Kirchhoff elastic rods in the three-sphere

Abstract

The Kirchhoff elastic rod is one of the mathematical models of thin elastic rods, and is a critical point of the energy functional with the effect of bending and twisting. In this paper, we study Kirchhoff elastic rods in the three-sphere of constant curvature. In particular, we give explicit expressions of Kirchhoff elastic rods in terms of elliptic functions and integrals. In addition, we obtain equivalent conditions for Kirchhoff elastic rods to be closed, and give an example of closed Kirchhoff elastic rods.

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

“…Also, |a| stands for the Euclidean norm of the (n − 1) × (n − 1) matrix a. The system of (2.1) and (22) is equivalent to the system of (2.1) and (2.2) in [20]. For the derivation of these equations, see [20,Section 2].…”

“…Kirchhoff rods were originally considered in three-dimensional Euclidean space, but they extend naturally to an arbitrary higher-dimensional Riemannian manifold (see, for example, [18][19][20]). That is, we generalise the energy by replacing ordinary differentiation by covariant differentiation, and define a Kirchhoff rod as a solution of the Euler-Lagrange equations associated to the generalised energy (see [20,Section 2]).…”

“…Kirchhoff rods in three-dimensional simply connected space forms R 3 , S 3 , H 3 are investigated in [17,18,20,21,30], for example. It is known that the Frenet curvature and torsion of the centrelines of all Kirchhoff rods in R 3 [31,38,39] or in S 3 , H 3 [20] are explicitly expressed in terms of Jacobi sn function. By using these explicit formulas, it follows immediately that a Kirchhoff rod of finite length in R 3 , S 3 , H 3 extends to that of infinite length.…”

Kirchhoff elastic rods in three-dimensional space forms

“…It is natural to ask if we can get such explicit expressions as [22], [27], even in the three-sphere S 3 or the three-dimensional hyperbolic space H 3 . In the case of the three-sphere, the author obtained explicit formulas of the centerlines of Kirchhoff elastic rods by Jacobi sn function and the elliptic integrals in terms of a system of coordinates analogous to the cylindrical coordinates (Theorem 6.1 of [13]). However, we cannot apply the same method as [13] to the case of the three-dimensional hyperbolic space.…”

“…Let M be S 3 or H 3 of constant sectional curvature G. In Section 2, according to [13], we give explicit expressions of the curvature and torsion of the centerline γ of a Kirchhoff elastic rod {γ, M } in M , and then parametrize the space of the congruence classes of Kirchhoff elastic rods by four real numbers, which we will write as α, η, p, w (Proposition 2.2).…”