Coupled Nonlinear Oscillators: Symmetries and Integrability

  • M. Lakshmanan
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)

Abstract

It is shown how the Lie’s method of invariance analysis involving extended, velocity-dependent vector fields can systematically identify integrable cases of nonlinear dynamical systems. The method is illustrated for the case of coupled nonlinear oscillators involving polynomial potentials with two and three degrees of freedom.

Keywords

Couple Oscillator Invariance Analysis Anharmonic Oscillator Integrable Case Polynomial Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V. I. Arnold, Dynamical Systems, Vol. III ( Springer-Verlag, New York, 1988 ).Google Scholar
  2. [2]
    V. V. Kozlev, Russ. Math. Surveys 38 (1983) 1.Google Scholar
  3. [3]
    B. Dorizzi, B. Grammaticos and A. Ramani, J. Math. Phys. 25 (1984) 481.CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    T. Bountis, H. Segur and F. Vivaldi, Phys. Rev. A25 (1982) 1257.Google Scholar
  5. [5]
    M. Lutzky, J. Phys. 11A (1978) 249; 12A (1979) 973.Google Scholar
  6. [6]
    L. S. Hall, Physica 8D (1983) 90.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    E. T. Whittker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies ( Cambridge University Press, London, 1937 ).Google Scholar
  8. [8]
    S. Chandrasekhar, Principles of Stellar Dynamics (Dover, New York, 1942 ), Ch. III.Google Scholar
  9. [9]
    M. J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys. 21 (1980) 715.CrossRefMATHADSMathSciNetGoogle Scholar
  10. [10]
    G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations ( Springer, New York, 1974 ).CrossRefMATHGoogle Scholar
  11. [11]
    L. V. Ovsiannikov, Group Analysis of Differential Equations ( Academic Press, New York, 1982 ).MATHGoogle Scholar
  12. [12]
    P. J. Olver, Applications of Lie Groups to Differential Equations ( Springer, New York, 1986 ).MATHGoogle Scholar
  13. [13]
    M. Lakshmanan and R. Sahadevan, Phys. Rev. A31 (1985) 861.CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    B. Grammaticos, B. Dorrizzi and A. Ramani, J. Math. Phys. 24 (1983) 2289.Google Scholar
  15. [15]
    R. Sahadevan and M. Lakshmanan, J. Phys. 19A (1986) L949.ADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • M. Lakshmanan
    • 1
  1. 1.Department of PhysicsBharathidasan UniversityTiruchirapalliIndia

Personalised recommendations