Fractional Nonlocal Newton’s Law of Motion and Emergence of Bagley-Torvik Equation

Abstract

We construct a fractional nonlocal Newton’s law by extending Suykens’s nonlocal-in-time kinetic energy approach to its fractional counterpart by using a nonlocal Taylor series expansion. We derive the corresponding fractional order derivative Euler-Lagrange equations and we discuss some of their main consequences mainly for the case of a free particle and the case of an oscillator. Surprisingly, for the case of a time-dependent oscillator potential, the Bagley-Torvik equation used in viscoelasticity problems is obtained from nonlocal arguments. Some interesting features are obtained and discussed accordingly.