Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform
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...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2011 |
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| Sprache: | English |
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Інститут математики НАН України
2011
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| Zitieren: | 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. |
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Ballesteros, A. Enciso, A. Herranz, F.J. Ragnisco, O. Riglioni, D. 2019-02-13T18:08:35Z 2019-02-13T18:08:35Z 2011 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. 1815-0659 2010 Mathematics Subject Classification: 37J35; 70H06; 81R12 DOI:10.3842/SIGMA.2011.048 https://nasplib.isofts.kiev.ua/handle/123456789/147172 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. 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform |
| spellingShingle |
Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform Ballesteros, A. Enciso, A. Herranz, F.J. Ragnisco, O. Riglioni, D. |
| title_short |
Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform |
| title_full |
Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform |
| title_fullStr |
Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform |
| title_full_unstemmed |
Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform |
| title_sort |
superintegrable oscillator and kepler systems on spaces of nonconstant curvature via the stäckel transform |
| author |
Ballesteros, A. Enciso, A. Herranz, F.J. Ragnisco, O. Riglioni, D. |
| author_facet |
Ballesteros, A. Enciso, A. Herranz, F.J. Ragnisco, O. Riglioni, D. |
| publishDate |
2011 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147172 |
| citation_txt |
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. |
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