A Hamilton-Jacobi Treatment of Dissipative Systems with one Degree of Freedom

  • Giacomo Della Riccia
Part of the International Centre for Mechanical Sciences book series (CISM, volume 261)


Until recently, it had been thought that the equation of motion of classical systems with dissipative forces could not be derived from a Lagrangian function by a pure variational procedure. The dissipative terms of the equation were usually introduced directly in the Lagrange’s equation through a Rayleigh dissipative function or by using complex coordinates and velocities in the Lagrangian. Recently,1,2 a pure Lagrangian description was found for the linearly damped oscillator and certain dynamical systems with velocity-dependent forces and a Hamilton-Jacobi treatment was developed3 for these systems. We extend these results to the general class of classical systems with one degree of freedom governed by a Lienard equation of motion which is of the form


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  1. 1.
    Havas, P., Phys. Rev., 83, 224, 1951; Nuovo Cimento Suppl., 5, 363, 1957.Google Scholar
  2. 2.
    Denman, H.H., On linear friction in Lagrange’s equation, Am. J. Phys., 34, 1147, 1966.ADSCrossRefGoogle Scholar
  3. 3.
    Denman, H.H., and Buch, L.H., Solution of the Hamilton-Jacobi equation for certain dissipative classical mechanical systems, J. Math. Phys., 14, 326, 1973.ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Smith, R.A., A simple non-linear oscillation, J. London Math. Soc., 36, 33, 1961.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Whittaker, E.T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Fourth edition, Cambridge University Press, 1965, 276.Google Scholar
  6. 6.
    Currie, D.G., and Saletan, E.J., q-Equivalent particle Hamiltonians I. The classical one-dimensional case, J. Math. Phys., 7, 967, 1966.ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1980

Authors and Affiliations

  • Giacomo Della Riccia
    • 1
    • 2
  1. 1.Istituto di Scienze dell’InformazioneUniversità di SalernoItaly
  2. 2.Department of MathematicsBen-Gurion University of the NegevIsrael

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