Applications of Fractional Calculus to Newtonian Mechanics Gabriele U. Varieschi - Loyola Marymount University

Introduction Generalizing Gravitational Force and FC Fractional Calculus (FC) is a natural generalization of Mechanics Generalized Riesz Orbital Circular Velocities calculus that studies the possibility of computing derivatives and integrals of any real (or complex) order, One-dimensional Newtonian mechanics for a Gravitational Potential An interesting consequence of these generalized i.e., not just of standard integer orders, such as point-particle of constant m is based upon gravitational potentials is the analysis of the resulting orbital Newton's second law of motion, As our first A possible generalization of Newton’s law of universal circular velocities, for the inner and outer solutions. From the first-derivative, second-derivative, etc. gravitation can be obtained by considering a generalized example of FC applied to mechanics, we generalize Lorem ipsum Riesz potentials, we can easily obtain the circular velocities, The history of FC started in 1695 when l'Hôpital raised Riesz potential: the previous equation by using derivatives of arbitrary the question as to the meaning of taking a fractional where 1/2 1/2 (real) order q, and by considering a constant force per derivative such as d y/dx and Leibniz replied: "...This is the distance between the infinitesimal source mass which are plotted in Figure 3, for the same values of the is an apparent paradox from which, one day, useful unit mass, f=F/m=const: element and the being considered. fractional order q used in Fig. 2. We also set G=M=R=a=1 consequences will be drawn." 12 Due to the fractional order q, a "length scale" a is as done previously. The q=1 case (red-solid curve) Since then, eminent mathematicians such as Fourier, q 1.00 q 1.75 q 2.50 needed for dimensional correctness. For a spherical represents the standard Newtonian situation. 10 = = = q 1.25 q 2.00 q 2.75  3.0 Abel, Liouville, Riemann, Weyl, Riesz, and many others = = = source of radius R and uniform density , the q 1.50 q 2.25 q 3.00 = = = contributed to the , but until lately FC has played a 8 Riesz potential can be evaluated analytically for any q: 2.5 negligible role in physics. x q 0.00 q 0.75 q 1.50 6 = = = However, in recent years, applications of FC to q 0.25 q 1.00 q 1.75 = = = q 0.50 q 1.25 q 2.00 physics have become more common in fields ranging 4 2.0 = = = from classical and quantum mechanics, nuclear physics, 2

hadron spectroscopy, and up to quantum field theory. circ 1.5 v

0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t Figure 1 1.0 In Figure 2, we illustrate the shape of these generalized gravitational potentials, for different values of the fractional order q ranging from zero to two. The q=1 0.5 case (red-solid curve) represents the standard Newtonian 0.0 gravitational potential. All these plots were obtained by 0.0 0.5 1.0 1.5 2.0 2.5 3.0 setting G = M = R = a = 1 for simplicity's sake, thus the r Figure 3 A brief review of FC The general solution of the (extraordinary) differential vertical grid line at r=1.0 denotes the boundary between It is interesting to note that, for values of q decreasing equation above is: the inner (0≤r≤R) and the outer (r>R) potentials. from one toward zero, the rotational velocity curves in the 0.0 Unlike standard calculus, there is no unique definition outer (r≥R) region show a definite "flattening" effect, which of derivation and integration in FC. Historically, becomes more pronounced for the lowest q values (for 0.5 several different definitions were introduced and - example, in the q=0.25 case, blue-dotted curve). This used. consideration might be of some interest in relation with the 1.0 All proposed definitions reduce to standard where we also used the fractional derivative of a - well-established problem of dark matter in galaxies, as derivatives and integrals for integer orders n, but they evidenced by the galactic rotation curves and their lack of constant quantity f, which is not zero in FC. RZ 1.5 V Newtonian behavior in the outer regions. might not be fully equivalent for non-integer orders of The l constants of integration, c , c, ... , c, can be - l Also, given possible connections between fractional differ-integration. For example, Riemann proposed to determined from the l initial conditions: x(t), x′(t),..., 2.0 q 0.00 q 0.75 q 1.50 calculus and fractal geometry, the self-similar patterns shown extend the simple recursive relation (l-1) - = = = x (t). For example, choosing for simplicity's sake q 0.25 q 1.00 q 1.75 = = = by mathematical fractals might also be present in the by using gamma functions: , where q is a  q 0.50 q 1.25 q 2.00 t =1, x(1)=x′(1)=x′′(1)=...=1, and also f=1, our general 2.5 = = = astrophysical structures of the Universe, such as galaxies or real (or complex) number. In this way, for any function - solution above can be easily plotted for different values others. The fractal dimension of these structures can be f(x) expanded in series of power functions, , of the order q, as shown in Figure 1 in the panel above. 3.0 directly connected to the fractional order of the related one can easily define a “fractional derivative” of - 0.0 0.5 1.0 1.5 2.0 2.5 3.0 This figure illustrates the resulting position vs. time r Figure 2 gravitational equations. arbitrary order q: . We note that, using functions, with the order q ranging from 1 to 3 with this definition of fractional derivative, the derivative of fractional increments. The standard Newtonian solution a constant C is not zero in FC: . is obviously recovered for q=2 (red-solid curve), for a More general definitions of fractional derivatives motion with constant acceleration. Two other solutions and integrals to any arbitrary order q exist in the for integer values of q are presented: the case for q=1 Conclusions literature, such as the Grünwald formula: (blue-solid line) represents a simple motion with In this , we have applied Fractional Calculus to some basic problems in constant velocity; the q=3 case (green-solid line) standard Newtonian mechanics. The main goal was to show that FC can be used which involves only evaluations of the function itself represents instead a motion with constant jerk. In and can be used for both positive (derivation) and as a pedagogical tool, even in introductory physics courses, to gain more insight Fig.1, we also show (dashed and dotted curves) the into basic concepts of physics, such as Newton's laws of motion and gravitation. negative values of q (integration). Another general position vs. time functions for some fractional values of definition is the Riemann-Liouville fractional integral, the order q. These additional curves interpolate well This work was supported by a grant from the Seaver College of Science and for negative q: between the integer-order functions described above. Engineering, LMU, Los Angeles. For more details, see the pre-print: which can also be extended to fractional derivatives. Gabriele U. Varieschi, arXiv:1712.03473 [physics.class-ph].