川久保 哲

The Kirchhoff elastic rod is a classical mathematical model 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. We consider the initial-value problem for the Euler-Lagrange equations in a Riemannian manifold. In a previous paper, the author proved the existence and uniqueness of global solutions of the initial-value problem in the case where the ambient space is a space form. In the present paper, we extend this result to the case where the ambient space is a general complete Riemannian manifold. This implies that an arbitrary Kirchhoff elastic rod of finite length in a complete Riemannian manifold extends to that of infinite length. (C) 2014 AIP Publishing LLC.

We give examples of Kirchhoff rod centerlines in five-dimensional space forms which are fully immersed and not helices. We also show that the natural curvatures of these Kirchhoff rod centerlines are expressed explicitly in terms of Jacobi sn function. (C) 2013 Elsevier B.V. All rights reserved.

The localized induction hierarchy in three-dimensional space forms is studied. In particular, we determine all the generating curves of congruence solutions to each evolution equation belonging to the localized induction hierarchy. Here, a congruence solution means a solution moving without changing shape. Also, we give the characterization of some low-order soliton curves.