Advertisement

Nonlinear Dynamics

, Volume 96, Issue 4, pp 2225–2239 | Cite as

The approximate Noether symmetries and approximate first integrals for the approximate Hamiltonian systems

  • R. NazEmail author
  • I. Naeem
Original Paper
  • 315 Downloads

Abstract

We provide the Hamiltonian version of the approximate Noether theorem developed for the perturbed ordinary differential equations (ODEs) (Govinder et al. in Phys Lett 240(3):127–131, 1998) for the approximate Hamiltonian systems. We follow the procedure adopted by Dorodnitsyn and Kozlov (J Eng Math 66(1–3):253–270, 2010) for the Hamiltonian systems of unperturbed ODEs. The approximate Legendre transformation connects the approximate Hamiltonian and approximate Lagrangian. The approximate Noether symmetries determining equation for the approximate Hamiltonian systems is defined explicitly. We provide a formula to establish an approximate first integral associated with an approximate Noether symmetry of the approximate Hamiltonian systems. We analyzed several physical models to elaborate the approach developed here.

Keywords

Approximate Noether symmetries Classification problem Phase space 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.

References

  1. 1.
    Govinder, K.S., Heil, T.G., Uzer, T.: Approximate Noether symmetries. Phys. Lett. A 240(3), 127–131 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dorodnitsyn, V., Kozlov, R.: Invariance and first integrals of continuous and discrete Hamiltonian equations. J. Eng. Math. 66(1–3), 253–270 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Approximate symmetries. Math. Sbornik, 136 (178)(3), 435–450 (1989). (Engl. Transl. Math. USSR Sb 64, 427–441, 1988)Google Scholar
  4. 4.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.K.: Perturbation methods in group analysis. J. Sov. Math. 55(1), 1450–1490 (1991)CrossRefzbMATHGoogle Scholar
  5. 5.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.H., Mahomed, F.M.: Closed orbits and their stable symmetries. J. Math. Phys. 35(12), 6525–6535 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Noether, E.: Invariante Variationsprobleme. Nachr. König. Gesell. Wissen., Göttingen, Math. Phys. Kl. 1918, Heft 2, 235–257 (1918). (English translation in Transport Theory and Statistical Physics 1(3), 186–207 1971)Google Scholar
  7. 7.
    Feroze, T., Kara, A.H.: Group theoretic methods for approximate invariants and Lagrangians for some classes of \(y^{\prime \prime }+\varepsilon F (t) y^{\prime }+ y= f (y, y^{\prime })\). Int. J. Nonlinear Mech. 37(2), 275–280 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Naeem, I., Mahomed, F.M.: Approximate partial Noether operators and first integrals for coupled nonlinear oscillators. Nonlinear Dyn. 57(1–2), 303–311 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Naeem, I., Mahomed, F.M.: Approximate first integrals for a system of two coupled van der Pol oscillators with linear diffusive coupling. Math. Comput. Appl. 15(4), 720–731 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kara, A.H., Mahomed, F.M., Naeem, I., Wafo Soh, C.: Partial Noether operators and first integrals via partial Lagrangians. Math. Methods Appl. Sci. 30(16), 2079–2089 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Naz, R., Mahomed, F.M., Chaudhry, A.: A partial Lagrangian method for dynamical systems. Nonlinear Dyn. 84(3), 1783–1794 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Naz, R., Freire, I.L., Naeem, I.: Comparison of different approaches to construct first integrals for ordinary differential equations. In: Torrisi, M. (ed.) Abstract and Applied Analysis, vol. 2014. Hindawi, Cairo (2014)Google Scholar
  13. 13.
    Dorodnitsyn, V., Kaptsov, E., Kozlov, R., Winternitz, P.: First integrals of ordinary difference equations beyond Lagrangian methods (2013). arXiv preprint arXiv:1311.1597
  14. 14.
    Dorodnitsyn, V., Kaptsov, E., Kozlov, R., Winternitz, P.: The adjoint equation method for constructing first integrals of difference equations. J. Phys. A Math. Theor. 48(5), 055202 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hojman, S.A.: A new conservation law constructed without using either Lagrangians or Hamiltonians. J. Phys. A Math. Gen. 25(7), L291 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Contopoulos, G.: On the existence of a third integral of motion. Astron. J. 68, 1 (1963)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gustavson, F.G.: Oil constructing formal integrals of a Hamiltonian system near ail equilibrium point. Astron. J. 71, 670 (1966)CrossRefGoogle Scholar
  18. 18.
    Lichtenberg, A.J., Lieberman, M.A.: Regular and Stochastic Motion, p. 277. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  19. 19.
    Naz, R., Naeem, I.: Generalization of approximate partial Noether approach in phase-space. Nonlinear Dyn. 88(1), 735–748 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ünal, G.: Approximate generalized symmetries, normal forms and approximate first integrals. Phys. Lett. A 269(1), 13–30 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    de Lagrange, J.L.: Méchanique analitique. Paris: Desaint 512 p.; in 8.; DCC. 4.403, 1 (1788)Google Scholar
  22. 22.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, vol. 107. Springer, Berlin (2012)Google Scholar
  23. 23.
    Campoamor-Stursberg, R.: Perturbations of Lagrangian systems based on the preservation of subalgebras of Noether symmetries. Acta Mech. 227(7), 1941–1956 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Centre for Mathematics and Statistical SciencesLahore School of EconomicsLahorePakistan
  2. 2.Department of Mathematics, School of Science and EngineeringLUMSLahore CanttPakistan

Personalised recommendations