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(Draft 3) A generalized differential operator on the real line is defined bymeans of a limiting process. These generalized derivatives include, as aspecial case, the classical derivative and current studies of fractionaldifferential operators. All such operators satisfy properties such as the sum,product/quotient rules, chain rule, etc. We study a Sturm-Liouville eigenvalueproblem with generalized derivatives and show that the general case is actuallya consequence of standard Sturm-Liouville Theory. As an application of thedevelopments herein we find the general solution of a generalized harmonicoscillator. We also consider the classical problem of a planar motion under acentral force and show that the general solution of this problem is stillgenerically an ellipse, and that this result is true independently of thechoice of the generalized derivatives being used modulo a time shift. Theprevious result on the generic nature of phase plane orbits is extended to theclassical gravitational n-body problem of Newton to show that the global natureof these orbits is independent of the choice of the generalized derivativesbeing used in defining the force law (modulo a time shift). Finally,restricting the generalized derivatives to a special class of fractionalderivatives, we consider the question of motion under gravity with and withoutresistance and arrive at a new notion of time that depends on the fractionalparameter. The results herein are meant to clarify and extend many knownresults in the literature and intended to show the limitations and use ofgeneralized derivatives and corresponding fractional derivatives.

On generalized and fractional derivatives and their applications to classical mechanics