Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerat...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2012 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2012
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148418 |
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| Cite this: | Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems / E.G. Kalnins, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862532421082152960 |
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| author | Kalnins, E.G. Miller Jr., W. |
| author_facet | Kalnins, E.G. Miller Jr., W. |
| citation_txt | Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems / E.G. Kalnins, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler-Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis (2011) showed that in the quantum case, if a second 4th order symmetry is added to the generators, the double commutators in the symmetry algebra close polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of classical extended Kepler-Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers (k₁,k₂) and reducing to the usual systems when k₁=k₂=1. We show these systems to be superintegrable of arbitrarily high order and work out explicitly the structure of the symmetry algebras determined by the 5 basis generators we have constructed. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations.
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| first_indexed | 2025-11-24T04:39:42Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148418 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T04:39:42Z |
| publishDate | 2012 |
| publisher | Інститут математики НАН України |
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| spelling | Kalnins, E.G. Miller Jr., W. 2019-02-18T12:09:46Z 2019-02-18T12:09:46Z 2012 Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems / E.G. Kalnins, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C35; 22E70; 37J35; 81R12 DOI: http://dx.doi.org/10.3842/SIGMA.2012.034 https://nasplib.isofts.kiev.ua/handle/123456789/148418 The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler-Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis (2011) showed that in the quantum case, if a second 4th order symmetry is added to the generators, the double commutators in the symmetry algebra close polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of classical extended Kepler-Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers (k₁,k₂) and reducing to the usual systems when k₁=k₂=1. We show these systems to be superintegrable of arbitrarily high order and work out explicitly the structure of the symmetry algebras determined by the 5 basis generators we have constructed. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations. This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html.
 This work was partially supported by a grant from the Simons Foundation (# 208754 to Willard Miller, Jr.). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems Article published earlier |
| spellingShingle | Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems Kalnins, E.G. Miller Jr., W. |
| title | Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems |
| title_full | Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems |
| title_fullStr | Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems |
| title_full_unstemmed | Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems |
| title_short | Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems |
| title_sort | structure theory for extended kepler-coulomb 3d classical superintegrable systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148418 |
| work_keys_str_mv | AT kalninseg structuretheoryforextendedkeplercoulomb3dclassicalsuperintegrablesystems AT millerjrw structuretheoryforextendedkeplercoulomb3dclassicalsuperintegrablesystems |