On the geometric flow of kirchhoff elastic rods

Chun Chi Lin*, Hartmut R. Schwetlick

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


Recently, rod theory has been applied to the mathematical modeling of bacterial fibers and biopolymers (e.g., DNA) to study their mechanical properties and shapes (e.g., supercoiling). In static rod theory, an elastic rod in equilibrium is the critical point of an elastic energy. This induces a natural question of how to find elasticae. In this paper, we focus on how to find the critical points by means of gradient flows. We relate a geometric function of curves to the Isotropic Kirchhoff elastic energy of rods so that the generalized elastic curves are the centerlines of elastic rods in equilibrium. Thus, the variational problem for rods is formulated in curve geometry. This problem turns out to be a generalization of curve-straightening flows, which induce nonlinear fourth-order evolution equations. We establish the long time existence of length-preserving gradient flow for the geometric energy. Furthermore, by studying the asymptotic behavior, we show that the limit curves are the centerlines of the Kirchhoff elastic rods in equilibrium.

Original languageEnglish
Pages (from-to)720-736
Number of pages17
JournalSIAM Journal on Applied Mathematics
Issue number2
Publication statusPublished - 2005


  • Fourth order
  • Geometric flows
  • Kirchhoff elastic rods

ASJC Scopus subject areas

  • Applied Mathematics


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