Skip to main content
SpringerLink
Log in

Symmetries and conservation laws of the damped harmonic oscillator

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

We work with a formulation of Noether-symmetry analysis which uses the properties of infinitesimal point transformations in the space-time variables to establish the association between symmetries and conservation laws of a dynamical system. Here symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of the damped harmonic oscillator representing it by an explicitly time-dependent Lagrangian and found that a five-parameter group of transformations leaves the action integral invariant. Amongst the associated conserved quantities only two are found to be functionally independent. These two conserved quantities determine the solution of the problem and correspond to a two-parameter Abelian subgroup.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E Noether, Invariante Variationsprobleme, Nachr. d. König. Gessellsch. d. Wiss. zu Göttingen, Math-phys. Klasse 235–257 (1918); English translation: M A Travel, Transport Theory and Statistical Physics 1(3), 183–207 (1971)

  2. H Goldstein, Classical mechanics (Narosa Publishing House, New Delhi, India, 1998)

    Google Scholar 

  3. C Lanczos, The variational principles of mechanics, 4th ed. (Dover, New York, 1970)

    MATH  Google Scholar 

  4. R M Santilli, Foundations of theoretical mechanics I (Springer-Verlag, New York, 1978)

    MATH  Google Scholar 

  5. P Caldirola, Nuovo Cimento 18, 393 (1941) E Kanai, Prog. Theor. Phys. 3, 440 (1948)

    Article  Google Scholar 

  6. G B Arfken and H J Weber, Mathematical method for physicists (Elsevier, New Delhi, India, 2004)

    Google Scholar 

  7. R L Kaushal, Ind. J. Pure Appl. Phys. 43, 479 (2005); Classical and quantum mechanics of noncentral potentials: A survey of two dimensional systems (Narosa, New Delhi/Springer, Heidelberg, 1998)

    Google Scholar 

  8. V K Chandrasekar, M Senthilvelan and M Lakshmanan, J. Math. Phys. 48, 032701 (2007)

  9. R de Ritis, G Marmo, G Plataniao and P Scudellaro, Int. J. Theor. Phys. 22, 931 (1983)

    Article  MATH  Google Scholar 

  10. Subrata Ghosh, J Shamanna and B Talukdar, Can. J. Phys. 82, 561 (2004)

    Article  ADS  Google Scholar 

  11. P J Olver, Applications of Lie groups to differential equations (Springer-Verlag, New York, 1993)

    MATH  Google Scholar 

  12. E L Hill, Rev. Mod. Phys. 23, 253 (1951) J Struckmeier and C Riedel, Phys. Rev. E66, 066605 (2002)

    Article  MATH  ADS  Google Scholar 

  13. V Tapia, An extended application of the Noether theorem, ICTP preprint number IC/96/161 (1996)

  14. M Lutzky, J. Phys. A11, 249 (1978)

    ADS  MathSciNet  Google Scholar 

  15. N A Lemos, Am. J. Phys. 47, 857 (1979)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Visva-Bharati University, Santiniketan, 731 235, India

    Amitava Choudhuri, Subrata Ghosh & B. Talukdar

Authors
  1. Amitava Choudhuri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Subrata Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Talukdar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Talukdar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Choudhuri, A., Ghosh, S. & Talukdar, B. Symmetries and conservation laws of the damped harmonic oscillator. Pramana - J Phys 70, 657–667 (2008). https://doi.org/10.1007/s12043-008-0027-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12043-008-0027-y

Keywords

PACS Nos

Search

Navigation