Fractional Variational Principle of Herglotz

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DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2014.19.2367 DYNAMICAL SYSTEMS SERIES B Volume 19, Number 8, October 2014 pp. 2367–2381

FRACTIONAL VARIATIONAL PRINCIPLE OF HERGLOTZ

Ricardo Almeida

Department of Mathematics, University of Aveiro 3810-193 Aveiro, Portugal Agnieszka B. Malinowska

Faculty of Computer Science, Bialystok University of Technology 15-351 Bia lystok, Poland

Abstract. The aim of this paper is to bring together two approaches to non- conservative systems – the generalized variational principle of Herglotz and the fractional calculus of variations. Namely, we consider functionals whose extrema are sought, by diﬀerential equations that involve Caputo fractional derivatives. The Euler–Lagrange equations are obtained for the fractional variational problems of Herglotz-type and the transversality conditions are derived. The fractional Noether-type theorem for conservative and non-conservative physical systems is proved.

1. Introduction. As it was pointed out by Cornelius Lánczos [15], frictional and non-conservative forces are beyond the usual macroscopic variational treatment and, consequently, beyond the most advanced methods of classical mechanics. Over the years, a number of methods have been presented to circumvent the discrimination against non-conservative systems. We can mention the Rayleigh dissipation function [10] – which is the best known, the Bateman–Caldirola–Kanai (BCK) Lagrangian [18], the generalized variational principle of Herglotz [11, 12] or the fractional calculus of variations [16, 21]. The aim of this paper is to marge the generalized variational principle of Herglotz with the fractional calculus of variations. In other words, we consider functionals whose extrema are sought, by differential equations that involve Caputo fractional derivatives. The generalized variational principle was proposed by Gustav Herglotz in 1930 (see [12]). It generalizes the classical variational principle by defining the functional through a differential equation and reduces to the classical variational in- tegral under appropriate conditions. The generalized variational principle gives a variational description of non-conservative processes. For a system, conservative or non-conservative, which can be described with the generalized variational principle, one can systematically derive conserved quantities, as shown in [7, 8, 9], by applying the Noether-type theorem. The fractional calculus of variations generalizes the classical variational calculus by considering fractional derivatives into the variational integrals to be extremized. Fred Riewe [21] showed that a Lagrangian involving fractional time derivatives leads to an equation of motion with non-conservative forces such as friction. After the

2010 Mathematics Subject Classiﬁcation. Primary: 49K05; Secondary: 26A33. Key words and phrases. Calculus of variations, fractional calculus, Caputo fractional derivative, Euler–Lagrange equation, Noether-type theorem.

2367 2368 RICARDO ALMEIDA AND AGNIESZKA B. MALINOWSKA seminal papers of Riewe, several diﬀerent approaches have been developed to generalize the least action principle and the Euler–Lagrange equations to include fractional derivatives. It has been proved, using the notion of Euler–Lagrange fractional extremal, a Noether-type theorem that combines conservative and non-conservative cases of dynamical systems (see [4] or [16] and references given there). The signiﬁcance of both approaches to non-conservative systems – the generalized variational principle of Herglotz and the fractional calculus of variations – motivated this work. The paper is organized as follows. In Section 2, for the reader’s convenience, we review the necessary notions of the fractional calculus. Our results are given in next sections: in Section 3 we derive the Euler-Lagrange equations and the transversality conditions for fractional variational problems of Herglotz-type with one independent variable (Theorems 3.4 and 3.9), with higher- order fractional derivatives (Theorems 3.11 and 3.12), and for problems with several independent variables (Theorem 3.13). An heuristic method for solving the Euler– Lagrange equations for the fractional Herglotz-type problem is proposed. Finally, in Section 4 we obtain the fractional Noether-type theorem (Theorem 4.3 and The- orem 4.5) that can be used for conservative and non-conservative physical systems. Throughout the paper we illustrate our results by examples.

2. Preliminaries. For the convenience of the reader, we present the definitions of fractional operators that will be used in the sequel. For more on the theory of fractional calculus we refer to [13, 19, 22]; and for general results on the fractional calculus of variations to [16] and references therein. Let x :[a,b] → R be a function, α be a positive real number and n = [α] + 1, where [α] denotes the integer part of α. In what follows, we assume that x satisfies appropriate conditions in order to the fractional operators to be well defined. The left and right Riemann–Liouville fractional integrals of order α are given by t 1 − Iαx(t)= (t − τ)α 1x(τ)dτ, t>a a t Γ(α) Za and b 1 − Iαx(x)= (τ − t)α 1x(τ)dτ, t < b, t b Γ(α) Zt respectively, where Γ represents the Gamma function: ∞ − − Γ(z)= tz 1e t dt. Z0 The left and right Riemann–Liouville fractional derivatives are given by n t 1 d − − Dαx(t)= (t − τ)n α 1x(τ)dτ, t>a a t Γ(n − α) dtn Za and n n b (−1) d − − Dαx(t)= (τ − t)n α 1x(τ)dτ, t < b, t b Γ(n − α) dtn Zt respectively. The left and right Caputo fractional derivatives are defined by t 1 − − C Dαx(t)= (t − τ)n α 1x(n)(τ)dτ, t>a a t Γ(n − α) Za and n b (−1) − − C Dαx(t)= (τ − t)n α 1x(n)(τ)dτ, t < b, t b Γ(n − α) Zt FRACTIONAL VARIATIONAL PRINCIPLE OF HERGLOTZ 2369 respectively. The integration by parts formula plays a crucial role in deriving the Euler–Lagrange equation. For the left Caputo fractional derivative, this formula is formulated the following way. Theorem 2.1. (cf. [14]) Let α> 0, and x, y :[a,b] → R be two functions of class Cn, with n = [α]+1. Then,

b C α y(t) · a Dt x(t)dt a Z − b n 1 b α α+j−n n−1−j (n−1−j) = x(t) · tDb y(t)dt + tDb y(t) · (−1) x (t) . a a j=0 Z X h i Partial fractional integrals and derivatives for functions of p independent variables are defined in a similar way as is done for integer order derivatives. Let p R x : i=1[ai,bi] → be a function, αi, i = 1,...,p be real numbers and define ni = [αi] + 1. Then, we define the partial Riemann–Liouville fractional integrals of Q order αi with respect to ti by

ti αi 1 α −1 I x(t ,...,t )= (t − τ ) i x(t ,...,t − , τ ,t ,...,t )dτ , ai ti 1 p Γ(α ) i i 1 i 1 i i+1 p i i Zai

bi αi 1 αi−1 t I x(t1,...,tp)= (τi − ti) x(t1,...,ti−1, τi,ti+1,...,tp)dτi. i bi Γ(α ) i Zti Partial Riemann–Liouville and Caputo derivatives are deﬁned by

ni ti αi 1 ∂ ni−αi−1 a D x(t1,...,tp)= (ti − τi) x(t1,...,ti−1,τi,ti+1,...,tp)dτi, i ti n − α ∂tni Γ( i i) i Zai

− ni ni bi αi ( 1) ∂ ni−αi−1 t D x(t1,...,tp)= (τi − ti) x(t1,...,ti−1,τi,ti+1,...,tp)dτi, i bi n − α ∂tni Γ( i i) i Zti

ti ni C αi 1 ni−αi−1 ∂ D x(t1,...,tp)= (ti − τi) x(t1,...,ti−1,τi,ti+1,...,tp)dτi, ai ti n − α ∂τ ni Γ( i i) Zai i

− ni bi ni C αi ( 1) ni−αi−1 ∂ t D x(t1,...,tp)= (τi − ti) x(t1,...,ti−1,τi,ti+1,...,tp)dτi. i bi n − α ∂τ ni Γ( i i) Zti i

3. The generalized fractional Euler–Lagrange equations. Consider the differential equation with dependence on Caputo fractional derivatives dz = L(t, x(t), C Dαx(t),z(t)), t ∈ [a,b], (1) dt a t with the initial condition z(a) = za, where t is the independent variable, x = (x1,...,xn) the dependent variable and

C α C α1 C αn a Dt x(t) := (a Dt x1(t),..., a Dt xn(t)). In what follows we assume that: n 1. x(a)= xa, x(b)= xb, xa, xb ∈ R ; 2. αj ∈ (0, 1), j =1,...,n; 1 Rn C α 1 Rn 3. x ∈ C ([a,b], ), a Dt x ∈ C ([a,b], ); 2370 RICARDO ALMEIDA AND AGNIESZKA B. MALINOWSKA

4. the Lagrangian L :[a,b] × R2n+1 → R is of class C1 and the maps t 7→

∂L αj ∂L λ(t) [x, z](t) are such that tD λ(t) [x, z](t) , j =1,..., ∂C Dαj x b ∂C Dαj x a t j a t j n, exist and are continuous on [a,b], where t ∂L [x, z](t) := (t, x(t), C Dαx(t),z(t)) and λ(t) := exp − [x, z](τ)dτ . a t ∂z Za For any arbitrary but fixed function x(t) and a fixed initial value z(a) = za the C α solution of the differential equation (1): z[x; t] = z(t, x(t), a Dt x(t)) exists, and C α depends on t, x(t) and a Dt x(t) (see, e.g., [1]). Moreover, under our assumptions, is a C2 function of its arguments. In order for equation (1) to define a functional z, of x(t), we must solve equation (1) with the same fixed initial condition z(a)= za for all argument functions x(t), and evaluate the solution z = z[x; t] at the same fixed final time t = b for all argument functions x(t). The fractional variational principle of Herglotz (FVPH) is as follows: Let the functional z = z[x; b] of x(t) be given by a differential equation of the form (1) and let the function η ∈ C1([a,b], Rn) satisfies the boundary conditions C α 1 Rn η(a) = η(b)=0 and such that a Dt η ∈ C ([a,b], ). Then the value of the functional z[x; b] is an extremum for function x(t) which satisfy the condition d d z[x + η; b]| = z b, x(b)+ η(b), C Dαx(b)+ C Dαη(b) =0. (2) d =0 d a b a b =0 Observe that, if we introduce a parameter in the equation (1), then the solution z still exists and depends also on , and it is differentiable with respect to (see, e.g., Section 2.6 in [1]). Moreover, under our assumptions, z is a C2 function of its arguments. Remark 3.1. Later we will consider the case where x(b) is free and deduce the respective natural boundary conditions. In this case, a variation η that we consider in the formulation of the FVPH will be such that η(a) = 0 but η(b) may take any value. Remark 3.2. In the case when α → 1, FVPH gives the classical variational principle of Herglotz (see [12]). Remark 3.3. In the fractional calculus of variations, L does not depend on z and we can integrate from a to b, to obtain the functional b z z[x]= L(t, x(t), C Dαx(t)) + a dt. a t b − a Za Theorem 3.4. Let x = (x1,...,xn) be such that z[x; b] defined by equation (1) attains an extremum. Then, x is a solution of

∂L αj ∂L λ(t) [x, z](t)+ tD λ(t) [x, z](t) =0, j =1,...n (3) ∂x b ∂C Dαj x j a t j on [a,b]. Proof. Let x be such that z[x; b] deﬁned by equation (1) attains an extremum. We will show that (3) is a consequence of condition (2). The rate of change of z, in the direction of η, is given by d d θ(t)= z[x + η; t]| = z t, x(t)+ η(t), C Dαx(t)+ CDαη(t) . (4) d =0 d a t a t =0

FRACTIONAL VARIATIONAL PRINCIPLE OF HERGLOTZ 2371

Applying the variation η to the argument function in equation (1) we get d z[x + η; t]= L(t, x(t)+ η(t), C Dαx(t)+ C Dαη(t),z[x + η; t]). dt a t a t Observe that, d d d θ˙(t)= θ(t)= z t, x(t)+ η(t), C Dαx(t)+ C Dαη(t) dt dt d a t a t =0 d d = z t, x(t)+ η(t), C Dαx(t)+ C Dαη(t) d dt a t a t =0 d = L t, x(t)+ η(t), C Dαx(t)+ C Dαη(t),z [x + η; t] . d a t a t =0 This gives a diﬀerential equation for θ(t):

∂L n ∂L ∂L θ˙ t x, z t θ t x, z t η t x, z t C Dαj η t ( ) − [ ]( ) ( )= [ ]( ) j ( )+ C αj [ ]( )a t j ( ) ∂z ∂xj ∂ D x j=1 a t j X whose solution is t n ∂L ∂L θ t λ t θ a x, z τ η τ x, z τ C Dαj η t λ τ dτ ( ) ( )− ( )= [ ]( ) j ( )+ C αj [ ]( )a τ j ( ) ( ) ∂xj ∂ Dτ x a j=1 a j Z X t ∂L with notation λ(t) = exp − a ∂z [x, z](τ)dτ . Observe that θ(a) = 0. Indeed, as explained earlier, we evaluate R the solution of the equation (1) with the same ﬁxed initial condition z(a), independently of the function x(t). At t = b, we get b n ∂L ∂L θ b λ b x, z t η t x, z t C Dαj η t λ t dt ( ) ( )= [ ]( ) j ( )+ C αj [ ]( )a t j ( ) ( ) ∂xj ∂ D x a j=1 a t j Z X Observe that θ(b) is the variation of z[x; b]. If x is such that z[x; b] deﬁned by equation (1) attains an extremum, then θ(b) is identically zero. Hence, we get b n ∂L ∂L λ t x, z t η t x, z t C Dαj η t dt . ( ) [ ]( ) j ( )+ C αj [ ]( )a t j ( ) =0 ∂xj ∂ D x a j=1 a t j Z X Integrating by parts (cf. Theorem 2.1) we get

b n ∂L αj ∂L λ(t) [x, z](t)+ tDb λ(t) αj [x, z](t) ηj (t)dt ∂x j ∂C D x a j=1 a t j Z X n b 1−αj ∂L + ηj (t)tI λ(t) [x, z](t) =0. b ∂C Dαj x j=1 a t j a X Since η(a)= η(b) = 0, and η is an arbitrary function elsewhere, equation (3) follows by the fundamental lemma of the calculus of variations. Remark 3.5. The function θ in Eq. (4) is well deﬁned (see, e.g., Section 2.6 in [1]).

Remark 3.6. Let αj goes to 1, for all j =1,...,n and the Lagrangian L is of class C2. Then, equations (3) become ∂L d ∂L λ(t) [x, z](t) − λ(t) [x, z](t) =0, j =1, . . . , n. ∂x dt ∂x˙ j j 2372 RICARDO ALMEIDA AND AGNIESZKA B. MALINOWSKA

∂L Then, since λ˙ (t)= − [x, z](t)λ(t), we get ∂z ∂L ∂L ∂L d ∂L λ(t) [x, z](t)+ [x, z](t) [x, z](t) − [x, z](t) =0 ∂x ∂z ∂x˙ dt ∂x˙ j j j if and only if ∂L ∂L ∂L d ∂L [x, z](t)+ [x, z](t) [x, z](t) − [x, z](t)=0, j =1, . . . , n. (5) ∂xj ∂z ∂x˙ j dt ∂x˙ j Equations (5) Herglotz called the generalized Euler–Lagrange equations [12]. Example 3.7. Consider the system 2 C 0.5 2 1.5 z˙(t)= 0 Dt x(t) − Γ(2.5) t , with t ∈ [0, 1],  z(0) =h 0, i  x(0) = 0, x(1) = 1.

We wish to find a curve x for which z[x; 1] attains the minimum value. Observe that for all x, we have that z is an increasing function and thus z(t) ≥ 0, for all 2 C 0.5 2 1.5 t ∈ [0, 1]. Let x(t)= t . Then 0 Dt x(t)= Γ(2.5) t (see e.g. [13]). For such choice of x we get the differential equationz ˙(t)=0. As z(0) = 0 the unique solution is z(t) = 0. Therefore z[x; 1] obtain the minimum value, which is 0, at x. In this case λ(t) = 1 and equation (3) takes the form 2 D0.5 C D0.5x(t) − t1.5 =0 t 1 0 t Γ(2.5) which is satisfied for x. Example 3.8. Consider the system with the Lagrangian depending on z 2 C 0.5 2 1.5 z˙(t)= 0 Dt x(t) − Γ(2.5) t exp(t)+ z, with t ∈ [0, 1],  z(0) =h 1, i  x(0) = 0, x(1) = 1.

In this case we have λ(t) = exp(−t) and equation (3) takes the form 2 D0.5 exp(−t) C D0.5x(t) − t1.5 exp(t) =0. (6) t 1 0 t Γ(2.5) Observe that the function x(t) = t2 is a solution to equation (6), but we cannot conclude that it is a minimizer (or maximizer) of the functional z[x; 1]. Note that in order to uniquely determine the unknown function x, which is a solution to equation (3), we must consider the system of diﬀerential equations dz = L[x, z](t), dt  ∂L αj ∂L λ(t) [x, z](t)+ tD λ(t) [x, z](t) =0,  ∂x b ∂C Dαj x j a t j  t ∂L with λ(t) = exp − a ∂z [x, z](τ)dτ , together with the boundary conditions: z(a)= R Rn za, x(a) = xa, x(bR) = xb, where za ∈ and xa, xb ∈ . In the case where xj (b), j ∈{1,...,n}, is not speciﬁed we need an additional condition known as the transversality condition. FRACTIONAL VARIATIONAL PRINCIPLE OF HERGLOTZ 2373

Theorem 3.9. Let x be such that z[x; b] defined by equation (1) attains an extremum. Then, x is a solution to the system of fractional differential equations (3). If xj (b), j ∈{1,...,n}, is not specified, then the transversality condition

1−αj ∂L tI λ(t) [x, z](t) =0 at t = b (7) b ∂C Dαj x a t j holds.

Proof. The only diﬀerence with respect to the proof of Theorem 3.4 is that ηj (b) may vanish or not. Assuming ﬁrst that ηj (b) = 0 we deduce equations (3). Therefore

n b 1−αj ∂L ηj (t)tI λ(t) [x, z](t) =0. b ∂C Dαj x j=1 a t j a X Since η(a) = 0, ηi(b)=0 for i 6= j and ηj (b) is arbitrary, equation (7) follows.

When dealing with fractional operators usually we do not need to assume that functions have a nice behavior in the sense of smoothness properties. From the other hand, the drawback is that equations of type (3) are in most cases impossible to be solved analytically, and to overcome this problem numerical methods are applied (see, e.g., [2, 3, 5, 6, 20]). One of these methods consists in replacing the fractional operator by an expansion that involves only integer order derivatives [22]. Let α ∈ (0, 1), (c, d) be an open interval in R, and [a,b] ⊂ (c, d) be such that for each t ∈ [a,b] the closed ball Bb−a(t), with center at t and radius b − a, lies in (c, d). If x is an analytic function in (c, d), then ∞ α (t − a)k−α α (−1)k−1αΓ(k − α) Dαx(t)= x(k)(t), where = . a t k Γ(k +1 − α) k Γ(1 − α)Γ(k + 1) k=0 X (8) Using the well-known relations between the two type of fractional derivatives,

x(a) − C Dαx(t)= Dαx(t) − (t − a) α, (9) a t a t Γ(1 − α) similar formula can be given for the Caputo fractional derivative. For simplicity, let us assume that n = 1 and ﬁx N ∈ N. If x ∈ C2N ([a,b], R), L ∈ CN+1([a,b] × R3, R), then by replacing the fractional derivative in equation (1) by formulas (8)–(9), with the approximation up to order N, we obtain

N k−α α (t − a) x − z˙(t)= L t, x(t), x(k)(t) − a (t − a) α,z(t) k Γ(k +1 − α) Γ(1 − α) k ! X=0 with t ∈ [a,b]. Deﬁne L(t, x(t), x˙(t),...,x(N)(t),z(t))

N k−α α (t − a) x − := L t, x(t), x(k)(t) − a (t − a) α,z(t) . k Γ(k +1 − α) Γ(1 − α) k ! X=0 Therefore, we get the higher-order variational problem of Herglotz (see [23]) z˙(t)= L(t, x(t), x˙(t),...,x(N)(t),z(t)), with t ∈ [a,b]. 2374 RICARDO ALMEIDA AND AGNIESZKA B. MALINOWSKA

1.0

0.75

0.5

0.25

0.0 0.25 0.5 0.75 1.0

Figure 1. Numerical solution (dot line) vs exact solution (continuous line) in Example 3.10.

As it was proved in [23], if xN is such that zN [xN ; b] attains an extremum, then (xN ,zN ) satisﬁes the Euler–Lagrange equation

N dk ∂L (−1)k λ (t) (t, x(t), x˙(t),...,x(N)(t),z(t)) =0, dtk N ∂x(k) k X=0 t ∂L on [a,b], where λ (t) = exp − (τ, x (τ), x˙ (τ),...,x(N)(τ),z (τ))dτ . N ∂z N N N N Za Example 3.10. Consider the system 2 C 0.5 2 1.5 z˙(t)= 0 Dt x(t) − Γ(2.5) t , with t ∈ [0, 1],  z(0) =h 0, i  x(0) = 0, x(1) = 1.

We replace the Caputo fractional derivative by the truncated sum N k−α α (t − a) x(a) − C Dαx(t) ≈ x(k)(t) − (t − a) α, a t k Γ(k +1 − α) Γ(1 − α) k X=0 where N ∈ N depends on the number of given boundary conditions. Since in the considered system x(0) = 0 and x(1) = 1, we take N = 1. Therefore, we get 2 1 α (t − a)k−α 2 z˙(t)= x(k)(t)) − t1.5 . k Γ(k +1 − α) Γ(2.5) "k # X=0 Applying the classical Euler–Lagrange equation we obtain a second order diﬀerential equation, its solution is shown on Figure 1. FRACTIONAL VARIATIONAL PRINCIPLE OF HERGLOTZ 2375

3.1. Higher-order fractional derivatives. In this subsection, we generalize the fractional Herglotz problem by considering higher-order fractional derivatives. Theorem 3.11. Consider the system C α z˙(t)= L(t, x(t), a Dt x(t),z(t)), with t ∈ [a,b], z(a)= za, (10)  (j) j (j) j  xi (a)= xia, xi (b)= xib, with for all i ∈{1,...,n} and j ∈{0,...,i−1}, where we assume that the following conditions:

1. αi ∈ (i − 1,i), n R C α 1 R 2. xi ∈ C ([a,b], ), a Dt xi ∈ C ([a,b], ), j j 3. za, xia, xib are fixed reals, 4. the Lagrangian L :[a,b] × R2n+1 → R is of class C1 and the maps t 7→ ∂L αi ∂L λ t x, z t i ,...,n tD λ(t) [x,z](t) ( ) C αi [ ]( ), ∈{1 }, are such that b C αi ∂a Dt xi ∂a Dt xi exist and are continuous on [a,b], where C α [x, z](t) := (t, x(t), a Dt x(t),z(t)) t ∂L and λ(t) = exp − [x, z](τ)dτ , ∂z Za hold. If x = (x1,...,xn) is such that z[x; b] defined by (10) attains an extremum, then (x1,...,xn,z) satisfies the system of equations ∂L ∂L λ(t) [x, z](t)+ Dαi λ(t) [x, z](t) =0, i =1, . . . , n, (11) ∂x t b ∂C Dαi x i a t i on [a,b]. Proof. Let x be such that z[x; b] defined by (10) attains an extremum. Consider (j) (j) variations of type x + η with η = (η1,...,ηn) satisfying ηi (a) = ηi (b) = 0, for all i ∈{1,...,n} and j ∈{0,...,i − 1}. The rate of change of z in the direction of η is given by d d θ(t)= z[x + η; t]| = z t, x(t)+ η(t), C Dαx(t)+ CDαη(t) . d =0 d a t a t =0 Applying η to differential equation in (10) we get d z[x + η; t]= L(t, x(t)+ η(t), C Dαx(t)+ C Dαη(t),z[x + η; t]). dt a t a t Then, as in the proof of Theorem 3, we obtain the linear ODE with respect to θ: n ∂L ∂L ∂L θ˙ t x, z t η t x t C Dαi η t x, z t θ t . ( )= [ ]( ) i( )+ C αi [ ]( )a t i( ) + [ ]( ) ( ) ∂xi ∂ D xi ∂z i=1 a t X Solving the linear ODE and using the fact that θ(a)= θ(b)=0 we get b n ∂L ∂L λ t x, z t η t x, z t C Dαi η t dt . ( ) [ ]( ) i( )+ C αi [ ]( )a t i( ) =0 ∂xi ∂ D xi a i=1 a t Z X Integrating by parts we obtain b n ∂L ∂L λ t x, z t Dαi λ t x, z t η t dt ( ) [ ]( )+ t b ( ) C αi [ ]( ) i( ) ∂xi ∂ D xi a i=1 a t Z X 2376 RICARDO ALMEIDA AND AGNIESZKA B. MALINOWSKA

b n i−1 − − − ∂L i−1−j η(i 1 j) t Dαi+j i λ t x, z t . + (−1) i ( )t b ( ) C αi [ ]( ) =0  ∂ D xi  i=1 j=0 a t X X a (j) (j)  Since ηi (a) = ηi (b) = 0 for all i ∈ {1,...,n} and j ∈ {0,...,i − 1}, and η is arbitrary elsewhere, the theorem is proven.

Next theorem generalizes transversality conditions given in Theorem 3.9.

(j) Theorem 3.12. Consider system (10) with an exception that xi (b) may take any value, for all i ∈{1,...,n} and j ∈{0,...,i − 1}. If x = (x1,...,xn) is such that z[x; b] attains an extremum, then (x1,...,xn,z) satisﬁes the system of equations (11) on [a,b], and the transversality conditions

− ∂L Dαi+j i λ(t) [x, z](t) =0 at t = b, t b ∂C Dαi x a t i for all i ∈{1,...,n} and all j ∈{0,...,i − 1}. 3.2. Several independent variables. The fractional variational principle of Her- glotz can be generalized to one involving several independent variables. In the following, the independent variables will be the time variable t ∈ [a,b] and the n N spacial coordinates x = (x1,...,xn) ∈ Ω = i=1[ai,bi], n ∈ . The argument function of the functional z[u; b] deﬁned by the variational principle will be Q u = (u1(t, x),...,um(t, x)), m ∈ N. In what follows, for α = (α1,...,αm) and βi = (βi,1,...,βi,m), i =1,...,n we will use the following notations:

C α C α1 C αm a Dt u = (a Dt u1,..., a Dt um),

C βi C βi,1 C βi,m ai Dxi u = (ai Dxi u1,..., ai Dxi um), i =1, . . . , n. The fractional variational principle with several independent variables is as follows: Let the functional z = z[u; b] of u = u(t, x) be given by an integro-diﬀerential equation of the form dz = L(t,x,u(t, x), C Dαu(t, x), C Dβ1 u(t, x),..., C Dβn u(t, x),z(t)) dnx, (12) dt a t a1 x1 an xn ZΩ n t ∈ [a,b], where d x = dx1 . . . dxn; and let the following conditions be satisﬁed 1. u(t, x) = g(t, x) for (t, x) ∈ ∂P , where P = [a,b] × Ω, ∂P is the boundary of P and g : ∂P → Rm is a given function; 2. αj ,βi,j ∈ (0, 1); 1 R C αj C βi,j 1 R 3. uj ∈ C (P, ), a Dt uj , ai Dxi uj ∈ C (P, ); 4. L : P × Rm(n+2)+1 → R is of class C1, and the maps ∂L t, x λ t u,z t, x ( ) 7→ ( ) C αj [ ]( ) ∂a Dt uj and ∂L (t, x) 7→ λ(t) [u,z](t, x) C βi,j ∂ai Dxi uj are such that

αj ∂L tD λ(t) [u,z](t, x) b ∂C Dαj u a t j FRACTIONAL VARIATIONAL PRINCIPLE OF HERGLOTZ 2377

and

βi,j ∂L x D λ(t) [u,z](t) i bi C βi,j ∂ai Dxi uj ! exist and are continuous on P , where

C α C β1 C βn [u,z](t, x) := (t,x,u(t, x), a Dt u(t, x), a1 Dx1 u(t, x),..., an Dxn u(t, x),z(t),

t ∂L λ(t) := exp − [u,z](τ, x)dnxdτ ∂z Za ZΩ for all i ∈{1,...,n} and j ∈{0,...,m}. 1 m Consider η ≡ (η1(t, x),...,ηm(t, x)) ∈ C ([a,b]×Ω, R ) that satisﬁes the bound- C αj C βi,j ary conditions η(t, x) = 0 for (t, x) ∈ ∂P and such that a Dt ηj , ai Dxi ηj ∈ C1(P, R). Then, the value of the functional z[u; b] is an extremum for function u which satisfy the condition d d z[u + η; b]| = z(b, [u](b, x))| =0, (13) d =0 d =0

C α C β1 C βn where [u](t, x) := (t,x,u(t, x), a Dt u(t, x), a1 Dx1 u(t, x),..., an Dxn u(t, x)). It should be pointed that when a variation η is applied to u the equation (12), defining the functional z, must be solved with the same fixed initial condition za at t = a and the solution evaluated at the same fixed final time t = b for all varied argument functions u + η. Theorem 3.13. If u is such that the functional z[u; b] defined by equation (12) attains an extremum, then u is a solution of the system of equations

∂L αj ∂L λ(t) [u,z](t, x)+ tD λ(t) [u,z](t, x) ∂u b ∂C Dαj u j a t j n βi,j ∂L + x D λ(t) [u,z](t, x) =0, (14) i bi C βi,j i=1 ∂a Dxi uj ! X i j =1,...,m. Proof. The idea of the proof is similar to that of the proof of Theorem 3.4. We show that equations (14) are a consequence of condition (13). The rate of change of z, in the direction of η, is given by d d θ(t)= z[u + η; t]| = z(t, [u](t, x))| . (15) d =0 d =0 Applying the variation η to the argument function in equation (12), diﬀerentiating with respect to and then setting = 0 we get a diﬀerential equation for θ(t):

dθ m ∂L m ∂L t u,z t, x η u,z t, x C Dαj η ( )= [ ]( ) j + C αj [ ]( )a t j dt  ∂uj ∂ D u Ω j=1 j=1 a t j Z X X  n m ∂L C βi,j ∂L n + [u,z](t, x)a Dx ηj + [u,z](t, x)θ(t) d x. C βi,j i i ∂z  i=1 j=1 ∂a Dxi uj X X i  2378 RICARDO ALMEIDA AND AGNIESZKA B. MALINOWSKA

Denoting

m ∂L ∂L A t u,z t, x η u,z t, x C Dαj η ( )= [ ]( ) j + C αj [ ]( )a t j ∂uj ∂ D u Ω j=1 a t j Z X n ∂L C βi,j n + [u,z](t, x)a Dx ηj d x C βi,j i i i=1 ∂a Dxi uj # X i and ∂L B(t)= [u,z](t, x)dnx. ∂z ZΩ we get dθ (t) − B(t)θ(t)= A(t). (16) dt Solving equation (16) and taking into consideration that θ(a) = θ(b) = 0 (see the proof of Theorem 3.4) we obtain b λ(t)A(t)dt =0. Za Integrating by parts we get b m ∂L ∂L λ t u,z t, x Dαj λ t u,z t, x ( ) [ ]( )+ t b ( ) C αj [ ]( ) ∂uj ∂ D u a Ω j=1 a t j Z Z X n βi,j ∂L n + x D λ(t) [u,z](t, x) ηj d xdt =0 i bi C βi,j i=1 ∂ai Dxi uj !# X as η satisﬁes the boundary conditions η(a, x)= η(b, x) =0for x ∈ Ω, and η(t, x)=0 for x ∈ ∂Ω, a ≤ t ≤ b. Finally, since ηj are arbitrary functions, it follows that for all j ∈{1,...,m} we have

∂L αj ∂L λ(t) [u,z](t, x)+ tD λ(t) [u,z](t, x) ∂u b ∂C Dαj u j a t j n βi,j ∂L + x D λ(t) [u,z](t, x) = 0 (17) i bi C βi,j i=1 ∂a Dxi uj ! X i on [a,b] × Ω. Remark 3.14. We note that it is straightforward to obtain the transversality conditions for Herglotz’s fractional variational problem with several variables (cf. [16]).

4. Noether-type theorem. Symmetric principles are key issues in mathematics and physics. There are close relationships between symmetries and conserved quantities. Noether ﬁrst proposed the famous Noether symmetry theorem which describes the universal fact that invariance of the action functional with respect to some family of parameter transformations gives rise to the existence of certain con- servation laws, i.e., expressions preserved along the solutions of the Euler-Lagrange equation. In this section we extend Noether’s theorem to one that holds for fractional variational problems of Herglotz-type with one (Theorem 4.3) or several independent variables (Theorem 4.5). FRACTIONAL VARIATIONAL PRINCIPLE OF HERGLOTZ 2379

Consider the one-parameter family of transformations

x¯j = hj (t,x,s), j =1, . . . , n, (18) 2 depending on a parameter s, s ∈ (−ε,ε), where hj are of class C and such that n hj(t, x, 0) = xj for all j ∈ {1,...,n} and for all (t, x) ∈ [a,b] × R . By Taylor’s formula we have

hj(t,x,s)= hj (t, x, 0)+ sξj (t, x)+ o(s)= xj + sξj (t, x)+ o(s), (19) ∂ provided that |s| ≤ ε, where ξj (t, x) = ∂s hj (t,x,s)|s=0. For |s| ≤ ε the linear approximation to transformation (18) isx ¯j ≈ xj + sξj (t, x). This transformation leads, for any fixed sufficiently small parameter s and any fixed function x(·) = (x1(·),...,xn(·)), to x(·) = (x1(·),..., xn(·)) given by

x¯j (t)= hj (t, x(t),s), j =1, . . . , n. Denote by θ = θ(t) the total variation produced by the family of transformations (19), that is d θ(t)= z¯[x + sξ; t]| . (20) ds s=0 We define the invariance of the functional z, defined by differential equation (1), under transformation (19) as follows. Definition 4.1. The transformation (19) leaves the functional z, defined by differential equation (1), invariant if θ(t) ≡ 0. Remark 4.2. When s = 0, the following holdsz ¯[¯x; t]= z[x; t]. Theorem 4.3. If the functional z defined by differential equation (1) is invariant in the sense of definition 4.1, then n αj ∂L D λ(t) [x, z](t), ξj (t, x) =0, (21) ∂C Dαj x j=1 a t j X t ∂L α C α α where λ(t) := exp − a ∂z [x, z](τ)dτ and D [f,g] := f · a Dt g − g · tDb f, holds along solutions of theR generalized fractional Euler–Lagrange equations (3). Proof. Applying transformations (19) to equation (1) we get d z¯ = L(t, x¯(t), C Dαx¯(t), z¯(t)). (22) dt a t Differentiating both sides of (22) with respect to s and then setting s = 0, we get a differential equation for θ(t) ∂L n ∂L ∂L θ˙ t x, z t θ t x, z t ξ t x, z t C Dαj ξ t ( ) − [ ]( ) ( )= [ ]( ) j ( )+ C αj [ ]( )a t j ( ) ∂z ∂xj ∂ D x j=1 a t j X whose solution is t n ∂L ∂L θ t λ t θ a x, z τ ξ τ x, z τ C Dαj ξ t λ τ dτ ( ) ( ) − ( )= [ ]( ) j ( )+ C αj [ ]( )a τ j ( ) ( ) ∂xj ∂ Dτ x a j=1 a j Z X t ∂L with notation λ(t) = exp − a ∂z [x, z](τ)dτ . Observe that θ(a) = 0, also by hypothesis θ(τ) = 0 for arbitrary R τ. Thus τ n ∂L ∂L x, z t ξ t x, z t C Dαj ξ t λ t dt 0= [ ]( ) j ( )+ C αj [ ]( )a t j ( ) ( ) ∂xj ∂ D x a j=1 a t j Z X 2380 RICARDO ALMEIDA AND AGNIESZKA B. MALINOWSKA for arbitrary τ. On the solutions of the generalized fractional Euler–Lagrange equations (3) we have n ∂L C αj αj ∂L 0= λ(t) a D ξj (t,x) −t D λ(t) ξj (t,x) . ∂C Dαj x t b ∂C Dαj x Xj=0 a t j a t j By definition of operator Dα we obtain equation (21). Example 4.4. Consider the Lagrangian in one spatial dimension C α C α L(t, x(t), a Dt x(t),z(t)) = f(a Dt x) − γz, (23) where f is a C1 function and γ is a positive constant. Obviously, the Lagrangian (23) is invariant under transformation x¯(t)= x(t)+ c, where c is a constant. Therefore, Theorem 4.3 indicates that α γt 0 C α D e f (a Dt x),c =0, (24) α γt 0 C α along any solution of tDb e f (a Dt x) = 0. Notice that equation (24) can be − written in the form d I1 α eγtf 0(C Dαx) = 0, that is, the quantity dt t b a t 1−α γt 0 C α tIb e f (a Dt x) following the classical approach, can be called a generalized fractional constant of motion. Now we generalize Theorem 4.3 to the case of several independent variables. For that consider the one-parameter family of transformations

u¯j = hj (t,x,u,s), j =1,...,m (25) 2 depending on a parameter s, s ∈ (−ε,ε), where hj are of class C and such that m hj(t, x, u, 0) = uj (t, x) for all j ∈ {1,...,m} and for all (t,x,u) ∈ [a,b] × Ω × R . As before, for |s|≤ ε, the linear approximation to transformation (25) isu ¯j ≈ uj + ∂ sξj (t,x,u), where ξj (t,x,u) = ∂s hj (t,x,u,s)|s=0. The following result generalizes Theorem 4.3, and can be proved in a similar way (cf. [17]). Theorem 4.5. Let the functional z, defined by differential equation (12), is invariant under the family of transformations (25). Then n αj ∂L D λ(t) [u,z](t, x), ξj (t,x,u) ∂C Dαj u j=1 a t j X n βj,i ∂L + D λ(t) [u,z](t, x), ξj (t,x,u) =0, C βi,j i=1 " ∂a Dxi uj #) X i t ∂L n α C α α where λ(t) := exp − a Ω ∂z [u,z](τ, x)d xdτ and D [f,g] := f ·a Dt g−g·tDb f, holds along solutions R ofR the generalized fractional Euler–Lagrange equations (14). Acknowledgments. Ricardo Almeida is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applica- tions, and the Portuguese Foundation for Science and Technology (FCT-Funda¸cão para a Ciência e a Tecnologia), within project PEst-OE/MAT/UI4106/2014. Ag- nieszka B. Malinowska is supported by the Bialystok University of Technology grant S/WI/02/2011. We thank the anonymous reviewer for his/her careful reading of our manuscript and his/her many insightful comments and suggestions. FRACTIONAL VARIATIONAL PRINCIPLE OF HERGLOTZ 2381

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