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On the Symmetries of a Liénard Type Nonlinear Oscillator Equation

  • R. Mohanasubha
  • V. K. Chandrasekar
  • M. Senthilvelan
  • M. LakshmananEmail author
Conference paper
  • 424 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 266)

Abstract

In the contemporary nonlinear dynamics literature, the nonlinear oscillator equation \(\ddot{x}+kx \dot{x}+\frac{k^2}{9}x^3+\tilde{\lambda } x=0\) is being analyzed in various contexts both classically and quantum mechanically. Classically this nonlinear oscillator equation has been shown to admit three different types of dynamics depending upon the sign and magnitude of the parameter \(\tilde{\lambda }\), namely (i) \(\tilde{\lambda }=0\), (ii) \(\tilde{\lambda }>0\) and (iii) \(\tilde{\lambda }<0\). By considering its importance, in this paper, we present the symmetries of its Lagrangian and underlying equation of motion for all the three cases. In particular, we present Lie point symmetries, \(\lambda \)-symmetries, Noether symmetries and telescopic symmetries of this equation. The utility of the symmetries for all the three cases is demonstrated explicitly.

Keywords

Nonlinear oscillators Lie point symmetries \(\lambda \)-symmetries Noether symmetries Telescopic vector fields 

Notes

Acknowledgements

RMS acknowledges the University Grants Commission (UGC), New Delhi, India, for the award of a Dr. D. S. Kothari Post Doctoral Fellowship [Award No. F.4-2/2006 (BSR)/PH/17-18/0004]. The work of M.S. forms part of CSIR research project under Grant No. 03(1397)/17/EMR-II. The work of V.K.C. was supported by the SERB-DST Fast Track scheme for young scientists under Grant No. YSS/2014/000175. The work of M.L. is supported by a DST- SERB Distinguished Fellowship program.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • R. Mohanasubha
    • 1
  • V. K. Chandrasekar
    • 2
  • M. Senthilvelan
    • 3
  • M. Lakshmanan
    • 3
    Email author
  1. 1.Department of PhysicsAnna UniversityChennaiIndia
  2. 2.School of Electrical and Electronics EngineeringCentre for Nonlinear Science and Engineering, SASTRA UniversityThanjavurIndia
  3. 3.Centre for Nonlinear Dynamics, Bharathidasan UniversityTiruchirappalliIndia

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