Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2012), Article ID 147921, 15 pages.

Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability

Muhammad Ayub, Masood Khan, and F. M. Mahomed

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Abstract

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.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2012), Article ID 147921, 15 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807339

Digital Object Identifier
doi:10.1155/2013/147921

Mathematical Reviews number (MathSciNet)
MR3039734

Zentralblatt MATH identifier
1266.34054

Citation

Ayub, Muhammad; Khan, Masood; Mahomed, F. M. Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability. J. Appl. Math. 2013, Special Issue (2012), Article ID 147921, 15 pages. doi:10.1155/2013/147921. https://projecteuclid.org/euclid.jam/1394807339


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