Table Of ContentA Formulation of Noether’s Theorem for
7
0 Fractional Problems of the Calculus of Variations∗
0
2
n Gast˜ao S. F. Frederico Delfim F. M. Torres
a [email protected] [email protected]
J
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Department of Mathematics
] University of Aveiro
C
3810-193 Aveiro, Portugal
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Abstract
m
Fractional(ornon-integer)differentiationisanimportantconceptboth
[
from theoretical and applicational pointsof view. The studyof problems
1 of the calculus of variations with fractional derivatives is a rather recent
v subject, the main result being the fractional necessary optimality con-
7 dition of Euler-Lagrange obtained in 2002. Here we use the notion of
8 Euler-LagrangefractionalextremaltoproveaNoether-typetheorem. For
1
that we propose a generalization of the classical concept of conservation
1
law, introducing an appropriate fractional operator.
0
7 Keywords: calculusofvariations; fractional derivatives;Noether’stheo-
0 rem.
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Mathematics Subject Classification 2000: 49K05; 26A33.
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1 Introduction
:
v
i The notion of conservation law – first integral of the Euler-Lagrangeequations
X
– is well-knownin Physics. One of the most important conservationlaws is the
r
a integral of energy, discovered by Leonhard Euler in 1744: when a Lagrangian
L(q,q˙) corresponds to a system of conservative points, then
−L(q,q˙)+∂ L(q,q˙)·q˙≡constant (1)
2
holds along all the solutions of the Euler-Lagrange equations (along the ex-
tremalsoftheautonomousvariationalproblem),where∂ L(·,·)denotethepar-
2
tial derivative of function L(·,·) with respect to its second argument. Many
other examples appear in modern physics: in classic, quantum, and optical
∗ResearchreportCM06/I-04,UniversityofAveiro. AcceptedforpublicationintheJournal
of Mathematical Analysis and Applications.
1
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 can
be used to reduce the order of the Euler-Lagrange differential equations, thus
simplifying 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, all
theconservationlawsthatappearinapplications. Inthelastdecades,Noether’s
principle has been formulated in various contexts (see [16, 17] and references
therein). In this work we generalize Noether’s theorem for problems having
fractional derivatives.
Fractionaldifferentiationplaysnowadaysanimportantroleinvariousseem-
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 image
processing, dynamics of earthquakes, control theory, and so on [3, 6, 8, 9]. Its
origin goes back more than 300 years, when in 1695 L’Hopital asked Leibniz
the meaning of dny for n = 1. After that, many famous mathematicians, like
dxn 2
J. Fourier, N. H. Abel, J. Liouville, B. Riemann, among others, contributed to
the development of the Fractional Calculus [6, 10, 14].
F. Riewe [12, 13] obtained a version of the Euler-Lagrange equations for
problemsoftheCalculusofVariationswithfractionalderivatives,thatcombines
the conservative and non-conservativecases. More recently, O. Agrawalproved
a formulation for variational problems with right and left fractional derivatives
in the Riemann-Liouville sense [1]. Then these Euler-Lagrange equations were
used by D. Baleanu and T. Avkar to investigate problems with Lagrangians
whicharelinearonthe velocities[4]. Hereweuse the resultsof[1]to generalize
Noether’s theorem for the more general context of the Fractional Calculus of
Variations.
The paper is organized in the following way. In Section 2 we recall the no-
tions of right and left Riemann-Liouville fractionalderivatives, that are needed
for formulating the fractional problem of the calculus of variations. There are
many different ways to approach classical Noether’s theorem. In Section 3 we
reviewthe only proofthat we areable to extend, with success,to the fractional
context. The methodisbasedonatwo-stepprocedure: itstartswithaninvari-
ancenotionoftheintegralfunctionalunderaone-parameterinfinitesimalgroup
of transformations,without changing the time variable; then it proceeds with a
time-reparameterization to obtain Noether’s theorem in general form. The in-
tendedfractionalNoether’stheoremisformulatedandprovedinSection4. Two
illustrative examples of application of our main result are given in Section 5.
We finish with Section 6 of conclusions and some open questions.
2
2 Riemann-Liouville fractional derivatives
In this section we collect the definitions of right and left Riemann-Liouville
fractional 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], the
left Riemann-Liouville fractional derivative Dαf(t), and the right Riemann-
a t
Liouville fractional derivative Dαf(t), of order α, are defined in the following
t b
way:
1 d n t
Dαf(t)= (t−θ)n−α−1f(θ)dθ, (2)
a t Γ(n−α) dt
(cid:18) (cid:19) Za
1 d n b
Dαf(t)= − (θ−t)n−α−1f(θ)dθ, (3)
t b Γ(n−α) dt
(cid:18) (cid:19) Zt
where n∈N, n−1≤α<n, and Γ is the Euler gamma function.
Remark 2.2. If α is an integer, then from (2) and (3) one obtains the standard
derivatives, that is,
d α d α
Dαf(t)= f(t), Dαf(t)= − f(t).
a t dt t b dt
(cid:18) (cid:19) (cid:18) (cid:19)
Theorem 2.3. Let f and g be two continuous functions on [a,b]. Then, for all
t∈[a,b], the following properties hold:
1. for p>0, Dp(f(t)+g(t))= Dpf(t)+ Dpg(t);
a t a t a t
2. for p≥q ≥0, Dp D−qf(t) = Dp−qf(t);
a t a t a t
3. for p>0, Dp D−(cid:0)pf(t) =f(cid:1)(t) (fundamental property of the Riemann-
a t a t
Liouville fractional derivatives);
(cid:0) (cid:1)
4. for p > 0, b( Dpf(t))g(t)dt = bf(t) Dpg(t)dt (we are assuming that
a a t a t b
Dpf(t) and Dpg(t) exist at every point t∈[a,b] and are continuous).
a t R t b R
Remark 2.4. In general, the fractional derivative of a constant is not equal to
zero.
Remark 2.5. Thefractionalderivativeoforderp>0offunction(t−a)υ,υ >−1,
is given by
Γ(υ+1)
Dp(t−a)υ = (t−a)υ−p.
a t Γ(−p+υ+1)
Remark 2.6. In the literature, when one reads “Riemann-Liouville fractional
derivative”, one usually means the “left Riemann-Liouville fractional deriva-
tive”. In Physics, if t denotes the time-variable, the right Riemann-Liouville
fractional 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
3
the present state of the process does not depend on the results of the future
development. From a mathematical point ofview, both derivatives appearnat-
urally in the fractional calculus of variations [1].
Wereferthereaderinterestedinadditionalbackgroundonfractionaltheory,
to the comprehensive book of Samko et al. [14].
3 Review of the classical Noether’s theorem
There exist several ways to prove the classical theorem of Emmy Noether. In
this section we review one of those proofs [7]. The proof is done in two steps:
we begin by proving Noether’s theorem without transformation of the time
(without transformation of the independent variable); then, using a technique
of time-reparameterization, we obtain Noether’s theorem in its general form.
This technique is not so popular while proving Noether’s theorem, but it turns
outto be the only one we succeedto extendfor the more generalcontextofthe
Fractional Calculus of Variations.
We begin by formulating the fundamental problem of the calculus of varia-
tions:
b
I[q(·)]= L(t,q(t),q˙(t))dt−→min (P)
Za
under the boundary conditions q(a) = q and q(b) = q , and where q˙ = dq.
a b dt
The Lagrangian L :[a,b]×Rn×Rn →R is assumed to be a C2-function with
respect to all its arguments.
Definition 3.1 (invariance without transforming the time). Functional (P) is
said to be invariant under a ε-parameter groupof infinitesimal transformations
q¯(t)=q(t)+εξ(t,q)+o(ε) (4)
if, and only if,
tb tb
L(t,q(t),q˙(t))dt= L(t,q¯(t),q¯˙(t))dt (5)
Zta Zta
for any subinterval [t ,t ]⊆[a,b].
a b
Along the work, we denote by ∂ L the partial derivative of L with respect
i
to its i-th argument.
Theorem3.2(necessaryconditionofinvariance). Iffunctional (P)isinvariant
under transformations (4), then
∂ L(t,q,q˙)·ξ+∂ L(t,q,q˙)·ξ˙=0. (6)
2 3
Proof. Equation (5) is equivalent to
L(t,q,q˙)=L(t,q+εξ+o(ε),q˙+εξ˙+o(ε)). (7)
Differentiating both sides of equation (7) with respect to ε, then substituting
ε=0, we obtain equality (6).
4
Definition3.3 (conservedquantity). QuantityC(t,q(t),q˙(t))issaidtobecon-
served if, and only if, dC(t,q(t),q˙(t)) =0 along all the solutions of the Euler-
dt
Lagrange equations
d
∂ L(t,q,q˙)=∂ L(t,q,q˙) . (8)
dt 3 2
Theorem 3.4 (Noether’s theorem without transforming time). If functional
(P) is invariant under the one-parameter group of transformations (4), then
C(t,q,q˙)=∂ L(t,q,q˙)·ξ(t,q) (9)
3
is conserved.
Proof. Using the Euler-Lagrange equations (8) and the necessary condition of
invariance (6), we obtain:
d
(∂ L(t,q,q˙)·ξ(t,q))
dt 3
d
= ∂ L(t,q,q˙)·ξ(t,q)+∂ L(t,q,q˙)·ξ˙(t,q)
dt 3 3
=∂ L(t,q,q˙)·ξ(t,q)+∂ L(t,q,q˙)·ξ˙(t,q)
2 3
=0.
Remark 3.5. Inclassicalmechanics,∂ L(t,q,q˙)isinterpretedasthegeneralized
3
momentum.
Definition3.6(invarianceof(P)). Functional(P)issaidtobeinvariantunder
the one-parameter group of infinitesimal transformations
t¯=t+ετ(t,q)+o(ε),
(10)
(q¯(t)=q(t)+εξ(t,q)+o(ε),
if, and only if,
tb t¯(tb)
L(t,q(t),q˙(t))dt= L(t¯,q¯(t¯),q¯˙(t¯))dt¯
Zta Zt¯(ta)
for any subinterval [t ,t ]⊆[a,b].
a b
Theorem 3.7 (Noether’stheorem). If functional (P) is invariant, in the sense
of Definition 3.6, then
C(t,q,q˙) = ∂ L(t,q,q˙) · ξ(t,q) + (L(t,q,q˙)−∂ L(t,q,q˙)·q˙)τ(t,q) (11)
3 3
is conserved.
5
Proof. Everynon-autonomousproblem(P)isequivalenttoanautonomousone,
considering t as a dependent variable. For that we consider a Lipschitzian one-
to-one transformation
[a,b]∋t7−→σ ∈[σ ,σ ]
a b
such that
b
I[q(·)]= L(t,q(t),q˙(t))dt
Za
σb dq(t(σ)) dt(σ)
= L t(σ),q(t(σ)), dσ dσ
dt(σ) dσ
Zσa dσ !
= σbL t(σ),q(t(σ)), qσ′ t′ dσ
t′ σ
Zσa σ!
=. σbL¯ t(σ),q(t(σ)),t′ ,q′ dσ
σ σ
. Zσa (cid:16) (cid:17)
=I¯[t(·),q(t(·))] ,
where t(σ )=a, t(σ )=b, t′ = dt(σ), and q′ = dq(t(σ)). If functional I[q(·)] is
a b σ dσ σ dσ
invariantinthesenseofDefinition3.6,thenfunctionalI¯[t(·),q(t(·))]isinvariant
in the sense of Definition 3.1. Applying Theorem 3.4, we obtain that
C t,q,t′ ,q′ =∂ L¯·ξ+∂ L¯τ (12)
σ σ 4 3
(cid:16) (cid:17)
is a conserved quantity. Since
∂ L¯ =∂ L(t,q,q˙) ,
4 3
′
q
∂ L¯ =−∂ L(t,q,q˙)· σ +L(t,q,q˙) (13)
3 3 t′
σ
=L(t,q,q˙)−∂ L(t,q,q˙)·q˙,
3
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 we
prove a Noether’s theorem for the fractional Euler-Lagrangeextremals.
Thefundamentalfunctionalofthefractionalcalculusofvariationsisdefined
as follows:
b
I[q(·)]= L t,q(t), Dαq(t), Dβq(t) dt−→min, (P )
a t t b f
Za (cid:16) (cid:17)
where the Lagrangian L : [a,b]×Rn ×Rn ×Rn → R is a C2 function with
respect to all its arguments, and 0<α,β ≤1.
6
Remark 4.1. In the case α=β =1, problem (P ) is reduced to problem (P):
f
b
I[q(·)]= L(t,q(t),q˙(t))dt−→min
Za
with
L(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 (P ), then it satisfies the
f
fractional Euler-Lagrangeequations:
∂ L t,q, Dαq, Dβq + Dα∂ L t,q, Dαq, Dβq
2 a t t b t b 3 a t t b
(cid:16) (cid:17) (cid:16) (cid:17)
+ Dβ∂ L t,q, Dαq, Dβq =0. (15)
a t 4 a t t b
(cid:16) (cid:17)
Definition 4.3 (cf. Definition 3.1). We say that functional (P ) is invariant
f
under the transformation (4) if, and only if,
tb tb
L t,q(t), Dαq(t), Dβq(t) dt = L t,q¯(t), Dαq¯(t), Dβq¯(t) dt
a t t b a t t b
Zta (cid:16) (cid:17) Zta (cid:16) (cid:17)
(16)
for any subinterval [t ,t ]⊆[a,b].
a b
Thenexttheoremestablishesanecessaryconditionofinvariance,ofextreme
importance for our objectives.
Theorem 4.4 (cf. Theorem 3.2). If functional (P ) is invariant under trans-
f
formations (4), then
∂ L t,q, Dαq, Dβq ·ξ(t,q)+∂ L t,q, Dαq, Dβq · Dαξ(t,q)
2 a t t b 3 a t t b a t
(cid:16) (cid:17) (cid:16) (cid:17)
+∂ L t,q, Dαq, Dβq · Dβξ(t,q)=0. (17)
4 a t t b t b
(cid:16) (cid:17)
Remark 4.5. Intheparticularcaseα=β =1,weobtainfrom(17)thenecessary
condition (6) applied to L (14).
Proof. Having in mind that condition (16) is valid for any subinterval [t ,t ]⊆
a b
[a,b], we can get rid off the integralsigns in (16). Differentiating this condition
with respect to ε, substituting ε = 0, and using the definitions and properties
7
of the Riemann-Liouville fractional derivatives given in Section 2, we arrive to
0=∂ L t,q, Dαq, Dβq ·ξ(t,q)
2 a t t b
(cid:16) (cid:17) d 1 d n t
+∂ L t,q, Dαq, Dβq · (t−θ)n−α−1q(θ)dθ
3 a t t b dε Γ(n−α) dt
(cid:16) ε (cid:17) d n(cid:20) t (cid:18) (cid:19) Za
+ (t−θ)n−α−1ξ(θ,q)dθ
Γ(n−α) dt
(cid:18) (cid:19) Za (cid:21)ε=0
d 1 d n b
+∂ L t,q, Dαq, Dβq · − (θ−t)n−β−1q(θ)dθ
4 a t t b dε Γ(n−β) dt
(cid:16) (cid:17) " (cid:18) (cid:19) Zt
ε d n b
+ − (θ−t)n−β−1ξ(θ,q)dθ . (18)
Γ(n−β) dt
(cid:18) (cid:19) Zt #ε=0
Expression(18) is equivalent to (17).
Thefollowingdefinitionisusefulinordertointroduceanappropriateconcept
of 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:
Dγ(f,g)=−g Dγf +f Dγg,
t t b a t
where t∈[a,b] and γ ∈R+.
0
Remark 4.7. If γ =1, the operator Dγ is reduced to
t
d
D1(f,g)=−g D1f +f D1g =f˙g+fg˙ = (fg).
t t b a t dt
In particular, D1(f,g)=D1(g,f).
t t
Remark 4.8. In the fractionalcase (γ 6=1), functions f and g do not commute:
in general Dγ(f,g)6=Dγ(g,f).
t t
Remark 4.9. The linearity of the operators Dγ and Dγ imply the linearity of
a t t b
the operator Dγ.
t
Definition 4.10 (fractional-conserved quantity – cf. Definition 3.3). We say
that C t,q, Dαq, Dβq is fractional-conserved if (and only if) it is possible
f a t t b
to write(cid:16)Cf in the form (cid:17)
m
C (t,q,d,d )= C1(t,q,d,d )·C2(t,q,d,d ) (19)
f l r i l r i l r
i=1
X
for some m ∈ N and some functions C1 and C2, i = 1,...,m, where each pair
i i
C1 and C2, i=1,...,m, satisfy
i i
Dγi Cji1 t,q, Dαq, Dβq ,Cji2 t,q, Dαq, Dβq =0 (20)
t i a t t b i a t t b
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
8
withγ ∈{α,β},j1 =1andj2 =2orj1 =2andj2 =1,alongallthefractional
i i i i i
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, like
the structure (19) we are assuming in Definition 4.10. For α = β = 1 (20) is
equivalent to the standard definition d [C (t,q(t),q˙(t),−q˙(t))]=0.
dt f
Example 4.12. Let C be a fractional-conservedquantity written in the form
f
(19)withm=1,thatis,letC =C1·C2 be afractional-conservedquantityfor
f 1 1
some given functions C1 and C2. Condition (20) of Definition 4.10 means one
1 1
of four things: or Dα C1,C2 =0, or Dα C2,C1 =0, or Dβ C1,C2 =0, or
t 1 1 t 1 1 t 1 1
Dβ C2,C1 =0.
t 1 1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Rem(cid:0)ark 4.1(cid:1)3. Given a fractional-conserved quantity Cf, the definition of Ci1
and C2, i = 1,...,m, is never unique. In particular, one can always choose
i
C1 to be C2 and C2 to be C1. Definition 4.10 is imune to the arbitrariness in
i i i i
defining the Cj, i=1,...,m, j =1,2.
i
Remark 4.14. Due to the simple fact that the same function can be written in
several different but equivalent ways, to a given fractional-conserved quantity
C itcorrespondsanintegervaluemin(19)whichis,ingeneral,alsonotunique
f
(see Example 4.15).
Example4.15. Letf,gandhbefunctionssatisfyingDα(g,f)=0,Dα(f,g)6=
t t
0,Dα(h,f)=0,Dα(f,h)6=0alongallthefractionalEuler-Lagrangeextremals
t t
ofagivenfractionalvariationalproblem. Onecanprovidedifferentproofstothe
fact that C = f(g+h) is a fractional-conserved quantity: (i) C is fractional-
conservedbecausewecanwriteC intheform(19)withm=2,C1 =g,C2 =f,
1 1
C1 = h, and C2 = f, satisfying (20) (Dα C1,C2 = 0 and Dα C1,C2 = 0);
2 2 t 1 1 t 2 2
(ii) C is fractional-conserved because we can wrire C in the form (19) with
(cid:0) (cid:1) (cid:0) (cid:1)
m=1,C1 =g+h,andC2 =f,satisfying(20)(Dα C1,C2 =Dα(g+h,f)=
1 1 t 1 1 t
Dα(g,f)+Dα(h,f)=0).
t t (cid:0) (cid:1)
Theorem 4.16 (cf. Theorem 3.4). If the functional (P ) is invariant under
f
the transformations (4), in the sense of Definition 4.3, then
C t,q, Dαq, Dβq
f a t t b
(cid:16) (cid:17)
= ∂ L t,q, Dαq, Dβq −∂ L t,q, Dαq, Dβq ·ξ(t,q) (21)
3 a t t b 4 a t t b
h (cid:16) (cid:17) (cid:16) (cid:17)i
is fractional-conserved.
Remark 4.17. In the particular case α = β = 1, we obtain from (21) the
conserved quantity (9) applied to L (14).
Proof. We use the fractional Euler-Lagrangeequations
∂ L t,q, Dαq, Dβq =− Dα∂ L t,q, Dαq, Dβq
2 a t t b t b 3 a t t b
(cid:16) (cid:17) (cid:16) (cid:17)
− Dβ∂ L t,q, Dαq, Dβq
a t 4 a t t b
(cid:16) (cid:17)
9
in (17), obtaining
− Dα∂ L t,q, Dαq, Dβq ·ξ(t,q)+∂ L t,q, Dαq, Dβq · Dαξ(t,q)
t b 3 a t t b 3 a t t b a t
(cid:16) (cid:17) (cid:16) (cid:17)
− Dβ∂ L t,q, Dαq, Dβq ·ξ(t,q)+∂ L t,q, Dαq, Dβq · Dβξ(t,q)
a t 4 a t t b 4 a t t b t b
(cid:16) (cid:17) (cid:16) (cid:17)
=Dα ∂ L t,q, Dαq, Dβq ,ξ(t,q) −Dβ ξ(t,q),∂ L t,q, Dαq, Dβq =0.
t 3 a t t b t 4 a t t b
(cid:16) (cid:16) (cid:17) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
The proof is complete.
Definition4.18(invarianceof(P )–cf. Definition3.6). Functional(P )issaid
f f
to be invariant under the one-parameter group of infinitesimal transformations
(10) if, and only if,
tb t¯(tb)
L t,q(t),aDtαq(t),tDbβq(t) dt= L t¯,q¯(t¯),aDt¯αq¯(t¯),t¯Dbβq¯(t¯) dt¯
Zta (cid:16) (cid:17) Zt¯(ta) (cid:16) (cid:17)
for any subinterval [t ,t ]⊆[a,b].
a b
ThenexttheoremprovidestheextensionofNoether’stheoremforFractional
Problems of the Calculus of Variations.
Theorem 4.19 (fractionalNoether’stheorem). If functional (P ) is invariant,
f
in the sense of Definition 4.18, then
C t,q, Dαq, Dβq = ∂ L t,q, Dαq, Dβq
f a t t b 3 a t t b
(cid:16) (cid:17) h (cid:16) (cid:17)
− ∂ L t,q, Dαq, Dβq ·ξ(t,q)
4 a t t b
(cid:16) (cid:17)i
+ L t,q, Dαq, Dβq −α∂ L t,q, Dαq, Dβq · Dαq
a t t b 3 a t t b a t
h (cid:16) (cid:17) (cid:16) (cid:17)
−β∂ L t,q, Dαq, Dβq · Dβq τ(t,q) (22)
4 a t t b t b
(cid:16) (cid:17) i
is fractional-conserved (Definition 4.10).
Remark 4.20. If α = β = 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) by
the Lipschitzian transformation
[a,b]∋t7−→σf(λ)∈[σ ,σ ]
a b
′
that satisfies t =f(λ)=1 if λ=0. Functional (P ) is reduced, in this way,to
σ f
an autonomous functional:
I¯[t(·),q(t(·))]= σbL t(σ),q(t(σ)), Dα q(t(σ)), Dβ q(t(σ)) t′ dσ, (23)
σa t(σ) t(σ) σb σ
Zσa (cid:16) (cid:17)
10
