Curvature Dependent Energy of Surface Curves in Minkowski Space

Abstract

In this paper, we firstly introduce kinematics properties of the moving particle lying on a surface S. We assume that the particle corresponds to a different type of surface curves such that they are characterized by using the Darboux vector field W in Minkowski spacetime. Based on this result, we present geometrical understanding of the energy of the particle in each Darboux vector fields whether they lie on a spacelike surface or a timelike surface. Then, we also determine the bending elastic energy functional for the same particle on a surface S by assuming the particle has a bending feature of elastica. Finally, we prove that bending energy formula can be represented by the energy of the particle in each Darboux vector field W.