no ether theorem calculus fractional derivatives

7/29/2019 No Ether Theorem Calculus Fractional Derivatives

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arXiv:math/0701187v1

[math.O

C]6Jan2007

A Formulation of Noethers Theorem for

Fractional Problems of the Calculus of Variations

Gastao S. F. Frederico

[email protected]

Delfim F. M. Torres

[email protected]

Department of Mathematics

University of Aveiro3810-193 Aveiro, Portugal

Abstract

Fractional (or non-integer) differentiation is an important concept bothfrom theoretical and applicational points of view. The study of problemsof the calculus of variations with fractional derivatives is a rather recentsubject, the main result being the fractional necessary optimality con-dition of Euler-Lagrange obtained in 2002. Here we use the notion ofEuler-Lagrange fractional extremal to prove a Noether-type theorem. Forthat we propose a generalization of the classical concept of conservationlaw, introducing an appropriate fractional operator.

Keywords: calculus of variations; fractional derivatives; Noethers theo-

rem.Mathematics Subject Classification 2000: 49K05; 26A33.

1 Introduction

The notion of conservation law first integral of the Euler-Lagrange equations is well-known in Physics. One of the most important conservation laws is theintegral of energy, discovered by Leonhard Euler in 1744: when a LagrangianL(q, q) corresponds to a system of conservative points, then

L(q, q) + 2L(q, q) q constant (1)

holds along all the solutions of the Euler-Lagrange equations (along the ex-tremals of the autonomous variational problem), where 2L(, ) denote the par-tial derivative of function L(, ) with respect to its second argument. Manyother examples appear in modern physics: in classic, quantum, and optical

Research report CM06/I-04, University of Aveiro. Accepted for publication in the Journalof Mathematical Analysis and Applications.

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mechanics; in the theory of relativity; etc. For instance, in classic mechanics,beside the conservation of energy (1), it may occur conservation of momentum

or angular momentum. These conservation laws are very important: they canbe used to reduce the order of the Euler-Lagrange differential equations, thussimplifying the resolution of the problems.

In 1918 Emmy Noether proved a general theorem of the calculus of varia-tions, that permits to obtain, from the existence of variational symmetries, allthe conservation laws that appear in applications. In the last decades, Noethersprinciple has been formulated in various contexts (see [16, 17] and referencestherein). In this work we generalize Noethers theorem for problems havingfractional derivatives.

Fractional differentiation plays nowadays an important role in various seem-ingly diverse and widespread fields of science and engineering: physics (clas-sic and quantum mechanics, thermodynamics, optics, etc), chemistry, biology,economics, geology, astrophysics, probability and statistics, signal and imageprocessing, dynamics of earthquakes, control theory, and so on [3, 6, 8, 9]. Itsorigin goes back more than 300 years, when in 1695 LHopital asked Leibnizthe meaning of d

nydxn

for n = 12 . After that, many famous mathematicians, likeJ. Fourier, N. H. Abel, J. Liouville, B. Riemann, among others, contributed tothe development of the Fractional Calculus [6, 10, 14].

F. Riewe [12, 13] obtained a version of the Euler-Lagrange equations forproblems of the Calculus of Variations with fractional derivatives, that combinesthe conservative and non-conservative cases. More recently, O. Agrawal proveda formulation for variational problems with right and left fractional derivativesin the Riemann-Liouville sense [1]. Then these Euler-Lagrange equations wereused by D. Baleanu and T. Avkar to investigate problems with Lagrangianswhich are linear on the velocities [4]. Here we use the results of [1] to generalize

Noethers theorem for the more general context of the Fractional Calculus ofVariations.The paper is organized in the following way. In Section 2 we recall the no-

tions of right and left Riemann-Liouville fractional derivatives, that are neededfor formulating the fractional problem of the calculus of variations. There aremany different ways to approach classical Noethers theorem. In Section 3 wereview the only proof that we are able to extend, with success, to the fractionalcontext. The method is based on a two-step procedure: it starts with an invari-ance notion of the integral functional under a one-parameter infinitesimal groupof transformations, without changing the time variable; then it proceeds with atime-reparameterization to obtain Noethers theorem in general form. The in-tended fractional Noethers theorem is formulated and proved in Section 4. Twoillustrative examples of application of our main result are given in Section 5.

We finish with Section 6 of conclusions and some open questions.

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2 Riemann-Liouville fractional derivatives

In this section we collect the definitions of right and left Riemann-Liouvillefractional derivatives and their main properties [1, 10, 14].

Definition 2.1 (Riemann-Liouville fractional derivatives). Let f be a con-tinuous and integrable function in the interval [a, b]. For all t [a, b], theleft Riemann-Liouville fractional derivative aD

t f(t), and the right Riemann-

Liouville fractional derivative tDb f(t), of order , are defined in the following

way:

aDt f(t) =

1

(n )

d

dt

n ta

(t )n1f()d , (2)

tDb f(t) =

1

(n ) d

dtn

b

t

( t)n1f()d , (3)

where n N, n 1 < n, and is the Euler gamma function.

Remark 2.2. If is an integer, then from (2) and (3) one obtains the standardderivatives, that is,

aDt f(t) =

d

dt

f(t) , tD

b f(t) =

d

dt

f(t) .

Theorem 2.3. Letf and g be two continuous functions on [a, b]. Then, for allt [a, b], the following properties hold:

1. for p > 0, aDpt (f(t) + g(t)) = aD

pt f(t) + aD

pt g(t);

2. for p q 0, aDp

taD

q

t f(t)

= aDpq

t f(t);

3. for p > 0, aDpt

aD

pt f(t)

= f(t) (fundamental property of the Riemann-

Liouville fractional derivatives);

4. for p > 0,ba

(aDpt f(t)) g(t)dt =

ba

f(t) tDpbg(t)dt (we are assuming that

aDpt f(t) and tD

pbg(t) exist at every point t [a, b] and are continuous).

Remark 2.4. In general, the fractional derivative of a constant is not equal tozero.

Remark 2.5. The fractional derivative of order p > 0 of function (ta), > 1,is given by

aDpt (t a)

=( + 1)

(p

+

+ 1)

(t a)p .

Remark 2.6. In the literature, when one reads Riemann-Liouville fractionalderivative, one usually means the left Riemann-Liouville fractional deriva-tive. In Physics, if t denotes the time-variable, the right Riemann-Liouvillefractional derivative of f(t) is interpreted as a future state of the process f(t).For this reason, the right-derivative is usually neglected in applications, when

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the present state of the process does not depend on the results of the futuredevelopment. From a mathematical point of view, both derivatives appear nat-

urally in the fractional calculus of variations [1].

We refer the reader interested in additional background on fractional theory,to the comprehensive book of Samko et al. [14].

3 Review of the classical Noethers theorem

There exist several ways to prove the classical theorem of Emmy Noether. Inthis section we review one of those proofs [7]. The proof is done in two steps:we begin by proving Noethers theorem without transformation of the time(without transformation of the independent variable); then, using a techniqueof time-reparameterization, we obtain Noethers theorem in its general form.This technique is not so popular while proving Noethers theorem, but it turns

out to be the only one we succeed to extend for the more general context of theFractional Calculus of Variations.

We begin by formulating the fundamental problem of the calculus of varia-tions:

I[q()] =

ba

L (t, q(t), q(t)) dt min (P)

under the boundary conditions q(a) = qa and q(b) = qb, and where q =dqdt

.The Lagrangian L : [a, b] Rn Rn R is assumed to be a C2-function withrespect to all its arguments.

Definition 3.1 (invariance without transforming the time). Functional (P) issaid to be invariant under a -parameter group of infinitesimal transformations

q(t) = q(t) + (t, q) + o() (4)if, and only if, tb

ta

L (t, q(t), q(t)) dt =

tbta

L (t, q(t), q(t)) dt (5)

for any subinterval [ta, tb] [a, b].

Along the work, we denote by iL the partial derivative of L with respectto its i-th argument.

Theorem 3.2 (necessary condition of invariance). If functional (P) is invariantunder transformations (4), then

2L (t,q, q) + 3L (t,q, q) = 0 . (6)

Proof. Equation (5) is equivalent to

L (t,q, q) = L(t, q+ + o(), q+ + o()) . (7)

Differentiating both sides of equation (7) with respect to , then substituting = 0, we obtain equality (6).

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Definition 3.3 (conserved quantity). Quantity C(t, q(t), q(t)) is said to be con-served if, and only if, d

dtC(t, q(t), q(t)) = 0 along all the solutions of the Euler-

Lagrange equationsd

dt3L (t,q, q) = 2L (t,q, q) . (8)

Theorem 3.4 (Noethers theorem without transforming time). If functional(P) is invariant under the one-parameter group of transformations (4), then

C(t,q, q) = 3L (t,q, q) (t, q) (9)

is conserved.

Proof. Using the Euler-Lagrange equations (8) and the necessary condition ofinvariance (6), we obtain:

ddt

(3L (t,q, q) (t, q))

=d

dt3L (t,q, q) (t, q) + 3L (t,q, q) (t, q)

= 2L (t,q, q) (t, q) + 3L (t,q, q) (t, q)

= 0 .

Remark 3.5. In classical mechanics, 3L (t,q, q) is interpreted as the generalizedmomentum.

Definition 3.6 (invariance of(P)). Functional (P) is said to be invariant under

the one-parameter group of infinitesimal transformationst = t + (t, q) + o() ,

q(t) = q(t) + (t, q) + o() ,(10)

if, and only if,

tbta

L (t, q(t), q(t)) dt =

t(tb)t(ta)

L (t, q(t), q(t)) dt


Theorem 3.7 (Noethers theorem). If functional (P) is invariant, in the sense

of Definition 3.6, then

C(t,q, q) = 3L (t,q, q) (t, q) + (L(t,q, q) 3L (t,q, q) q) (t, q) (11)

is conserved.

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Proof. Every non-autonomous problem (P) is equivalent to an autonomous one,considering t as a dependent variable. For that we consider a Lipschitzian one-

to-one transformation[a, b] t [a, b]

such that

I[q()] =

ba

L (t, q(t), q(t)) dt

=

ba

L

t(), q(t()),

dq(t())d

dt()d

dt()

dd

=

ba

L

t(), q(t()),

q

t

t

d

.=ba

L

t(), q(t()), t

, q

d.

= I[t(), q(t())] ,

where t(a) = a, t(b) = b, t

=dt()d

, and q

=dq(t())

d. If functional I[q()] is

invariant in the sense of Definition 3.6, then functional I[t(), q(t())] is invariantin the sense of Definition 3.1. Applying Theorem 3.4, we obtain that

C

t ,q ,t

, q

= 4L + 3L (12)

is a conserved quantity. Since

4L = 3L (t,q, q) ,

3L = 3L (t,q, q) q

t

+ L(t,q, q)

= L(t,q, q) 3L (t,q, q) q ,

(13)

substituting (13) into (12), we arrive to the intended conclusion (11).

4 Main Results

In 2002 [1], a formulation of the Euler-Lagrange equations was given for prob-lems of the calculus of variations with fractional derivatives. In this section weprove a Noethers theorem for the fractional Euler-Lagrange extremals.

The fundamental functional of the fractional calculus of variations is defined

as follows:

I[q()] =

ba

L

t, q(t), aDt q(t), tD

b q(t)

dt min , (Pf)

where the Lagrangian L : [a, b] Rn Rn Rn R is a C2 function withrespect to all its arguments, and 0 < , 1.

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Remark 4.1. In the case = = 1, problem (Pf) is reduced to problem (P):

I[q()] =

b

a

L (t, q(t), q(t)) dt min

withL (t,q, q) = L(t,q, q, q) . (14)

Theorem 4.2 summarizes the main result of [1].

Theorem 4.2 ([1]). If q is a minimizer of problem (Pf), then it satisfies thefractional Euler-Lagrange equations:

2L

t,q, aDt q, tD

b q

+ tDb 3L

t,q, aD

t q, tD

b q

+ aD

t

4

Lt,q, aDt

q, tD

b

q = 0 . (15)Definition 4.3 (cf. Definition 3.1). We say that functional (Pf) is invariantunder the transformation (4) if, and only if,

tbta

L


b q(t)

dt =

tbta

L


b q(t)

dt

(16)


The next theorem establishes a necessary condition of invariance, of extremeimportance for our objectives.

Theorem 4.4 (cf. Theorem 3.2). If functional (Pf) is invariant under trans-formations (4), then

2L

t,q, aDt q, tD

b q

(t, q) + 3L

t,q, aDt q, tD

b q

aDt (t, q)

+ 4L

t,q, aDt q, tD

b q

tDb (t, q) = 0 . (17)

Remark 4.5. In the particular case = = 1, we obtain from (17) the necessarycondition (6) applied to L (14).

Proof. Having in mind that condition (16) is valid for any subinterval [ta, tb] [a, b], we can get rid off the integral signs in (16). Differentiating this conditionwith respect to , substituting = 0, and using the definitions and properties

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of the Riemann-Liouville fractional derivatives given in Section 2, we arrive to

0 = 2L

t,q, aDt q, tDb q

(t, q)

+ 3L

t,q, aDt q, tD

b q

d

d

1

(n )

d

dt

n ta

(t )n1q()d

+

(n )

d

dt

n ta

(t )n1(, q)d

=0

+ 4L

t,q, aDt q, tD

b q

d

d

1

(n )

d

dt

n bt

( t)n1q()d

+

(n )

d

dt

n bt

( t)n1(, q)d

=0

. (18)

Expression (18) is equivalent to (17).

The following definition is useful in order to introduce an appropriate conceptof fractional conserved quantity.

Definition 4.6. Given two functions f and g of class C1 in the interval [a, b],we introduce the following notation:

Dt (f, g) = g tDb f + faD

t g ,

where t [a, b] and R+0 .

Remark 4.7. If = 1, the operator Dt is reduced to

D1t (f, g) = g tD1bf + faD

1t g = f g + fg =

d

dt

(f g) .

In particular, D1t (f, g) = D1t (g, f).

Remark 4.8. In the fractional case ( = 1), functions f and g do not commute:in general Dt (f, g) = D

t (g, f).

Remark 4.9. The linearity of the operators aDt and tD

b imply the linearity of

the operator Dt .

Definition 4.10 (fractional-conserved quantity cf. Definition 3.3). We say

that Cf

t,q, aD

t q, tD

b q

is fractional-conserved if (and only if) it is possible

to write Cf in the form

Cf (t ,q ,dl, dr) =m

i=1

C1i (t ,q ,dl, dr) C2i (t ,q ,dl, dr) (19)

for some m N and some functions C1i and C2i , i = 1, . . . , m, where each pair

C1i and C2i , i = 1, . . . , m, satisfy

Dit

Cj1ii

t,q, aD

t q, tD

b q

, Cj2ii

t,q, aD

t q, tD

b q

= 0 (20)

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with i {, }, j1i = 1 and j

2i = 2 or j

1i = 2 and j

2i = 1, along all the fractional

Euler-Lagrange extremals (i.e. along all the solutions of the fractional Euler-

Lagrange equations (15)).

Remark 4.11. Noether conserved quantities (11) are a sum of products, likethe structure (19) we are assuming in Definition 4.10. For = = 1 (20) isequivalent to the standard definition d

dt[Cf (t, q(t), q(t), q(t))] = 0.

Example 4.12. Let Cf be a fractional-conserved quantity written in the form(19) with m = 1, that is, let Cf = C

11 C

21 be a fractional-conserved quantity for

some given functions C11 and C21 . Condition (20) of Definition 4.10 means one

of four things: or Dt

C11 , C21

= 0, or Dt

C21 , C

11

= 0, or Dt

C11 , C

21

= 0, or

Dt

C21 , C11

= 0.

Remark 4.13. Given a fractional-conserved quantity Cf, the definition of C1i

and C2i , i = 1, . . . , m, is never unique. In particular, one can always choose

C1i to be C

2i and C

2i to be C

1i . Definition 4.10 is imune to the arbitrariness in

defining the Cji , i = 1, . . . , m, j = 1, 2.

Remark 4.14. Due to the simple fact that the same function can be written inseveral different but equivalent ways, to a given fractional-conserved quantityCf it corresponds an integer value m in (19) which is, in general, also not unique(see Example 4.15).

Example 4.15. Let f, g and h be functions satisfying Dt (g, f) = 0, Dt (f, g) =

0, Dt (h, f) = 0, Dt (f, h) = 0 along all the fractional Euler-Lagrange extremals

of a given fractional variational problem. One can provide different proofs to thefact that C = f(g + h) is a fractional-conserved quantity: (i) C is fractional-conserved because we can write C in the form (19) with m = 2, C11 = g, C

21 = f,

C12 = h, and C22 = f, satisfying (20) (D

t C

11 , C

21 = 0 and D

t C

12 , C

22 = 0);

(ii) C is fractional-conserved because we can wrire C in the form (19) withm = 1, C11 = g + h, and C21 = f, satisfying (20) (D

t

C11 , C

21

= Dt (g + h, f) =

Dt (g, f) + Dt (h, f) = 0).

Theorem 4.16 (cf. Theorem 3.4). If the functional (Pf) is invariant underthe transformations (4), in the sense of Definition 4.3, then

Cf

t,q, aD

t q, tD

b q

=

3L

t,q, aDt q, tD

b q

4L

t,q, aDt q, tD

b q

(t, q) (21)

is fractional-conserved.

Remark 4.17. In the particular case = = 1, we obtain from (21) theconserved quantity (9) applied to L (14).

Proof. We use the fractional Euler-Lagrange equations

2L

t,q, aDt q, tD

b q

= tDb 3L

t,q, aD

t q, tD

b q

aDt 4L

t,q, aD

t q, tD

b q

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in (17), obtaining

tDb 3L

t,q, aDt q, tD

b q

(t, q) + 3L

t,q, aDt q, tD

b q

aDt (t, q)

aDt 4L

t,q, aD

t q, tD

b q

(t, q) + 4L

t,q, aDt q, tD

b q

tDb (t, q)

= Dt

3L

t,q, aD

t q, tD

b q

, (t, q)

Dt

(t, q), 4L

t,q, aD

t q, tD

b q

= 0 .

The proof is complete.

Definition 4.18 (invariance of (Pf) cf. Definition 3.6). Functional (Pf) is saidto be invariant under the one-parameter group of infinitesimal transformations(10) if, and only if,

tbta

Lt, q(

t)

,a

Dt

q(

t)

,tD

bq

(t) dt

=t(tb)t(ta)

Lt,q

(t)

,a

Dt

q(

t)

,tD

b q

(t)dt


The next theorem provides the extension of Noethers theorem for FractionalProblems of the Calculus of Variations.

Theorem 4.19 (fractional Noethers theorem). If functional (Pf) is invariant,in the sense of Definition 4.18, then

Cf

t,q, aD

t q, tD

b q

=

3L

t,q, aDt q, tD

b q

4L

t,q, aD

t q, tD

b q

(t, q)

+

L

t,q, aDt q, tDb q

3L

t,q, aDt q, tDb q

aDt q

4L

t,q, aDt q, tD

b q

tDb q

(t, q) (22)

is fractional-conserved (Definition 4.10).

Remark 4.20. If = = 1, the fractional conserved quantity (22) gives (11)applied to L (14).

Proof. Our proof is an extension of the method used in the proof of Theo-rem 3.7. For that we reparameterize the time (the independent variable t) bythe Lipschitzian transformation

[a, b] t f() [a, b]

that satisfies t

= f() = 1 if = 0. Functional (Pf) is reduced, in this way, toan autonomous functional:

I[t(), q(t())] =

ba

L

t(), q(t()), aDt()q(t()), t()D

b

q(t())

t

d, (23)

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where t(a) = a, t(b) = b,

aDt()q(t())

=1

(n )

d

dt()

n f()a

f()

(f() )n1 q

f1()

d

=(t

)

(n )

d

d

n a

(t

)2

( s)n1q(s)ds

= (t

)

a

(t

)2

Dq() ,

and, using the same reasoning,

t()Db

q(t()) = (t

)

D

b

(t

)2

q() .

We have

I[t(), q(t())]

=

ba

L

t(), q(t()), (t

)

a

(t

)2

Dq(), (t

)

D

b

(t

)2

q()

t

d

.=

ba

Lf

t(), q(t()), t

, a(t

)2

Dq(t()), D

b

(t

)2

q(t())

d

=

ba

L


b q(t)

dt

= I[q()] .

By hypothesis, functional (23) is invariant under transformations (10), and itfollows from Theorem 4.16 that

Cf

t(), q(t()), t

, a(t

)2

D q(t()), D

b

(t

)2

q(t())

=

4Lf 5Lf

+

tLf (24)

is a fractional conserved quantity. For = 0,

a

(t

)2

Dq(t()) = aDtq(t) ,

D

b

(t

)2

q(t()) = tDbq(t) ,

and we get4Lf 5Lf = 3L 4L , (25)

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and

tLf = 4Lf

t

(t)(n )

d

d

n

a

(t

)2

( s)n1q(s)ds

t+ 5Lf

t

(t

)

(n )

d

d

n b(t

)2

(s )n1q(s)ds

t

+ L

= 4Lf

(t)1

(n )

d

d

n a

(t

)2

( s)n1q(s)ds

+ 5Lf

(t

)1

(n )

d

d

n b(t

)2

(s )n1q(s)ds

+ L

= 3

L a

Dt

q 4

L tD

bq+ L .

(26)

Substituting (25) and (26) into equation (24), we obtain the fractional-conservedquantity (22).

5 Examples

In order to illustrate our results, we consider in this section two problems from[4, 3], with Lagrangians which are linear functions of the velocities. In bothexamples, we have used [5] to compute the symmetries (10).

Example 5.1. Let us consider the following fractional problem of the calculusof variations (Pf) with n = 3:

I[q()] = I[q1(), q2(), q3()]

=

ba

((aDt q1) q2 (aD

t q2) q1 (q1 q2)q3) dt min .

The problem is invariant under the transformations (10) with

(, 1, 2, 3) = (ct, 0, 0, cq3) ,

where c is an arbitrary constant. We obtain from our fractional Noetherstheorem the following fractional-conserved quantity (22):

Cf (t,q, aDt q) = [(1 )((aD

t q1) q2 (aD

t q2) q1) (q1 q2)q3] t . (27)

We remark that the fractional-conserved quantity (27) depends only on thefractional derivatives ofq1 and q2 if ]0, 1[. For = 1, we obtain the classicalresult:

C(t, q) = (q1 q2) q3t

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is conserved for the problem of the calculus of variations

ba

(q1q2 q2q1 (q1 q2)q3) dt min .

Example 5.2. We consider now a variational functional (Pf) with n = 4:

I[q()] = I[q1(), q2(), q3(), q4()]

=

ba

tD

b

q1

q2 +

tD

b

q3

q4

1

2

q24 2q2q3

dt .

The problem is invariant under (10) with

(, 1, 2, 3, 4) =

2c

3t,cq1, cq2,

c

3q3,

c

3q4

,

where c is an arbitrary constant. We conclude from (22) that

Cf

t,q, tD

b q

=

( 1)

tDb q1

q2 +

tD

b q3

q4

+

1

2

q24 2q2q3

23

t + q1q2 +q3q4

3(28)

is a fractional conserved quantity. In the particular case = 1, the classicalresult follows from (28):

C(t, q) = q1q2 +q3q4

3+

1

3

q24 2q2q3

t

is preserved along all the solutions of the Euler-Lagrange differential equations(8) of the problem

ba

q1q2 + q3q4 +1

2

q24 2q2q3

dt min .

6 Conclusions and Open Questions

The fractional calculus is a mathematical area of a currently strong research,with numerous applications in physics and engineering. The theory of the cal-culus of variations for fractional systems was recently initiated in [1], with theproof of the fractional Euler-Lagrange equations. In this paper we go a stepfurther: we prove a fractional Noethers theorem.

The fractional variational theory is in its childhood so that much remainsto be done. This is particularly true in the area of fractional optimal control,where the results are rare. A fractional Hamiltonian formulation is obtained in

[11], but only for systems with linear velocities. A study of fractional optimalcontrol problems with quadratic functionals can be found in [2]. To the best ofthe authors knowledge, there is no general formulation of a fractional version ofPontryagins Maximum Principle. Then, with a fractional notion of Pontryaginextremal, one can try to use the techniques of [15] to extend the present resultsto the more general context of the fractional optimal control.

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Acknowledgements

GF acknowledges the financial support of IPAD (Portuguese Institute for Devel-opment); DT the support from the Control Theory Group (cotg) of the Centrefor Research in Optimization and Control (http://ceoc.mat.ua.pt). The authorsare grateful to Paulo Gouveia and Ilona Dzenite for several suggestions.

References

[1] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional vari-ational problems, J. Math. Anal. Appl. 272 (2002), no. 1, 368379.

[2] O. P. Agrawal, A general formulation and solution scheme for fractionaloptimal control problems, Nonlinear Dynam. 38 (2004), no. 1-4, 323337.

[3] O. P. Agrawal, J. A. Tenreiro Machado and J. Sabatier, Introduction [Spe-cial issue on fractional derivatives and their applications], Nonlinear Dy-nam. 38 (2004), no. 1-4, 12.

[4] D. Baleanu and Tansel Avkar, Lagrangians with linear velocities withinRiemann-Liouville fractional derivatives, Nuovo Cimento, 119 (2004), 7379.

[5] P. D. F. Gouveia and D. F. M. Torres, Automatic computation of con-servation laws in the calculus of variations and optimal control, Comput.Methods Appl. Math., 5 (2005), no. 4, 387409.

[6] R. Hilfer, Applications of fractional calculus in physics, World Sci. Publish-ing, River Edge, NJ, 2000.

[7] J. Jost and X. Li-Jost, Calculus of variations, Cambridge Univ. Press,Cambridge, 1998.

[8] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applicationsof fractional differential equations, North-Holland Mathematics Studies,Vol. 204, Elsevier, 2006.

[9] M. Klimek, Stationarity-conservation laws for fractional differential equa-tions with variable coefficients, J. Phys. A 35 (2002), no. 31, 66756693.

[10] K. S. Miller and B. Ross, An introduction to the fractional calculus andfractional differential equations, Wiley, New York, 1993.

[11] S. I. Muslih and D. Baleanu, Hamiltonian formulation of systems withlinear velocities within Riemann-Liouville fractional derivatives, J. Math.Anal. Appl. 304 (2005), no. 2, 599606.

[12] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys.Rev. E (3) 53 (1996), no. 2, 18901899.

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[13] F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E (3) 55 (1997),no. 3, part B, 35813592.

[14] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals andderivatives theory and applications, Translated from the 1987 Russianoriginal, Gordon and Breach, Yverdon, 1993.

[15] D. F. M. Torres, On the Noether Theorem for Optimal Control, EuropeanJournal of Control 8 (2002), no. 1, 5663.

[16] D. F. M. Torres, Quasi-invariant optimal control problems, Port. Math.(N.S.) 61 (2004), no. 1, 97114.

[17] D. F. M. Torres, Proper extensions of Noethers symmetry theorem fornonsmooth extremals of the calculus of variations, Commun. Pure Appl.Anal. 3 (2004), no. 3, 491500.

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no ether theorem calculus fractional derivatives

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