Minimizing energy of the dynamical force fields via parallel vectors in the 3D space

Abstract

In this paper, first we analyze the geometry of a moving particle in a 3-dimensional ordinary space in terms of parallel adapted frame elements. Then, we compute energy on each parallel vector fields in the space. We also obtain a necessary condition of having a critical point for the energy functional of unit parallel vector fields by only using the geometric description of the curvature of the worldline belonging to the particle. Finally, we give the condition of the minimizing energy for frictional, gravitational, and normal force vector fields by considering the critical curve among the class of all curves in space.