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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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2006-01-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://www.emis.de/journals/SIGMA/2006/Paper010/ |
| _version_ | 1811253550625325056 |
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| author | Francisco José Herranz Ángel Ballesteros |
| author_facet | Francisco José Herranz Ángel Ballesteros |
| author_sort | Francisco José Herranz |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-04-12T16:52:43Z |
| format | Article |
| id | doaj.art-54e1562a5b534a19babb5578d99fb8eb |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-04-12T16:52:43Z |
| publishDate | 2006-01-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-54e1562a5b534a19babb5578d99fb8eb2022-12-22T03:24:21ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592006-01-012010Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant CurvatureFrancisco José HerranzÁngel BallesterosA 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.http://www.emis.de/journals/SIGMA/2006/Paper010/integrable systemscurvaturecontractionharmonic oscillatorKepler-Coulombhyperbolicde Sitter |
| spellingShingle | Francisco José Herranz Ángel Ballesteros Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature Symmetry, Integrability and Geometry: Methods and Applications integrable systems curvature contraction harmonic oscillator Kepler-Coulomb hyperbolic de Sitter |
| title | 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_short | Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_sort | superintegrability on three dimensional riemannian and relativistic spaces of constant curvature |
| topic | integrable systems curvature contraction harmonic oscillator Kepler-Coulomb hyperbolic de Sitter |
| url | http://www.emis.de/journals/SIGMA/2006/Paper010/ |
| work_keys_str_mv | AT franciscojoseherranz superintegrabilityonthreedimensionalriemannianandrelativisticspacesofconstantcurvature AT angelballesteros superintegrabilityonthreedimensionalriemannianandrelativisticspacesofconstantcurvature |