International Journal of Theoretical Physics

, Volume 45, Issue 9, pp 1769–1782 | Cite as

Lie Symmetries for Hamiltonian Systems Methodological Approach

  • Rodica Cimpoiasu
  • Radu Constantinescu


This paper proposes an algorithm for the Lie symmetries investigation in the case of a 2D Hamiltonian system. General Lie operators are deduced firstly and, in the the next step, the associated Lie invariants are derived. The 2D Yang-Mills mechanical model is chosen as a test model for this method.


Lie symmetries invariants Yang-Mills mechanical model 


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  1. Ablowitz, M. J., Ramani, A., and Segur, H. A. (1980). A connection between nonlinear evolution equations and ordinary differential equations of P-type. Journal of Mathematical Physics 21, 715.MATHMathSciNetCrossRefADSGoogle Scholar
  2. Cimpoiasu, R. (2005). Integrability features for the non-abelian gauge field. Romanian Reports in Physics 57(2), 167.Google Scholar
  3. Cimpoiasu, R., Constantinescu, R., and Cimpoiasu, V. (2005). Integrability of dynamical systems with polynomial Hamiltonians. Romanian Journal of Physics 50(3-4), 317.Google Scholar
  4. Ciraolo, G., Vittot, M., and Pettini, M. (2003). Control of chaos in Hamiltonian systems. Reviews of Mathematical Physics 8.Google Scholar
  5. Gandarias, M. L. (2001). New symmetries for a model of fast fast diffusion. Physics Letters A 286, 153.MATHMathSciNetCrossRefADSGoogle Scholar
  6. Geronimi, C., Feix, M., and Leach, P. (2001). Exponential nonlocal symmetries and nonnormal reduction of order. Journal of Physics A: Mathematical and General 34, 10109.MATHMathSciNetCrossRefADSGoogle Scholar
  7. Hietarinta, J. (1987). Direct methods for the search of the second invariant. Physics Reports 147, 101.MathSciNetCrossRefGoogle Scholar
  8. Kasperczuk, S. (1994). Integrability of the Yang-Mills Hamiltonian system. Celestial Mechanics and Dynamical Astronomy 58, 387.MathSciNetCrossRefADSGoogle Scholar
  9. Lakshmanan, M. (1992). Direct integration of generalized Lie summetries of nonlinear Hamiltonian systems with two degrees of freedom: Integrability and separability, M. S. Velan. Journal of Physics A: Mathematical and General, 1264.Google Scholar
  10. Lichtenberg, A. J. and Lieberman, M. A. (1994). Regular and haotic dynamics. Applied Mathematical Sciences 38.Google Scholar
  11. Oloumi, A. and Teychenné, D. (1999). Controlling Hamiltonian chaos via Gaussian curvature. Physical Review E 60, R 6279.Google Scholar
  12. Olshanetsky, M. A. and Perelomov, A. M. (1981). Classical integrable finite-dimensional systems related to Lie-algebras. Physics Reports 71, 313.MathSciNetCrossRefADSGoogle Scholar
  13. Olver, P. J. (1993). Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 110.Google Scholar
  14. Sirko, L. and Koch, P. M. (2002). Control of common resonances in bichromatically driven hydrogen atoms. Physical Review Letters 89, 274101.CrossRefADSGoogle Scholar
  15. Struckmeier, J. J. and Riedel, C. (2000). Exact invariants for a class of three-dimensional time-dependent Classical Hamiltonians. Physical Review Letters 85 (18), 3830.CrossRefADSGoogle Scholar
  16. Struckmeier, J. and Riedel, C. (2002). Noether's theorem and Lie symmetries for time-dependent Hamilton-Lagrange systems. Physical Review E 66, 066605.MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Rodica Cimpoiasu
    • 1
    • 2
  • Radu Constantinescu
    • 1
  1. 1.University of CraiovaCraiovaRomania
  2. 2.University of CraiovaCraiovaRomania

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