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Hamiltonian dynamics of an elastica and the stability of solitary waves

Abstract

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.

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Communicated by B. D. Coleman

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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

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Keywords

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