Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials
The structure theory for the quadratic algebra generated by first and second order constants of the motion for 2D second order superintegrable systems with nondegenerate (3-parameter) and or 2-parameter potentials is well understood, but the results for the strictly 1-parameter case have been incomp...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2009-01-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2009.008 |
| _version_ | 1819139471066529792 |
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| author | Ernest G. Kalnins Jonathan M. Kress Willard Miller Jr. Sarah Post |
| author_facet | Ernest G. Kalnins Jonathan M. Kress Willard Miller Jr. Sarah Post |
| author_sort | Ernest G. Kalnins |
| collection | DOAJ |
| description | The structure theory for the quadratic algebra generated by first and second order constants of the motion for 2D second order superintegrable systems with nondegenerate (3-parameter) and or 2-parameter potentials is well understood, but the results for the strictly 1-parameter case have been incomplete. Here we work out this structure theory and prove that the quadratic algebra generated by first and second order constants of the motion for systems with 4 second order constants of the motion must close at order three with the functional relationship between the 4 generators of order four. We also show that every 1-parameter superintegrable system is Stäckel equivalent to a system on a constant curvature space. |
| first_indexed | 2024-12-22T11:23:11Z |
| format | Article |
| id | doaj.art-c1da21cd506a45df94532fef1b194269 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-22T11:23:11Z |
| publishDate | 2009-01-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-c1da21cd506a45df94532fef1b1942692022-12-21T18:27:48ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592009-01-015008Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter PotentialsErnest G. KalninsJonathan M. KressWillard Miller Jr.Sarah PostThe structure theory for the quadratic algebra generated by first and second order constants of the motion for 2D second order superintegrable systems with nondegenerate (3-parameter) and or 2-parameter potentials is well understood, but the results for the strictly 1-parameter case have been incomplete. Here we work out this structure theory and prove that the quadratic algebra generated by first and second order constants of the motion for systems with 4 second order constants of the motion must close at order three with the functional relationship between the 4 generators of order four. We also show that every 1-parameter superintegrable system is Stäckel equivalent to a system on a constant curvature space.http://dx.doi.org/10.3842/SIGMA.2009.008superintegrabilityquadratic algebras |
| spellingShingle | Ernest G. Kalnins Jonathan M. Kress Willard Miller Jr. Sarah Post Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials Symmetry, Integrability and Geometry: Methods and Applications superintegrability quadratic algebras |
| title | Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials |
| title_full | Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials |
| title_fullStr | Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials |
| title_full_unstemmed | Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials |
| title_short | Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials |
| title_sort | structure theory for second order 2d superintegrable systems with 1 parameter potentials |
| topic | superintegrability quadratic algebras |
| url | http://dx.doi.org/10.3842/SIGMA.2009.008 |
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