Scholarly article on topic 'Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics'

Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics Academic research paper on "Physical sciences"

Share paper
Academic journal
Open Physics
OECD Field of science

Academic research paper on topic "Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics"

Cent. Eur. J. Phys. • 9(5) • 2011 • 1267-1279 DOI: 10.2478/s11534-011-0040-5


Central European Journal of Physics

Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics

Research Article

Dumitru Baleanu1*, Sergiu I. Vacaru2t,

1 Department of Mathematics and Computer Sciences, Qankaya University, 06530, Ankara, Turkey

2 Science Department, University "AI. I. Cuza" Iasi, 54, Lascar Catargi street, Iasi, Romania, 700107

Received OS March 2011; accepted OS April 2011

Abstract: We present a study of fractional configurations in gravity theories and Lagrange mechanics. The approach

is based on a Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists of a proof that, for corresponding classes of nonholonomic distributions, a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.

PACS C200B): 02.30.Ik, 45.10Hj, 45.10.Na, 05.45.Yv, 02.40.Yy

Keywords: fractional geometry • fractional gravity • generalized Finsler geometry • nonlinear connection • nonholo-

nomic manifold

© Versita Sp. zo.o.

1. Introduction

It Is well known that both the modified Korteweg - de Vrles equation, (mKdV), and the sine-Gordon, (SG , (solitonic) equations can be encoded as flows of the invariant curvatures invariant of plane curves in Euclidean plane geometry [7, 8, 10-12, 14]. Such constructions were developed [1, 3, 15] for curve flows in Riemannian manifolds of constant curvature [9, 16]. These which gave rise, for instance, to a vector generalization of the mKdV equa-

*E-mail:, ^E-mail:

tion, and encode its bi-Hamiltonlan structure of multi-component mKdV equations.

A crucial condition behind the above mentioned geometric models wasis that the frame curvature matrix ismust be constant for a series of examples of Riemann symmetric spaces, with group symmetries etc., for which certain classes of nonlinear/solitonic equations can be associated. It was considered that this approach would have a limited importance, for instance, for study of more general classes of curved spaces (pseudo-Riemanian ones, Einstein manifolds etc.) if such geometries do not include constant curvature matrix coefficients. Nevertheless, it was possible to elaborate a similar method of solitonic encoding of geometric data, for generalized geometries with non-constant


curvatures, using certain Ideas and methods from the geometry of nonholonomic distributions, and modelling of Lagrange-F'insler geometries [18]. Such models with nontrivial nonlinear connection structure were elaborated on (co) tangent bundles (see [13] and references therein) and our proposal was to apply such methods on (pseudo) Rie-mannian and Einstein manifolds endowed with nonholo-nomic distributions.

We developed a geometric formalism when nonholonomic deformations of geometric structures determined by a fundamental Lagrange/Finsler/Hamilton generating function (or, for instance, and Einstein metric) induce a canonical connection, adapted to a necessary type nonlinear connection structure, for which the matrix coefficients of curvature are constant [2, 19]. For such an auxiliary connection, it is possible to define a bi-Hamiltonian structure and derive the corresponding solitonic hierarchy. As it is known the fractional calculus which deals with derivative and integrals of arbitrary orders [22-26] is an emerging field which has many applications in various fields of science and engineering [26-28, 30-35].

In a series of our works [4, 5, 20, 21] (we recommend the reader to see the details, discussions and bibliography, including notation conventions etc.), we proposed a program of geometrization of fractional calculus applications in classical and quantum fractional mechanics, field theories and Ricci flow evolution theories with fractional/noncommutative/stochastic etc. derivatives. It should be emphasized here that it is not possible to elaborate an unified geometric formalism for all types of existing definitions of fractional derivatives. Different such operators have their specific properties and minusespros and cons, and result in very different geometric (integro-differential) and, for instance, physical implications. Nevertheless, there is a class of fractional derivatives which resulting in zero when for actingons on constants. This property is crucial for the construction ofng geometric models of physical and mechanical theories with fractional calculus. These models which (from many formal points of view) are very similar to those ofn integer dimensional spaces, but with corresponding nonholonomic distributions selecting a prescribed type of fractional dynamics and/or evolution and, in general, nonholonomic constraints. Such an analogy allows us to involve in our study a number of very powerful geometric methods which have already been used, for instance, in deformation quantization, geometric mechanics, Lagrange-Finsler geometry etc. Via nonholo-nomic deformations/transforms, the constructions based on Caputo fractional calculus seem to have generalizations to include other types of fractional derivatives. The main goal of this paper is to provide such a similar geometrization of fractional calculus formalism [22-

30, 32, 33] extended to certain applications/modifications of gravity and Lagrange mechanics theories. This will be done, in a fractional calculus manner, which will allow in the second of our partner works [6] to encode such these geometric constructions into (fractional) bi-Hamiltonians and associated solitonic hierarchies. In brief, two of our works (this paper and [6]) present a fractional versions of articles [2, 19].

The paper is organized as follows:

In Section 2, we outline the geometry of N-adapted fractional spaces. In section 3, we provide an introduction into the theory of fractional gravity, show how the fundamental equations in such theories can be integrated in general form and provide a geometrization for fractional Lagrange mechanics. Section 4 is devoted to the geometry of fractional curve flows and related models of fractional N-anholonomic Klein spaces. The Appendix contains necessary definitions and formulae on Caputo fractional derivatives and related nonholonomic fractional differential geometry (with nonlinear connections).

2. Fractional manifolds with constant curvature coefficients

Let us consider a "prime" nonholonomic manifold V is of integer dimension dim V = n + m, n > 2, m > 1,1 Its frac-

a a a a

tional extension V is modelled by a quadruple (V, N, d, I),

where N is a nonholonomic distribution stating a nonlinear connection (N-connection) structure (for details, see Appendix A with explanations for formula (A.4). The frac-

tional differential structure d is determined by Caputo fractional derivative (A.1) following formulae (A.2) and

(A.3). The non-integer integral structure I is defined by rules of type (A.2).

The goal of this section is to prove that it is possible to construct a metric compatible linear connection D with constant matrix coefficients of curvature, computed with respect to "N-adapted" frames, for any fractional metric g

1 A nonholonomic manifold is a manifold endowed with a non-integrable (equivalently, nonholonomic, or anholo-nomic) distribution. There are three useful (for our considerations) examples when: 1) V is a (pseudo) Riemannian manifold; 2) V = E(M), or 3) V = TM, for a vector, or tangent, bundle on a base manifold M. We also emphasize that in this paper we follow the conventions from Refs. [18, 20, 21] when left indices are used as labels and right indices may be abstract ones or running certain values.

(A.7) (equivalently, (A.9)) on a nonholonomic manifold V.

2.1. N-adapted frame transforms and fractional metrics

For any respective frame and co-frame (dual) structures, aea, = (ae',aea,) and ae?' = (ae'', ae0') on V, we can consider frame transforms

aea = Aa(x,y) aea' and a e? = (x,y) ae?'. (1)

A subclass of frame transforms Equations in (1), for fixed "prime" and "target" frame structures, is called N-adapted if such nonholonomic transformations preserve the splitting defined by a N-connection structure N = {N°}. Under (in general, nonholonomic) frame transforms, the

metric coefficients of any metric structure g on V are recomputed using the following formulae

aga? (x, y) = Aa "' (X, y) A??' (X, y) ag a'?' (x, y).

For any fixed g and N, there are N-adapted frame transforms when

g = agu(x,y) ae' ® V + ahob(x,y)ae0 ®a eb, = agn'(x, y) aei'® aej' + ah„,b,(x,y) ae0'® aeb',

where ae0 and ae0 are elongated followingas in the formulae denoted (A.7), respectively by aN0j and

aN°'f = Ao0'(x, y)A'j'(x,y) aN)(x,y), (2)

or, inversely,

aN0j = A0'0 (x, y)AJ'j(x, y) aNj(x, y)

with prescribed aN"j'.

We preserve the N-connection splitting for any frame transform of type (1) when

ag.,., = A' Aj ag■■ ah , = A0 Ab ah j.

g/j' — A ¡'A j' gí¡, h0'b' — A 0'A b' h0b,

for Ai ' constrained to get holonomic ae' = Ai ' ae', i.e. [ae'', aej'] = 0 and ae0' = dy0' + aNdxj', for certain x'' = x''(x',y0) and y0' = y0'(x',y0), with aN computed following formulae (2). Such conditions can be satisfied by initially prescribing from the very beginning a nonholonomic distribution of the necessary type. The constructions can be equivalently inverted, when aga? and aN° are computed from aga'?' and aN°' , if both the metric

and N-connection splitting structures are fixed on V.

2.2. d-connections with constant curvature coefficients

From the class of metric-compatible fractional d-connections2 uniquely defined by a fractional metric structure g, we chose such an N-connection splitting with nontrivial coefficients "N"(x, y) when with respect to an N-adapted frame the canonical d-connection (A.14) has constant coefficients.

Any fractional metric g on V defines a set of metric-compatible fractional d-connections of type

apy _ la I i' _ n a I a' _ a i a'

0 ' a'p ~ I Lj'k' — Lb'k' 0 Lb'k'

= const, aq:C' = o, acaa'C' = 0) (3)

with respect to correspondingly constructed N-adapted frames (A.5) and (A.7), when aN = {aNa,'(x, y)} is a nontrivial solution of the system of equations

2 "oLa'k = db'(aNa') - 0aha'c' 0ahd'b' dc' aNf' (4)

for any nondegenerate constant-coefficients symmetric matrix 0ahd'b' and its inverse 0aha c. The Caputo frac-

tional derivative db, allows us to dub such constructions for fractional spaces [2, 19].

For both integer and non-integer dimensions, the coefficients arYa,p' of the corresponding to g Levi-Civita connection gV are not constant with respect to N-adapted frames.3

The curvature tensor of fractional d-connection 0pYa>p> (3) defined by a metric g has constant coefficients with respect to N-adapted frames aea' = [aei', aea'] and aea' = [aei', aea'] with aNdk' subjected to conditions (4). Introducing constant coefficients 0pY a>p> (3) into formulae (A.10), we get

apa' _ iapi _ n apa'

0™ p'y'5' — (0n h'j'k' 0, 0n b'j'k'

_ a I c' a I a' a i c' a i a'

— 0 L b'j' 0 L c'k' 0 L b'k' 0 L c'j'

= const, 00P'h'J'a' = 0, 0Pc'b7a'

= 0 Sij'b'c' = 0, 0 Sab'k'c' = 0).

Such proprieties hold for a class of prescribed nonholonomic distributions. Of course, in general, with respect

2 see definitions and main formulae in Appendix A

3 In explicit form, such coefficients are computed following

formulae (A.15).

to local coordinate (or other N-adapted) frames, the curvature d-tensor aRapvS does not have constant coefficients. Nevertheless, we can always chose certain non-holonomic N-connection configurations when the corresponding adaptationsing will generate d-connections with constant matrix curvature coefficients.4 Using formulae (A.11) and (A.12), we compute

a „a'& a d a „1i'

og & oRa'& = °g

0 Ri'j'

0J a'b'

[ S = const.

We can conclude that a fractional d-connectlon 0r~Ya>p' (3) has constant scalar curvature.

3. Fractional Einstein and/or Lagrange spaces

The goal of this section is to show how the integer dimensional Einstein gravity and Lagrange mechanics can be encoded and/or generalized in terms of fundamental geometric objects on nonholonomic fractional manifolds. We review and develop our constructions from [4, 5, 20, 21].

3.1. Fractional gravity

An unified approach to Einstein-Lagrange/Finsler gravity for arbitrary integer and non-integer dimensions is possible for the fractional canonical d-connection aD. The fractional gravitational field equations are formulated for the Einstein d-tensor (A.13), following the same principle of constructing the matter source aY&& as in general relativity but for fractional metrics and d-connections,

type (6) for fractional dimensions, but they havere is a certain physical motivation if we develop models in which in integer limits result in the general relativity theory.

3.1.1. Separation of equations for fractional and integer dimensions

In previous works we studied [21] what type of conditions must satisfy the coefficients of a metric (A.9) for generating exact solutions of the fractional Einstein equations (5). For simplicity, we can use a "prime" dimension splitting of type 2 + 2 when coordinatesd are labelled in the form u& = (xj, y3 = v,y4), for i,j,... = 1, 2. and the metric ansatz has one Killing symmetry when the coefficients do not depend explicitly on variable y4. The solutions of equations can be constructed for a general source of type5

aYa & = diag[aYy; aYi = aY2 = aY2(xk,v); aY3 = aY4 = aY4(xk)].

For such sources and ansatz with Killing symmetries for metrics, the Einstein equations (5) transform into a system of partial differential equations with separation of equations which can be integrated in general form,

aR2 = -

2 agi ag2

'g'l ag2 - (agi 2 agi 2 ag2

x [°g2*

a g' a g' g1 g2


___\ / ] = — a Y

2 ag2 2 agi 4

aR4 = - 2 ah3 ah4 \["h"

- ("hrf - ah*3 ah4 - aY

2 ah4 2 ah3 J 2'

Such a system of integro-differential equations for generalized connections can be restricted to fractional nonholonomic configurations for aV if we impose the additional constraints

aRaj = aea( aNc), ac;b = 0, aQaji = 0. (6)

There areis not theoretical or experimental evidences that for fractional dimensions we must impose conditions of

4 Using the deformation relation in (A.15), we can compute the corresponding Ricci tensor a RapYs for the Levi-Civita connection °V, which is a general one with "non-constant" coefficients with respect to any local frames.

R3k = aWk ah** -h4 (ah4)2 ah*3 ah*4

2 ah4 2 ah4 2 ah3

a h* h4 I Jk i ah3 dk ah4\ a ixkdxk ah* h4

4 ah4 1 • h3 ah4 2 ah4

= k RR4 ah4 an*k*

1 ah4 ah* - 3 ah4) an*k 0,

\ ah3 2 ah3

(9) = 0,

5 such parametrizations of energy-momentum tensors are quite general ones for various types of matter sources

where the partial derivatives are

aa' = "d1a = dxiaa, aa' = d2a

= ix2 dx2 a0, a0* = 3V0 = lV3va0,

being usedusing the left Caputo fractional derivatives (A.2).

Configurations with fractional Levi-Civita connection a"V, of type (6), can be extracted by imposing additional constraints

aw* = % In |ah4\, aekawi = aeiawk,

an* = 0,diank = 3kan'. (11)

We can construct "non-Killing" solutions depending on all coordinates when

ag = agi(xk) adx' ® adx'

+ ato2(xj, v, y4) ah0(xk, v) ae0® ae0, ae3 = ady3 + aw(xk, v) adx', ae4

= ady4+ ant(xk,v) adx', (12)

for any aw for which

3.1.2. Solutions with ah3,A = 0 and aY24 = 0

For simplicity, in this paper we show how to construct an exact solution with metrics of type (12) when ah?i4 = 0 (in Ref. [21], there are analyzed all possibilities for coefficients are analyzed6) We consider the ansatz

ag = ea^(xk> akxi ® akxi + h3(xk, v) ae3® ae3

+h4(xk,v) ae4® ae4, ae3 = akv + awi(xk, v) akxi, ae4 = aky4

+ an(xk,v) akxi (13)

aßaWi + a Gi = 0,

'n** + aY an* = 0,

where аф = In

VI"hi h

(In I ah4\3/2l\a h3\)* ,

a ai = ah*4dk aj>, a? = ah*4 ap .

For ah*4 = 0;aY2 = 0, we have a<p* = 0. The exponent e a^(x ' is thea fractional analog of the "integer" exponential functions and called the Mittag-Leffler function, Ea[(x - 1x)a]. For a^(x) = Ea[(x - 1 x)a], we have

dEa = Ea, see (for instance) [17].

Choosing any nonconstant = arf>(x', v) as a generating function, we can construct exact solutions of (14)-(17). We have to solve respectively the two- dimensional fractional Laplace equation respectively, for ag-\ = ag2 = e a^(xk). Then we integrate on v, in order to determine ah3, ah4 and ani, and solving algebraic equations, for "w. We obtain (computing consequently for a chosen a4>(xk, v))

agi = ag2 = e \ ah3 = ±

ah4 = 0h4(xk) ± 2J1

aY2 , (exp[2 аф(хк,у)])*

Wi = -д^фГф*, ani = 1 nk (xi) + 2nk (xi)

. Iv[ah3l(V\h\)3],

where 0h4(xk), 1 nk[xl) and 2nk(x') are Integration

functions, and 1VIV is the fractional integral on variables v. variables.

We have to constrain the coefficients in (19) to satisfy the conditions in (11) in order to construct exact solutions for the Levi-Civita connection aV. To select such classes of solutions, we can fix a nonholonomic distribution when 2nk (x') = 0 and 1 nk (x') are any functions satisfying

the conditions di 1 nk (xj) = dk 1 nt (xj) . The constraints on a^)(xk, v) are related to the N-connection coefficients

aWi = aф/ с'ф* following relations

аф+ аф" = 2 aYA(xk), (14)

ah*4 = 2 Ч3 "h^ aY2(xi, v)Гф*, (15)

6 by nonholonomic transforms, various classes of solutions can be transformed from one to another

W a Ф])* + aWi[a ф]

'h<[ аф])* + д< ah4[aф] = 0,

hi aWk[aф] = дк aWi[aф], (20)

where, for instance, we denoted by ah4[ a$] as the functional dependence on a$. Such conditions are always satisfied for = a^(v) or if = const when aW'(xk, v) can be any functions as follows from (16) with zero a? and a ah see (19)).

ek aw = dkaw + aWk aw* + "nkd,A aw = 0

3.2. A geometric model for fractional Lagrange spaces

Any solution ag = {aga>&>(ua )} of fractional Einstein equations (5) can be parametrized in a form derived in previous section. Using frame transforms of type aea = ea'a aea,, with aga& = ea'ae&& aga&,, for any aga& (A.9), we relate the class of such solutions, for instance, to the family of metrics of type (13). Such solutions can also be related also to analogous fractional models, via corresponding eaa to a Sasaki type metric, see below in (21). A fractional Lagrange space is defined on a fractional

tangent bundle TM of fractional dimension a e (0,1)

a a a a

by a couple L" = (M_,L), for a regular real function L :

TM —.> R, when the fractional Hessian is

a 1 l a a a a

Lgj = -4 rn + did^L = °.

3. the canonical metrical d-connection

ad = (haD,vaD) = {ar>a& = (aVjk, ^

is a metric compatible, aD = 0, for

a r = a ri a eY =R ek + R i a ec c1 j — c1 jY Le _ L jke + {-jc Le ,

with R = I a R i = R a :n a ra = a ra a eY = with L jk — L bk, Cjc — Cbc in c 1 b — c ' bY L e —

R"bkek + C'bc aec,7 and generalized Christoffel indices

aRi — - a ir (aa a„ , a„ a„ a ^ \

Ljk = 2 Lg \L ek Lgjr + L ej Lgkr - L erLg.jk), 1

a R a a „ad ta „ a „ a „ a „ \

Rbc = 2 Lg ( ec Lgbd + ec Lgcd - ed Lgbc) .

Any LI can be associated to a prime "integer" Lagrange space L".

Let us consider values yk(t) = dxk(r)/dr, for x(t)

parametrizing smooth curves on a manifold M with t e

[0,1]. The fractional analogs of such configurations are

determined by changing d/dT into the fractional Caa a a

puto derivative dT = 1T dTwhen ayk(t) = d_Txk(t). Any

L defines the fundamental geometric objects of Lagrange spaces when

1.the fractional Euler-Lagrange equations

a a a a a

dT (iyidiL) — -xi dL = 0 are equivalent to the fractional "nonlinear geodesic" (equivalently,

la \ 2 a

semi-spray) equations I dT I xk +2Gk(x, ay) = 0, where

4. Nonholonomic fractional curve flows

We formulate a model of the geometry of curve flows adapted to an N-connection structure on a fractional manifold V.

4.1. Non-stretching and N-adapted fractional curve flows

A canonical d-connection operator aD (A.14) acts in the forms

aDX aea = (XJ araY) aey and (22)

aDY aea = (YJ afa") aey,

aa k 1 k G = 4 Lg

yJ 1yjdj I-xidiL) — -xidiL

defines the canonical N-connection aN" =

iyjdjGk(x, ay);

2. the canonical (Sasaki type) metric structure is

Lg = Igkj(x,y) aek ® aej + îgcb(x.y) aLec ® Leb, (21)

where the frame structure "ev = ( "ei, ea) is linear

on aLNa;

where "J" denotes the interior product. The value aDX = X& aD& is a covariant derivation operator along curve y(t, l). It is convenient to fix the N-adapted frame to be parallel to curve y(l) adapted in the form

where h ag(hX, aei) =0, = vX, for a = n + 1, and aes where v ag(vX, aea) =0,

7 for integer dimensions, we contract "horizontal" and "vertical" indices following the rule: i = 1 is a = " + 1;

i = 2 is a = n + 2; ... i = n is a = n + n"

for i = 2, 3,... n and a = n + 2, n + 3,..., n + m. The co-variant derivative of each "normal" d-vector's aea results into d-vectors adapted to y(r, l),

aDx aea = -pi(u) X and aDhxhX = pi(u) aea,(24) aDx aea = -pb(u) X and aDvxvX = pb(u) aed,

which holds for certain classes of functions p'(u) and pa(u). The formulae (23) and (24) are distinguished-split into h- and v-components for X =hX + vX and aD = (hD, vD) for aD = {a?yap}, hD = {alik, aHabk} and

vd = {aqc, aCbc}.

A non-stretching curve y(r, l) on V, (where т is a real pa-

rameter and l is the arclength of the curve on V,) is defined with such evolution d-vector Y = ут and tangent d-vector X = yl such that ag(X, X) =1. Such a curve у(т, l) sweep-

spt out a two-dimensional surface in T_Y(Tl)V С TV. Along y(l), we can movechange differential forms into a parallel N-adapted form. For instance, aa"X = XJ afae. Such fractional spaces can be characterized algebraically if we perform a frame transform preserving the decomposition (A.4) to an orthonormalized basis aeai, when

is an orthonormal d-basis. In this case, the coefficients of the d-metric (A.9) transform into the (pseudo) Euclidean one aga'&' = na'&' by encoding the fractional configuration into the structure of local d-bases. We obtain two skew matrices

a r-i'f I


hX a —a'b'

hXJ a f' =2 a vXJ a—a'b' = 2 ea pb

p] and

g(hX,e') = [1, 0.....0] and

h(vX,ea') = [1, 0.....0],

0 / -Pi' 0[h]

and a-vXa' b' =

0 pb' -Pa' 0[v]

with 0[b] and 0[v] being respectively (" — 1) x (" — 1) and (m — 1) x (m — 1) matrices.8

8 The above presented row-matrices and skew-matrices show that locally an N-anholonomic fractional mani-

The torsion and curvature tensors (A.10) can be written in orthonormalized component form with respect to (24) mapped into a distinguished orthonormalized dual frame

a П^ (X, Y) =

The values

jdy aг

- aD Y

a- a— Dx I v

- с _

X/ ' Xß'

rv0'У arv„'a'_ arv' arYy'a'. (27)

c g(Y,

- a' ' в' ~

ag(aea , aDY aee) define respectively the N-adapted orthonormalized frame row-matrix and the canonical d-connection skew-matrix in the flow directs, and a(X, Y) = ag(aea', [aDX, aDY] aee) is the curvature matrix.

4.2. N-anholonomic fractional manifolds with constant matrix curvature

The geometry of integer dimensional Einstein and Lagrange-Finsler spaces can be encoded into bi-Hamilton structures and associated solitonic hierarichies [2, 19]. The goal of this section is to show that there is a geometric background for extending the constructions to the case of fractional spaces. We shall elaborate the concept of fractional N-anholonomic Klein space which in [6] will be applied for constructing fractional solitonic hierarchies.

4.2.1. Symmetric fractional nonholonomic manifolds

For trivial N-connection curvature and torsion but constant matrix curvature on spaces of integer dimension, we get a holonomic Riemannian manifold and the equations (26) and (27) directly encode a bi-Hamiltonian structure [1, 15]. A well known class of Riemannian manifolds for which the frame curvature matrix constant consists of the symmetric spaces M = G/H for compact semisimple Lie groups G D H. A complete classification and summary of

fold V are related to prime spaces of integer-dimension n + m. With respect to distinguished orthonormalized frames the constrained fractional dynamics is characterized algebraically by couplespairs of unit vectors in Rn and Rm, preserved respectively by the SO(n — 1) and SO(m — 1) rotation subgroups of the local N-adapted frame structure group SO(n) ф SO(m). The connection matrices arhX v ' and arvX a, b belong to the orthogonal complements of the corresponding Lie subalgebras and algebras, so(n — 1) С so(n) and so(m — 1) С so(m).

a- ' '

main results on such Integer dimension spaces are given In Ref. [9]. Using the Caputo partial derivative, such a classification can be provided for fractional spaces of constant matrix curvature. The derived algebraic classification is that for a used "prime" integer dimension space but the differential and integral calculus are those for fractional nonholonomic distributions.

Algebraically, our aim is to solderattatch, in a canonic way, the horizontally and vertically symmetric Rieman-nian spaces of dimension n and m with a (total) symmetric Riemannian space V of dimension n + m, when V = G/SO(n + m) with the isotropy group H = SO(n + m) D O(n + m) and G = SO(n + m + 1). The Caputo fractional derivative is encoded for constructing tangent spaces and related geometric objects. For the above mentioned horizontal, vertical and total symmetric Riemannian spaces one exists natural settings to Klein geometry. A prime fractional metric tensor h ag = {адц} on h aV is defined by the Cartan-Killing inner product < •, • >h on T_xhG — hg restricted to the Lie algebra quotient spaces hp =hg/hh, with TxhH — hh, where hg = hh © hp is stated such that there is an involutive automorphism of hG under hH is fixed, i.e. [hh,hp] С hp and [hp,hp] С hh. In a similar form, we can define the group spaces and related inner products and Lie algebras,

for vg = {hab}, < •, • >v, LyvG — vg, vp = vg/vh, with TyVH — vh,vg =vh © vp, where [vh,vp] С vp,[vp,vp] С vh;

for g = {дав Ь < ^ • ^ T(x,y)G - 0, p = в/h, with Tix,y)H - h, 0 = h © p, where [h, p] С p, [p, p] С h.

on hG = SO(n + 1). We can also define an equivalent d-metric structure of type (A.9)

aei = (dxi)a,

Sab aêa ® aèb,

= (dya)a + a Na(dxi)a.

We note that, for integer dimensions, such trivial parametrizations define algebraic classifications of symmetric Riemannian spaces of dimension n + m with constant matrix curvature admitting splitting (by certain algebraic constraints) into symmetric Riemannian subspaces of dimension n and m, also (both withalso have constant matrix curvature), and introducing the concept of N-anholonomic Riemannian space of type aV = [hG = SO(n + 1), vG = SO(m + 1), a Nf\. Such spaces of constant distinguished curvature are constructed as trivially N-anholonomic group spaces which possess a Lie d-algebra symmetry soN(n + m) = so(n) ф so(m). A fractional generalization of constructions is to consider nonholonomic distributions on V = G/SO(n + m) defined locally by arbitrary N-connection coefficients aN"(x,y) with nonvanishing and aQ"- but with constant d-

metric coefficients when the fractional metric ag is of type agare> = [agi'j', ahaibi] (A.9) with constant coefficients a9i'ï = 0gvi' = a Nij and aha,b, = 0ha,b, = ahab (in this section induced by the corresponding Lie d-algebra structure S0N(n + m)). Such spaces transform into N-anholonomic fractional manifolds aVN = [hG = SO(n + 1), vG = SO(m + 1), aNf] with nontrivial N-connection curvature and induced d-torsion coefficients of the canonical d-connection9.

Similar formulae in [2, 19] are for usual partial derivatives with T not underlined. with not underlined symbols T. We parametrize the metric structure with constant coefficients on V = G/SO(n + m) and fractional differentials in the form

ag= agm(kuY-)a ® (kup)a,

where the coefficients of a fractional metric of type (A.7) are parametrized in the form

Naß =

aNa a Nb aNa

aNe aNa

4.2.2. Fractional N-anholonomic Klein spaces

We can characterize curve flows (both in integer and fractional dimensions) by two Hamiltonian variables given by the principal normals hv and Vv, respectively, in the horizontal and vertical subspaces, defined by the canonical d-connection aD = (hD,vD), hv = aDhXhX = V aeç and vv = aDvXvX = va aea, (see formulae (24) and (24). This normal fractional d-vector av = (hv, Vv), with components of type ava = (vi, va) = (v1, vi,vn+1,va), encoding the Caputo fractional derivative is in the tangent direction of curve y. It can be also considered the principal normal d-vector aQ = (hQ, vQ) with components of type aOa = (Oi, Oa) = (Q\ ai, On+1, Os) in the flow

with trivial, constant, N-connection coefficients computed aN't = aheb agjb for any given sets aheb and agjb, i.e. from the inverse metrics coefficients defined respectively

9 see formulae (A.10) computed for constant d-metric coefficients and the canonical d-connection coefficients in (A14)

a Ne ah

direction, with

ha = aDhyhX =Qi aeh va = aDvyvX = a" aed,

representing a fractional Hamiltonian d-covector field. We argue that the normal part of the flow d-vector hx = Yx = h' aer + h" ae" represents a fractional Hamiltonian d-vector field and use parallel N-adapted frames aea' = ( ae-', aear) when the h-variables v', a' , h' are respectively encoded in the top row of the horizontal canonical d-connection matrices a'hX1 and arhy 1 and in the row matrix |ey j = ey —g eX, where g = ag(hY,hX) is the tangential h-part of the fractional flow d-vector.10 An N-connection structure (in particular, a fractional La-grangian) induces an N-anholonomic Klein space (stated by two left-invariant hg- and vg-valued Maurer-Cartan forms on the Lie d-group G = (hG, vG)) is identified

with the zero-curvature canonical d-connection 1-form

a r _ r a ra' 1

G' IG1 p'},

ara' _ ara' y' _ a i'' a k' . a r'' a k'

G1 PJ — G1 p'y'e — hGLJ'k' e -r vG^- j'k' e ■

For n = m, and canonical d-objects (N-connection, d-metric, d-connection, ...) derived from (A.9), and any N-anholonomic space with constant d-curvatures, the Cartan d-connection transforms just in the canonical d-connection (A.14). Using the Lie d-algebra decompositions g = hg ® vg, for the horizontal splitting: hg = so(n) ® hp, when [hp, hp] C so(n) and [so(n), hp] C hp; for the vertical splitting vg = so(m) ® vp, when [vp, vp] C so(m) and [so(m), vp] C vp, the Cartan d-connection determines an N-anholonomic Riemannian structure on the nonholo-nomic bundle aE = [hG = SO(n + 1), vG = SO(m + 1), Nf]^ It is possible to consider a quotient space with distinguished structure group aVN = G/SO(n)® SO(m) regarding G as a principal (SO(n) ® SO(m))-bundle over aE, which is an N-anholonomic bundle. In this case, we

10 It is possible to encode v-variables v" , a" , h" in the top row of the vertical canonical d-connection matrices arvXa' b' and arvya' b' and in the row matrix (ey'j =

e" — h eX where h = ah(vy,vX) is the tangential v-part of the flow d-vector. In a compact form of notations, we shall write va and aa where the primed small Greek indices a', p', ■■■ will denotes both N-adapted and then orthonormalized components of geometric objects Id-vectors, d-covectors, d-tensors, d-groups, d-algebras, d-matrices) admitting further decompositions into h- and v-components.

can always fix a local section of this bundle and pullback G' to give a (hg ® vg)-valued 1-form g' in a point u <E aE■ 11

It is possible to generate a G( = hG ® vG)-invariant fractional d-derivative aD with restriction to the tangent space T aVN for any N-anholonomic curve flow y(r, l) in aVn = G/SO(n)® SO(m) is

aDx ae = [ae, yiJ ar] and aDy ae = [ae, yTJ ar],

admitting further h- and v-decompositions. The derivatives aDX and aDy are equivalent to (23) and obey the Cartan structure equations (26) and (27). We consider a N-adapted orthonormalized coframe

aea' = (ae'', ae"') identified with the (hp ® vp)-valued

coframe ae in a fixed orthonormal basis for p =hp ® vp Chg ® vg^ For the kernel/ cokernel of Lie algebra multiplications in the h- and v-subspaces, respectively, [ aehX, •]hg and [ aevX, •]vg , we can decompose the coframes into parallel and perpendicular parts with respect to aeX, ae = (ec = heC + veC, eC± = heC± + veC±), for p( = hp ® vp)-valued mutually orthogonal d-vectors eC and eC±, when there are satisfied the conditions [eX, eC]g = 0 but [eX, eC±]g = 0;are satisfied;such conditions can be stated in h- and v-component form, respectively, [hex, heC]hg = 0, [hex, he^]hg = 0 and [veX, veC]vg = 0, [veX, veC±]vg = 0^ There are decompositions

TuVn ^ p =hp ® vp = g =hg ® vg/so(n) ® so(m) and p = p C ® pC± = (hpC ® vpC) ® (hpC±® vpC±),

11 There are involutive automorphisms ha = ±1 and vo = ±1, respectively, of hg and vg, defined that so(n) (or so(m)) is eigenspace ho = +1 (or vo = +1) and hp (or vp) is eigenspace ho = —1 (or vo = —1)^ We construct a N-adapted fractional decomposition taking into account the exisOing etgenspaces, when the symmetric parts ar= 2 (gr+o ( gr) j , with respective h- and v-splitting

aL= 2 OhgL+ho thg'tt and aC= 1(aBC + ho£„£))■ This defines a (so(n) ® so(m))-valued d-connection fractional 1-form. The antisymmetric part e =2(gl~—o(gr)), with respective h- and v-splitting h ae = 1 (ahge — ho ) and

v ae = 2(age — ho(age)), defines a (hp ® vp)-valued N-adapted coframe for the Cartan-Killing inner product < •, • >p on TuG ~ hg ® vg restricted to TuVN ^ p^ This inner product, distinguished into h- and v-components, provides a d-metric structure of type ag = [ag, ah] (A.9), where ag =< h ae®h ae >hp and ah =< v ae®v ae >vp on aVn = G/SO(n)® SO(m).

with p C pc and pCi C p±, where [p, pc] = 0, < pCi,pc >= 0, but [p,pCi] = 0 (i.e. pc is the cen-tral'izer of eX in p =hp © vp chg © v0);in h- and v-components, one have hp C hpC and hpC± C hp1, where [hp, hpC] = 0, < hpCi, hpC >= 0, but [hp, hpCi] = 0 (i.e. hpC is the centralizer of ehX in hp chg) and vp C vpC and vpCi C vp^ where [vp, vpC] = 0, < vpCi, vpC >= 0, but [vp,vpCi] = 0 (i.e. vpC is the centralizer of evX in

vp Cvg).

Acting with the canonical d-connection derivative "DX of a d-covector perpendicular (or parallel) to aeX, we get a new d-vector which is parallel (or perpendicular) to aeX, i.e. aDeC e pCi (or aDeCi e pC). In bland v-components, such formulae are written aDhXheC e hpCi (or "DhxheCi e hpC) and aDvxvec e vpCi (or aDyXveCi e vpC). All such d-algebraic relations can be written in N-anholonomic manifolds and canonical d-connection settings, for instance, using certain relations of type

aDx(e"')c = v?(e?)ci and aDx(ea')ci = -v?(e?)c,

for some antisymmetric d-tensors va ? = -v? a . This is an N-adapted (SO(n) © SO(m))-parallel frame defining a generalization of the concept of parallel frame on N-adapted fractional manifolds whenever pC is larger than p. If we substitute aea' = (ae'', aea') into the last formulae and considering h- and v-components, we construct SO(n)-parallel and SO(m)-parallel frames.

5. Conclusions

The use of the fractional calculus techniques in differential geometry is still at the beginning ofearly in its application. The fractional operators reveal a complex structure and possess lessfewer properties that the classical ones. For these reasons the fractional operators are not easy to be used within the differential geometry and its applications. In this manuscript the calculations were done within the Caputo derivative. It was shown that for corresponding classes of nonholonomic distributions, a large class of physical theories are modelled as nonholonomic manifolds possessing constant matrix curvature. As a result we encoded the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.


For spaces of fractional dimension, it is possible to construct models of fractional differential geometry similarly to certain corresponding integer dimension geometries, if the Caputo fractional derivative is used [20, 21]. Applying nonholonomic deformations, the constructions can be generalized for another types of fractional derivatives.

1. Caputo fractional derivatives, local (co) bases and integration

The fractional left, respectively,and right Caputo derivatives are defined respectively by formulae

Jj(*) :=n-T) I (* - x'Y-- (£ )Sf (x')dx';; (A.1)

xd 2Xf(x) := r—) f(*'- *)S-a-1 (-d )Sf(*')d*' .

We can introduce d := (dxj)a 03j for the fractional

a i j |1— a

absolute differential, where dxj = (dxj)a *{2-a) if 1*' = 0. Such formulae allow us to elaborate the concept of a

fractional tangent bundle T_M, for a e (0,1), associated to a manifold M of necessary smooth class and integer dim M = n.12

Using denote by Lz(1x, 2x) to denote the set of those Les-

begue measurable functions f on [1x, 2x] when ||f\\z =

(f\f(x)\zdx)Vz < to, and let Cz[1x, 2x] be the space of

functions which are z times continuously differentiable on this interval. For any real-valued function f(x) defined on a closed interval [1x, 2x], there is a function

F(x) =1X IX f(x) defined by the fractional Riemanna *

Liouville integral lXIXf(x) := J(x — *')a-1 f(x')dx',

when f(x) =1 X dXF(x), for all x e [1x, 2x], satisfies the conditions

id ^Xai*f(X)J = f(X), a > 0,

i*aix [ixixF(x)) = F(x) - F(ix), 0 < a < 1.

12 For simplicity, we may write both the integer and fractional local coordinates in the form u? = (xj, y"). We underlined the symbol T in order to emphasize that we shall associate thise approach towith a fractional Caputo derivative.

We can consider fractional (co) frame bases on T_M. For

a o> a

instance, a fractional frame basis e_ß = eßß(uß)3ßl is connected via a vierlbein transform eßß(uß) with a fractional

local coordinate basis

dß' = № = ^ dr ,öb, = ^ d„I , (A.2)

for j' = 1, 2,..., n and b' = n + 1, n + 2,..., n + n. The fractional co-bases are related via e = e? (u?)du?, where

du= ((dx'' )a, (dya' f}. (A.3)

2. N- and d-connections and metrics

A nonlinear connection (N-connection) N for a fractional

space V is defined by a nonholonomic distribution (Whit-

ney sum) with conventional h- and v-subspaces, hV and

a a a a

TV = hV©vV. (A.4)

A fractional N-connection is defined by its local coeffi-

cients N = {aN?}, when

N = aN?(u)(dx')a « da.

The fractional absolute differential d is written in the form

d := (dx')a 0d,, where dx' = (dx')a

X )1-a

where we consider 1x' = 0. The differentials dxj =

(dxj)a=1 are used as local coordinate co-bases/-frames

or the "integer" calculus. For 0 < a < 1, we have

dx = (dx)1-a(dx)a. The "fractional" symbol (dxj)a is rea

lated to dxj and can be used instead of "integer" dx' for elaborating a co-vector/differential form of calculus. Following the above system of notation, the exterior fractional differential is

For a N-connection N, we can always construct a class of fractional (co) frames (N-adapted) linearly depending on aN?,

'ei = d - aN"3a, aeb = 3b

= [ae' = (dx')a, aeb = (dyb)° + aNb(dxk )a ].

(A.5) (A.6)

The nontrivial nonholonomy coefficients are computed as

aWb = db aN? and aW« = a^ = aef aN^ - ae, aN? for

d =^\~(2 - a)(xj)a-1 dxj 0dj. j=1

The fractional integration for differential forms on interval L = [ 1x, 2x] is performed following formulaby

Lai[x] 1xdxf(X) = f( 2X) - f( 1X),

when the fractional differential of a function f(x) is

,xdxf(x) = [...], when

[aea, aeß] = aea aeß - aeß aea = aWvaß aeY.

In the above formulae, the values aQ? values are called the coefficients of N-connection curvature. A nonholo-

nomic manifold defined by a structure N is called, in brief, a N-anholonomic fractional manifold.

We introduce a metric structure g = {agaß} on V as a symmetric second rank tensor with coefficients determined locally with respect to a corresponding tensor product of fractional differentials,

g = agYß(u)(duY)a ® (duß)a.

r(a)( 2x - xy

[(dx')a ,xâx" f (x'')] = f(x) - f ( ix ).

The nonholonomic geometry of a fractional tangent bundle depends on the type of chosen fractional derivative chosen. We also emphasize that the above formulasfor-mulae can be generalized for an arbitrary vector bundle E and/or nonholonomic manifold V.

For N-adapted constructions, It Is Important to use the property that any fractional metric g can be represented equ'ivalently as a distinguished metric (d-metric), g = [a9kj, agcb] , when

g = agki (x, y) aek «

+ agcb(x, y) aec

= nk'i'

a '' a c' a b'

e + nc'b' e « e ,

where matrices rik'j' = diag[±1, +1,..., ±1] and ria'b' = diag[±1, ±1,..., ±1], (for the signature of a "prime" spacetime V,), are obtained by frame transforms rjk'j' =

7gkj and ria'b' = eaa, ebb, agab.

A distinguished connection (d-connection) D on V is defined as a linear connection preserving under parallel transport the Whitney sum (A.4). We can associate an N-adapted differential 1-form

There is a unique canonical metric compatible fractional d-connection "D = {a=

b%, aL°bk. "Cjc, °Cabc)}, when aD (ag) = 0, satisfying the conditions aC'jk = 0 and aTabc = 0, but aT)a, aCaji and aCabi are not zero. The N-adapted coefficients are explicitly determined by the coefficients of (A.9),

a\~r _ ar~T a aY

' ß~ ' ßY e '

ni = Lit- —

parametrizing the coefficients (with respect to (A.7) and (A.5)) in the form ar\p = (aL\k. aL«bk, aC>c, aCb. .

The absolute fractional differential ad = ixdx + iydy acts on fractional differential forms in N-adapted form; the value ad := aep aep splits into exterior h- and v-derivatives when

iXdx := (dxi)a a =

J, := (dya)a a ,*da =

The torsion and curvature of a fractional d-connection D = {arTpy} can be defined and computed, respectively, as fractional 2-forms,

aTt — D aeT = ad aeT + arTß A aeß and

t pv ai-T _ aj ar-T ar~Y A ar~T

K, ß — L/ I ß— d I ß — I ß A I y

= aRTßYöaeY A ae5.


There are two another important geometric objects: the fractional Ricci tensor aUic = {aRap — aRra^r} with components

an a ryk a d a ryk

Rii — R iik' Rib — — R ikb'

ID a Db a d a ne /A11\

Rai — R aib, Rab — R abe (AJI)

and the scalar curvature of fractional d-connectlon D,

ar — agTß aRTß = aR + aS,

a R = agij aRij, aS = agabaRab, (A.12)

with agTß being the inverse coefficients to a d-metric (A.9). We can introduce the Einstein tensor a£ns = {aGaß},

a a 1 a a

Gaß := Raß — 2 gaß s R. (A.13)

For various applications, we can considered more special classes of d-connections:

g [ ek gjr + ej gkr — er g^) ,

a ia _ a. /a\/a\ i a^aei a_ a_

Lbk = eb( Nk 1+2 g ( ek gbe

a „ a. a\id a_ a. a\id\

— gde eb Nk — gdb ee Nk ), 1

a ni a „ik a ^ a „

Cje = 2 g gjk'


an a — ' a „ad / a a a ^ a a a ^ \

Cbe = 2 g ' ee gbd + ee ged — ed gbe) .

• The fractional Levi-Civita connection "^V = {TYap} can be defined in standard from but for the fractional Caputo left derivatives acting on the coefficients of a fractional metric (A.7).

On spaces with nontrivial nonholonomic structure, it is

preferrableed to work on V with aD = {ar~yT^} instead of aV (the last onelatter is not adapted to the N-connection splitting (A.4)). The torsion afT (A.10) of aD is uniquely induced nonholonomically by off-diagonal coefficients of the d-metric (A.9).

With respect to N-adapted fractional bases (A.5) and (A.7), the coefficients of the fractional Levi-Civita and canonical d-connection satisfy the distorting relations

ar~Y _ anY

1 rvq — 1

Z aß,


where the N-adapted coefficients of distortion tensor ZYaf are computed

a~7i r\ a~7a a fi a „ a „ab a r\a

Zik = Zik =- Lib gik g - t ilik,

ZhL- —

ZhL- —

ZL-h —

Zhr- -

<-jk — jb yik y 2 jk

ar\e a - a-ji i Xi X^ a - a-ih\ a r-i

2 ujk geb g — 2 j5k— gjk g ) Chb,

I a b ab e e

2(°a °b + ged g )[ Lbk — eb( Nk)\ ,

a j i i h ih j

ujk g eb gj + 2(djôk— gjk g ) Chb,

Znh —

2 jk 1 2 0,

— 1(5ac5db — agebagad) [aLdj — aed(aNe)],

{[aLeaj — aea(aN)] ageb

+ [ aLebj — aeb(a N )] agea}.


[1] S. Anco, J. Phys. A: Math. Gen. 39, 2043 (2006)

[2] S. Anco, S. Vacaru, J. Geom. Phys. 59, 79 (2009)

[3] C. Athorne, J. Phys. A: Math. Gen. 21, 4549 (1998)

[4] D. Baleanu, S. Vacaru, Nonlin. Dyn. 64, 365 (2011)

[5] D. Baleanu, S. Vacaru, Int. J. Phys. 50, 233 (2001)

[6] D. Baleanu, S. Vacaru, arXiv:1007.2866v3 [math-ph]

[7] K.S. Chou, C. Qu, J. Phys. Soc. Japan 70, 1912 (2001)

[8] R. E. Goldstein, D. M. Petrich, Phys. Rev. Lett. 67, 3203 (1991)

[9] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Amer. Math. Soc., Providence, 2001)

[10] J. Langer, R. Perline, Phys. Lett. A 239, 36 (1998)

[11] G.L. Lamb, Jr., J. Math. Phys. 18, 1654 (1977)

[12] G. Mari Beffa, J. Sanders, J.P. Sanders, J. Nonlinear. Sci. 12, 143 (2002)

[13] R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, FTPH no. 59, (Kluwer Academic Publishers, Dordrecht, Boston, London, 1994)

[14] K. Nakayama, H. Segur, M. Wadati, Phys. Rev. Lett. 69, 2606 (1992)

[15] J. Sanders, J. P. Wang, Mosc. Math. J. 3, 1369 (2003)

[16] R.W. Sharpe, Differential Geometry, (SpringerVerlag, New York, 1997)

[17] V. E. Tarasov, Ann. Phys. (NY) 323, 2756 (2008)

[18] S. Vacaru, Int. J. Geom. Meth. Mod. Phys. 5, 473 (2008)

[19] S. Vacaru, Acta Appl. Math. 110, 73 (2010)

[20] S. Vacaru, arXiv:1004.0625v1 [math.DG]

[21] S. Vacaru, arXiv:1004.0628v1 [math-ph]

[22] A.A. Kilbas, H. H. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, (Elsevier, Amsterdam, 2006)

[23] S.G. Samko, A. A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives - Theory and Applications (Gordon and Breach, Linghorne, P.A., 1993)

[24] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego CA, (1999)

[25] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Inc. Connecticut, (2006)

[26] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continoum Mechanics, Springer Verlag, Wien and New York (1997)

[27] R. Metzler, J. Klafter, J. Phys. A: Math. Gen. 37, 1505 (2004)

[28] I.M. Sokolov, J. Klafter, A. Blumen, Physics Today 55, 48 (2002)

[29] F. Mainardi, Chaos Sol. Frac. 7, 1461 (1996)

[30] O.P. Agrawal, J. Math. Anal. Appl. 272, 368 (2002)

[31] M. Klimek, Czech. J. Phys. 52, 1247 (2002)

[32] E.M. Rabei, K.I. Nawafleh, R.S. Hijjawi, S.I. Muslih, D. Baleanu, J. Math. Anal. Appl. 327, 891 (2007)

[33] D. Baleanu, S.I. Muslih, Phys. Scripta 72, 119 (2005)

[34] R.L. Magin, X. Feng, D. Baleanu, Concept. Magn. Re-son. A 34A, 16 (2009)

[35] M.F. Silva, J.A. Tenreiro Machado, A.M. Lopes, Robot-ica 23, 595 (2005)