On motion of an elastic wire in a Riemannian manifold and singular perturbation

Abstract

R.E. Caflish and J.H. Maddocks analyzed the dynamics of a planar slender elastic rod. We consider a thin elastic rod $\gamma$ in an $N$-dimensional riemannian manifold. The former model represents an elastic rod with positive thickness, and the equation becomes a semilinear wave equation. Our model represents an infinitely thin elastic rod, and the equation becomes a 1-dimensional semilinear plate equation. We prove the short time existence of solutions. We also discuss the behaviour of the solution when the resistance goes to infinity, and find that the solution converges to a solution of a gradient flow equation.