Fractional Geometric Calculus: Towar A Unifie Mathematical Language for Physics an Engineering Xiong Wang Center of Chaos an Complex Network, Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China. (e-mail: wangxiong8686@gmail.com). Abstract: This paper iscuss the longstaning problems of fractional calculus such as too many efinitions while lacking physical or geometrical meanings, an try to exten fractional calculus to any imension. First, some ifferent efinitions of fractional erivatives, such as the Riemann-Liouville erivative, the Caputo erivative, Kolwankar s local erivative an Jumarie s moifie Riemann-Liouville erivative, are iscusse an conclue that the very reason for introucing fractional erivative is to stuy nonifferentiable functions. Then, a concise an essentially local efinition of fractional erivative for one imension function is introuce an its geometrical interpretation is given. Base on this simple efinition, the fractional calculus is extene to any imension an the Fractional Geometric Calculus is propose. Geometric algebra provie an powerful mathematical framework in which the most avance concepts moern physic, such as quantum mechanics, relativity, electromagnetism, etc., can be expresse in this framework graciously. At the other han, recent evelopments in nonlinear science an complex system suggest that scaling, fractal structures, an nonifferentiable functions occur much more naturally an abunantly in formulations of physical theories. In this paper, the extene framework namely the Fractional Geometric Calculus is propose naturally, which aims to give a unifying language for mathematics, physics an science of complexity of the 21st century. Keywors: geometric algebra, geometric calculus, fractional calculus, nonifferentiable function 1. INTRODUCTION It seems a general pattern of our unerstaning of nature to evolve from integer to fraction. In number theory, we iscovery integer number firstly, then we introuce fractional number in orer to o ivision an finally we introuce real number to form a soli base for analysis. In geometry, for a long time we only know the typical geometry objects like point, line, surface an volume with integer number imensions. But recent iscovery of fractal geometry has completely change an enlarge people s unerstaning about the geometry. Fractal is even consiere as the true geometry of nature, see Manelbrot (1983). In calculus, it also follow this path. At the beginning, the motivation of this extension from integer to fractional ifferential operator is just formal, but lack of substantial physical meaning. From the beauty of the mathematical formality, we hope that the semigroup of powers D α will form a continuous semigroup with parameter α, insie which the original iscrete semigroup of D n for integer n can be recovere as a subgroup, see Samko et al. (1993); Miller an Ross (1993). Moreover, for centuries we can only o calculus to a relatively rare an special kin of sufficiently well-behave functions, i.e. the ifferentiable function or even smooth function. But later we come to realize that the typical an prevalent continuous functions are nowhere-ifferentiable functions like the Weierstrass function. The classical calculus with integer orer is just powerless on this type of function. In this paper, First, some ifferent efinitions of fractional erivatives, such as the Riemann-Liouville, the Caputo erivative, Kolwankar s local erivative Kolwankar an Gangal (1997); Kolwankar et al. (1996)an Jumarie s moifie Riemann-Liouville erivative Jumarie (26), are iscusse. Then it s argue that the very reason for introucing fractional erivative is to stuy nonifferentiable functions. A concise an essentially local efinition of fractional erivative for one imension function is introuce an its geometrical interpretation is given. Base on this simple efinition, the fractional calculus is extene to any imension an the Fractional Geometric Calculus is propose. the paper is submitte for consieration of the best stuent paper awar.
2. REVIEW AND REMARKS OF SOME DEFINITIONS 2.1 The Riemann-Liouville approach Fractional integration Accoring to the Riemann-Liouville approach to fractional calculus, the notion of fractional integral of orer α (α > ) for a function f(x), is a natural generalization of the well known Cauchy formula for repeate integration, which reuces the calculation of the n fol primitive of a function f(x) to a single integral of convolution type. In our notation the Cauchy formula reas I n f(x) := f ( n) (x) = 1 (n 1)! x where N is the set of positive integers. Taking avantage of the Gamma function Γ(z) := which has the property (x t) n 1 f(t) t, n N (2.1) e u u z 1 u, Re {z} > (2.2) Γ(z + 1) = z Γ(z), (n 1)! = Γ(n), it s very natural to exten the above formula from positive integer values of the inex to any positive real values an efines the Fractional Integral of orer α > : Definition 1. I α f(x) := f ( α) (x) = 1 x (x t) α 1 f(t) t. (2.3) Γ(α) where α is in the set of positive real numbers. Fractional ifferentiation The efinition of fractional ifferentiation is somehow more subtle than the fractional integration. A straightforwar way to efine fractional ifferentiation is to o enough fractional integration firstly, then take integer ifferentiation (of course, one can o in another orer, just as the Caputo approach). That is to introuce a positive integer m such that m 1 < α m, then efine the fractional erivative of orer α > : Definition 2. D α f(t) := D m I m α f(t), namely f (α) (x) := α x α I α α f (2.4) where enotes the Ceiling function which map a real number to the smallest following integer. For complementation, ifferentiation of any real number orer α coul be efine as, α f (α) (x) := x α I α α f(x) α > f(x) α = I α f(x) α <. (2.5) (1) The funamental relations hol x Iα+1 f(x) = I α f(x), I α (I β f) = I β (I α f) = I α+β f, (2.6) the latter of which is a semigroup property fractional for integration operator. But unfortunately the erivative operator D α is quite ifferent an significantly more complex, for D α is neither commutative nor aitive in general. (2) Take a constant function f(x) = 1 for example: x (I 1 2 f)(x) = x x x ( 1 x ) Γ( 1 2 ) (x t) 1 2.1t ( x ) (x t) 1 2 t x (2 x) x ([ 2(x t) 1 2 The fractional ifferentiation of constant functions is not necessary zero from this efinition, which somehow obscures the physical meaning of these fractional erivatives as some kin of velocity. (3) The omain an bounary conitions of the functions must be explicitly chosen. Information about the omain an bounary conition of the functions is neee for calculation, so the fractional erivative is not a local property of the function. Therefore, unlike the integral orer erivative, the fractional orer erivative at a point x is not etermine by an arbitrarily small neighborhoo of x. This obscures the geometric meaning of these fractional erivatives. (4) To partially solve the above problem, many people use as the base point, instea of : I α f(x) := f ( α) (x) = 1 x (x t) α 1 f(t) t. Γ(α) (2.7) 2.2 The Caputo approach Alternatively, we can take integer ifferentiation firstly, then o fractional integration. The Caputo fractional erivative: Definition 3. ( f (α) (x) := I α α α ) f x α ] x ) (2.8) (1) This Caputo fractional erivative prouces zero from constant functions. (2) But it is easy to see that the Caputo efinition for the fractional erivative is in a sense, more restrictive than the Riemann-Liouville efinition because it require the function to have higher orer ifferentiability, but it is often the case that f (α) (x) may exist whilst f (x) oes not. These troublesome features greatly reuce its scope of application.
2.3 The Jumarie approach Moifie Riemann Liouville Jumarie s moifie Riemann Liouville fractional erivative is efine by Definition 4. f (α) 1 x (x) := (x t) α (f(t) f()) t, (2.9) Γ(1 α) x where α a real number on the interval (, 1). (1) If f() =, then f (α) is equal to the Riemann Liouville fractional erivative of f of orer α. (2) has the avantages of both the stanar Riemann Liouville an Caputo fractional erivatives. The fractional erivative of a constant is zero, as esire. (3) It is efine for arbitrary continuous (nonifferentiable) functions. (4) But this efinition is still an nonlocal efinition. Information about the omain an bounary conition of the functions is still neee, thus has no goo geometric interpretation. (3) It shoul be applicable to a large class of functions (i.e. continuous but nonifferentiable, or even generalize function) which are not ifferentiable in the classical sense. (4) It shoul have reasonable geometric interpretation, like the classical one. (5) The calculation shoul be easy. 3. THE VERY REASON FOR INTRODUCING FRACTIONAL DERIVATIVE 3.1 Discussion on semigroup property It is generally believe that the motivation for introucing fractional calculus is that we want the semigroup of powers D a to form a continuous semigroup with parameter a, insie which the original iscrete semigroup of D n for integer n can be recovere as a subgroup. But we argue that this is ue to we limit our scope of iscussion in the very special smooth function. Of course, we can calculate the fractional erivative of the common smooth function, like the following: 2.4 Kolwankar-Gangal approach Kolwankar-Gangal local fractional erivative Kolwankar an Gangal (henceforth K-G) has propose a local fractional erivative by introucing Definition 5. If, for a function f : [, 1] IR, the limit ID q q (f(x) f(y)) f(y) = lim x y (x y) q (2.1) exists an is finite, then we say that the local fractional erivative (LFD) of orer q ( < q < 1), at x = y, exists. (1) The operator in the RHS is the Riemann-Liouville erivative, so the K-G local fractional erivative can be equivalently expresse as f (α) (x) x=x := 1 Γ(1 α) lim x x x x f(t) f(x ) x (x t) α t, (2.11) where α a real number on the interval (, 1). (2) The fractional erivative of a constant is zero because the subtraction of f(x ). (3) The erivative is efine by the limiting process x x, so only local information near x is neee. (4) Although this efinition in local in nature, the geometric interpretation is still vague. One can harly have any geometric intuition irectly from this abstract efinition. 2.5 Final remarks We have reviewe several ifferent efinitions about fractional erivative, an each has some avantages an isavantages. But we still hope to have a efinition which can have all the following merits: (1) It shoul be local in nature, thus oes not rely on information of omain an bounary conitions. (2) The erivative of constant function shoul be zero. 1 2 x 1 2 ( 1 2 x) = 1 2 2π 1 1 x 1 2 x 1 2 x 2 2 = 2π Γ(1 + 1 1 2 2 ) Γ( 1 2 1 2 + 2 1 2 1)x1 = 2π 1 2 Γ( 3 2 ) Γ(1) x = 2 πx 2 π! = 1, It s simple an elegant from the pure mathematical point of view, but the semigroup property is not always satisfie for erivative operator. Moreover, it s har to give any geometrical interpretation of this kin 1 2 erivative. 3.2 Differentiability class of function The most basic functions x n, e x, sin x are smooth function. For smooth function we can use power series to approximate an express any smooth function. In mathematical analysis, a ifferentiability class is a classification of functions accoring to the properties of their erivatives. Higher orer ifferentiability classes correspon to the existence of more erivatives. Functions that have erivatives of all orers are calle smooth. Definition 6. The function f is sai to be of class C k if the erivatives f, f,..., f (k) exist an are continuous (except for a finite number of points or a measure zero set of points). The function f is sai to be of class C, or smooth, if it has erivatives of all orers. Uner the above efinition, the class C consists of all continuous functions. The class C 1 consists of all ifferentiable functions whose erivative is continuous; such functions are calle continuously ifferentiable. Thus, a C 1 function is exactly a function whose erivative exists an is of class C. The relation hol: C... C n... C 2 C 1 C G, where G enote the generalize function. In
particular, C k is containe in C k 1 for every k, an there are examples to show that this containment is strict. One interesting property is if f C m an g C n, then f + g C min(m,n). 3.3 Non-ifferentiable fuction It s interesting that, in the above relation C k+1 is always a proper (in fact, a measure zero) subset of C k. So coul there be some non-trivial examples which are C but not C 1 almost everywhere? The answer is yes, Weierstrass type of functions are typical such examples, see Berry an Lewis (198). What s more, the Weierstrass function is far from being an isolate or special example. In fact, it is the ominant an typical continuous functions, in the math sense. In physics sense, it s also very important to stuy this kin of fractal curve. Observe path of a quantum mechanical particle are known to be fractals an are continuous but non-ifferentiable, see Abbott an Wise (1981); Sornette (199); Cannata an Ferrari (1988). A simple example of such a parametrization is the well known Weierstrass function given by Wλ(t) s = λ (D 2)k sin λ k t k=1 where λ > 1 an 1 < D < 2. The graph of Wλ s (t) is known to be a fractal curve with box-imension D. (1) C is the set of all continuous functions. (2) C α for < α < 1 is the set functions which the α erivatives exist an are continuous. (3) C k for any positive integer k is the set of all ifferentiable functions whose 1 erivative is in C k 1 (4) C a for any positive real number a = k + α is the set of all functions whose α erivative is in C k (5) C is the intersection of the sets C k So we argue that, the semigroup property may not be a necessary an essential conition when introucing fractional calculus. The classical calculus is enough to eal with smooth function an the geometrical interpretation is natural an intuitive. So why o we nee to introuce some other erivative but has no intuitive interpretation for smooth function? For smooth function, there is no nee for something more. For nonifferentiable functions, things are quite ifferent an classical calculus is not enough. One of the main reason to introuce something new in math is when we want to enlarge our scope of iscussion an fin the ol tool is not applicable. We argue that he very reason for introucing fractional erivative is to stuy nonifferentiable functions. 4. GEOMETRICAL MEANING OF FRACTIONAL DERIVATIVE 4.1 Geometrical meaning of orinary erivative The geometrical meaning of orinary erivative is simple an intuitive: For smooth function f which is ifferentiable at x, the local behavior of f aroun point x can be approximate by using the orinary erivative: f(x + h) f(x) + f (x)h (4.1) Fig. 1. Example of Weierstrass function with D = 1.8 The orinary erivative gives the linear approximation of smooth function. Using Taylor series an higher orer erivative, we have f(x+h) f(x)+f (x)h+ f (x) 2! h 2 + f (3) (x) h 3 +... (4.2) 3! 3.4 Fractional ifferentiability class of function So like the question of introucing fractional erivative, here we ask the following questions: (1) Coul there be fractional ifferentiability class of function? (2) What s the structure of the space C 1 2, or generally the space of C a? (3) What s the representation of these kin of fractional ifferentiability function space? (4) What s the relation between fractional ifferentiability function space an fractional erivative. In general, the classes C a for any non-negative real number a can be efine recursively as following: Fig. 2. The orinary erivative gives the linear approximation. Here we expect the fractional erivative to have the similar geometrical meaning. We hope for non-ifferentiable
functions, the fractional erivative coul give some kin approximation of its local behavior. 5. A SIMPLE DEFINITION DIRECTLY FROM GEOMETRICAL MEANING We expect that the fractional erivative coul give nonlinear (power law) approximation of the local behavior of non-ifferentiable functions: f(x + h) f(x) + f (α) (x) Γ(1 + α) hα (5.1) in which the function f is not ifferentiable because f (x) α so the classical erivative f/x will iverge. Note that the purpose of aing the coefficient Γ(1 + α) is just to make the formal consistency with the Taylor series. We can give a very simple efinition irectly from the above meaning: Definition 7. For function f C α, < α < 1, the fractional erivative is efine as f (α) (x) := Γ(1 + α) lim h + f(t + h) f(t) h α (5.2) Also the fractional integral can formally efine as: Definition 8. For function f, the fractional integral of orer < α < 1 is a function f ( α) (x) := g(x) which satisfy g (α) (x) = Γ(1 + α) lim h + g(t + h) g(t) h α = f(x) (5.3) (1) The erivative of constant function is zero. (2) It coul be applicable to a large class of functions C α < α < 1 which are not ifferentiable in the classical sense. (3) It o have reasonable geometric interpretation, similar to the classical one. The fractional erivative coul give nonlinear (power law) approximation of the local behavior of non-ifferentiable functions. (4) The calculation of erivative is easy. (5) Last but not least, this efinition is local by nature. This property facilitate the generalization of one variable fractional erivative to vector erivative, which is very har if the nonlocal omain an bounary conition of the function is neee for calculation. 6. GEOMETRIC CALCULUS In orer to generalize the one variable fractional erivative to vector erivative, we make use of the language of geometric calculus (GC) instea of Gibbs vector calculus. Geometric algebra provie an powerful mathematical framework in which the most avance concepts moern physic, such as quantum mechanics, relativity, electromagnetism, etc., can be expresse in this framework graciously. There are many great avantages of the GC formalism. For example, the cross prouct only applies in three imensional space, because in four imensions there is an infinity of perpenicular vectors. But in GC formalism, the inner prouct an outer prouct are unifie an both have perfect generalization. Gibbs vector calculus is able to reuce Maxwell s twelve equations own to four, but in GC formalism Maxwell equation can be written in one single elegant equation with a spacetime operator which is reversible. Please see Hestenes an Sobczyk (1984); Hestenes (1966, 1999); Hestenes et al. (21); Gull et al. (1993); Doran et al. (1993, 1996); Lasenby et al. (1998)for further information about geometric calculus. The funamental ifferential operator is the erivative with respect to the position vector x (any imension). This is given the symbol, an is efine in terms of its irectional erivatives, with the erivative in the a irection of a general multivector M efine by M(x + ɛa) M(x) a M(x) = lim. (6.1) ɛ + ɛ Similarly we can efine the Fractional Geometric Calculus(FGC) as following: Definition 9. The Fractional Geometric Calculus is given the symbol α, an is efine in terms of its fractional irectional erivatives, with the fractional erivative in the a irection of a general multivector M efine by a α M(x) = lim ɛ + M(x + ɛa) M(x) ɛ α. (6.2) The α acts algebraically as a vector, as well as inheriting a calculus from its irectional erivatives. As an explicit example, consier the {γ µ } frame introuce above. In terms of this frame we can write the position vector x as x µ γ µ, with x = t, x 1 = x etc. an {x, y, z} a usual set of Cartesian components for the restframe of the γ vector. From the efinition it is clear that γ µ α = α x µ (6.3) which we abbreviate to α µ. From the efinition we can now write α = γ µ α µ = γ α t + γ 1 α x + γ 2 α y + γ 3 α z. (6.4) 7. CONCLUSION In this paper, some ifferent efinitions of fractional erivatives are iscusse an conclue that the very reason for introucing fractional erivative is to stuy nonifferentiable functions. Then, a concise an essentially local efinition of fractional erivative for one imension function is introuce an its geometrical interpretation is given. Base on this simple efinition, the fractional calculus is extene to any imension an the Fractional Geometric Calculus is propose. There are still many open questions: (1) Is it possible to give an explicit expression of f ( α) (t) through f(t) an α. That means a efinition of some kin integration operator corresponing to the new kin of fractional ifferential operator. (2) For Weierstrass function like the above, it has a fixe s, thus coul o α erivative with a fixe α. How to eal with nowhere ifferentiable function which has variable α(t)? (3) For what kin of f(t), there coul be non-trivial f (α) (t) an f ( α) (t)?
Fig. 3. Family tree for Fractional Geometric Calculus, extene from figure in http://geocalc.clas.asu.eu/html/evolution.html (4) Weierstrass function are C but not C 1 almost everywhere. An interesting question is how to give explicit expression of all the continuous everywhere but nowhere ifferentiable functions, just like we can use power series to approximate an express any smooth function. (5) Can we express some basic nowhere ifferentiable functions, an fin the fractional erivative of these basic functions? Is it possible to give a fractional calculus table for nowhere ifferentiable functions, just like the classical calculus? (6) Is it possible to give an example function f C 1 but not C 2 almost everywhere? Does the integral of Weierstrass function belong to this class? There shoul be such functions, but it s har to image it. How it looks like? Finally, we summarize the history of Fractional Geometric Calculus in the following figure 3. Doran, C., Lasenby, A., Gull, S., Somaroo, S., an Challinor, A. (1996). Spacetime algebra an electron physics. Avances in imaging an electron physics, 95, 271 386. Gull, S., Lasenby, A., an Doran, C. (1993). Imaginary numbers are not realthe geometric algebra of spacetime. Founations of Physics, 23(9), 1175 121. Hestenes, D. (1966). Space-time algebra, volume 1. Goron an Breach Lonon. Hestenes, D. (1999). New founations for classical mechanics, volume 99. Springer. Hestenes, D., Li, H., an Rockwoo, A. (21). New algebraic tools for classical geometry. Geometric computing with Cliffor algebras, 4, 3 23. Hestenes, D. an Sobczyk, G. (1984). Cliffor algebra to geometric calculus: a unifie language for mathematics an physics, volume 5. Springer. Jumarie, G. (26). Moifie riemann-liouville erivative an fractional taylor series of nonifferentiable functions further results. Computers & Mathematics with Applications, 51(9-1), 1367 1376. Kolwankar, K. an Gangal, A. (1997). Höler exponents of irregular signals an local fractional erivatives. Pramana, 48(1), 49 68. Kolwankar, K., Gangal, A., et al. (1996). Fractional ifferentiability of nowhere ifferentiable functions an imensions. Chaos An Interisciplinary Journal of Nonlinear Science, 6(4), 55. Lasenby, A., Doran, C., an Gull, S. (1998). Gravity, gauge theories an geometric algebra. Philosophical Transactions of the Royal Society of Lonon. Series A: Mathematical, Physical an Engineering Sciences, 356(1737), 487 582. Manelbrot, B. (1983). The fractal geometry of nature. Wh Freeman. Miller, K. an Ross, B. (1993). An introuction to the fractional calculus an fractional ifferential equations. Samko, S., Kilbas, A., an Marichev, O. (1993). Fractional integrals an erivatives: theory an applications. Sornette, D. (199). Brownian representation of fractal quantum paths. European Journal of Physics, 11, 334. ACKNOWLEDGEMENTS Thanks my supervisor for his selfless support! REFERENCES Abbott, L. an Wise, M. (1981). Dimension of a quantum mechanical path. Am. J. Phys, 49(1), 37 39. Berry, M. an Lewis, Z. (198). On the weierstrassmanelbrot fractal function. Proceeings of the Royal Society of Lonon. A. Mathematical an Physical Sciences, 37(1743), 459 484. Cannata, F. an Ferrari, L. (1988). Dimensions of relativistic quantum mechanical paths. Am. J. Phys, 56(8), 721 725. Doran, C., Lasenby, A., an Gull, S. (1993). States an operators in the spacetime algebra. Founations of physics, 23(9), 1239 1264.