Fractional almost Kaehler - Lagrange geometry

7/30/2019 Fractional almost Kaehler - Lagrange geometry

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Nonlinear Dyn (2011) 64:365373

DOI 10.1007/s11071-010-9867-3

O R I G I N A L P A P E R

Fractional almost KhlerLagrange geometry

Dumitru Baleanu Sergiu I. Vacaru

Received: 29 June 2010 / Accepted: 7 October 2010 / Published online: 24 November 2010

Springer Science+Business Media B.V. 2010

Abstract The goal of this paper is to encode equiv-

alently the fractional Lagrange dynamics as a non-

holonomic almost Khler geometry. We use the frac-

tional Caputo derivative generalized for nontrivial

nonlinear connections (N-connections) originally in-

troduced in Finsler geometry, with further develop-

ments in Lagrange and Hamilton geometry. For fun-

damental geometric objects induced canonically by

regular Lagrange functions, we construct compatible

almost symplectic forms and linear connections com-

pletely determined by a prime Lagrange (in par-

ticular, Finsler) generating function. We emphasize

the importance of such constructions for deformation

quantization of fractional Lagrange geometries and

applications in modern physics.

D. Baleanu is on leave of absence from Institute of Space

Sciences, P.O. Box, MG-23, R 76900, MagureleBucharest,

Romania.

D. Baleanu ()

Department of Mathematics and Computer Sciences,

ankaya University, 06530 Ankara, Turkey

e-mail: [email protected]


S.I. Vacaru

Science Department, University Al. I. Cuza Iasi, 54,

Lascar Catargi street, Iasi 700107, Romania



Keywords Fractional derivatives and integrals

Fractional Lagrange mechanics Nonlinear

connections Almost Khler geometry

1 Introduction

The fractional calculus and geometric mechanics are

in areas of many important researches in modern

mathematics and applications. During the last years,

various developments of early approaches to fractionalcalculus [119] were reported in various branches of

science and engineering. The fractional calculus is

gaining importance due to its superior results in de-

scribing the dynamics of complex systems [6, 8, 12].

One important branch of fractional calculus is devoted

to the fractional variational principles and their appli-

cations in physics and control theory [2024]. One of

the open problems in the area of fractional calculus is

to find an appropriate geometrization of the fractional

Lagrange mechanics.

In our paper, we provide a geometrization of

the fractional Lagrange mechanics using methods of

LagrangeFinsler and (almost) Khler geometry fol-

lowing the ideas and constructions from two partner

works [28, 29] which initiated a new direction of frac-

tional differential geometries and related mathematical

relativity/gravity theory and nonholonomic Ricci flow

evolution. Quantization of such theories is an interest-

ing and challenging problem connected to fundamen-

tal issues in modern quantum gravity and fundamental

interaction/evolution theories.
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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366 D. Baleanu, S.I. Vacaru

There were published different papers related to

quantum fractional theories (see, for instance, [25

27]). Nevertheless, there is not a general/systematic

approach to quantization of fractional field/evolution

theories. The aim of our third partner work [30] is

to prove that the strategy used in [3134] allows us

to quantize also a class of fractional geometries fol-lowing Fedosov method [35, 36]. To perform such

a program is necessary to elaborate (in this paper)

a general fractional version of almost Khler geom-

etry and then to apply (in the next paper [30]) the

KarabegovSchlichenmaier deformation quantization

scheme [37]. For the canonical models (with the so

called canonical CartanFinsler distinguished connec-

tions [38]) of Finsler and Lagrange geometries [39],

there is a standard scheme of encoding such spaces

into almost Khler geometries (the result is due to

Matsumoto [40], for Finsler geometry, and then ex-tended to Lagrange spaces [41]; see further develop-

ments in [42, 43]).

In brief, a geometrization of a physical theory in

terms of an almost Khler geometry means a nat-

ural encoding of fundamental physical objects into

terms of basic geometric objects of an almost sym-

plectic geometry (with certain canonical one- and two-

forms and connections, almost complex structure, etc.

induced in a unique form, for instance, by a Lagrange

or Hamilton function). In general, physical theories

are with nonholonomic (equivalently, anholonomic,or nonintegrable) structures which does not allow us

to transform realistic physical models into (complex)

symplect/Khler geometries. Nevertheless, the bulk of

physical theories can be modeled as almost symplectic

geometries for which there were elaborated advanced

mathematical approaches and formalism of geomet-

ric/deformation quantization, spinor and noncommu-

tative generalizations, unification schemes, etc. Our

main idea is to show that certain versions of frac-

tional calculus (based on Caputo derivative), and re-

lated physical models, admit almost Khler type frac-tional geometric encoding which provides an unifica-

tion scheme with integer and noninteger derivatives for

classical and quantum physical theories.

In our papers [2830], we work with the Caputo

fractional derivative. The advantage of using this frac-

tional derivative (resulting in zero values for actions

on constants) is that it allows us to develop a frac-

tional differential geometry which is quite similar to

standard (integer dimensions) models and to use stan-

dard boundary conditions of the calculus of varia-

tions. There is also a series expansion formalism for

the fractional calculus and fractional differential equa-

tions [44] allowing us to compute fractional modifica-

tions of formulas in quite simple forms. This explains

why such constructions are popular in engineering and

physics.1

The plan of this paper is as follows: In Sect. 2,

we outline the necessary formulas on fractional Ca-

puto calculus and related geometry of tangent bundles.

A brief formulation of fractional LagrangeFinsler

geometry is presented in Sect. 3. The main results on

almost Khler models of fractional Lagrange spaces

are provided in Sect. 4. Finally, conclusions are pre-

sented in Sect. 5.

2 Preliminaries: fractional Caputo calculus ontangent bundles

We follow the conventions established in [28, 29].2 In

our approach, we try to elaborate fractional geomet-

ric models with nonholonomic distributions when the

constructions are most closed to integer calculus.

Let f(x) be a derivable function f : [1x, 2x] R,

for R > 0, and x = /x be the usual partial

derivative. By definition, the left RiemannLiouville

(RL) derivative is

1x

x f(x) :=1

(s )

x

s

x1x

(x x)s1f (x) dx,

where is the Eulers gamma function. It is also con-

sidered the left fractional RiemannLiouville deriva-

tive of order , where s 1 < < s, with respect to

coordinate x is

x f(x) := lim1x 1x

x f (x).

1Perhaps, it is not possible to introduce a generally accepted

definition of fractional derivative. For constructions with vari-

ous types of distributions, in different theories of physics, eco-

nomics, biology, etc., when the geometric picture is less impor-

tant, it may be more convenient to use the RiemannLiouville

(RL) fractional derivative.

2All left, up and/or low indices will be considered as labels

for some geometric objects but the right ones as typical abstract

or coordinate indices which may rung values for corresponding

space/time dimensions. Boldface letters will be used for spaces

endowed with nonlinear connection structure.


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Fractional almost KhlerLagrange geometry 367

Similarly, the right RL derivative is introduced by

x

2x f(x) :=1

(s )

x

s

2x

x

(x x)s1f (x) dx ,

when the right fractional Liouville derivative is com-

puted x

f(xk ) := lim2x x

2x f (x).

Integro-differential constructions based only on RL

derivatives seem to be very cumbersome for the pur-

pose to elaborate (fractional) differential geometric

models and has a number of properties which are very

different from similar ones for the integer calculus.

This problem is a consequence of the fact that the frac-

tional RL derivative of a constant C is not zero, but for

instance, 1x

x C = C (x1x)

(1) . We shall use a differenttype of fractional derivative.

The fractional left, respectively, right Caputo deriv-

atives are defined

1x

x f(x) :=1

(s )

x1x

(x x)s1

x

sf (x) dx;

x

2x f(x) :=1

(s )

2x

x

(x x)s1

x

sf (x) dx.

(1)

In the above formulas, we underline the partial deriv-

ative symbol, , in order to distinguish the Caputo dif-

ferential operators from the RL ones with usual . For

a constant C, for instance, 1x

x C = 0.

The fractional absolute differential

d is written inthe form

d :=

dxj

0

j, where

dx j =

dxj (xj)1

(2 ),

where we consider 1xi = 0. For the integer cal-

culus, we use as local coordinate cobases/frames

the differentials dx j = (dx j)=1. If 0 < < 1,

we have dx = (dx)1(dx). The fractional sym-

bol (dx j), related to

dx j, is used instead of usual

integer differentials (dual coordinate bases) dx i

for elaborating a covector/differential form calculus.

We consider the exterior fractional differential

d =

nj=1 (2 )(x

j)1

dx j 0

j and write the exact

fractional differential 0-form as a fractional differ-

ential of the function1x

dx f(x) := (dx)

1x

x f (x

).

The formula for the fractional exterior derivative is

1x

dx :=

dxi

1x

i . (2)

For a fractional differential 1-form with coeffi-

cients {i (xk )}, we may write

=

dx i

i

xk

(3)

when the exterior fractional derivatives of such a frac-

tional 1-form

results a fractional 2-form, 1x

dx (

) =

(dx i ) (dx j) 1x

j i (xk ). The fractional exterior

derivative (2) can be also written in the form

1x

dx :=1

( + 1) 1x

dx

xi 1xi

1x

i

when the fractional 1-form (3) is = 1

(+1)

1x

dx (xi 1x

i ) Fi (x).

The above formulas introduce a well-defined ex-

terior calculus of fractional differential forms on flat

spaces Rn; we can generalize the constructions for a

real manifold M, dim M = n. Any charts of a cov-

ering atlas of M can be endowed with a fractional

derivative-integral structure of Caputo type as we ex-

plained above. Such a space

M (derived for a prime

integer dimensional M of necessary smooth class) is

called a fractional manifold. Similarly, the concept

of tangent bundle can be generalized for fractional

dimensions, using the Caputo fractional derivative.

A tangent bundle T M over a manifold M is canon-

ically defined by its local integer differential struc-

ture i . A fractional version is induced if instead of

i we consider the left Caputo derivatives 1xi

i of

type (1), for every local coordinate xi on M. A frac-

tional tangent bundle

T M for (0, 1) (the symbol

T is underlined in order to emphasize that we shall as-

sociate the approach to a fractional Caputo derivative).

For simplicity, we shall write both for integer and frac-

tional tangent bundles the local coordinates in the form

u = (xj, yj).


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Any fractional frame basise = e

(u )

on

T M

is connected via a vielbein transform e

(u ) with a

fractional local coordinate basis

=

j = 1xj

j ,

b = 1yb

b

, (4)

for j = 1, 2, . . . , n and b = n + 1, n + 2, . . . ,

n + n. The corresponding fractional cobases aree

=

e

(u )

du, where the fractional local coordinate

cobasis is written

du

=

dxi

,

dya

, (5)

with h- and v-components, correspondingly, (dx i)

and (dya), being of type (3).

3 Fractional LagrangeFinsler geometry

A Lagrange space [45] Ln = (M,L), of integer di-

mension n, is defined by a Lagrange fundamental

function L(x,y), i.e., a regular real function L :

T M R, for which the Hessian

Lgij = (1/2)2L/yi y j

is not degenerate.3

We can consider that a Lagrange space Ln is aFinsler space Fn if and only if its fundamental func-

tion L is positive and two homogeneous with respect

to variables yi , i.e., L = F2. For simplicity, we shall

work with Lagrange spaces and their fractional gener-

alizations, considering the Finsler ones to consist of a

more particular, homogeneous, subclass.

Definition 3.1 A (target) fractional Lagrange space

Ln = (

M,

L) of fractional dimension (0, 1), for

a regular real function

L :

T M R, when the frac-tional Hessian is

Lgij =

1

4

i

j +

j

i

L = 0. (6)

3The construction was used for elaborating a model of geomet-

ric mechanics following methods of Finsler geometry in a num-

ber of works on LagrangeFinsler geometry and generalizations

[39]; see our developments for modern gravity and noncommu-

tative spaces [46, 47].

In our further constructions, we shall use the coef-

ficients L

gij being inverse to Lgij (6).

4 Any

Ln can

be associated to a prime integer Lagrange space Ln.

The concept of nonlinear connection (N-connec-

tion) on

Ln can be introduced similarly to that on

nonholonomic fractional manifold [28, 29] consider-ing the fractional tangent bundle

T M.

Definition 3.2 A N-connection

N on

T M is defined

by a nonholonomic distribution (Whitney sum) with

conventional h- and v-subspaces, h

T M and v

T M,

when

T

T M = h

T M v

T M. (7)

Locally, a fractional N-connection is defined by a

set of coefficients,N = { Nai }, when

N = Nai (u)

dx i

a , (8)

see local bases (4) and (5).

By an explicit construction, we prove the following

proposition.

Proposition 3.1 There is a class of N-adapted frac-

tional (co) frames linearly depending on Nai ,

e = ej = j Naj a , eb = b, (9)

e =

ej =

dx j

, eb =

dyb

+ Nbk

dx k

.

(10)

The above bases are nonholonomical (equivalently,

nonintegrable/anholonomic) and characterized by the

property that e,

e

= ee

ee =

W

e,

where the nontrivial nonholonomy coefficients W

are computed Waib =

bNai and

Waij = aj i =

ei Naj

ejNai ; the values

aj i define the coeffi-

cients of the N-connection curvature.

4We shall put a left label L to certain geometric objects if it is

necessary to emphasize that they are induced by Lagrange gen-

erating function. Nevertheless, such labels will be omitted (in

order to simplify the notations) if that will not result in ambigu-

ities.


5/9


Let us consider values yk ( ) = dxk()/d, for

x() parameterizing smooth curves on a manifold M

with [0, 1]. The fractional analogs of such configu-

rations are determined by changing d/d into the frac-

tional Caputo derivative

= 1

whenyk( ) =

xk(). For simplicity, we shall omit the label for

y

T M if that will not result in ambiguities and/or we

shall do not associate to it an explicit fractional deriv-

ative along a curve.

By straightforward computations, following the

same schemes as in [39, 46, 47] but with fractional

derivatives and integrals, we prove the following the-

orem.

Theorem 3.1 Any

L defines the fundamental geo-

metric objects determining canonically a nonholo-

nomic fractional RiemannCartan geometry on T M

being satisfied the properties:

1. The fractional EulerLagrange equations

1yi

i

L

1x

i

i

L = 0

are equivalent to the fractional nonlinear geo-

desic (equivalently, semi-spray) equations

2

xk + 2

Gkx,y = 0,

where

Gk =1

4L

gkj

yj1y

j

j

1xi

i

L

1x

i

i

L

defines the canonical N-connection

LN

aj = 1yj

j

Gk

x, y

. (11)

2. There is a canonical (Sasaki type) metric structure,

Lg = Lgkj(x,y)

ek ej + Lgcb (x,y)Le

c Leb,

(12)

where the preferred frame structure (defined lin-

early by LNaj ) is

Le = (

Lei , ea ).

5

5A (fractional) general metric structureg = { g } is

defined on

T M by a symmetric second rank tensor

3. There is a canonical metrical distinguished con-

nection

c D =

h c D, v

c D

=

c

=

Lij k, Cij c,

(in brief, d-connection), which is a linear connec-

tion preserving under parallelism the splitting (7)

and metric compatible, i.e., c D (Lg) = 0, when

c

ij =

c

ij

Le

=Lij kek + Cij c Lec,forLij k =Labk ,Cij c = Cabc in c ab = c ab Le =Labk ek + Cabc Lec, andLij k = 12 Lgir Lek Lgj r + Lej Lgkr

LerLgj k

,

Cabc = 12 Lgadec Lgbd + ec Lgcd ed

Lgbc

(13)

are just the generalized Christoffel indices.6

g = g (u)(du

) (du ) . For N-adapted construc-

tions, see the details in [2830, 46, 47]. It is im-

portant to use the property that any fractional metricg can be represented equivalently as a distinguished metric

(d-metric),g = [gkj,

gcb], when

g = gkj(x,y) ek ej + gcb(x,y) ec eb

= kjek

ej

+ cbec

eb

,

where matrices kj = diag[1, 1, . . . , 1] and a b =

diag[1, 1, . . . , 1] (reflecting signature of prime space

T M) are obtained by frame transforms k j = ekk

ej

jgkj and

a b = eaa

ebb

gab . For fractional computations, it is conve-

nient to work with constants kj and a b because the Ca-

puto derivatives of constants are zero. This allows us to keep

the same tensor rules as for the integer dimensions even the

rules for taking local derivatives became more sophisticate be-

cause of N-coefficients Nai (u) and additional vierbein trans-

forms ek

k (u) and e

a

a (u). Such coefficients mix fractional deriv-

atives

a computed as a local integration (1). If we work with

RL fractional derivatives, the computation become very sophis-

ticate with nonlinear mixing of integration, partial derivatives,

etc. For some purposes, just RL may be very important. Our

ideas, for such cases, is to derive a geometric model with the

Caputo factional derivatives, and after that we can redefine the

nonholonomic frames/distributions in a form to extract certain

constructions with the RL derivative.

6For integer dimensions, we contract horizontal and vertical

indices following the rule: i = 1 is a = n + 1; i = 2 is a = n + 2;

. . . i = n is a = n + n.


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Finally, in this section, we note that:

Remark 3.1 We note that c D is with nonholonomi-

cally induced torsion structure defined by 2-forms

LT

i =

Cij c

ei Lec,

LT

a = 12

Laij

ei ej (14)

+

ebLN

ai

Labi ei Lebcomputed from the fractional version of Cartans

structure equations

dei ek c ik =

LT

i ,

dLea Le

b c ab =

LT

a , (15)

dc ij

c

kj

c

ik =

LR

ij

in which the curvature 2-form is denoted LRij.

In general, for any d-connection on

T M, we

can compute respectively the N-adapted coefficients

of torsion T = { } and curvatureR

=

{R } as it is explained for general fractional non-

holonomic manifolds in [28, 29].

4 Almost Khler models of fractional Lagrange

spaces

The goals of this section is to prove that the canoni-

cal N-connection LN (11) induces an almost Khler

structure defined canonically by a fractional regular

L(x, y).

Definition 4.1 A fractional nonholonomic almost

complex structure is defined as a linear operator

J act-

ing on the vectors on

T M following formulas

J

Lei

= ei andJ ei= Lei ,

where the superposition

J

J = I, for I being the

unity matrix.

The fractional operator

J reduces to a complex

structure J if and only if the distribution (7) is inte-

grable and the dimensions are taken to be some integer

ones.

Lemma 4.1 A fractional

L induces a canonical

1-form L =12 (1yi

i

L) ei and a metric Lg (12)

induces a canonical 2-form

L= L

gij

x, y

Le

i ej (16)

associated toJ following formulas L(X,Y)

Lg(

JX, Y) for any vectors X andY on

T M.

Using a fractional N-adapted form calculus,

dL =1

6

(ijk)

Lgis

L

sj k

ei ej ek

+1

2

L

gijk L

gik j

Le

i ej ek

+

1

2 ekL gij ei L gkj Lek Le

i ej,

where (ijk) means symmetrization of indices and

Lgijk :=

Lek L

gij

LBsik Lgsj

LBsj k Lgis ,

for LBsik =ei

LN

sk , we prove the results.

Proposition 4.1

1. A regular L defines on T M an almost Hermitian(symplectic) structure L for which d

L =

L;

2. The canonical N-connection LNaj (11) and its cur-

vature have the propertiesijk

Lgl(i

L

lj k) = 0, L

gijk L

gikj = 0,

ek

Lgij

ej

Lgik

= 0.

Conclusion 4.1 The above properties state an al-

most Hermitian model of fractional Lagrange space

Ln = ( M , L) as a fractional almost Khler manifold

with dL= 0. The triadK2n = (

T M, L

g,

J), whereT M :=

T M\{0} for {0} denoting the null-sections un-

der

M, defines a fractional, and nonholonomic, almost

Khler space.

Our next purpose is to construct a canonical (i.e.,

uniquely determined by

L) almost Khler distin-


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guished connection (d-connection)

D being com-

patible both with the almost Khler (L,

J) and N-

connection structures LN when

DX Lg = 0 and

DX

J = 0, (17)

for any vector X = Xi Lei + Xa ea . By a straightfor-

ward computation, we prove (see similar integer de-

tails in [31, 39, 46]) the following theorem.

Theorem 4.1 (Main result) The fractional canonical

metrical d-connection c D, possessing N-adapted co-

efficients (Labk ,Cabc ) (13), defines a (unique) canon-ical fractional almost Khler d-connection c

D = c D

satisfying the conditions (17).

Finally, we note the following remark.

Remark 4.1 There are two important particular cases:

1. If

L = (

F )2, for a Finsler space of an integer di-

mension, we get a canonical almost Khler model

of Finsler space [40], when c

D = c D transforms in

the socalled CartanFinsler connection [38] (for

integer dimensional Lagrange spaces, a similar re-

sult was proven in [41], see details in [39]).

2. We get a Khlerian model of a Lagrange, or Finsler,

space if the respective almost complex structure Jis integrable. This property holds also for respec-

tive spaces of fractional dimension.

Finally, we emphasize that the geometric back-

ground for fractional LagrangeFinsler spaces pro-

vided in this section has been used in our partner

work [30] for deformation quantization of such the-

ories. This is an explicit proof that fractional physical

models can be quantized in general form that it is pos-

sible to elaborate new types/models/theories of quan-

tum fractional calculus and geometries stating a newunified formalism to fractional classical and quantum

geometries and physics and propose various applica-

tions in mechanics and engineering.

5 Conclusions

The fractional Caputo derivative allows us to for-

mulate a self-consistent nonholonomic geometric ap-

proach to fractional calculus [28, 29] when various

important mathematical and physical problems for

spaces and processes of noninteger dimension can be

investigated using various methods originally elabo-

rated in modern LagrangeHamiltonFinsler geome-

try and generalizations. In this paper, we provided

a geometrization of regular fractional Lagrange me-chanics in a form similar to that in Finsler geome-

try but, in our case, extended to noninteger dimen-

sions. This way, applications of fractional calculus be-

come a direction of non-Riemannian geometry with

nonholonomic distributions and generalized connec-

tions.

Nonholonomic fractional mechanical interactions

are described by a sophisticate variational calculus

and derived cumbersome systems of nonlinear equa-

tions and less defined types of symmetries. To quan-

tize such mechanical systems is a very difficult prob-lem related to a number of unsolved fundamental

problems of nonlinear functional analysis. Neverthe-

less, for certain approaches based on fractional Caputo

derivative, or those admitting nonholonomic deforma-

tions/transforms to such fractional configurations, it is

possible to provide a geometrization of fractional La-

grange mechanics. In this case, the problem of quanti-

zation can be approached following powerful methods

of geometric and/or deformation quantization.

Having a similarity between fractional Lagrange

geometry and certain generalized almost Khler

Finsler models, for which the Fedosov quantization

was performed in our previous works [3133], our

main goal was to prove Theorem 4.1. As a result, we

obtained a suitable geometric background for Fedosov

quantization of fractional LagrangeFinsler spaces

which will be performed in our partner work [30].

Finally, we conclude that the examples of frac-

tional almost Khler spaces considered in this works

can be redefined for another types of nonholonomic

distributions and applied, for instance, as a geometric

scheme for deformation/A-brane quantization of frac-

tional Hamilton and Einstein spaces and various gen-

eralizations, similarly to our former constructions per-

formed for integer dimensions in [34, 42, 43]. Such

developments consist a purpose of our further re-

search.

Acknowledgement S.V. is grateful to ankaya University for

support of his research on fractional calculus, geometry and ap-

plications.


8/9


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Fractional almost Kaehler - Lagrange geometry

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