$ and the hyperbolic plane $H^2$.Comment: 35 pages, 7 figure

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Motion On the Sphere: Integrability and Families of Orbits

Cited by 9 publications

References 13 publications

Motion on a given surface: Monoparametric families of orbits sufficient for separability

Motion on a given surface: Monoparametric families of orbits sufficient for separability

Symmetries of Differential equations with applications in Relativistic Physics

The quantum harmonic oscillator on the sphere and the hyperbolic plane

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