Global solutions of the equation of the Kirchhoff elastic rod in space forms (Q2847598)

The elastica and the Kirchhoff elastic rod (or simply Kirchhoff rod) are both classical mathematical models of equilibrium configurations of thin elastic rods. The elastica is probably the simplest model, and is characterised as a critical curve of the energy of bending only. The Kirchhoff rod is a more complicated model, and is characterised as a critical framed curve of the energy with the effects of both bending and twisting. In this paper, the author considered the initial-value problem for the Euler-Lagrange equations, which are a pair of nonlinear ordinary differential equations of fourth and first orders. In particular, when the ambient space is a space form, that is, a complete connected Riemannian manifold of constant sectional curvature, the author proves that there exists a unique global solution of this initial-value problem (Theorem 2.1). Here, a global solution stands for a solution defined on the whole \(\mathbb R\). Since a Kirchhoff rod is, by definition, arclength-parametrised, this result implies that a Kirchhoff rod of finite length extends to that of infinite length. We note that in general, solution curves of a variational problem do not necessarily extend to global solutions. The outline of the proof of the main theorem (Theorem 2.1) is as follows. The point is to use the natural curvatures of an arclength-parametrised curve, which are different from the ordinary Frenet curvatures. The author reduces the Euler-Lagrange equations to the equation for the natural curvature vector, which is an \(\mathbb R^{n-1}\)-valued second-order ordinary differential equation. Then, a first integral of the equation is constructed and an estimate for solutions of the equation is derived. By using this estimate, the existence of a global solution of the initial-value problem for the Euler-Lagrange equations is shown.