Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature
A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of th...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2006 |
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| Sprache: | English |
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Інститут математики НАН України
2006
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| Zitieren: | Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature / F.J. Herranz, Á. Ballesteros // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 43 назв. — англ. |
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Herranz, F.J. Ballesteros, Á 2019-02-09T17:13:53Z 2019-02-09T17:13:53Z 2006 Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature / F.J. Herranz, Á. Ballesteros // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 43 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37J35; 22E60; 37J15; 70H06 https://nasplib.isofts.kiev.ua/handle/123456789/146443 A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are explicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maximally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz system. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented. This work was partially supported by the Ministerio de Educaci´on y Ciencia (Spain, Project FIS2004-07913) and by the Junta de Castilla y Le´on (Spain, Projects BU04/03 and VA013C05). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| spellingShingle |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature Herranz, F.J. Ballesteros, Á |
| title_short |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_full |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_fullStr |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_full_unstemmed |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_sort |
superintegrability on three-dimensional riemannian and relativistic spaces of constant curvature |
| author |
Herranz, F.J. Ballesteros, Á |
| author_facet |
Herranz, F.J. Ballesteros, Á |
| publishDate |
2006 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are explicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maximally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz system. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146443 |
| citation_txt |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature / F.J. Herranz, Á. Ballesteros // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 43 назв. — англ. |
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AT herranzfj superintegrabilityonthreedimensionalriemannianandrelativisticspacesofconstantcurvature AT ballesterosa superintegrabilityonthreedimensionalriemannianandrelativisticspacesofconstantcurvature |
| first_indexed |
2025-11-27T17:05:54Z |
| last_indexed |
2025-11-27T17:05:54Z |
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1850852578376548352 |