  (Draft 3) A generalized differential operator on the real line is defined by
means of a limiting process. These generalized derivatives include, as a
special case, the classical derivative and current studies of fractional
differential operators. All such operators satisfy properties such as the sum,
product/quotient rules, chain rule, etc. 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
previous result on the generic nature of phase plane orbits 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 (modulo a time shift). 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.
