Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability
We present a systematic procedure for the determination of a complete set of kth-order (k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/147921 |
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| author | Muhammad Ayub Masood Khan F. M. Mahomed |
| author_facet | Muhammad Ayub Masood Khan F. M. Mahomed |
| author_sort | Muhammad Ayub |
| collection | DOAJ |
| description | We present a systematic procedure for the determination of a complete set of kth-order
(k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case of k = 2 and 31 classes for
the case of k≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of two kth-order (k≥3) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories. |
| format | Article |
| id | doaj-art-8f4fefe31c834318b6136dc8a90db626 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-8f4fefe31c834318b6136dc8a90db6262025-02-03T06:06:35ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/147921147921Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and IntegrabilityMuhammad Ayub0Masood Khan1F. M. Mahomed2Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, PakistanDepartment of Mathematics, Quaid-i-Azam University, Islamabad 45320, PakistanCentre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South AfricaWe present a systematic procedure for the determination of a complete set of kth-order (k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case of k = 2 and 31 classes for the case of k≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of two kth-order (k≥3) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.http://dx.doi.org/10.1155/2013/147921 |
| spellingShingle | Muhammad Ayub Masood Khan F. M. Mahomed Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability Journal of Applied Mathematics |
| title | Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability |
| title_full | Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability |
| title_fullStr | Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability |
| title_full_unstemmed | Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability |
| title_short | Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability |
| title_sort | second order systems of odes admitting three dimensional lie algebras and integrability |
| url | http://dx.doi.org/10.1155/2013/147921 |
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