Mathematical mechanics of one-dimensional filaments in three-dimensional space

Abstract

A one-dimensional filament is a useful mathematical object for modeling a variety of physical phenomena such as vortex filaments. Through a description of the filament using the tangent, normal, and binormal vectors, we get a more meaningful description of the curve in terms of curvature and torsion. Hasimoto's transformation defines a mapping between a kinematic evolution of a space curve and nonlinear scalar equations evolving its intrinsic curve geometry. Through this transformation, the problem of understanding the time evolution of curves can be greatly simplified, yielding useful equations which arise in other areas of nonlinear physics. In our work, we generalize this transformation on arbitrary flows and test against several existing kinematic flows. We consider the time dynamics of length and bending energy to see that binormal flows are generally length-preserving, and bending energy is fragile and unlikely to be conserved in the general case. By describing a space curve in terms of curve geometry, we can perform a transformation that yields a useful description of the time evolution of the curve. Through a generalization of this transformation, we can better understand how the kinematic equation dictates the behavior of the curve, and how this relates to the modeling of physical phenomena.