Table Of ContentVariational problems with fractional derivatives: Invariance
conditions and N¨other’s theorem
1
1 Teodor M. Atanackovi´c ∗
0
2 Sanja Konjik †
Stevan Pilipovi´c ‡
n
a Srboljub Simi´c §
J
5
1
Abstract
]
A Avariational principlefor Lagrangian densities containingderivativesof real orderis for-
F mulated and the invariance of this principle is studied in two characteristic cases. Necessary
. and sufficient conditions for an infinitesimal transformation group (basic N¨other’s identity)
h
are obtained. These conditions extend the classical results, valid for integer order deriva-
t
a tives. Ageneralization of N¨other’stheorem leading toconservation laws for fractional Euler-
m Lagrangianequationisobtainedaswell. Resultsareillustratedbyseveralconcreteexamples.
[ Finally, an approximation of a fractional Euler-Lagrangian equation by a system of inte-
ger order equations is used for the formulation of an approximated invariance condition and
1
corresponding conservation laws.
v
2 Mathematics Subject Classification (2000): Primary: 49K05; secondary: 26A33
6
PACS numbers: 02.30 Xx,45.10 Hj
9
2 Keywords: variational problem, Riemann-Liouville fractional derivatives, variational sym-
. metry,infinitesimal criterion, N¨other’s theorem, conservation laws, approximations
1
0
1
1 Introduction
1
:
v
There are two distinct approachesin the formulationof fractionaldifferential equations ofmodels
i
X in various branches of science, e.g. physics. In the first one, differential equations containing
r integer order derivatives are modified by replacing one or more of them with fractional ones
a
(derivatives of real order). In the second approach, which we will follow in the sequel, one starts
with a variationalformulation of a physical process in which one modifies the Lagrangiandensity
by replacing integer order derivatives with fractional ones. Then the action integral in the sense
of Hamilton is minimized and the governing equation of a physical process is obtained. Hence,
one is faced with the following problem: find minima (or maxima) of a functional
B
L[u]= L(t,u(t), Dαu)dt, 0<α<1, (1)
a t
ZA
where Dαu is the left Riemann-Liouville fractional derivative, under certain assumptions on a
a t
Lagrangian L, as well as on functions u among which minimizers are sought. After this, the
∗FacultyofTechnicalSciences,InstituteofMechanics,UniversityofNoviSad,TrgDositejaObradovi´ca6,21000
Novi Sad, Serbia. Electronic mail: [email protected]. Work supported byProjects 144016 and 144019 of the
SerbianMinistryofScienceandSTART-projectY-237oftheAustrianScienceFund.
†Faculty of Agriculture, Department of Agricultural Engineering, University of Novi Sad, Trg Dositeja
Obradovi´ca8,21000NoviSad,Serbia. Electronicmail: [email protected].
‡Faculty of Sciences, Department of Mathematics, University of Novi Sad, Trg Dositeja Obradovi´ca 4, 21000
NoviSad,Serbia. Electronicmail: [email protected]
§FacultyofTechnicalSciences,InstituteofMechanics,UniversityofNoviSad,TrgDositejaObradovi´ca6,21000
NoviSad,Serbia. Electronicmail: [email protected]
1
Euler-Lagrange equations are formed for the modified Lagrangian, which leads to equations that
intrinsically characterize a physical process. This approach has a more sound physical basis (see
e.g. [8, 9, 10, 31, 34]). It provides the possibility of formulating conservation laws via No¨ther’s
theorem. Conservationlawsareveryimportantinparticlereactionphysicsforexample,wherethey
are often postulated as conservation principles [37]. The central point is the fact that equations
of a model, i.e., Euler-Lagrange equations, give minimizers of a functional. As a rule, fractional
differential equations obtained throughEuler-Lagrangeequations containleft and rightfractional
derivatives, which makes this approachmore delicate.
In the last years fractional calculus has become popular as a useful tool for solving problems
from various fields [21, 41] (see also [6, 17, 18, 19, 20, 22, 23, 25, 26, 35, 38]).
The study of fractional variational problems also has a long history. F. Riewe [32, 33] inves-
tigated nonconservative Lagrangian and Hamiltonian mechanics and for those cases formulated
a version of the Euler-Lagrange equations. O. P. Agrawal continued the study of the fractional
Euler-Lagrangeequations [1, 2, 3], for generalfractionalvariationalproblems involving Riemann-
Liouville, Caputo and Riesz fractional derivatives. G. S. F. Frederico and D. F. M. Torres [13]
studied invariance properties of fractional variational problems with No¨ther type theorems by in-
troducinganewconceptoffractionalconservedquantitywhichisnotconstantintime,sotheterm
conserved quantity is not clear. Conservation laws and Hamiltonian type equations for the frac-
tionalactionprinciples havealsobeen derivedin[39]. Thereare nowadaysnumerousapplications
of fractional variational calculus, see e.g. [5, 7, 8, 9, 10, 11, 12, 16, 24, 25, 27, 28, 30].
Wecansummarizethenoveltiesofourpaperasfollows. First,wederivedinfinitesimalcriterion
for a local one-parameter group of transformations to be a variational symmetry group for the
fractional variational problem (1). In previous work [13, 14, 15] this was done only in a special
case. Moreover,weseparatelyconsidercaseswhenthelowerboundaintheleftRiemann-Liouville
fractional derivative is not transformed (Subsection 2.1), and when a is transformed (Subsection
2.2). Both cases have their physical interpretation. In the case when fractional derivatives model
memory effects the lowerbound a in the definition of derivativeshould not be varied. However,if
fractionalderivativesmodel nonlocalinteractions(e.g. innonlocalelasticity)the lowerbound has
to be varied. Second, we give a No¨ther type theorem in terms of conservation laws as it is done
in the classical theory. This approach preserves the essential property of conserved quantities
to be constant in time. Third novelty is the approximation procedure which is given for the
Euler-Lagrangeequations and infinitesimal criterion, therefore also for No¨ther’s theorem. By the
use of additional assumptions we obtain that appropriate sequences of classical Euler-Lagrange
equations,infinitesimalcriteriaandconservationlawsconvergeinthesenseofanappropriatespace
ofgeneralizedfunctionstothecorrespondingEuler-Lagrangeequations,infinitesimalcriterionand
conservationlawsofthefractionalvariationalproblem(1). Therearealsoanumberofillustrative
examples which complete the theory.
The paper is organized as follows. To the end of this introductory section we provide basic
notions and definitions from the calculus with derivatives and integrals of any real order, and
recall the Euler-Lagrange equations of (1) for (A,B)⊆(a,b) (so far only the case (A,B)=(a,b)
was treated, cf. [1, 2]). Section 2 is devoted to the study of variational symmetry groups and
invariance conditions. In Section 3 we derive a version of No¨ther’s theorem which establishes a
relation between fractional variational symmetries and conservation laws for the Euler-Lagrange
equation, which have the important property of being constant in time. We use in Section 4 the
approximationoftheRiemann-Liouvillefractionalderivativebyafinitesumofclassicalderivatives.
In this way and by introducing a suitable space of analytic functions we obtain approximated
Euler-Lagrangeequations, approximated local Lie group actions with corresponding infinitesimal
criteria and No¨ther theorems.
1.1 Preliminaries
In the sequel we briefly recall some basic facts from the fractional calculus.
Foru∈L1([a,b]),α>0andt∈[a,b]wedefinetheleft,resp.rightRiemann-Liouvillefractional
2
integral of order α, as
1 t 1 b
Iαu= (t−θ)α−1u(θ)dθ, resp. Iαu= (θ−t)α−1u(θ)dθ.
a t Γ(α) t b Γ(α)
Za Zt
Left, resp. right Riemann-Liouville fractional derivative of order α, 0 ≤α <1, is well defined for
an absolutely continuous function u in [a,b], i.e. u∈AC([a,b]), and t∈[a,b] as
d 1 d t u(θ)
Dαu= I1−αu= dθ, (2)
a t dta t Γ(1−α)dt (t−θ)α
Za
resp.
d 1 d b u(θ)
Dαu= − I1−αu= − dθ.
t b dt t b Γ(1−α) dt (θ−t)α
(cid:16) (cid:17) (cid:16) (cid:17)Zt
If f,g ∈AC([a,b]) and 0≤α<1 the following fractional integration by parts formula holds:
b b
f(t) Dαgdt= g(t) Dαfdt. (3)
a t t b
Za Za
For the left Riemann-Liouville fractional derivative (2) of order α (0≤α<1) we have:
1 t u˙(θ) 1 u(a)
Dαu= dθ+ , t∈[a,b], (4)
a t Γ(1−α) (t−θ)α Γ(1−α)(t−a)α
Za
The integral on the right hand side is called the left Caputo fractional derivative of order α and
is denoted by cDαu. Similarly, the right Caputo fractional derivative cDαu is defined.
a t t b
Itfollowsfrom(4)thattheleftRiemann-LiouvilleequalstotheleftCaputofractionalderivative
in the case u(a)=0 (the analogueholds for the rightderivatives under the assumption u(b)=0).
The same condition, i.e. u(a)=0, provides that d Dαu= Dα du.
dta t a t dt
Weintroducethenotationwhichwillbeusedthroughoutthispaper: derivativesofLagrangian
∂L
L = L(t,u(t), Dαu) with respect to the first, second and third variable will be denoted by ,
a t ∂t
∂L ∂L
and , or by ∂ L, ∂ L and ∂ L respectively.
∂u ∂ Dαu 1 2 3
a t
1.2 Euler-Lagrange equations
As stated in Introduction, we solve a fractional variational problem
B
L[u]= L(t,u(t), Dαu)dt→min, 0<α<1. (5)
a t
ZA
(A,B) is a subinterval of (a,b), and LagrangianL is a function in (a,b)×R×R such that
L∈C1((a,b)×R×R)
and (6)
t7→∂ L(t,u(t), Dαu)∈AC([a,b]), for every u∈AC([a,b])
3 a t
Solutions to (5) are sought among all absolutely continuous functions in [a,b],which in addition
satisfy condition u(a)=a , for a fixed a ∈R.
0 0
OnecanconsidermoregeneralproblemswithLagrangiansdependingalsoontherightRiemann-
Liouville fractional derivative. Our results can be easily formulated in that case, and because of
simplicity we shall consider only Lagrangians of the form (5). Moreover, in (5) one can consider
the case α>1, but this also does not give any essentialnovelty and because of that it is skipped.
We will recallresults about the Euler-Lagrangeequations obtained in [1, 2, 4]. As we said,we
consider the fractional variational problem defined by (5).
3
Fractional Euler-Lagrange equations, which provide a necessary condition for extremals of a
fractional variational problem, have been recently studied in [1, 2, 4]. Let A = a and B = b.
Euler-Lagrangeequations are obtained in [1, 2]:
∂L ∂L
+ Dα =0. (7)
∂u t b ∂ Dαu
a t
(cid:16) (cid:17)
In terms of the Caputo fractional derivative, the Euler-Lagrange equation (7) is given in [4] and
reads:
∂L ∂L ∂L 1 1
+cDα + =0, (8)
∂u t b ∂ Dαu ∂ Dαu Γ(1−α)(b−t)α
a t a t t=b
(cid:16) (cid:17) (cid:12)
Euler-Lagrangeequations for (5) are derived in(cid:12)[4]:
(cid:12)
∂L ∂L ∂L 1 1
+cDα + = 0, t∈(A,B) (9)
∂u t B ∂ Dαu ∂ Dαu Γ(1−α)(B−t)α
a t a t t=B
(cid:16) (cid:17) (cid:12)
∂L(cid:12) ∂L
tDBα ∂ D(cid:12)αu −tDAα ∂ Dαu = 0, t∈(a,A). (10)
a t a t
(cid:16) (cid:17) (cid:16) (cid:17)
Equation (9) is equivalent to
∂L ∂L
+ Dα =0, t∈(A,B). (11)
∂u t B ∂ Dαu
a t
(cid:16) (cid:17)
Remark 1. If we replace the Lagrangian in (5) by L(t,u(t), Dαu,u˙(t)) then Euler-Lagrange
a t
d ∂L
equations will be also equipped with a ‘classical’ term − on the left hand side. It can be
dt∂u˙
shownthat in this caseinfinitesimal criterionand conservationlaw canbe obtainedby combining
the results of the present analysis (see Theorems 5, 11 and 15) and the classical theory (see e.g.
[29]).
2 Infinitesimal invariance
Let G be a local one-parameter group of transformations acting on a space of independent and
dependent variables as follows: (t¯,u¯) = g ·(t,u) = (Ξ (t,u),Ψ (t,u)), for smooth functions Ξ
η η η η
and Ψ , and g ∈G. Let
η η
∂ ∂
v=τ(t,u) +ξ(t,u)
∂t ∂u
be the infinitesimal generator of G. Then we also have
t¯ = t+ητ(t,u)+o(η)
(12)
u¯ = u+ηξ(t,u)+o(η).
Weintroducethefollowingnotation(cf., e.g.,[40]): ∆t= d | (t¯−t)and∆u= d | (u¯(t¯)−
dη η=0 dη η=0
u(t)). More precisely, the notations ∆t and ∆u denote limη→0 t¯(ηη)−t and limη→0 u¯(t¯,η)η−u(t),
respectively. It follows from (12) that ∆t = τ, and writing the Taylor expansion of u¯(t¯) =
u¯(t+ητ(t,u)+o(η)) at η =0 yields that ∆u=ξ.
On the other hand, if we write Taylor expansion of u¯(t¯) at t¯=t we obtain
d
∆u= (u¯(t)−u(t))+u˙∆t,
dη
η=0
(cid:12)
(cid:12)
or, if we introduce the Lagrangianvariati(cid:12)on
d
δu:= (u¯(t)−u(t)) (13)
dη
η=0
(cid:12)
(cid:12)
(cid:12)
4
we obtain
∆u=δu+u˙∆t.
Thus, we have
δu=ξ−τu˙.
Note that δt=0.
In the same way we can define ∆F and δF of an arbitrary absolutely continuous function
F =F(t,u(t),u˙(t)):
d ∂F ∂F ∂F
∆F = F(t¯,u¯(t¯),u¯˙(t¯))−F(t,u(t),u˙(t)) = ∆t+ ∆u+ ∆u˙
dη ∂t ∂u ∂u˙
η=0
(cid:12) (cid:16) (cid:17)
(cid:12) d ∂F ∂F
δF =(cid:12) F(t,u¯(t),u¯˙(t))−F(t,u(t),u˙(t)) = δu+ δu˙,
dη ∂u ∂u˙
η=0
(cid:12) (cid:16) (cid:17)
and (cid:12)
(cid:12) ∆F =δF +F˙∆t.
Itwillbeapparentintheforthcomingsectionsthat,ifwewanttofindaninfinitesimalcriterion,
we need to know ∆ Dαu and ∆L. Therefore, we have to transform the left Riemann-Liouville
a t
fractionalderivative ofu under the actionofa localone-parametergroupoftransformations(12).
Therearetwodifferentcaseswhichwillbeconsideredseparately. Inthefirstone,thelowerbound
a in Dαu is not transformed, while in the second case a is transformed in the same way as the
a t
independent variable t. Physically, the first case is important when Dαu represents memory
a t
effects, and the second one is important when action on a distance is involved.
2.1 The case when a in Dαu is not transformed
a t
In this section we consider a local group of transformations G which transforms t ∈ (A,B) into
t¯∈(A¯,B¯) sothat both intervalsremainsubintervalsof (a,b),but the actionofG has no effecton
the lower bound a in Dαu, i.e., τ(a,u(a))=0. So, suppose that G acts on t,u and Dαu in the
a t a t
following way:
g ·(t,u, Dαu):=(t¯,u¯, Dαu¯),
η a t a t¯
where t¯and u¯ are defined by (12). In this case we have:
Lemma 2. Let u∈AC([a,b]) and let G be a local one-parameter group of transformations given
by (12). Then
d
∆ Dαu= Dαδu+ Dαu·τ(t,u(t)),
a t a t dta t
where
d
δ Dαu= Dαu¯− Dαu = Dαδu.
a t dη a t a t a t
η=0
(cid:12) (cid:16) (cid:17)
Proof. To prove that δ Dαu = Dαδ(cid:12)u it is enough to apply the definition of the Lagrangian
a t a t (cid:12)
variation (13). Also by definition we have
d
∆ Dαu= Dαu¯− Dαu .
a t dη a t¯ a t
η=0
(cid:12) (cid:16) (cid:17)
Thus, (cid:12)
(cid:12)
1 d d t¯ u¯(θ) d t u¯(θ) d t u(θ)
∆ Dαu = dθ± dθ− dθ
a t Γ(1−α)dη dt¯ (t¯−θ)α dt (t−θ)α dt (t−θ)α
(cid:12)η=0" Za Za Za #
d(cid:12)
= Dαδu+ (cid:12) Dαu¯− Dαu¯
a t dη a t¯ a t
η=0
d (cid:12)(cid:12)dt¯ (cid:16) (cid:17)
= Dαδu+ (cid:12) Dαu¯− Dαu¯
a t dt¯dη a t¯ a t
η=0
(cid:12) (cid:16) (cid:17)
= aDtαδu+ ddtaD(cid:12)(cid:12)tαu·τ(t,u(t)).
5
2
Lemma 3. Let L[u] be a functional of the form L[u] = BL(t,u(t), Dαu)dt, where u is an
A a t
absolutely continuous function in [a,b], (A,B) ⊆ (a,b) and L satisfies (6). Let G be a local
R
one-parameter group of transformations given by (12). Then
B
∆L=δL+(L∆t) ,
A
(cid:12)
(cid:12)
where δL= bδLdt. (cid:12)
a
Proof. AgaRin it is clear that δL= bδLdt. For ∆L we have
a
d B¯ R B
∆L = L(t¯,u¯(t¯), Dαu¯)dt¯− L(t,u(t), Dαu)dt
dη(cid:12)η=0(cid:18) ZA¯ a t¯ ZA a t (cid:19)
d (cid:12) B B
= (cid:12) L(t¯,u¯(t¯), Dαu¯)(1+ητ˙(t,u(t)))dt− L(t,u(t), Dαu)dt
dη a t¯ a t
(cid:12)η=0(cid:18) ZA ZA (cid:19)
d (cid:12) B B
= (cid:12) L(t¯,u¯(t¯), Dαu¯)dt− L(t,u(t), Dαu)dt
dη a t¯ a t
(cid:12)η=0(cid:18) ZA ZA (cid:19)
(cid:12) B
+(cid:12) L(t,u(t), Dαu)τ˙(t,u(t))dt.
a t
ZA
This further yields
B B
∆L = ∆L(t,u(t), Dαu)dt+ L(t,u(t), Dαu)τ˙(t,u(t))dt
a t a t
ZA ZA
B B d
= δL(t,u(t), Dαu)dt+ L(t,u(t), Dαu)τ(t,u(t))dt
a t dt a t
ZA ZA
B
+ L(t,u(t), Dαu)τ˙(t,u(t))dt
a t
ZA
B B
= δLdt+(Lτ)
ZA (cid:12)A
(cid:12)
and the claim is proved. (cid:12) 2
We define a variational symmetry group of the fractional variational problem (5), which we
call a fractional variational symmetry group:
Definition 4. A local one-parameter group of transformations G (12) is a variational symmetry
groupofthefractionalvariationalproblem(5)ifthefollowingconditionsholds: forevery[A′,B′]⊂
(A,B), u=u(t)∈AC([A′,B′]) and g ∈G such that u¯(t¯)=g ·u(t¯) is in AC([A¯′,B¯′]), we have
η η
B¯′ B′
L(t¯,u¯(t¯), Dαu¯)dt¯= L(t,u(t), Dαu)dt. (14)
a t¯ a t
ZA¯′ ZA′
We are now able to prove the following infinitesimal criterion:
Theorem 5. Let L[u] be a fractional variational problem (5) and let G be a local one-parameter
transformation group (12) with the infinitesimal generator v =τ(t,u)∂ +ξ(t,u)∂ . Then G is a
t u
variational symmetry group of L if and only if
∂L ∂L d ∂L
τ +ξ + Dα(ξ−u˙τ)+ Dαu τ +Lτ˙ =0. (15)
∂t ∂u a t dta t ∂ Dαu
a t
(cid:16) (cid:16) (cid:17) (cid:17)
6
Proof. SupposethatGisavariationalsymmetrygroupofL. Then(14)holdsforallsubintervals
(A′,B′) of (A,B) with closure [A′,B′]⊂(A,B). We have:
B′ B′
∆L = δLdt+(L∆t)
ZAB′′ ∂L ∂(cid:12)(cid:12)LA′ B′
= δu+ (cid:12) δ( Dαu) dt+(L∆t)
∂u ∂ Dαu a t
ZA′ (cid:18) a t (cid:19) (cid:12)A′
B′ ∂L ∂L (cid:12) B′
= (∆u−u˙∆t)+ Dα(∆u−u˙(cid:12)∆t) dt+(L∆t)
∂u ∂ Dαua t
ZA′ (cid:18) a t (cid:19) (cid:12)A′
B′ ∂L ∂L (cid:12)
= ∆u+ Dα∆u dt (cid:12)
∂u ∂ Dαua t
ZA′ (cid:18) a t (cid:19)
B′ ∂L ∂L ∂L B′
− (u˙∆t)+ Dα(u˙∆t)± ∆t dt+(L∆t)
∂u ∂ Dαua t ∂t
= ∗ ZA′ (cid:18) a t (cid:19) (cid:12)A′
(cid:12)
(cid:12)
ApplyingtheLeibnitzrulefor d(L∆t)wereplace ∂L∆t+∂Lu˙∆tby d(L∆t)− ∂L (d Dαu)∆t−
dt ∂t ∂u dt ∂aDtαu dta t
L∆t(1). Then
B′ ∂L ∂L ∂L
∗ = ∆t+ ∆u+ Dα∆u
∂t ∂u ∂ Dαua t
ZA′ (cid:18) a t
∂L d
+ Dαu ∆t− Dα(u˙∆t) +L∆t(1) dt.
∂ Dαu dta t a t
a t (cid:16)(cid:16) (cid:17) (cid:17) (cid:19)
Since ∆L has to be zero in all (A′,B′) with [A′,B′]⊂(A,B), the above integrand has also to be
equal zero, i.e.,
∂L ∂L d ∂L
τ +ξ + Dα(ξ−u˙τ)+ Dαu τ +Lτ˙ =0.
∂t ∂u a t dta t ∂ Dαu
a t
(cid:16) (cid:16) (cid:17) (cid:17)
wherewehaveusedthat∆u=ξand∆t=τ. Hence,necessityofthestatementisproved. Toprove
that condition (15) is also sufficient, we first realize that if (15) holds on every [A′,B′] ⊂ (A,B),
then ∆L = 0 in every [A′,B′] ⊂ (A,B). Thus, integrating ∆L from 0 to η we obtain (14), for η
near the identity. The proof is now complete. 2
In the following example we calculate the transformation of the fractional derivative Dαu
a t
under the group of translations. In Subsection 2.2 we shall perform such calculation also in the
case when a in Dαu is transformed.
a t
Example 6. Let G be a local one-parameter translation group: (t¯,u¯) = (t+η,u+η), with the
infinitesimalgeneratorv=∂ +∂ . Then u¯(t¯)=u(t¯−η)+η andbya straightforwardcalculation
t u
it can be shown that
1 d a u(s)+η
Dαu¯= Dα(u+η)+ ds.
a t¯ a t Γ(1−α)dt (t−s)α
Za−η
2.2 The case when a in Dαu is transformed
a t
Now let the action of a local one-parameter group of transformations (12) be of the form
g ·(t,u, Dαu):=(t¯,u¯, Dαu¯).
η a t a¯ t¯
Thus, the one-parameter group also acts on a and transforms it to a¯, where a¯ = g ·t| . This
η t=a
will also influence the calculation of ∆ Dαu and ∆L:
a t
7
Lemma 7. Let u∈AC([a,b]) and let G be a local one-parameter group of transformations (12).
Then
d α u(a)
∆ Dαu= Dαδu+ Dαu·τ(t,u(t))+ τ(a,u(a)), (16)
a t a t dta t Γ(1−α)(t−a)α+1
where
d
δ Dαu= Dαu¯− Dαu = Dαδu.
a t dη a t a t a t
η=0
(cid:12) (cid:16) (cid:17)
Proof. Again using (13) we check th(cid:12)(cid:12)at δaDtαu = aDtαδu. To prove (16) we start with the
definition of ∆ Dαu:
a t
d
∆ Dαu= Dαu¯− Dαu .
a t dη a¯ t¯ a t
η=0
(cid:12) (cid:16) (cid:17)
Thus, (cid:12)
(cid:12)
1 d d t¯ u¯(θ) d t u¯(θ) d t u(θ)
∆ Dαu = dθ± dθ− dθ
a t Γ(1−α)dη dt¯ (t¯−θ)α dt (t−θ)α dt (t−θ)α
(cid:12)η=0" Za¯ Za Za #
(cid:12)(cid:12) 1 d d a u¯(θ) d t¯ u¯(θ)
= Dαδu+ dθ+ dθ
a t Γ(1−α)dη dt¯ (t¯−θ)α dt¯ (t¯−θ)α
(cid:12)η=0" Za¯ Za
(cid:12)
d t u¯(θ) (cid:12)
− dθ
dt (t−θ)α
Za #
d 1 d d a u¯(θ)
= Dαδu+ Dαu·τ(t,u(t))+ dθ.
a t dta t Γ(1−α)dη dt¯ (t¯−θ)α
(cid:12)η=0 Za¯
(cid:12)
Note that in the last term we can interchange the order o(cid:12)f integration and differentiation with
respect to t¯. This gives
−α d a u¯(θ)
dθ.
Γ(1−α)dη (t¯−θ)α
(cid:12)η=0Za¯
If we differentiate this integral with respect t(cid:12)o η at η =0 we eventually obtain (16). 2
(cid:12)
Lemma 8. Let L[u] be a functional of the form L[u] = BL(t,u(t), Dαu)dt, where u is an
A a t
absolutely continuous function in (a,b), (A,B) ⊆ (a,b) and L satisfies (6). Let G be a local
R
one-parameter group of transformations given by (12). Then
B α u(a) B ∂L
∆L=δL+(L∆t) + τ(a,u(a)) dt, (17)
Γ(1−α)(t−a)α+1 ∂ Dαu
(cid:12)A ZA a t
(cid:12)
where δL= BδLdt. (cid:12)
A
Proof. CleaRrly, δL= bδLdt. On the other hand we have
a
d R B¯ B
∆L = L(t¯,u¯(t¯), Dαu¯)dt¯− L(t,u(t), Dαu)dt
dη(cid:12)η=0(cid:18) ZA¯ a¯ t¯ ZA a t (cid:19)
d (cid:12) B B
= (cid:12) L(t¯,u¯(t¯), Dαu¯)(1+ητ˙(t,u(t)))dt− L(t,u(t), Dαu)dt
dη a¯ t¯ a t
(cid:12)η=0(cid:18) ZA ZA (cid:19)
B(cid:12) B
= (cid:12)∆Ldt+ L(t,u(t), Dαu)τ˙(t,u(t))dt
a t
ZA ZA
B ∂L ∂L ∂L B
= ∆t+ ∆u+ ∆ Dαu dt+ L(t,u(t), Dαu)τ˙(t,u(t))dt.
∂t ∂u ∂ Dαu a t a t
ZA (cid:16) a t (cid:17) ZA
8
We now apply (16) to obtain
B ∂L ∂L ∂L d
∆L = ∆t+ (δu+u˙∆t)+ Dαδu+ Dαu·τ(t,u(t))
∂t ∂u ∂ Dαu a t dta t
ZA (cid:18) a t (cid:16)
α u(a) B
+ τ(a,u(a)) dt+ L(t,u(t), Dαu)τ˙(t,u(t))dt
Γ(1−α)(t−a)α+1 a t
(cid:17)(cid:19) ZA
B B α u(a) B ∂L
= δLdt+(L∆t) + τ(a,u(a)) dt,
Γ(1−α)(t−a)α+1 ∂ Dαu
ZA (cid:12)A ZA a t
B (cid:12) α u(a) B ∂L
= δL+(L∆t) + (cid:12) τ(a,u(a)) dt.
Γ(1−α)(t−a)α+1 ∂ Dαu
(cid:12)A ZA a t
(cid:12) 2
(cid:12)
Remark 9. It should be emphasized that the last summand on the right-hand side of (16), as
well as of (17), appears only as a consequence of the fact that the action of a transformation
group affects also a, on which the left Riemann-Liouville fractional derivative depends. In the
case when u(a) = 0 or α = 1 that term is equal to zero (since limα→1−Γ(1 − α) = ∞) and
B
∆L=δL+(L∆t) .
A
(cid:12)
Nowwedefinea(cid:12) variationalsymmetrygroupofthefractionalvariationalproblem(5)asfollows:
(cid:12)
Definition 10. A local one-parameter group of transformations G (12) is a variational symmetry
groupofthefractionalvariationalproblem(5)ifthefollowingconditionsholds: forevery[A′,B′]⊂
(A,B), u=u(t)∈AC([A′,B′]) and g ∈G such that u¯(t¯)=g ·u(t¯) is in AC([A¯′,B¯′]), we have
η η
B¯′ B′
L(t¯,u¯(t¯), Dαu¯)dt¯= L(t,u(t), Dαu)dt.
a¯ t¯ a t
ZA¯′ ZA′
Recallthatwearesolvingthefractionalvariationalproblem(5)amongallabsolutelycontinuous
functions in [a,b], which additionally satisfy u(a) = 0. Therefore, the last term in both (16) and
(17) vanishes, and the infinitesimal criterion reads the same as in Theorem 5:
Theorem 11. Let L[u] be a fractional variational problem defined by (5) and let G be a local one-
parameter transformation group defined by (12) with the infinitesimal generator v = τ(t,u)∂ +
t
ξ(t,u)∂ . Assumethat u(a)=0, for all admissible functions u. Then Gis a variational symmetry
u
group of L if and only if
∂L ∂L d ∂L
τ +ξ + Dα(ξ−u˙τ)+ Dαu τ +Lτ˙ =0. (18)
∂t ∂u a t dta t ∂ Dαu
a t
(cid:16) (cid:16) (cid:17) (cid:17)
Proof. See the proof of Theorem 5. 2
Example 12. Let v = ∂ +∂ be the infinitesimal generator of the translation group (t¯,u¯) =
t u
(t+η,u+η). Then u¯(t¯)=u(t¯−η)+η and
Dαu¯= Dα(u+η).
a¯ t¯ a t
3 N¨other’s theorem
Inthe formulationofNo¨ther’stheoremweshallneeda generalizationofthe fractionalintegration
by parts.
9
Lemma 13. Let f,g ∈AC([a,b]). Then, for all t∈[a,b] the following formula holds:
t t
f(s)· Dαgds= Dαf ·g(s)ds
s b a s
Za t 1Za g(b) g(t) b g˙(σ) (19)
+ f(s)· − − dσ ds.
Γ(1−α) (b−s)α (t−s)α (σ−s)α
Za (cid:20) Zt (cid:21)
Proof. In order to derive (19) we shall use the representation of the right (resp. left) Riemann-
Liouville fractional derivative via the right (resp. left) Caputo fractional derivative (4), as well as
(3):
t t 1 g(b) b g˙(σ)
f(s)· Dαgds = f(s)· − dσ ds
s b Γ(1−α) (b−s)α (σ−s)α
Za Za (cid:20) Zs (cid:21)
t 1 g(b) t g˙(σ)
= f(s)· − dσ
Γ(1−α) (b−s)α (σ−s)α
Za (cid:20) Zs
b g˙(σ) g(t)
− dσ± ds
(σ−s)α (t−s)α
Zt (cid:21)
t 1 g(b)
= f(s)· Dαg+f(s)·
s t Γ(1−α) (b−s)α
Za " (cid:20)
g(t) b g˙(σ)
− − dσ ds
(t−s)α (σ−s)α
Zt (cid:21)#
t 1 g(b)
= Dαf ·g(s)+f(s)·
a s Γ(1−α) (b−s)α
Za " (cid:20)
g(t) b g˙(σ)
− − dσ ds.
(t−s)α (σ−s)α
Zt (cid:21)#
2
Remark 14. If we put t=b in (19), we obtain (3).
The main goal of symmetry group analysis in the calculus of variations are the first integrals
of Euler-Lagrangeequations of a variational problem, that is a No¨ther type result.
Analogouslytotheclassicalcase,anexpression dP(t,u(t), Dαu)=0iscalledafractionalfirst
dt a t
integral(orafractionalconservationlaw)forafractionaldifferentialequationF(t,u(t), Dαu)=0,
a t
if it vanishes along all solutions u(t) of F.
Inthestatementwhichistofollow,weproveaversionofthefractionalNo¨thertheorem. Aswe
will show, a fractional conserved quantity will also contain integral terms, which is unavoidable,
duetothepresenceoffractionalderivatives. Ifα=1then(20)reducestothewell-knownclassical
conservation laws for a first-order variational problem.
Theorem 15. (No¨ther’s theorem) Let G be a local fractional variational symmetry group defined
by (12) of the fractional variational problem (5), and let v = τ(t,u(t))∂ +ξ(t,u(t))∂ be the
t u
infinitesimal generator of G. Then
t ∂L ∂L
Lτ + Dα(ξ−u˙τ) −(ξ−u˙τ) Dα ds= const. (20)
a s ∂ Dαu s B ∂ Dαu
Za (cid:18) a s (cid:16) a s (cid:17)(cid:19)
is a fractional first integral (or fractional conservation law) for the Euler-Lagrange equation (9).
The fractional conservation law (20) can equivalently be written in the form
t 1 ∂ L(B,u(B), Dαu) ∂ L(t,u(t), Dαu)
Lτ − (ξ−u˙τ)· 3 a B − 3 a t
Γ(1−α) (B−s)α (t−s)α
Za (cid:20)
b d ∂ L(σ,u(σ), Dαu)
− dσ 3 a σ dσ ds= const. (21)
(σ−s)α
Zt (cid:21)
10
