A New Characterization on the Energy of Elastica with the Energy of Bishop Vector Fields in Minkowski Space

Abstract

In this paper, we firstly introduce kinematics properties of moving particles lying on Minkowski space E-1(4). Then we assume that particles correspond to different type of space curves. Thus, we characterize them by using Bishop frame equations. Guided by these, we present geometrical understanding of an energy on the particle in each Bishop vector fields depending on the particle corresponds to a spacelike or timelike curve in E-1(4). Then we also determine the bending elastic energy functional for the same particle in E-1(4) by assuming the particle has a bending feature of elastica. Finally, we prove that bending energy formula can be represented by the energy on the particle in each Bishop vector field.