Abstract
We investigate some basic applications of Fractional Calculus (FC) to Newtonian mechanics. After a brief review of FC, we consider a possible generalization of Newton's second law of motion and apply it to the case of a body subject to a constant force. In our second application of FC to Newtonian gravity, we consider a generalized fractional gravitational potential and derive the related circular orbital velocities. This analysis might be used as a tool to model galactic rotation curves, in view of the dark matter problem. Both applications have a pedagogical value in connecting fractional calculus to standard mechanics and can be used as a starting point for a more advanced treatment of fractional mechanics.
Highlights
IntroductionFractional Calculus (FC) is a natural generalization of calculus that studies the possibility of computing derivatives and integrals of any real (or complex) order [1] [2] [3], i.e., not just of standard integer orders, such as first-derivative, second-derivative, etc
Fractional Calculus (FC) is a natural generalization of calculus that studies the possibility of computing derivatives and integrals of any real order [1] [2] [3], i.e., not just of standard integer orders, such as first-derivative, second-derivative, etc.The history of FC started in 1695 when l’Hôpital raised the question as to the meaning of taking a fractional derivative such as d1 2 y dx1 2 and Leibniz replied [2]: “...This is an apparent paradox from which, one day, useful consequences will be drawn.”Since eminent mathematicians such as Fourier, Abel, Liouville, Riemann, Weyl, Riesz, and many others contributed to the field, but until lately FC has played a negligible role in physics
We investigate some basic applications of Fractional Calculus (FC) to Newtonian mechanics
Summary
Fractional Calculus (FC) is a natural generalization of calculus that studies the possibility of computing derivatives and integrals of any real (or complex) order [1] [2] [3], i.e., not just of standard integer orders, such as first-derivative, second-derivative, etc. In theoretical physics we can study the fractional equivalent of many standard physics equations [4]: frictional forces, harmonic oscillator, wave equations, Schrödinger and Dirac equations, and several others. In applied physics [5], FC methods can be used in the description of chaotic systems and random walk problems, in polymer material science, in biophysics, and other fields. We will apply these concepts to some basic problems in Newtonian mechanics, such as possible generalizations of Newton’s second law of motion and applications of FC to Newtonian gravity
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