Global Solutions of the Equation of the Kirchhoff Elastic Rod in Space Forms

Abstract

The Kirchhoff elastic rod is one of the mathematical models of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler–Lagrange equations associated to the energy with the effect of bending and twisting. In this paper, we consider Kirchhoff elastic rods in a space form. In particular, we give the existence and uniqueness of global solutions of the initial-value problem for the Euler–Lagrange equations. This implies that an arbitrary Kirchhoff elastic rod of finite length extends to that of infinite length.

Extendability of Kirchhoff elastic rods in complete Riemannian manifolds

“…Here, a global solution stands for a solution defined on the whole of R. 26). On the other hand, when M is a general complete Riemannian manifold, we cannot expect the existence of such a first integral.…”