Ermakov's Superintegrable Toy and Nonlocal Symmetries
We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient, and the algebra is unsuitable for the complete specification of the system...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2005 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2005
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209342 |
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| Zitieren: | Ermakov's Superintegrable Toy and Nonlocal Symmetries / P.G.L. Leach, A. Karasu (Kalkanli), M.C. Nucci, K. Andiopoulos // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 50 назв. — англ. |
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Leach, P.G.L. Karasu (Kalkanli), A. Nucci, M.C. Andriopoulos, K. 2025-11-19T12:23:52Z 2005 Ermakov's Superintegrable Toy and Nonlocal Symmetries / P.G.L. Leach, A. Karasu (Kalkanli), M.C. Nucci, K. Andiopoulos // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 50 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B80; 22E70; 34C14; 37C80 https://nasplib.isofts.kiev.ua/handle/123456789/209342 https://doi.org/10.3842/SIGMA.2005.018 We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient, and the algebra is unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal, and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties that can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion. PGLL thanks the University of KwaZulu-Natal for its continuing support. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ermakov's Superintegrable Toy and Nonlocal Symmetries Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Ermakov's Superintegrable Toy and Nonlocal Symmetries |
| spellingShingle |
Ermakov's Superintegrable Toy and Nonlocal Symmetries Leach, P.G.L. Karasu (Kalkanli), A. Nucci, M.C. Andriopoulos, K. |
| title_short |
Ermakov's Superintegrable Toy and Nonlocal Symmetries |
| title_full |
Ermakov's Superintegrable Toy and Nonlocal Symmetries |
| title_fullStr |
Ermakov's Superintegrable Toy and Nonlocal Symmetries |
| title_full_unstemmed |
Ermakov's Superintegrable Toy and Nonlocal Symmetries |
| title_sort |
ermakov's superintegrable toy and nonlocal symmetries |
| author |
Leach, P.G.L. Karasu (Kalkanli), A. Nucci, M.C. Andriopoulos, K. |
| author_facet |
Leach, P.G.L. Karasu (Kalkanli), A. Nucci, M.C. Andriopoulos, K. |
| publishDate |
2005 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient, and the algebra is unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal, and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties that can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209342 |
| citation_txt |
Ermakov's Superintegrable Toy and Nonlocal Symmetries / P.G.L. Leach, A. Karasu (Kalkanli), M.C. Nucci, K. Andiopoulos // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 50 назв. — англ. |
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| first_indexed |
2025-12-07T20:04:55Z |
| last_indexed |
2025-12-07T20:04:55Z |
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