Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Hamiltonian dynamics of an elastica and the stability of solitary waves


A method is presented for deriving unconstrained Hamiltonian systems of partial differential equations equivalent to given constrained Lagrangian systems. The method is applied to the theory of planar, finite-amplitude motions of inextensible and unshearable elastic rods. The constraints of inextensibility and unshearability become integrals of motion in the Hamiltonian formulation.

It is known that in the theory of uniform, inextensible, unshearable rods of infinite length there arise solitary-wave solutions with the property that each profile can move at arbitrary speed. The Hamiltonian formulation is exploited to analyze the stability properties of these solitary waves. The wave profiles are first characterized as critical points of an appropriate time-invariant functional. It is then shown that for a certain range of wave speeds the solitary-wave profiles are actually nonisolatedminimizers of the functional, a fact with implications for nonlinear stability.

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


  1. 1.

    Antman, S. S. &T.-P. Liu. Travelling waves in hyperelastic rods.Quart. Appl. Math. 36 (1979) 377–399.

  2. 2.

    Arnold, V. I., V. V. Kozlov &A. I. Neishtadt. Mathematical aspects of classical and celestial mechanics, inDynamical Systems III, Encyclopœdia of the Mathematical Sciences, Vol. 3, ed. V. I. Arnold (Springer-Verlag, 1988).

  3. 3.

    Beliaev, A. &A. Il'ichev. Conditional stability of solitary waves propagating in elastic rods,Physica D 90 (1996) 107–118.

  4. 4.

    Benjamin, T. B. The stability of solitary waves.Proc. Roy. Soc. Lond. A. 328 (1972) 153–183.

  5. 5.

    Benjamin, T. B. Impulse, flow force and variational principles.IMA J. Applied Math. 32 (1984) 3–68.

  6. 6.

    Caflisch, R. E. &J. H. Maddocks. Nonlinear dynamical theory of the elastica.Proc. Roy. Soc. Edinburgh 99A (1984) 1–23.

  7. 7.

    Coleman, B. D. &E. H. Dill. Flexure waves in elastic rods.J. Acoustical Soc. America 91 (1992) 2663–2673.

  8. 8.

    Coleman, B. D., E. H. Dill, M. Lembo, Z. Lu &I. Tobias. On the dynamics of rods in the theory of Kirchhoff and Clebsch.Arch. Rational Mech. Anal. 121 (1993) 339–359.

  9. 9.

    Coleman, B., E. Dill &D. Swigon. On the dynamics of flexure and stretch of elastic rods.Arch. Rational Mech. Anal. 129 (1995) 147–174.

  10. 10.

    Coleman, B. D. &Xu, J-M. On the interaction of solitary waves of flexure in elastic rods.Acta Mech. 110 (1995) 173–182.

  11. 11.

    Dichmann, D. J. Hamiltonian Dynamics of an Elastica and Stability of Solitary Waves. Ph. D. thesis, University of Maryland (1994).

  12. 12.

    Domokos, G. &A. Ruina. A circle construction based on elastostatics and hydrodynamics.Mechanics Research Communications,20 (1993) 181–185.

  13. 13.

    Drazin, P. Solitons, (Cambridge University Press, 1983).

  14. 14.

    Euler, L. Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes, Bousquent, Lausanne, 1744.Opera Omnia I/24 (Füssli, 1960).

  15. 15.

    Fleming, W. H. &R. W. Rishel.Optimal Control Theory, (Springer-Verlag, 1975).

  16. 16.

    Falk, R. S. &J.-M. Xu. Convergence of a second-order scheme for the nonlinear dynamical equations of elastic rods.SIAM J. Num. Anal. 32 (1995) 1185–1209.

  17. 17.

    Henry, D. B., J. F. Perez &W. F. Wreszinski. Stability theory for solitary wave solutions of scalar field equations.Comm. Math. Phys. 85 (1982) 351–361.

  18. 18.

    Ichikawa, Y. H., K. Konno &M. Wadati. Nonlinear transverse oscillation of elastic beams under tension.J. Phys. Soc. Japan 50 (1981) 1799–1802.

  19. 19.

    Kirchhoff, G. R. Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes.Gesammelte Abhandlungen (Leipzig, 1882).

  20. 20.

    Konno, K., Y. H. Ichikawa &M. Wadati. A loop soliton propagating along a stretched rope.J. Phys. Soc. Japan 50 (1981) 1025–1026.

  21. 21.

    Konno, K. &A. Jeffrey. Some remarkable properties of two loop soliton solutions.J. Phys. Soc. Japan 52 (1983) 1–3.

  22. 22.

    Konno, K. &A. Jeffrey. The loop soliton, inAdvances in Nonlinear Waves. Vol. 1, ed.L. Debnath (Pitman, 1984) 162–183.

  23. 23.

    Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity, (Dover, 1944).

  24. 24.

    Maddocks, J. H. &D. J. Dichmann. Conservation laws in the dynamics of rods.J. Elasticity 34 (1994) 83–96.

  25. 25.

    Maddocks, J. H. &R. L. Pego. An unconstrained Hamiltonian formulation for incompressible fluid flow.Comm. Math. Phys. 170 (1995) 207–217.

  26. 26.

    Morse, P. &H. Feshbach.Methods of Theoretical Physics, Part I. (McGraw-Hill, 1953).

  27. 27.

    Olver, P. J. Applications of Lie Groups to Differential Equations. (Springer-Verlag, 1986).

  28. 28.

    Simo, J. C., J. E. Marsden &P. S. Krishnaprasad. The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods and plates.Arch. Rational Mech. Anal. 104 (1988) 125–183.

  29. 29.

    Wadati, M., K. Konno &Y. H. Ichikawa. New integrable nonlinear evolution equations.J. Phys. Soc. Japan 47 (1979) 1698–1700.

  30. 30.

    Xu, J.-M. An Analysis of the Dynamical Equations of Elastic Rods and Their Numerical Approximation. Ph. D. thesis, Rutgers University (1992).

  31. 31.

    Xu, J-M. Private communication.

Download references

Author information

Additional information

Communicated by B. D. Coleman

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dichmann, D.J., Maddocks, J.H. & Pego, R.L. Hamiltonian dynamics of an elastica and the stability of solitary waves. Arch. Rational Mech. Anal. 135, 357–396 (1996). https://doi.org/10.1007/BF02198477

Download citation


  • Neural Network
  • Complex System
  • Partial Differential Equation
  • Nonlinear Dynamics
  • Solitary Wave