Equilibrium Shapes and Vibrations of Thin Elastic Rod

Static equilibrium equations and dynamical equations for a thin elastic rod are studied in detail. The relation between the local induction equtrtion for the vortex filament and the equilibrium equations of the elastic rod is clarified. The static equations are solved exactly and the shapes of the rod in a real space are obtained. The localized solution is a solitary wave. It deviates from a plane when the torsion exists. When the rod is strained initially, vibrational modes are coupled and the dispersion relations are modified. Linear stability of the twisted circular rod is also examined.