Commun Nonlinear Sci Numer Simulat

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1 Commun Nonlinear Sci Numer Simulat 18 (2013) Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: 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 article info abstract 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 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 ¼ðqðtÞÞ t2½a;bš (resp. Q ¼ðQ k Þ k¼0;...;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: CqðtÞ ð Þ ¼ c; 8t 2½a; bš: ðresp: CðQ k Þ¼c; 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. addresses: bourdin.l@etud.univ-pau.fr (L. Bourdin), jacky.cresson@univ-pau.fr (J. Cresson), isabelle.greff@univ-pau.fr (I. Greff) /$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.

2 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 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 R d where d 2 N is the dimension (N denotes the set of positive integer). The trajectories of these systems are curves q in C 0 ð½a; bš; R d Þ where a < b are two reals 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; bš with values in R d. 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 2Ša; bš; I a f ðtþ ¼ 1 CðaÞ Z t a ðt yþ a1 f ðyþdy; ð2þ respectively: 8t 2½a; b½; I a f ðtþ ¼ 1 CðaÞ Z b t ðy tþ a1 f ðyþdy; ð3þ where C denotes the Euler s Gamma function and provided the right side terms are defined. For a ¼ 0, let us define I 0 f ¼ I0 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 2Ša; bš; D a f ðtþ ¼ d dt I 1a f ðtþ; ð4þ respectively: 8t 2½a; b½; D a f ðtþ ¼d dt I 1a f ðtþ; ð5þ provided the right side terms are defined. A fractional Lagrangian functional is an application defined by: L a : C 2 ð½a; bš; R d ÞR q # Z b a LqðtÞ; D a qðtþ; t ð6þ dt; where L is a Lagrangian i.e. a C 2 application defined by: L : R d R d ½a; bš R ðx;v; tþ # Lðx;v; tþ: ð7þ 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 L a if and only if q is a solution of the fractional Euler Lagrange equation q; Da q; t D ¼ 0: ðel a q; Da q; t A dynamical system governed by an equation shaped as (EL a ) is called fractional Lagrangian system. For a ¼ 1; D 1 ¼D1 ¼ d=dt and then ðel1 ) coincides with the classical Euler Lagrange equation (see [3]).

3 880 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) Symmetries We first review the definition of a one parameter group of diffeomorphisms: Definition 1. For any real s, let /ðs; Þ : R d R d be a diffeomorphism. Then, U ¼f/ðs; Þg s2r is a one parameter group of diffeomorphisms if it satisfies 1. /ð0; Þ ¼ Id R d, 2. 8s; s 0 2 R; /ðs; Þ /ðs 0 ; Þ ¼ /ðs s 0 ; Þ, 3. / is of class C 2. 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 ¼f/ðs; Þg s2r be a one parameter group of diffeomorphisms and let L be a Lagrangian. L is said to be D a -invariant under the action of U if it satisfies: 8q solution of ðel a Þ; 8s 2 R; L /ðs; qþ; D a ð /ðs; qþ Þ; t ¼ Lq; D a 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 D a -invariant under the action of a one parameter group of diffeomorphisms U ¼f/ðs; Þg s2r. Then, the following equality holds for any solution q of (EL a ): D @v ðq; Da q; ðq; Da q; tþ ¼ 0: 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 ¼f/ðs; Þg s2r. Then, the following equality holds for any solution q of ðel 1 ): ð0; ðq; _q; tþ ¼ 0; 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 2C 1 ð½a; bš; R d Þ assuming the following Condition (C): the sequences of functions ði pa f gðpþ Þ p2n and ðf ðpþ I pa gþ p2nconverge uniformly to 0 on ½a; bš. Then, the following equality holds: D a f g f Da g ¼ d X 1 ð1þ r I r1a f g ðrþ f ðrþ I r1a g : ð11þ dt Proof. Let f ; g 2C 1 ð½a; bš; R d 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: D a f g ¼ð1Þp I pa f gðpþ d dt X p1 ð1þ r I r1a f g ðrþ ð12þ and f D a g ¼ f ðpþ I pa g d X p1 f ðrþ I r1a g : ð13þ dt

4 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) For any p 2 N, we define u p the following function: u p ¼ Xp1 ð1þ r I r1a f g ðrþ : ð14þ According to (12), for any p 2 N ; _u p ¼ D a f g ð1þp I pa f gðpþ. Hence, the assumption made on the sequence of functions ði pa f gðpþ Þ p2n implies that the sequence ð _u p Þ p2n converges uniformly to D a f g on ½a; bš. Moreover, let us note that ðu pþ p2n point-wise converges in t ¼ a. Finally, we can conclude that the sequence ðu p Þ p2n converges uniformly on ½a; bš to a function u equal to u ¼ X1 ð1þ r I r1a f g ðrþ ð15þ and satisfying _u ¼ D a f g. Similarly, one can prove that: d X 1 f ðrþ I r1a g ¼f D a dt g; 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 D a -invariant under the action of a one parameter group of diffeomorphisms U ¼f/ðs; Þg s2r. Let q be a solution of (EL a ) and let f and g denote: ð0; qþ and ðq; Da q; tþ: ð17þ If f and g satisfy Condition (C), then the following equality holds: d X 1 ð1þ r I r1a f g ðrþ f ðrþ I r1a g ¼ 0: dt 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. ð18þ 2.4. Comments 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 D a 0 the bilinear operator given by: D a 0 ðf ; gþ ¼Da f g f Da g; authors give the following definition: a function C : Rd R d ½a; bš R is a fractional-conserved quantity of a fractional system if it is possible to write C in the form Cðx; v; tþ ¼C 1 ðx; v; tþc 2 ðx; v; tþ with D a 0 C1 ðq; D a q; tþ; C2 ðq; D a q; tþ ¼ 0 for any ðx; v; tþ # Cðx; v; tþ solution q of the considered system. They deduce easily xþ@l=@vðx; v; tþ is then a fractional-conserved quantity of (EL a ). Two important difficulties appear: Although all these notions and results coincide with the classical ones when a ¼ 1, we do not have that D a 0ðf ; gþ ¼0 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 D a 0 is not symmetric Hence, in the result of Torres et al., we have: Cðx;v; ð0; ðx; v; tþ ðx;v; tþ@/ ð0; ð20þ ð19þ is a fractional-conserved quantity of (EL a ) but we do not have:

5 882 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) D Da q; tþ; ð0; qþ ¼ Let us note that the work developed in [23] is similar to this previous one. In[4], Atanackovic` 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 (EL a ), let f qþ and g D a q; tþ, the following element t # R t a Da f g f Da gdy is constant on ½a; bš. 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: ð21þ D a f g f Da 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 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 2C 1 ð½a; bš; R d Þ. 1. If f and g satisfy the two following conditions: ðb tþ p1 max kf ðpþ ðtþk 0 and max t2½a;bš ðp 1Þ p1 t2½a;bš ðt aþ p1 kg ðpþ ðtþk ðp 1Þ p1 ð23þ then f and g satisfy Condition (C). 2. If there exists M > 0 such that: 8p 2 N ; 8t 2½a; bš; kf ðpþ ðtþk 6 M and kg ðpþ ðtþk 6 M ð24þ then f and g satisfy Condition (C). Proof. 1. Let us prove that the sequence of functions ði pa f gðpþ Þ p2n converges uniformly to 0. The proof is similar for the sequence ðf ðpþ I pa gþ p2n. Let us denote M ¼ maxðkf ðtþkþ. For any p 2 N and any t 2½a; bš: t2½a;bš Z I pa f ðtþgðpþ ðtþ g ðpþ ðtþ t ¼ Cðp aþ ðt yþ pa1 f ðyþdy 6 M ðt aþpa Cðp 1 aþ kgðpþ ðtþk: ð25þ Finally, since Cðp 1 aþ P ðp 1ÞCð2 aþ: I pa f ðtþgðpþ ðtþ ðt aþ 1a 6 M Cð2 aþ which concludes the proof. 2. It follows from the first result. h a ðt aþ p1 kg ðpþ ðb aþ1a ðtþk 6 M ðp 1Þ Cð2 aþ ðt aþ p1 kg ðpþ ðtþk; ðp 1Þ 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 ¼1=t on ½a; bš 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. ð26þ 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.

6 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 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 ¼ðb aþ=n the step size of the discretization and s ¼ðt k Þ k¼0;...;n ¼ða khþ k¼0;...;n the usual partition of the interval ½a; bš. Let us define D a and Da the following discrete analogous of Da and Da respectively: D a : ðrd Þ N1 ðr d Þ N Q # 1 X k h a a r Q kr ; k¼1;...;n ð27þ D a : ðrd Þ N1 ðr d Þ N Q # 1 X Nk h a a r Q kr k¼0;...;n1 where the elements ða r Þ r2n are defined by a 0 ¼ 1 and 8r 2 N ; a r ¼ ; ðaþð1 aþ...ðr 1 aþ : ð29þ r 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 2ðR d Þ N1, is ðq; Da ðq; Da Q; sþ ¼ 0; ðel a h 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 (EL a h ) correspond to discrete critical points of the discrete fractional Lagrangian functional defined by: L a h : ðrd Þ N1 R Q # h XN LQ k ; ðd a QÞ k ; t ð30þ k : k¼1 We refer to [10] for a proof. In the case a ¼ 1, the discrete operators D 1 correspond to the implicit and explicit Euler approximations of d=dt and (ELa h ) 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 ðrðqþ; Da ðrðqþ; Da Q; sþ ¼ 0; where rðqþ k ¼ Q k1 ; k ¼ 0;...; N 1. These definitions differ at least in two directions: ð31þ 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]) 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:

7 884 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) Definition 9. Let U ¼f/ðs; Þg s2r be a one parameter group of diffeomorphisms and let L be a Lagrangian. L is said to be D a -invariant under the action of U if it satisfies: 8Q solution of ðel a h Þ; 8s 2 R; L /ðs; QÞ; Da ð /ðs; QÞ Þ; s ¼ LQÞ; D a Q; s : ð32þ Then, we can prove the following discrete version of Lemma 3: Lemma 10. Let L be a Lagrangian D a -invariant under the action of a one parameter group of diffeomorphisms U ¼f/ðs; Þg s2r. Then, the following equality holds for any solution Q of (EL a h ): D @v ðq; Da Q; ðq; Da Q; sþ ¼ 0: ð33þ Proof. This proof is a direct adaptation to the discrete case of the proof of Lemma 3. Let Q 2ðR d Þ N1 be a solution of (EL a h ). Let us differentiate Eq. (32) with respect to s and invert the operators D a We finally obtain for any s 2 R and any k 2f1;...; N 1g: D /ðs; Q kþ; D a ð /ðs; kþþ; t /ðs; Q kþ; D a ð /ðs; kþþ; t ðs; Q kþ¼0: Since /ð0; Þ ¼ Id R d and Q is a solution of (EL a h ), taking s ¼ 0in(34) leads to (33). h ð34þ 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: 8F 2 R N1 ; D 1 F ¼ 0 )9c 2 R; 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 D 1 of an explicit quantity). We first introduce some notations and definitions. The shift operator denoted by r is defined by r : ðr d Þ N1 ðr d Þ N1 ð36þ Q # rðqþ ¼ðQ k1 Þ k¼0;...;n with the convention Q N1 ¼ 0. We also introduce the following square matrices of length ðn 1Þ. First, A 1 ¼Id N1 and then, for any r 2f2;...; N 1g, the square matrices A r 2M N1 defined by: 8 >< 0 if i ¼ 0 8i; j ¼ 0;...; N; ða r Þ i;j ¼ d fj¼0g d fr6ig d f06ij6r1g d f16j6nrg if 1 6 i 6 N 1 ð37þ >: ða r Þ N1;j if i ¼ N where d is the Kronecker symbol. For example, for N ¼ 5, the matrices A r are given by: A 1 ¼Id 6 and A 2 ¼ ; A 3 ¼ ; A 4 ¼ : B C B C B A Considering these previous elements, we state the following result: Theorem 11 (A discrete fractional Noether s theorem). Let L be a Lagrangian D a -invariant under the action of a one parameter group of diffeomorphisms U ¼f/ðs; Þg s2r. Then, the following equality holds for any solution Q of (EL a h ): " X N1 # a r A r ðq; Da Q; sþ ¼ 0: ð38þ r¼1

8 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 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 a r ¼ 0 for any r P 2 in this case, we obtain: Theorem 12 (Discrete classical Noether s theorem). Let L be a Lagrangian D 1 -invariant under the action of a one parameter group of diffeomorphisms U ¼f/ðs; Þg s2r. Then, the following equality holds for any solution Q of ðel 1 h ): ðq; D1 Q; sþ ¼ 0: 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 2ðR d Þ N1 ; D 1 ðf rðgþþ ¼ D1 F G F D1 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 QÞ and G D a Q; sþ. Let us multiply (33) by ha and obtain the following equality for any k ¼ 1;...; N 1: X k X a Nk r F kr G k F k a r G kr ¼ 0; ð41þ and since a 0 ¼ 1: a 1 ðf k1 G k F k G k1 Þ k Xk a r F kr G k F k X Nk a r G kr ¼ 0; ð42þ then: " a 1 D 1 ðf rðgþþ k ¼ 1 X k X # a Nk r F kr G k F k a r G kr : ð43þ h 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 Atanackovic` in [4]. For any k ¼ 1;...; N 1, we obtain J k ¼ðD 1 HÞ k where H i :¼ h P i j¼1 J j for any i ¼ 0;...; N with the convention H 0 ¼ J N ¼ 0. Let us provide an explicit formulation of the element H ¼ðH i Þ i¼0;...;n. Case 1 6 i 6 N 1: H i ¼ Xi X j a r F jr G j XNj a r F j G jr ¼ Xi X j a r F jr G j Xi X Nj a r F j G jr : ð44þ j¼1 As P i P j j¼2 ¼ P i i P j¼r, we have: X i X j j¼2 a r F jr G j ¼ Xi X i j¼r P Ni Then, since P i P Nj j¼1 ¼ P i j¼1 H i ¼ Xi X ir j¼0 Finally, we have: H i ¼ XN1 X N j¼0 a r ðf r r ðgþ a r F jr G j ¼ Xi P i j¼1 Þ j XNi X i j¼1 j¼0 j¼2 X ir a r F j G jr ¼ Xi P Nj r¼n1i ¼ P Ni a r ðf r r ðgþþ j XN1 j¼0 j¼1 X ir P i j¼1 P N1 X Nr r¼n1i j¼1 a r ðf r r ðgþþ j : ð45þ r¼n1i P Nr j¼1 : a r ðf r r ðgþþ j : ð46þ a r A r ði; jþðf r r ðgþþ j ; ð47þ where the elements ða r ði; jþþ are defined for r ¼ 2;...; N 1 and j ¼ 0;...; N as the real coefficients in front of a r ðf r r ðgþþ j. 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: A r ði; jþ ¼d fr6ig d f06j6irg d fr6nig d f16j6ig d fn1i6rg d f16j6nrg : For example, for j ¼ 0, we have: ð48þ

9 886 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) r ¼ 2;...; N 1; A r ði; 0Þ ¼d fr6ig : ð49þ Let r 2f2;...; N 1g and j 2f1;...; Ng. Let us prove that: A r ði; jþ ¼d f06ij6r1g d fj6nrg ¼d fj6ig d fij6r1g d fj6nrg : ð50þ Let us see the four following cases: Ifj > i, then j > i r. Moreover, if N 1 i 6 r then j > N r. In this case, A r ði; jþ ¼0 0 0 ¼ 0. Ifi j > r 1 then r 6 i; j 6 i r; j 6 i and j 6 N r. Then, A r ði; jþ ¼1 1 0 ¼ 0orA r ði; jþ ¼1 0 1 ¼ 0 depending on r 6 N i or N 1 i 6 r. Finally, in this case, A r ði; jþ ¼0. Ifj > N r then j > i r. Moreover, if r 6 N i then j > i. In this case, A r ði; jþ ¼0 0 0 ¼ 0. Ifj 6 i; i j 6 r 1 and j 6 N r, then i r < j. In this case, A r ði; jþ ¼0 1 0 ¼1orA r ði; jþ ¼0 0 1 ¼1 depending on r 6 N i or N 1 i 6 r. Finally, in this case, A r ði; jþ ¼1. Consequently, (50) holds for any r 2f2;...; N 1g and any j 2f1;...; Ng. Finally, from (50) and (49), we have: 8r ¼ 2;...; N 1; 8j ¼ 0;...; N; A r ði; jþ ¼d fj¼0g d fr6ig d f06ij6r1g d f16j6nrg : ð51þ Case i ¼ 0ori¼N. AsH 0 ¼ 0, for any r ¼ 2;...; N 1 and any j ¼ 0;...; N, we define A r ð0; jþ ¼0. As H N ¼ H N1, for any r ¼ 2;...; N 1 and any j ¼ 0;...; N, we define A r ðn; jþ ¼A r ðn 1; jþ. Hence, for any r ¼ 2;...; N 1, the elements ða r ði; jþþ are defined for i ¼ 0;...; N and j ¼ 0;...; N. Then, we denote by A r the matrix ða r ði; jþþ 06i;j6N 2M N1. Finally, from (43), we have proved that D 1 a ð 1F rðgþhþ ¼ 0 where H ¼ P N1 a r A r ðf r r ðgþþ. Finally, denoting A 1 ¼Id N1 2M N1, we conclude the proof. h 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 D p but this operator approximates the operator ðd=dtþ p only for t k with k P p. For the p first terms, we obtain in general a numerical blow up. Secondly, the use of operator D p implies the use of hp. Hence, for a large enough p, we exceed the machine precision Example: the discrete fractional harmonic oscillator We consider the classical bi-dimensional example (d ¼ 2) of the quadratic Lagrangian Lðx; v; tþ ¼ðx 2 v 2 Þ=2 with ½a; bš ¼½0; 1Š and a ¼ 1=2. Then, L is D a -invariant under the action of the rotations given by: / : R R 2 R 2 cosðsþ sinðsþ x1 ðs; x 1 ; x 2 Þ # : sinðsþ cosðsþ We choose N ¼ 600; Q 0 ¼ð1; 2Þ and Q N ¼ð2; 1Þ. Let F and G denote: x 2 ð52þ Q1 Q constant of motion Fig. 1. Computation of (EL a h ).

10 L. Bourdin et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) ð0; QÞ ¼ðQ 2 ; Q 1 Þ and ðq; Da Q; sþ ¼ðDa Q 1 ; D a Q 2 Þ: The computation of (EL a h ) gives the following graphics: ð53þ Fig. 1a represents the discrete solutions of (EL a h ) and Fig. 1b represents the explicit computable quantity P N1 r¼1 a r A r ðf r r ðgþþ. As expected from Theorem 11, this quantity is a constant of motion of (EL a h ). 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. References [1] Agrawal OP. Formulation of Euler Lagrange equations for fractional variational problems. J Math Anal Appl 2002;272(1): [2] Almeida R, Malinowska AB, Torres DFM. A fractional calculus of variations for multiple integrals with application to vibrating string. J Math Phys 2010;51(3:033503):12. [3] Arnold VI. Mathematical methods of classical mechanics. Graduate texts in mathematics, vol. 60. New York: Springer-Verlag; [4] Atanacković TM, Konjik S, Pilipović S, Simić S. Variational problems with fractional derivatives: invariance conditions and Noether s theorem. Nonlinear Anal 2009;71(5 6): [5] Bagley RL, Calico RA. Fractional order state equations for the control of viscoelastically damped structures. J Guid Control Dynam 1991;14: [6] Baleanu D, Muslih SI. Lagrangian formulation of classical fields within Riemann Liouville fractional derivatives. Phys Scripta 2005;72(2 3): [7] Bonilla B, Rivero M, Rodríguez-Germá L, Trujillo JJ. Fractional differential equations as alternative models to nonlinear differential equations. Appl Math Comput 2007;187(1): [8] Bastos NRO, Ferreira RAC, Torres DFM. Discrete-time fractional variational problems. Signal Proces 2011;91(3): [9] Bastos NRO, Ferreira RAC, Torres DFM. Necessary optimality conditions for fractional differences problems of the calculus of variations. Discrete Contin Dynam Syst (DCDS-A) 2011;29: [10] Bourdin L, Cresson J, Greff I, Inizan P. Variational integrators on fractional Lagrangian systems in the framework of discrete embeddings. preprint arxiv: v1 [math.ds]. [11] Comte F. Opérateurs fractionnaires en économétrie et en finance. Prépublication MAP5, [12] Cresson J. Fractional embedding of differential operators and Lagrangian systems. J Math Phys 2007;48(3:033504):34. [13] Cresson J, Malinowska AB, Torres DFM. Differential, integral and variational time-scales embedding of Lagrangian systems. Comput Math Appl 2012;64(7): [14] Frederico GSF, Torres DFM. A formulation of Noether s theorem for fractional problems of the calculus of variations. J Math Anal Appl 2007;334(2): [15] Hairer E, Lubich C, Wanner G. Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Springer series in computational mathematics, 2nd ed., vol. 31. Berlin: Springer-Verlag; [16] Hilfer R. Applications of fractional calculus in physics. River Edge, New Jersey: World Scientific; [17] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. North-Holland mathematics studies, vol Amsterdam: Elsevier Science B.V.; [18] Tenreiro Machado J, Kiryakova Virginia, Mainardi Francesco. Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul 2011;16(3): [19] Magin RL. Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl 2010;59(5): [20] Malinowska AB. A formulation of the fractional Noether-type theorem for multidimensional Lagrangians. Appl Math Lett [21] Malinowska AB. On fractional variational problems which admit local transformations. J Vib Control [22] Marsden JE, West M. Discrete mechanics and variational integrators. Acta Numer 2001;10: [23] S. Muslih. A formulation of noether s theorem for fractional classical fields. preprint arxiv: v1 [math-ph]. [24] Oldham KB, Spanier J. The fractional calculus. Theory and applications of differentiation and integration to arbitrary order. Mathematics in science and engineering, vol New York, London: Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers]; With an annotated chronological bibliography by Bertram Ross. [25] Podlubny I. Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in science and engineering, vol San Diego, CA: Academic Press Inc.; [26] Riewe F. Mechanics with fractional derivatives. Phys. Rev. E (3) 1997;55(3, part B): [27] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives. Yverdon: Gordon and Breach Science Publishers; Theory and applications, Translated from the 1987 Russian original. [28] Zoia A, Néel M-C, Cortis A. Continuous-time random-walk model of transport in variably saturated heterogeneous porous media. Phys. Rev. E 2010;81(3):