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Symmetries of second-order systems of ODEs and integrability

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Abstract

We present a method for finding a complete set of kth-order (k≥2) differential invariants including bases of invariants corresponding to vector fields in three variables of four-dimensional real Lie algebras. As a consequence, we provide a complete list of second-order differential invariants and canonical forms for vector fields of four-dimensional Lie algebras and their admitted regular systems of two second-order ODEs. Moreover, we classify invariant representations of these canonical forms of ODEs into linear, partial linear, uncoupled, and partial uncoupled cases. In addition, we give an integration procedure for invariant representations of canonical forms for regular systems of two second-order ODEs admitting four-dimensional Lie algebras.

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References

  1. Aminova, A.V., Aminov, N.A.M.: Projective geometry of systems of second-order differential equations. Mat. Sb. 197(7), 3–28 (2006)

    Article  MathSciNet  Google Scholar 

  2. Ayub, M., Khan, M., Mahomed, F.M.: Second-order systems of ODEs admitting three-dimensional Lie algebras and integrability. J. Appl. Math. 2013, 147921 (2013)

    Article  MathSciNet  Google Scholar 

  3. Ayub, M., Khan, M., Mahomed, F.M.: Algebraic linearization criteria for systems of ordinary differential equations. Nonlinear Dyn. 67(3), 2053–2062 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bagderina, Y.Y.: Linearization criteria for a system of two second-order ordinary differential equations. J. Phys. A, Math. Theor. 43, 465201 (2010)

    Article  MathSciNet  Google Scholar 

  5. Bianchi, L.: Lezioni Sulla Teoria dei Gruppi Continui Finiti di Transformazioni. Enirco Spoerri, Pisa (1918)

    Google Scholar 

  6. Damianou, P.A., Sophocleous, C.: Symmetries of Hamiltonian systems with two degrees of freedom. J. Math. Phys. 40, 210–235 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Douglas, J.: Solution of the inverse problem of the calculus of variations. Trans. Am. Math. Soc. 50, 71–128 (1941)

    Article  Google Scholar 

  8. Gaponova, O., Nesterenko, M.: Systems of second-order ODEs invariant with respect to low-dimensional Lie algebras. Physics Am. Univ. Cairo 16(II), 238–256 (2006)

    Google Scholar 

  9. Gorringe, V.M., Leach, P.G.L.: Kepler’s third law and the oscillator isochronism. Am. J. Phys. 61, 991–995 (1993)

    Article  Google Scholar 

  10. Ibragimov, N.H., Mahomed, F.M., Mason, D.P., Sherwell, D.: Differential Equations and Chaos. New Age International (P) Limited, New Delhi (1996)

    Google Scholar 

  11. Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956)

    Google Scholar 

  12. Leach, P.G.L.: A further note on the Hénon-Heiles problem. J. Math. Phys. 22, 679–682 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Leach, P.G.L., Gorringe, V.M.: The relationship between the symmetries of and the existence of conserved vectors for the equation \(\ddot{x}+ f(r)L + g(r)\hat{r}= 0 \). J. Phys. A, Math. Gen. 23, 2765–2774 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lie, S.: Vorlesungen Uber Differentialgleichungen mit bekannten infinitesimalen Transformationen. B.G. Teubner, Leipzig (1891)

    Google Scholar 

  15. Lie, S.: Uber differentiation. Math. Ann. 24, 537–578 (1927). Gesammetle Abhandlungen, 6 Leipzig, B.G. Teubner, 95–138 (1927)

    Article  MathSciNet  Google Scholar 

  16. Mellin, C.M., Mahomed, F.M., Leach, P.G.L.: Solution of generalized Emden–Fowler equations with two symmetries. Int. J. Non-Linear Mech. 29, 529 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mubarakzyanov, G.M.: On solvable Lie algebras. Izv. Vysš. Učebn. Zaved., Mat. 1(32), 114–123 (1963) (in Russian)

    Google Scholar 

  18. Ovsiannikov, L.V.: In: Ames, W.F. (ed.) Group Analysis of Differential Equations. Academic Press, New York (1982)

    Google Scholar 

  19. Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Graduate Text in Mathematics. Springer, New York (1993)

    Book  MATH  Google Scholar 

  20. Olver, P.J.: Equivalence, Invariants and Symmetry. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  21. Popovych, R.O., Boyko, V.M., Nesterenko, M.O., Lutfullin, M.W.: Realizations of real low-dimensional Lie algebras. J. Phys. A, Math. Gen. 36, 7337–7360 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wafo Soh, C., Mahomed, F.M.: Canonical forms for systems of two second-order ordinary differential equations. J. Phys. A, Math. Gen. 34, 2883–2911 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wafo Soh, C., Mahomed, F.M.: Linearization criteria for a system of second-order ordinary differential equations. Int. J. Non-Linear Mech. 36(4), 671–677 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wafo Soh, C., Mahomed, F.M.: Reduction of order for systems of ordinary differential equations. J. Nonlinear Math. Phys. 11(1), 13–20 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

FMM thanks the National Research Foundation of South Africa for an enabling research grant. We are grateful to the referees for useful comments.

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Authors and Affiliations

  1. Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan

    Muhammad Ayub

  2. Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits, 2050, South Africa

    F. M. Mahomed

  3. Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

    Masood Khan

  4. Department of Mathematics, Azad Jammu and Kashmir University, Muzaffarabad, Pakistan

    M. N. Qureshi

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  1. Muhammad Ayub
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  2. F. M. Mahomed
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  3. Masood Khan
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Correspondence to Muhammad Ayub.

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Ayub, M., Mahomed, F.M., Khan, M. et al. Symmetries of second-order systems of ODEs and integrability. Nonlinear Dyn 74, 969–989 (2013). https://doi.org/10.1007/s11071-013-1016-3

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  • DOI: https://doi.org/10.1007/s11071-013-1016-3

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