Hamiltonian dynamics of a spatial elastica and the stability of solitary waves

A novel approach to constrained Lagrangian systems is introduced in order to derive an unconstrained Hamiltonian system of partial differential equations whose solutions model the large amplitude, time evolution of an inextensible and unshearable elastic rod. The inextensibility and unshearability conditions are not imposed as constraints, but instead arise as integrals of motion for the Hamiltonian system. The new formulation also has applications outside rod dynamics.;For a uniform rod of infinite length it is known that solitary waves arise. The solitary waves have the unusual property that a given profile can move at any speed. In this thesis the Hamiltonian formulation of the dynamics is exploited to analyze the stability properties of the planar solitary waves. The analysis begins with a model of the planar dynamics. In the usual manner in Hamiltonian systems, the solitary wave profiles are characterized as critical points of an appropriate variational principle. For the case of rods this variational principle involves a previously unrecognized conservation law. Furthermore, for subsonic wave speeds the solitary wave profiles can actually be characterized as nonisolated minimizers of a time invariant functional F amongst planar perturbations. The implications of this fact upon nonlinear dynamic stability properties are described.;The model is then generalized to consider three-dimensional motion. A variational analysis of planar solitary wave profiles on an isotropic rod shows that there is a range of subsonic wave speeds for which the invariant F has index one amongst admissible spatial perturbations; outside of that range F is infinitely indefinite. The analysis is then generalized to two related anisotropic rod problems.