Novel relationships between some coordinate systems and their effects on mechanics of an intrinsically curved filament

Abstract

The fixed frame, Frenet-Serret frame and generalized Frenet-Serret frame are commonly used coordinate systems in the study of a filament or a moving rigid body. In terms of Eulerian angles, we derive some relations in these frames and apply these relations to find some significant results. Especially, we find the angle between the normal of centerline of a filament and the line of nodes which is the crossover between the horizontal plane of fixed frame and normal plane of centerline. We prove that the general solution of a set of nonlinear differential equations represents a circular helix or the corresponding filament has a unique helical ground-state configuration. We show that the effective description of a planar filament depends on the value of its torsional modulus. Finally, we find the expression of energy for a three-dimensional intrinsically curved filament when its cross-section area vanishes, and show that under an applied force the finite intrinsic curvature alone can induce a discontinuous transition in extension.