On generalized and fractional derivatives and their applications to classical mechanics

Abstract

A generalized differential operator on the real line is defined by means of a limiting process. When an additional fractional parameter is introduced, this process leads to a locally defined fractional derivative. The study of such generalized derivatives includes, as a special case, basic results involving the classical derivative and current research involving fractional differential operators. All our operators satisfy properties such as the sum, product/quotient rules, and the chain rule. We study a Sturm–Liouville eigenvalue problem with generalized derivatives and show that the general case is actually a consequence of standard Sturm–Liouville Theory. As an application of the developments herein we find the general solution of a generalized harmonic oscillator. We also consider the classical problem of a planar motion under a central force and show that the general solution of this problem is still generically an ellipse, and that this result is true independently of the choice of the generalized derivatives being used modulo a time shift. The generic nature of phase plane orbits modulo a time shift is extended to the classical gravitational n-body problem of Newton to show that the global nature of these orbits is independent of the choice of the generalized derivatives being used in defining the force law. Finally, restricting the generalized derivatives to a special class of fractional derivatives, we consider the question of motion under gravity with and without resistance and arrive at a new notion of time that depends on the fractional parameter. The results herein are meant to clarify and extend many known results in the literature and intended to show the limitations and use of generalized derivatives and corresponding fractional derivatives.