Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
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Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier.com/locate/cnsns
A continuous/discrete fractional Noether’s theorem
Loïc Bourdin, Jacky Cresson ⇑, Isabelle Greff
Laboratoire de Mathématiques et de leurs Applications – Pau (LMAP), UMR CNRS 5142, Université de Pau et des Pays de l’Adour, France
a r t i c l e
i n f o
Article history:
Received 12 December 2011
Received in revised form 3 September 2012
Accepted 4 September 2012
Available online 21 September 2012
Keywords:
Lagrangian systems
Noether’s theorem
Symmetries
Fractional calculus
a b s t r a c t
We prove a fractional Noether’s theorem for fractional Lagrangian systems invariant under
a symmetry group both in the continuous and discrete cases. This provides an explicit
conservation law (first integral) given by a closed formula which can be algorithmically
implemented. In the discrete case, the conservation law is moreover computable in a finite
number of steps.
Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction
Fractional calculus is the emerging mathematical field dealing with the generalization of the derivative to any real order.
During the last two decades, it has been successfully applied to problems in economics [11], computational biology [19] and
several fields in physics [5,16,2,28]. Fractional differential equations are also considered as an alternative model to non-linear
differential equations, see [7]. We refer to [24,27,25,17] for a general theory and to [18] for more details concerning the recent
history of fractional calculus. Particularly, a subtopic of the fractional calculus has recently gained importance: it concerns the
variational principles on functionals involving fractional derivatives. This leads to the statement of fractional Euler–Lagrange
equations, see [26,1,6].
A conservation law for a continuous or a discrete dynamical system is a function which is constant on each solution.
Precisely, let us denote by q ¼ ðqðtÞÞt2½a;b (resp. Q ¼ ðQ k Þk¼0;...;N ) the solutions of a continuous (resp. discrete) system taking
values in a set denoted by E. Then, a function C : E ! R is a conservation law (or first integral) if, for any solution q (resp. Q),
it exists a real constant c such that:
C ðqðtÞÞ ¼ c;
8t 2 ½a; b: ðresp: CðQ k Þ ¼ c;
8k ¼ 0; . . . ; N:Þ
ð1Þ
Conservation laws are generally associated to some physical quantities like total energy or angular momentum in mechanical systems and sometimes, they also can be used to reduce or integrate by quadrature the equation.
In this paper, we study the existence of explicit conservation laws for continuous and discrete fractional Lagrangian
systems introduced in [1,10] respectively. We follow the usual strategy of the classical Noether’s theorem providing an
explicit conservation law for classical Lagrangian systems invariant under symmetries. Precisely, we prove a fractional
Noether’s theorem providing an explicit conservation law for fractional Lagrangian systems invariant under symmetries both
in the continuous and discrete cases. The given formula is algorithmic so that arbitrary hight order approximations can be
computed. Moreover, the algorithm is finite in the discrete case.
⇑ Corresponding author.
E-mail addresses: [email protected] (L. Bourdin), [email protected] (J. Cresson), [email protected] (I. Greff).
1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.cnsns.2012.09.003
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
879
Previous results in this direction (in the continuous case) have been obtained by several authors, see [12,14,4,23,20,21].
However, in each of these papers, the conservation law is not explicitly derived in terms of symmetry group and Lagrangian
but implicitly defined by functional relations (see [14,23]) or integral relations (see [4]).
The paper is organized as follows. In Section 2, we first remind classical definitions and results concerning fractional
derivatives and fractional Lagrangian systems. Then, we prove a transfer formula leading to the statement of a fractional
Noether’s theorem based on a result of Cresson in [12]. Finally, we give some remarks concerning the previous studies on
the subject and we give some precisions about our result. In Section 3, we first remind the definition of discrete fractional
Lagrangian systems in the sense of [10]. Then, introducing the notion of discrete symmetry for these systems, we state a discrete fractional Noether’s theorem. We conclude with some remarks and a numerical implementation of the discrete fractional harmonic oscillator.
2. A fractional Noether’s theorem
In this paper, we consider fractional differential systems in Rd where d 2 N is the dimension (N denotes the set of
positive integer). The trajectories of these systems are curves q in C0 ð½a; b; Rd Þ where a < b are two reals.
2.1. Reminder about fractional Lagrangian systems
We first review classical definitions extracted from [27,25,17] of the fractional derivatives of Riemann–Liouville. Let a > 0
and f be a function defined on ½a; b with values in Rd . The left (resp. right) fractional integral in the sense of Riemann–Liouville
with inferior limit a (resp. superior limit b) of order a of f is given by:
8t 2a; b;
Ia f ðtÞ ¼
1
CðaÞ
Z
t
ðt yÞa1 f ðyÞ dy;
ð2Þ
a
respectively:
8t 2 ½a; b½;
Iaþ f ðtÞ ¼
1
CðaÞ
Z
b
ðy tÞa1 f ðyÞ dy;
ð3Þ
t
where C denotes the Euler’s Gamma function and provided the right side terms are defined. For a ¼ 0, let us define
I0 f ¼ I0þ f ¼ f .
From now, let us consider 0 < a 6 1. The left (resp. right) fractional derivative in the sense of Riemann–Liouville with
inferior limit a (resp. superior limit b) of order a of f is given by:
8t 2a; b;
Da f ðtÞ ¼
d 1a I f ðtÞ;
dt ð4Þ
respectively:
8t 2 ½a; b½;
Daþ f ðtÞ ¼ d 1a I f ðtÞ;
dt þ
ð5Þ
provided the right side terms are defined.
A fractional Lagrangian functional is an application defined by:
La : C2 ð½a; b; Rd Þ ! R
Z b
q#
L qðtÞ; Da qðtÞ; t dt;
ð6Þ
a
where L is a Lagrangian i.e. a C2 application defined by:
L : Rd Rd ½a; b ! R
ð7Þ
ðx; v ; tÞ # Lðx; v ; tÞ:
It is well-known that extremals of a fractional Lagrangian functional can be characterized as solutions of a fractional differential system, see [1]. Precisely, q is a critical point of La if and only if q is a solution of the fractional Euler–Lagrange equation
given by:
@L @L q; Da q; t þ Daþ
q; Da q; t ¼ 0:
@x
@v
ðELa Þ
A dynamical system governed by an equation shaped as (ELa ) is called fractional Lagrangian system.
For a ¼ 1; D1 ¼ D1þ ¼ d=dt and then ðEL1 ) coincides with the classical Euler–Lagrange equation (see [3]).
880
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
2.2. Symmetries
We first review the definition of a one parameter group of diffeomorphisms:
Definition 1. For any real s, let /ðs; Þ : Rd ! Rd be a diffeomorphism. Then, U ¼ f/ðs; Þgs2R is a one parameter group of
diffeomorphisms if it satisfies
1. /ð0; Þ ¼ IdRd ,
2. 8s; s0 2 R; /ðs; Þ /ðs0 ; Þ ¼ /ðs þ s0 ; Þ,
3. / is of class C2 .
Classical examples of one parameter groups of diffeomorphisms are given by translations and rotations.
The action of a one parameter group of diffeomorphisms on a Lagrangian allows to define the notion of a symmetry for a
fractional Lagrangian system:
Definition 2. Let U ¼ f/ðs; Þgs2R be a one parameter group of diffeomorphisms and let L be a Lagrangian. L is said to be
Da -invariant under the action of U if it satisfies:
8q solution of ðELa Þ;
8s 2 R;
L /ðs; qÞ; Da ð/ðs; qÞÞ; t ¼ L q; Da q; t :
ð8Þ
2.3. A fractional Noether’s theorem
Cresson [12] and Torres et al. [14] proved the following result:
Lemma 3. Let L be a Lagrangian Da -invariant under the action of a one parameter group of diffeomorphisms U ¼ f/ðs; Þgs2R .
Then, the following equality holds for any solution q of (ELa ):
Da
@/
@L
@/
@L
ð0; qÞ ð0; qÞ Daþ
ðq; Da q; tÞ ðq; Da q; tÞ ¼ 0:
@s
@v
@s
@v
ð9Þ
In the classical case a ¼ 1, the classical Leibniz formula allows to rewrite (9) as the derivative of a product. Precisely, for
a ¼ 1, Lemma 3 leads to the classical Noether’s theorem given by:
Theorem 4 (Classical Noether’s theorem). Let L be a Lagrangian d=dt-invariant under the action of a one parameter group of
diffeomorphisms U ¼ f/ðs; Þgs2R . Then, the following equality holds for any solution q of ðEL1 ):
d @/
@L
_ tÞ ¼ 0;
ð0; qÞ ðq; q;
dt @s
@v
ð10Þ
where q_ is the classical derivative of q, i.e. dq=dt.
Theorem 4 provides an explicit constant of motion for any classical Lagrangian systems admitting a symmetry. In the fractional case, such a simple formula allowing to rewrite (9) as a total derivative with respect to t is not known yet. Nevertheless, the next theorem provides such a formula in an explicit form:
Theorem 5 (Transfer formula). Let f ; g 2 C1 ð½a; b; Rd Þ assuming the following Condition (C): the sequences of functions
a
ðpÞ
ðIp
Þp2N and
f g
a
ðf ðpÞ Ip
þ gÞp2N converge uniformly to 0 on ½a; b.
Then, the following equality holds:
Da f g f Daþ g ¼
"
#
1
d X
a
rþ1a
ðrÞ
ðrÞ
ð1Þr Irþ1
f
g
þ
f
I
g
:
þ
dt r¼0
ð11Þ
Proof. Let f ; g 2 C1 ð½a; b; Rd Þ. By induction and using the classical Leibniz formula at each step, we prove that for any p 2 N ,
the following equalities both hold:
a
D f g ¼
a
ð1Þp Ip
f
g
ðpÞ
"
#
p1
d X
r rþ1a
ðrÞ
þ
ð1Þ I
f g
dt r¼0
ð12Þ
and
a
f Dþ g ¼ f
ðpÞ
a
Ip
þ g
"
#
p1
d X ðrÞ rþ1a
þ
f Iþ
g :
dt r¼0
ð13Þ
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
881
For any p 2 N , we define up the following function:
up ¼
p1
X
a
ð1Þr Irþ1
f g ðrÞ :
ð14Þ
r¼0
a
ðpÞ
According to (12), for any p 2 N ; u_ p ¼ Da f g ð1Þp Ip
. Hence, the assumption made on the sequence of functions
f g
pa
ðpÞ
_
ðI f g Þp2N implies that the sequence ðup Þp2N converges uniformly to Da f g on ½a; b. Moreover, let us note that ðup Þp2N
point-wise converges in t ¼ a. Finally, we can conclude that the sequence ðup Þp2N converges uniformly on ½a; b to a function
u equal to
u¼
1
X
a
ð1Þr Irþ1
f g ðrÞ
ð15Þ
r¼0
and satisfying u_ ¼ Da f g. Similarly, one can prove that:
"
#
1
d X
rþ1a
ðrÞ
f Iþ
g ¼ f Daþ g;
dt r¼0
ð16Þ
which completes the proof. h
Thus, combining Lemma 3 and this transfer formula, we prove:
Theorem 6 (A fractional Noether’s theorem). Let L be a Lagrangian Da -invariant under the action of a one parameter group of
diffeomorphisms U ¼ f/ðs; Þgs2R . Let q be a solution of (ELa ) and let f and g denote:
f ¼
@/
@L
ðq; Da q; tÞ:
ð0; qÞ and g ¼
@s
@v
ð17Þ
If f and g satisfy Condition (C), then the following equality holds:
"
#
1
d X
a
rþ1a
ðrÞ
ðrÞ
ð1Þr Irþ1
f
g
þ
f
I
g
¼ 0:
þ
dt r¼0
ð18Þ
This theorem provides an explicit algorithmic way to compute a constant of motion for any fractional Lagrangian systems
admitting a symmetry. An arbitrary closed approximation of this quantity can be obtained with a truncature of the infinite
sum.
2.4. Comments
2.4.1. About previous studies
Previous results in this direction have been obtained by several authors, see [14,4,23]. Let us note that these authors consider symmetries modifying also the time variable in contrary to this paper. Nevertheless, considering symmetries not
involving a time transformation, Lemma 3 and their results coincide. Then, let us make the following comments explaining
the interest of Theorem 6 in this case:
– In [14], Torres et al. define a fractional conservation law for fractional Lagrangian systems. Precisely, denoting by Da0 the
bilinear operator given by:
Da0 ðf ; gÞ ¼ Da f g f Daþ g;
ð19Þ
d
d
C : R R ½a; b ! R
is a fractional-conserved quantity of a fractional sysðx; v ; tÞ # Cðx; v ; tÞ
tem if it is possible to write C in the form Cðx; v ; tÞ ¼ C 1 ðx; v ; tÞ C 2 ðx; v ; tÞ with Da0 C 1 ðq; Da q; tÞ; C 2 ðq; Da q; tÞ ¼ 0 for any
authors give the following definition: a function
solution q of the considered system. They deduce easily that @/=@sð0; xÞ @L=@ v ðx; v ; tÞ is then a fractional-conserved quantity of (ELa ). Two important difficulties appear:
Although all these notions and results coincide with the classical ones when a ¼ 1, we do not have that Da0 ðf ; gÞ ¼ 0
implies f g is constant.
The decomposition C ¼ C 1 C 2 is not unique and it leads to many issues. Indeed, the scalar product is commutative and
then C ¼ C 1 C 2 ¼ C 2 C 1 . Nevertheless, the operator Da0 is not symmetric! Hence, in the result of Torres et al., we have:
Cðx; v ; tÞ ¼
@/
@L
@L
@/
ð0; xÞ ð0; xÞ
ðx; v ; tÞ ¼
ðx; v ; tÞ @s
@v
@v
@s
is a fractional-conserved quantity of (ELa ) but we do not have:
ð20Þ
882
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
Da0
@L
@/
ð0; qÞ ¼ 0:
ðq; Da q; tÞ;
@v
@s
ð21Þ
Let us note that the work developed in [23] is similar to this previous one.
– In [4], Atanackovic̀ et al. obtain in Theorem 15 [4] a formulation of a constant of motion for a fractional Lagrangian systems admitting a symmetry: for any solution q of (ELa ), let f ¼ @/=@sð0; qÞ and g ¼ @L=@ v ðq; Da q; tÞ, the following element
Rt
t # a Da f g f Daþ gdy is constant on ½a; b. This result is then unsatisfactory because the constant of motion is not
explicit.
Hence, these definitions and results define implicitly a constant of motion by functional relations (see [14,23]) or integral
relations (see [4]). Theorem 6 is then interesting because it derives explicitly a constant of motion in terms of symmetry
group and Lagrangian.
Let us make the following remark concerning the time transformation. In each of these papers [14,4,23], authors finally
prove that a symmetry of a fractional Lagrangian system (with time transformation) implies an equality of the following type:
Da f g f Daþ g ¼ 0;
ð22Þ
where f depends on the symmetry group and g depends on the Lagrangian L. Consequently, the transfer formula given in
Theorem 5 is also relevant and then an explicit constant of motion can be obtained. Finally, the study developed in this paper
is also available in the case of time transformation and it will be done in a forthcoming paper.
2.4.2. About condition (C)
Condition (C) could seem strong or too particular. In order to make it more concrete and understand what classes of functions satisfy it, we give the two following sufficient conditions:
Proposition 7. Let f ; g 2 C1 ð½a; b; Rd Þ.
1. If f and g satisfy the two following conditions:
!
!
ðb tÞp1 ðpÞ
ðt aÞp1 ðpÞ
max
kf ðtÞk ! 0 and max
kg ðtÞk ! 0
p!1
p!1
t2½a;b
t2½a;b
ðp 1Þ!
ðp 1Þ!
ð23Þ
then f and g satisfy Condition (C).
2. If there exists M > 0 such that:
8p 2 N ;
8t 2 ½a; b;
kf ðpÞ ðtÞk 6 M
and kg ðpÞ ðtÞk 6 M
ð24Þ
then f and g satisfy Condition (C).
a
ðpÞ
Þp2N converges uniformly to 0. The proof is similar for the
Proof. 1. Let us prove that the sequence of functions ðIp
f g
pa
ðpÞ
sequence ðf I gÞp2N . Let us denote M ¼ maxðkf ðtÞkÞ. For any p 2 N and any t 2 ½a; b:
t2½a;b
Z t
pa
pa
ðpÞ
pa1
6 M ðt aÞ
I f ðtÞ g ðpÞ ðtÞ ¼ g ðtÞ ðt
yÞ
f
ðyÞ
dy
kg ðpÞ ðtÞk:
Cðp aÞ
Cðp þ 1 aÞ
a
ð25Þ
Finally, since Cðp þ 1 aÞ P ðp 1Þ!Cð2 aÞ:
1a
pa
ðt aÞp1 ðpÞ
ðb aÞ1a ðt aÞp1 ðpÞ
I f ðtÞ g ðpÞ ðtÞ 6 M ðt aÞ
kg ðtÞk 6 M
kg ðtÞk;
Cð2 aÞ ðp 1Þ!
Cð2 aÞ ðp 1Þ!
ð26Þ
which concludes the proof.
2. It follows from the first result. h
For example, if f is polynomial and g is the exponential function, one can prove that f and g satisfy the point 2 of
Proposition 7 and then Condition (C). If gðtÞ ¼ 1=t on ½a; b with a > 0, one can notice that the point 2 of Proposition 7 is
not satisfied anymore. Nevertheless, the point 1 is true and consequently the functions f and g satisfy Condition (C). In
general, every couple of analytic functions satisfy Condition (C) and consequently (11) is true.
3. A discrete fractional Noether’s theorem
We study the existence of discrete conservation laws for discrete fractional Lagrangian systems in the sense of [10]. Using
the same strategy as in the continuous case, we introduce the notion of discrete symmetry and prove a discrete fractional
Noether’s theorem providing an explicit computable discrete constant of motion.
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
883
3.1. Reminder about discrete fractional Lagrangian systems
We follow the definition of discrete fractional Lagrangian systems given in [10] to which we refer for more details.
Let us consider N 2 N ; h ¼ ðb aÞ=N the step size of the discretization and s ¼ ðtk Þk¼0;...;N ¼ ða þ khÞk¼0;...;N the usual partition of the interval ½a; b. Let us define Da and Daþ the following discrete analogous of Da and Daþ respectively:
Da : ðRd ÞNþ1 ! ðRd ÞN
!
k
1X
Q#
ar Q kr
;
a
h r¼0
k¼1;...;N
ð27Þ
Daþ : ðRd ÞNþ1 ! ðRd ÞN
!
Nk
1X
Q#
ar Q kþr
;
a
h r¼0
k¼0;...;N1
ð28Þ
where the elements ðar Þr2N are defined by a0 ¼ 1 and
8r 2 N ;
ar ¼
ðaÞð1 aÞ . . . ðr 1 aÞ
:
r!
ð29Þ
These discrete fractional operators are approximations of the continuous ones. We refer to [25] for more details.
A discrete fractional Euler–Lagrange equation, of unknown Q 2 ðRd ÞNþ1 , is defined by:
@L
@L
ðQ ; Da Q; sÞ ¼ 0;
ðQ ; Da Q ; sÞ þ Daþ
@x
@v
ðELah Þ
where L is a Lagrangian. In such a case, we speak of a discrete Lagrangian system.
The terminology is justified by the fact that solutions of (ELah ) correspond to discrete critical points of the discrete
fractional Lagrangian functional defined by:
Lah : ðRd ÞNþ1 ! R
N
X
Q # h L Q k ; ðDa Q Þk ; tk :
ð30Þ
k¼1
We refer to [10] for a proof.
In the case a ¼ 1, the discrete operators D1 correspond to the implicit and explicit Euler approximations of d=dt and (ELah )
is just the discrete Euler–Lagrange equation obtained in [22,15].
Remark 8. Different definitions for discrete fractional Lagrangian systems are given in [8,9] in which the authors derive
discrete fractional Euler–Lagrange equations. The main difference is the use of a forward jump operator acting on the
variable Q in the definition of discrete fractional functional (30) (see Section 3.2 in [8] and Section 3.2 in [9]). As a
consequence, when a ¼ 1 the corresponding discrete fractional Euler–Lagrange equation reduces to (see Corollary 3.1 and
Remark 3.1 in [8])
@L
@L
ðrðQ Þ; Da Q ; sÞ Da
ðrðQ Þ; Da Q ; sÞ ¼ 0;
@x
@v
ð31Þ
where rðQ Þk ¼ Q kþ1 ; k ¼ 0; . . . ; N 1.
These definitions differ at least in two directions:
– The use of the forward jump operator r seems artificial from the point of view of embedding formalisms developed in
[10]. This was already observed in ([13, Section 7], in the setting of the time-scale embedding.
– The discrete Euler–Lagrange Eq. (31) can be considered as a discretisation of a continuous Lagrangian systems. From this
point of view, the numerical approximation provided by (31) is not satisfying and does not correspond to the classical
discretisation of Lagrangian systems leading to variational integrators (see [22,15]). It follows from the fact that the second
order derivative is only approximated to the order 1 in (31) instead of the order 2 in our case. Moreover, such a scheme
has a very bad behavior with respect to the conservation of energy (see [15]).
3.2. Discrete symmetries
Here again, a discrete symmetry of a discrete fractional Lagrangian system is based on the action of a one parameter
group of transformations on the associated Lagrangian:
884
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
Definition 9. Let U ¼ f/ðs; Þgs2R be a one parameter group of diffeomorphisms and let L be a Lagrangian. L is said to be
Da -invariant under the action of U if it satisfies:
8Q solution of ðELah Þ;
8s 2 R;
L /ðs; QÞ; Da ð/ðs; Q ÞÞ; s ¼ L Q Þ; Da Q ; s :
ð32Þ
Then, we can prove the following discrete version of Lemma 3:
Lemma 10. Let L be a Lagrangian Da -invariant under the action of a one parameter group of diffeomorphisms U ¼ f/ðs; Þgs2R .
Then, the following equality holds for any solution Q of (ELah ):
Da
@/
@L
@/
@L
ð0; Q Þ ð0; Q Þ Daþ
ðQ; Da Q ; sÞ ðQ ; Da Q; sÞ ¼ 0:
@s
@v
@s
@v
ð33Þ
Proof. This proof is a direct adaptation to the discrete case of the proof of Lemma 3. Let Q 2 ðRd ÞNþ1 be a solution of (ELah ). Let
us differentiate Eq. (32) with respect to s and invert the operators Da and @=@s. We finally obtain for any s 2 R and any
k 2 f1; . . . ; N 1g:
Da
@L @/
@/
@L ðs; Q Þ /ðs; Q k Þ; Da ð/ðs; Q k ÞÞ; t k ðs; Q k Þ ¼ 0:
/ðs; Q k Þ; Da ð/ðs; Q k ÞÞ; t k þ
@s
@x
@s
k @v
ð34Þ
Since /ð0; Þ ¼ IdRd and Q is a solution of (ELah ), taking s ¼ 0 in (34) leads to (33). h
3.3. A discrete fractional Noether’s theorem
Let us remind that our aim is to express explicitly a discrete constant of motion for discrete fractional Lagrangian systems
admitting a discrete symmetry. As in the continuous case, our result is based on Lemma 10. Let us note that the following
implication holds:
D1 F ¼ 0 ) 9c 2 R;
8F 2 RNþ1 ;
8k ¼ 0; . . . ; N; F k ¼ c:
ð35Þ
Namely, if the discrete derivative of F vanishes, then F is constant. Our aim is then to write (33) as a discrete derivative (i.e. as
D1 of an explicit quantity).
We first introduce some notations and definitions. The shift operator denoted by r is defined by
r : ðRd ÞNþ1 ! ðRd ÞNþ1
Q # rðQÞ ¼ ðQ kþ1 Þk¼0;...;N
ð36Þ
with the convention Q Nþ1 ¼ 0. We also introduce the following square matrices of length ðN þ 1Þ. First, A1 ¼ IdNþ1 and then,
for any r 2 f2; . . . ; N 1g, the square matrices Ar 2 MNþ1 defined by:
8i; j ¼ 0; . . . ; N;
8
>
<0
ðAr Þi;j ¼ dfj¼0g dfr6ig df06ij6r1g df16j6Nrg
>
: ðA Þ
r N1;j
if i ¼ 0
if 1 6 i 6 N 1
if i ¼ N
ð37Þ
where d is the Kronecker symbol.
For example, for N ¼ 5, the matrices Ar are given by: A1 ¼ Id6 and
0
0
0
0
0
0 0
B 0 1 0
0 0
B
B
B 1 1 1 0 0
A2 ¼ B
B 1 0 1 1 0
B
B
@1 0
0 1 0
1
0
0
1
0C
C
C
0C
C;
0C
C
C
0A
1 0 0
0
0
0
0
0 0 0
B 0 1 0 0 0
B
B
B 0 1 1 0 0
A3 ¼ B
B 1 1 1 0 0
B
B
@ 1 0 1 0 0
1
0
1
0C
C
C
0C
C;
0C
C
C
0A
1 0 0 0
0
0
B0
B
B
B0
A4 ¼ B
B0
B
B
@1
0
0 0 0 0
1
1 0 0 0 0 C
C
C
1 0 0 0 0 C
C:
1 0 0 0 0 C
C
C
1 0 0 0 0 A
1 1 0 0 0 0
Considering these previous elements, we state the following result:
Theorem 11 (A discrete fractional Noether’s theorem). Let L be a Lagrangian Da -invariant under the action of a one parameter
group of diffeomorphisms U ¼ f/ðs; Þgs2R . Then, the following equality holds for any solution Q of (ELah ):
D1
"
N 1
X
ar Ar
r¼1
#
@/
a
r @L
ð0; Q Þ r
ðQ ; D Q; sÞ
¼ 0:
@s
@v
ð38Þ
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
885
Combining (35) and (38), Theorem 11 provides a constant of motion for discrete fractional Lagrangian systems admitting
a symmetry. Let us note that the discrete conservation law is not only explicit but computable in finite time. We give an
example in the next section.
Before giving its proof, we remark that Theorem 11 takes a particular simple expression when a ¼ 1. Indeed, since ar ¼ 0
for any r P 2 in this case, we obtain:
Theorem 12 (Discrete classical Noether’s theorem). Let L be a Lagrangian D1 -invariant under the action of a one parameter group
of diffeomorphisms U ¼ f/ðs; Þgs2R . Then, the following equality holds for any solution Q of ðEL1h ):
D1
@/
@L
ð0; Q Þ r
ðQ ; D1 Q ; sÞ
¼ 0:
@s
@v
ð39Þ
This result is a reformulation of the discrete Noether’s theorem proved in [22,15]. It corresponds to our Lemma 10 with
a ¼ 1 using the following discrete Leibniz formula:
8F; G 2 ðRd ÞNþ1 ;
D1 ðF rðGÞÞ ¼ D1 F G F D1þ G:
ð40Þ
Now, let us prove Theorem 11:
Proof of Theorem 11 According to Lemma 10, Eq. (33) holds for any k ¼ 1; . . . ; N 1. Let F ¼ @/=@sð0; Q Þ and
a
G ¼ @L=@ v ðQ ; Da Q ; sÞ. Let us multiply (33) by h and obtain the following equality for any k ¼ 1; . . . ; N 1:
!
k
X
!
Nk
X
ar F kr Gk F k ar Gkþr ¼ 0;
r¼0
ð41Þ
r¼0
and since a0 ¼ 1:
!
k
X
a1 ðF k1 Gk F k Gkþ1 Þk þ
!
Nk
X
ar F kr Gk F k ar Gkþr ¼ 0;
r¼2
ð42Þ
r¼2
then:
1
1 D ðF
a
1
rðGÞÞk ¼
h
"
!
k
X
ar F kr Gk F k r¼2
!#
Nk
X
ar Gkþr
ð43Þ
:
r¼2
Let us denote J k the right term of (43) for any k ¼ 1; . . . ; N 1. We are going to write J k as the discrete derivative of its discrete
anti-derivative: it corresponds to the discrete version of the method of Atanackovic̀ in [4]. For any k ¼ 1; . . . ; N 1, we obtain
P
J k ¼ ðD1 HÞk where Hi :¼ h ij¼1 J j for any i ¼ 0; . . . ; N with the convention H0 ¼ J N ¼ 0. Let us provide an explicit formulation of
the element H ¼ ðHi Þi¼0;...;N .
Case 1 6 i 6 N 1:
Hi ¼
"
j
i
X
X
Pi Pj
j¼2
¼
r¼2
Nj
X
r¼2
j¼1
As
ar F jr Gj Pi
r¼2
r¼2
Pi
j¼r ,
i X
i
X
j¼2 r¼2
r¼2 j¼r
ar F jr Gj ¼
Hi ¼
Pi PNj
j¼1
r¼2
¼
j
i X
X
Nj
i X
X
j¼2 r¼2
j¼1 r¼2
ar F jr Gj ar F j Gjþr :
ð44Þ
ar ðF rr ðGÞÞj :
ð45Þ
we have:
j
i X
X
Then, since
#
ar F j Gjþr ¼
ar F jr Gj ¼
Pi PNi
j¼1
r¼2
þ
i X
ir
X
r¼2 j¼0
r¼2 j¼0
ar F j Gjþr ¼
Pi PNj
j¼1
i X
ir
X
Ni X
i
X
r¼2 j¼0
r¼2 j¼1
ar ðF rr ðGÞÞj i X
ir
X
r¼Nþ1i
¼
PNi Pi
ar ðF rr ðGÞÞj r¼2
N1
X
j¼1
þ
PN1
Nr
X
r¼Nþ1i
PNr
j¼1
ar ðF rr ðGÞÞj :
:
ð46Þ
r¼Nþ1i j¼1
Finally, we have:
Hi ¼
N 1 X
N
X
ar Ar ði; jÞðF rr ðGÞÞj ;
ð47Þ
r¼2 j¼0
where the elements ðAr ði; jÞÞ are defined for r ¼ 2; . . . ; N 1 and j ¼ 0; . . . ; N as the real coefficients in front of ar ðF rr ðGÞÞj .
Our aim is then to express the values of these elements. From (46), we have for any r ¼ 2; . . . ; N 1 and any j ¼ 0; . . . ; N:
Ar ði; jÞ ¼ dfr6ig df06j6irg dfr6Nig df16j6ig dfNþ1i6rg df16j6Nrg :
For example, for j ¼ 0, we have:
ð48Þ
886
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
8r ¼ 2; . . . ; N 1;
Ar ði; 0Þ ¼ dfr6ig :
ð49Þ
Let r 2 f2; . . . ; N 1g and j 2 f1; . . . ; Ng. Let us prove that:
Ar ði; jÞ ¼ df06ij6r1g dfj6Nrg ¼ dfj6ig dfij6r1g dfj6Nrg :
ð50Þ
Let us see the four following cases:
– If j > i, then j > i r. Moreover, if N þ 1 i 6 r then j > N r. In this case, Ar ði; jÞ ¼ 0 0 0 ¼ 0.
– If i j > r 1 then r 6 i; j 6 i r; j 6 i and j 6 N r. Then, Ar ði; jÞ ¼ 1 1 0 ¼ 0 or Ar ði; jÞ ¼ 1 0 1 ¼ 0 depending on
r 6 N i or N þ 1 i 6 r. Finally, in this case, Ar ði; jÞ ¼ 0.
– If j > N r then j > i r. Moreover, if r 6 N i then j > i. In this case, Ar ði; jÞ ¼ 0 0 0 ¼ 0.
– If j 6 i; i j 6 r 1 and j 6 N r, then i r < j. In this case, Ar ði; jÞ ¼ 0 1 0 ¼ 1 or Ar ði; jÞ ¼ 0 0 1 ¼ 1 depending
on r 6 N i or N þ 1 i 6 r. Finally, in this case, Ar ði; jÞ ¼ 1.
Consequently, (50) holds for any r 2 f2; . . . ; N 1g and any j 2 f1; . . . ; Ng. Finally, from (50) and (49), we have:
8r ¼ 2; . . . ; N 1;
8j ¼ 0; . . . ; N;
Ar ði; jÞ ¼ dfj¼0g dfr6ig df06ij6r1g df16j6Nrg :
ð51Þ
Case i ¼ 0 or i ¼ N. As H0 ¼ 0, for any r ¼ 2; . . . ; N 1 and any j ¼ 0; . . . ; N, we define Ar ð0; jÞ ¼ 0. As HN ¼ HN1 , for any
r ¼ 2; . . . ; N 1 and any j ¼ 0; . . . ; N, we define Ar ðN; jÞ ¼ Ar ðN 1; jÞ.
Hence, for any r ¼ 2; . . . ; N 1, the elements ðAr ði; jÞÞ are defined for i ¼ 0; . . . ; N and j ¼ 0; . . . ; N. Then, we denote by Ar
matrix ðAr ði; jÞÞ06i;j6N 2 MNþ1 . Finally, from (43), we have proved that D1 ða1 F rðGÞ þ HÞ ¼ 0
PN1
H ¼ r¼2 ar Ar ðF rr ðGÞÞ. Finally, denoting A1 ¼ IdNþ1 2 MNþ1 , we conclude the proof. h
the
where
Remark 13. Using the discrete Leibniz formula given by (40), another choice in order to give a discrete version of (18) from
Lemma 10 would be to apply the discrete version of the method used in Section 2.3. Nevertheless, we would encounter many
numerical difficulties. Firstly, such a method would imply the use of the operator Dp but this operator approximates the
operator ðd=dtÞp only for tk with k P p. For the p first terms, we obtain in general a numerical blow up. Secondly, the use of
p
operator Dp implies the use of h . Hence, for a large enough p, we exceed the machine precision.
3.4. Example: the discrete fractional harmonic oscillator
We consider the classical bi-dimensional example (d ¼ 2) of the quadratic Lagrangian Lðx; v ; tÞ ¼ ðx2 þ v 2 Þ=2 with
½a; b ¼ ½0; 1 and a ¼ 1=2. Then, L is Da -invariant under the action of the rotations given by:
/ : R R2 ! R2
x1
cosðsÞ sinðsÞ
:
ðs; x1 ; x2 Þ #
x2
sinðsÞ cosðsÞ
ð52Þ
We choose N ¼ 600; Q 0 ¼ ð1; 2Þ and Q N ¼ ð2; 1Þ. Let F and G denote:
2
−0.15
Q1
Q2
1.8
1.6
−0.152
1.4
−0.153
1.2
−0.154
1
−0.155
0.8
−0.156
0.6
−0.157
0.4
−0.158
0.2
−0.159
0
0
0.2
0.4
0.6
0.8
constant of motion
−0.151
1
−0.16
0
Fig. 1. Computation of (ELah ).
0.2
0.4
0.6
0.8
1
L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 878–887
F¼
@/
@L
ð0; QÞ ¼ ðQ 2 ; Q 1 Þ and G ¼
ðQ ; Da Q; sÞ ¼ ðDa Q 1 ; Da Q 2 Þ:
@s
@v
887
ð53Þ
The computation of (ELah ) gives the following graphics:
– Fig. 1a represents the discrete solutions of (ELah )
P
r
– and Fig. 1b represents the explicit computable quantity N1
r¼1 ar Ar ðF r ðGÞÞ. As expected from Theorem 11, this quantity
a
is a constant of motion of (ELh ).
4. Conclusion
In this paper, we have proved a fractional Noether’s theorem for fractional Lagrangian systems invariant under a symmetry group both in the continuous and discrete cases. In contrary to previous results in this direction (in the continuous case),
it provides an explicit conservation law. Moreover, the given formula can be algorithmically implemented and in the discrete
case, the conservation law is moreover computable in a finite number of steps.
As explained in Section 2.4.1, symmetries with time transformation can be considered and the transfer formula given by
Theorem 5 can be used in order to obtain an explicit constant of motion. Hence, this study is also available in this case and it
will be done in a forthcoming paper.
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