Spatial Rotation of the Fractional Derivative in Two Dimensional Space

The transformation of the partial fractional derivatives under spatial rotation in $R^2$ are derived for the Riemann-Liouville and Caputo definitions. These transformation properties link the observation of physical quantities, expressed through fractional derivatives, with respect to different coordinate systems (observers). It is the hope that such understanding could shed light on the physical interpretation of fractional derivatives. Also it is necessary to able to construct interaction terms that are invariant with respect to equivalent observers.


Introduction
Fractional calculus deals with differentiation and integration to arbitrary real or complex orders. The idea is as old as the integer-order calculus.
Extensive mathematical discussion of fractional calculus can be found in Refs. [1]- [5] and references therein. The techniques of fractional calculus have been applied to wide range of fields, such as physics, engineering, chemistry, biology, economics, control theory, signal image processing, groundwater problems, and many others.
Physical applications of fractional calculus span a wide range of topics and problems (for a review see Refs. [6]- [9] and references therein). Generalizing fractional calculus to several variables, multidimensional space, and generalization of fractional vector calculus has been reported [10]- [13]. Also, progress has been reported on generalization of Lagrangian and Hamiltonian systems [14,15].
Despite the significant progress in applying fractional calculus to a wide range of physical problems, there is still a lack of satisfactory geometric and physical interpretation of fractional calculus, in comparison with the simple interpretations of their integer-order counterparts (see Ref. [16] and references therein). Effort has been devoted to relate fractional calculus and fractal geometry [17,18]. A different approach to geometric interpretation of fractional calculus is based on the idea of the contact of α−th order [19].
However, a satisfactory interpretation is still missing [16].
A study of the symmetry of physical systems described by fractional cal- Ref. [20,21], even though the class of variables substitution preserving the form of the fractional derivative is found to be very narrow. Generalizing the definition of fractional derivative of a function with respect to another function can be used to study general transformation of point transformation [22].
The question that we are trying to address in this work is basically the following: Given a quantity that is described by a fractional derivative of some function in a specific coordinate system (x, y), how can this quantity be expressed in a rotated coordinate system (x ′ , y ′ )? Thus, we try to relate the same fractional quantity in two equivalent coordinate systems. In this work, we only consider the cartesian space coordinates in two dimensions, (x, y), and adhere to the notion of time as invariant in all coordinate systems (nonrelativistic). In other words, we are looking into the transformation of fractional derivative operators and their effect on physical quantities under the rotation group SO(2). We are not interested in constructing the irreducible representations of SO(2), as done in the standard integer-order differentiation case [23], rather we focus on the orthonormal basis of the two dimesnional cartesian space of R 2 . Intuitively, quantities expressed through fractional derivative operators are expected to behave differently from scalar, vector, and tensor quantities.
In this work we investigate the transformation properties of the fractional derivative and its action on an invariant scalar field, under space rotation in two dimensions, and using both the Riemann-Liouville and Caputo definitions. We compare between the Riemann-Liouville and Caputo definitions in relating the same physical quantity under rotation. We check if there is a major difference in the transformation properties between the two definitions. In Ref. [24], the transformation properties of the Riemann-Liouville fractional derivative of a scalar field under infinitesimal transformation of SO(2) are derived. The current work is more general and can be applied to any function and is not bounded to infinitesimal transformation.
Also, we include the case of the Caputo definition of fractional derivative.
Their special result agrees with our general results. In section 2, we give a brief introduction to fractional calculus to lay out our notation. In section 3, we lay out the SO(2) transformation properties of fields and their derivatives expressed in Cartesian coordinates. In section 4, we apply our method to the Riemann-Liouville and Caputo fractional derivatives and layout their corresponding transformations under SO (2). Finally, in section 5 we give a brief discussion of our results.

Fractional Calculus
For mathematical properties of fractional derivatives and integrals one can consult Refs.[1]- [5] and the references therein. In this section we lay out the notation used in the rest of this work. We consider real analytic functions on R 2 and only discuss the Riemann-Liouville and Caputo definitions of the fractional derivative. Let f (x, y) to be a real analytic function in a specific domain in the Euclidian space R 2 ; f : R 2 → R. The x-partial fractional derivative of order α (keeping y constant) and where a is the lower limit of x is written as a D α x f (x, y). Similarly for the y-partial fractional derivative of order α (keeping x constat) and where b is the lower limit of y is written as b D α y f (x, y). Since the f (x, y) is analytic then the partial fractional derivatives are assumed to commute, i.e., a D α x , b D α y = 0. . .
A similar relation for the nth-order integration of the function f (x, y) Definition 2.2. The Riemann-Liouville fractional integration of order α < 0 and along x, keeping y constant, is defined as and along y, keeping x constant, is where Γ(.) is the Gamma function.
We consider a = b = 0 and drop them from the notation henceforth.
Definition 2.3. The Riemann-Liouville partial fractional derivatives of the order α > 0, where n − 1 < α < n and n ∈ N , are defined as where to unify notation we used D n x ≡ d n dx n and D n y ≡ d n dx n .
The leibniz rule applied to the Riemann-Liouville partial fractional derivatives can be written as [1]- [5], Definition 2.4. The Caputo partial fractional derivatives of order α > 0, where n − 1 < α < n and n ∈ N , are defined as The Riemann-Liouville and Caputo definitions of the fractional derivative As an example, for the function The Caputo partial fractional derivatives give the same result for x β y λ , except for β, λ ∈ N < n, where in this case the Caputo partial fractional derivatives vanish, as expected from Eqs. (9, 10).

Transformation of Functions and Integer-Order Derivatives Under Spatial Rotation
We consider the Euclidian space R 2 and work with orthonormal basis of the Cartesian coordinate system. The distinction between covariant and contravariant quantities disappears and we drop this notation in our discus- For the active transformation case the angle φ would be replaced by (−φ).
The coordinates (x ′ , y ′ ) are related to the coordinates (x, y) through We present some of the properties of the functional transformations of the Lie group as applied to function f (x, y). The generator of the SO (2) rotation (angular momentum), around the z axis, L is written as, The group elements of the rotation group are written, in the exponential form, as e φL , where the exponential is defined as an infinite sum of differential operators, explicitly where φ is the angle of rotation. Since L x = y and L y = −x, it is easy to derive the action of the element e φL on the coordinates x and y, explicitly Thus we retrieve the standard result in Eq. (15) .
Remark 3.1. In general, for n, m ∈ Z + , x ′ n y ′ m = e φL x n e φL y m = e φL (x n y m ) .
This result emerges from proving the following two Lemmas.
Lemma I: Given any two functions A(x, y) and B(x, y) that are analytic (of Class C ∞ ) in their common domain, then where n ∈ Z + . This is just the general Leibniz rule applied to the differential operator L and the proof is similar using mathematical induction. The Proof. We first write Noting that 1/(n − k)! vanishes for k > n we rewrite the above results as Shifting the order of the sum and then shifting the index n → n + k, we Noting that 1/n! vanishes for n = −k, −k + 1, . . . , −1, we reach the final conclusion e φL (A(x, y) B(x, y)) = = e φL A(x, y) e φL B(x, y) . In the x ′ y ′ coordinates frame, the transformed function Ψ ′ (x ′ , y ′ ) can also be written as If the constants a nm are independent of the coordinates, then we develop the following important result.
Remark 3.2. A function Ψ(x, y) that is analytic in a specific domain, where its dependence on the rotation angle φ comes only through its arguments (x, y), is transformed as A scalar function Ψ(x, y) that is invariant under spatial rotation, must assuming the dependence on the rotation angle φ occur only through the argument (x, y).
Next we investigate the transformation of derivatives, one notes the following D 1 x ′ x ′ n = n x ′ n−1 = n e φ L x n−1 = e φL D 1 x x n (27) = e φL D 1 x e −φL e φL x n = e φL D 1 x e −φL x ′ n .
Thus we derive the important result and similarly for The following commutation relations are easy to derive Transformation of higher derivatives can be derived, for example In general we conclude Remark 3.3. The partial derivatives transform under spatial rotation as where n, m ∈ N .
As an illustration, one can show that Thus, indicating that the Laplace operator D 2 x + D 2 y is a scalar operator, as expected.
An important remark is to check the transformation of the operator L itself under spatial rotation.
Thus the operator L is a scalar itself, a result that can be verified easily. This is an important point as it allows for the definition of inverse transformation between the xy and x ′ y ′ coordinate frames.

Transformation of the Fractional Derivative Under Spatial Rotation
In this section we generalize the results of the previous section to fractional transforms as where β, λ ∈ R. Next we look at the transformation of the partial fractional derivatives. Using the Riemann-Liouville and Caputo definition of the fractional derivative we note the following Considering an infinitesimal angle transformation, φ << 1, and keeping only linear terms of φ, we write

Remark 4.3.
Given an analytic function f (x, y), its partial fractional derivatives transform as Considering the infinitesimal angle approximation, φ << 1, and keeping only linear terms of φ, we write Thus we derived the transformation of the partial fractional derivatives according to the Riemann-Liouville and Caputo definitions. Next we consider the transformation of the partial fractional derivatives as applied to a scalar field Ψ(x, y) and using both Riemann-Liouville and Caputo definitions.

Transformation of the Riemann-Liouville Partial Fractional Derivatives of a Scalar Field
Consider a scalar function Ψ(x, y) that is invariant under spatial rotation, i.e., LΨ(x, y) = 0, then according to Eq. (40) where Since y and D 1 y commute with D α x we can write the above equation as Rewriting the above equation using commutation relations, we find The last term vanishes since LΨ = 0. Thus we arrive at Note that from the properties of the fractional derivative, D m x D α x = D m+α x , while D 1 x , D α x = 0 except for α ∈ Z + . We use the known relation [1]- [5] D α Thus, we conclude Using Leibniz rule in Eq. (7), substituting f (x) = Ψ(x, y) and g(x) = x, one can show that Thus we conclude that A similar relation holds for y Note that for α ∈ Z + , the last term in the above result vanishes. Also for 0 < α < 1, the term D α−1 y represents a fractional integral

Transformation of the Caputo Partial Fractional Derivatives of a Scalar Field
To derive the transformation of the Caputo partial fractional derivatives we consider 0 < α < 1, and use the relation between the Riemann-Liouville and Caputo definitions, as expressed in Eqs. (11,12). Also we use the results of Eqs. (48, 49). One can show that Therefore, the Caputo partial fractional derivative transform, under infinitesimal angle φ << 1, as . (57) Note that α−1 < 0 and thus D α−1 x and D α−1 y represent fractional integrals.

Discussion and Conclusions
In section 4, we derived the transformation of the Riemann-Liouville and Caputo partial fractional derivatives under spatial rotation. We also de- One can easily check that the result is as expected, for example as required. It is easy to show that the following quantities are invariant, y −α D α x (1) − x −α D α y (1) , x 2+α D α x (1) + y 2+α D α y (1) .
One can use (dx) α and (dy) α instead of x α and y α in the above invariant forms. The Caputo partial fractional derivatives of a constant field vanish.
Consider the scalar function Ψ(x, y) = x 2 + y 2 . The Riemann-Liouville and Caputo x-partial fractional derivatives are, respectively Again it is easy to check that the transformations of the partial fractional derivatives, for infinitesimal angle φ << 1, agree with Eqs. (50, 56) for both the Riemann-Liouville and Caputo fractional derivatives. A similar transformation holds for D α y and C D α y with x and y interchanged and φ → −φ. Once can easily check that the quantities x α D α y Ψ(x, y) + y α D α x Ψ(x, y) , x α y α D α y D α y Ψ(x, y) , x α C D α y Ψ(x, y) + y α C D α x Ψ(x, y) .
are invariant under spatial rotation. Again, one can use (dx) α and (dy) α instead of x α and y α in the above invariant forms.
For the function Ψ(x, y) = (x 2 +y 2 ) 2 , one can check the transformations correct. However, the quantities in Eq. (62) are not invariant separately.
Nevertheless, it is possible to find a linear combination of the two quantities to form an invariant quantity, namely, x α D α y Ψ(x, y) + y α D α x Ψ(x, y) + A x α y α D α y D α y Ψ(x, y) .
where A is some constant to make the combination invariant. Again, one can use (dx) α and (dy) α instead of x α and y α in the above invariant forms.
In Ref. [24], the authors derived the infinitesimal transformation of the partial fractional derivatives of a scalar field, by using the function's Taylor expansion and then applying the Riemann-Liouville fractional derivative to each term of the expansion. Their result agrees with our general result.