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Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
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- Фізико-технічні та математичні науки
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- Відділення математики
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7, випуск за цей рік
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Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
Інший ID:
2010 Mathematics Subject Classification: 37J35; 70H06; 81R12
DOI:10.3842/SIGMA.2011.048
DOI:10.3842/SIGMA.2011.048
Посилання:
Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform / A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, D. Riglioni // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 48 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html.
This work was partially supported by the Spanish MICINN under grants MTM2010-18556 and FIS2008-00209, by the Junta de Castilla y Le´on (project GR224), by the Banco Santander–UCM (grant GR58/08-910556) and by the Italian–Spanish INFN–MICINN (project ACI2009-1083). F.J.H. is deeply grateful to W. Miller Jr. for very helpful suggestions on the St¨ackel transform as well on superintegrability.
Дата:
2011
Переглядів:
548
Завантажень:
250
Анотація:
The Stäckel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler-Coloumb potentials, in order to obtain maximally superintegrable classical systems on N-dimensional Riemannian spaces of nonconstant curvature. By one hand, the harmonic oscillator potential leads to two families of superintegrable systems which are interpreted as an intrinsic Kepler-Coloumb system on a hyperbolic curved space and as the so-called Darboux III oscillator. On the other, the Kepler-Coloumb potential gives rise to an oscillator system on a spherical curved space as well as to the Taub-NUT oscillator. Their integrals of motion are explicitly given. The role of the (flat/curved) Fradkin tensor and Laplace-Runge-Lenz N-vector for all of these Hamiltonians is highlighted throughout the paper. The corresponding quantum maximally superintegrable systems are also presented.
