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Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to Physics discussed.


Introduction
The calculus of variations is a beautiful and useful field of mathematics that deals with problems of determining extrema (maxima or minima) of functionals [39,40,58]. It starts with the simplest problem of finding a function extremizing (minimizing or maximizing) an integral J (y) = b a F (t, y(t), y ′ (t))dt subject to boundary conditions y(a) = y a and y(b) = y b . In the literature many generalizations of this problem were proposed, including problems with multiple integrals, functionals containing higher-order derivatives, and functionals depending on several functions [35,37,45]. Of our interest is an extension proposed by Riewe in 1996Riewe in -1997, where fractional derivatives (real or complex order) are introduced in the Lagrangian [50,51].
During the last decade, fractional problems have increasingly attracted the attention of many researchers. As mentioned in [9], Science Watch of Thomson Reuters identified the subject as an Emerging Research Front area. Fractional derivatives are nonlocal operators and are historically applied in the study of nonlocal or time dependent processes [46]. The first and well established application of fractional calculus in Physics was in the framework of anomalous diffusion, which is related to features observed in many physical systems. Here we can mention the report [38] demonstrating that fractional equations works as a complementary tool in the description of anomalous transport processes. Within the fractional approach it is possible to include external fields in a straightforward manner. As a consequence, in a short period of time the list of applications expanded. Applications include chaotic dynamics [60], material sciences [33], mechanics of fractal and complex media [12,32], quantum mechanics [25,31], physical kinetics [61], long-range dissipation [54], long-range interaction [53,55], just to mention a few. One of the most remarkable applications of fractional calculus appears, however, in the fractional variational calculus, in the context of classical mechanics. Riewe [50,51] shows that a Lagrangian involving fractional time derivatives leads to an equation of motion with nonconservative forces such as friction. It is a remarkable result since frictional and nonconservative forces are beyond the usual macroscopic variational treatment and, consequently, beyond the most advanced methods of classical mechanics [30]. Riewe generalizes the usual variational calculus, by considering Lagrangians that dependent on fractional derivatives, in order to deal with nonconservative forces. Recently, several different approaches have been developed to generalize the least action principle and the Euler-Lagrange equations to include fractional derivatives. Results include problems depending on Caputo fractional derivatives, Riemann-Liouville fractional derivatives and others [3,4,10,11,14,19,20,34,36,[41][42][43]52].
A more general unifying perspective to the subject is, however, possible, by considering fractional operators depending on general kernels [1,28,44]. In this work we follow such an approach, developing a generalized fractional calculus of variations. We consider very general problems, where the classical integrals are substituted by generalized fractional integrals, and the Lagrangians depend not only on classical derivatives but also on generalized fractional operators. Problems of the type considered here, for particular kernels, are important in Physics [18]. Here we obtain general necessary optimality conditions, for several types of variational problems, which are valid for rather arbitrary operators and kernels. By choosing particular operators and kernels, one obtains the recent results available in the literature of Mathematical Physics [8,[15][16][17][18]24].
The paper is organized as follows. In Section 2 we introduce the generalized fractional operators and prove some of its basic properties. Section 3 is dedicated to prove integration by parts formulas for the generalized fractional operators. Such formulas are then used in later sections to prove necessary optimality conditions (Theorems 14 and 23). In Sections 4, 5 and 6 we study three important classes of generalized variational problems: we obtain fractional Euler-Lagrange conditions for the fundamental (Section 4) and generalized isoperimetric problems (Section 6), as well as fractional natural boundary conditions for generalized free-boundary value problems (Section 5). Finally, two illustrative examples are discussed in detail in Section 7, while applications to Physics are given in Section 8: in Section 8.1 we obtain the damped harmonic oscillator in quantum mechanics, in Section 8.2 we show how results from FALVA Physics can be obtained. We end with Section 9 of conclusion, pointing out an important direction of future research.

Preliminaries
In this section we present definitions and properties of generalized fractional operators. As particular cases, by choosing appropriate kernels, these operators are reduced to standard fractional integrals and fractional derivatives. Other nonstandard kernels can also be considered as particular cases. For more on the subject of generalized fractional calculus and applications, we refer the reader to the book [28]. Throughout the text, α denotes a real number between zero and one. Following [5], we use round brackets for the arguments of functions, and square brackets for the arguments of operators. By definition, an operator receives and returns a function.
Definition 1 (Generalized fractional integral). The operator K α P is given by where P = a, x, b, p, q is the parameter set (p-set for brevity), x ∈ [a, b], p, q are real numbers, and k α (x, t) is a kernel which may depend on α. The operator K α P is referred as the operator K (K-op for simplicity) of order α and p-set P , while K α P [f ] is called the operation K (or K-opn) of f of order α and p-set P .
Note that if we define then the operator K α P can be written in the form This is a particular case of one of the oldest and most respectable class of operators, so called Fredholm operators [23,47].
Theorem 2 (cf. Example 6 of [23]). Let α ∈ (0, 1) and P = a, x, b, p, q . If k α is a square integrable function on the square is well defined, linear, and bounded operator.
is a well defined bounded and linear operator. Proof. Obviously, the operator is linear. Let α ∈ (0, 1), P = a, t, b, p, q , and f ∈ L 1 ([a, b]). Define It follows from Fubini's theorem that F is integrable on the square ∆. Moreover, Remark 4. The K-op reduces to the left and the right Riemann-Liouville fractional integrals from a suitably chosen kernel k α (x, t) and p-set P .
is the standard left Riemann-Liouville fractional integral of f of order α; • if P = a, x, b, 0, 1 , then is the standard right Riemann-Liouville fractional integral of f of order α.
) are well defined, linear and bounded. The generalized fractional derivatives A α P and B α P are defined in terms of the generalized fractional integral K-op.
Definition 6 (Generalized Riemann-Liouville fractional derivative). Let P be a given parameter set and 0 < α < 1. The operator A α P is defined by where D denotes the standard derivative operator, and is referred as the operator A (A-op) of order α and p-set P , while A α , is called the operation A (A-opn) of f of order α and p-set P . Remark 8. The standard Riemann-Liouville and Caputo fractional derivatives are easily obtained from the generalized operators A α P and B α P , respectively. Let is the standard left Riemann-Liouville fractional derivative of f of order α, while is the standard left Caputo fractional derivative of f of order α; is the standard right Riemann-Liouville fractional derivative of f of order α, while is the standard right Caputo fractional derivative of f of order α.

On generalized fractional integration by parts
We now prove integration by parts formulas for generalized fractional operators.
Hence, we can use again Fubini's theorem to change the order of integration: , then the operator K α P satisfies the integration by parts formula (1). Proof. Define Hence, we can use Fubini's theorem to change the order of integration in iterated integrals.

The generalized fundamental variational problem
By ∂ i F we denote the partial derivative of a function F with respect to its ith argument. We consider the problem of finding a function y = t → y(t), t ∈ [a, b], that gives an extremum (minimum or maximum) to the functional when subject to the boundary conditions where α, β, γ ∈ (0, 1), P 1 =< a, b, b, 1, 0 > and P j =< a, t, b, p j , q j >, j = 2, 3. For simplicity of notation we introduce the operator {·} β,γ P2,P3 defined by With the new notation one can write (3) simply as has kernel k α (x, t), and operators B β P2 and K γ P3 have kernels h 1−β (t, τ ) and h γ (t, τ ), respectively. In the sequel we assume that: (H4) kernels k α (x, t), h 1−β (t, τ ) and h γ (t, τ ) are such that we are in conditions to use Theorems 9, 10 and 11.
for all t ∈ (a, b).
Proof. Suppose that y is an extremizer of J . Consider the value of J at a nearby function y = y + εη, where ε ∈ R is a small parameter, and η ∈ C 1 ([a, b]; R) is an arbitrary function with continuous B-op and K-op. We require that η(a) = η(b) = 0. Let A necessary condition for y to be an extremizer is given by Using classical and generalized fractional integration by parts formulas (Theorems 9, 10 and 11), We obtain (5) by application of the fundamental lemma of the calculus of variations (see, e.g., [22,Section 2.2]).
The next corollary gives an extension of the main result of [19].
Corollary 15. If y is a solution to the problem of minimizing or maximizing in the class y ∈ C 1 ([a, b]; R) subject to the boundary conditions where α, β ∈ (0, 1), for all t ∈ (a, b). The following result is the Caputo analogous to the main result of [7] done for the Riemann-Liouville fractional derivative.
holds for all t ∈ [a, b].

Generalized free-boundary variational problems
Assume now that in problem (3)-(4) the boundary conditions (4) are substituted by y(a) is free and y(b) = y b .
Theorem 18. If y is a solution to the problem of extremizing functional (3) with (11) as boundary conditions, then y satisfies the Euler-Lagrange equation (5). Moreover, the extra natural boundary condition holds.
Proof. Under the boundary conditions (11), we do not require η in the proof of Theorem 14 to vanish at t = a. Therefore, following the proof of Theorem 14, we obtain for every admissible η ∈ C 1 ([a, b]; R) with η(b) = 0. In particular, condition (13) holds for those η that fulfill η(a) = 0. Hence, by the fundamental lemma of the calculus of variations, equation (5) is satisfied. Now, let us return to (13) and let η again be arbitrary at point t = a. Inserting (5), we obtain the natural boundary condition (12).

Corollary 19.
Let J be the functional given by Let y be a minimizer of J satisfying the boundary condition y(b) = y b . Then, y satisfies the Euler-Lagrange equation (14) and the natural boundary condition Proof. Let functional (3) be such that it does not depend on the classical (integer) derivative y ′ (t) and on the K-op.  (14) and (15) from (5) and (12), respectively.
Corollary 20. Let J be the functional given by If y is a minimizer to J satisfying the boundary condition y(b) = y b , then y satisfies the Euler-Lagrange equation (16) and the natural boundary condition Proof. Choose, in the problem defined by (3) and (11), k α (x, t) ≡ 1. Then, equations (16) and (17) follow from (5) and (12), respectively.
In the next example we make use of the Mittag-Leffler function of two parameters: if α, β > 0, then the Mittag-Leffler function is defined by .
This function appears naturally in the solution of fractional differential equations, as a generalization of the exponential function [27].

Applications to Physics
If the functional (3) does not depend on B-op and K-op, then Theorem 14 gives the following result: if y is a solution to the problem of extremizing subject to y(a) = y a and y(b) = y b , where α ∈ (0, 1), then We recognize on the right hand side of (30) the generalized weak dissipative parameter

Quantum mechanics of the damped harmonic oscillator
As a first application, let us consider kernel k α (b, t) = e α(b−t) and the Lagrangian where V (y) is the potential energy and m stands for mass. The Euler-Lagrange equation (30) gives the following second order ordinary differential equation: Equation (31) coincides with (14) of [24], obtained by modification of Hamilton's principle.
We study two particular kernels.
1. If we choose kernel defined in [26], then the Euler-Lagrange equation is In particular, when ρ → 0, (34) becomes the kernel of the Riemann-Liouville fractional integral, and equation (35) gives which is the Euler-Lagrange equation proved in [16]. For ρ = 0, we have Therefore, both at the very early time and at very large time, dissipation disappears. Moreover, if ρ → 0, then This shows that at the origin of time, the time-dependent dissipation becomes stationary, and that at very large time no dissipation, of any kind, exists.
We note that there is a small inconsistence in [16], regarding to the coefficient ofẏ(t) in (33), and a small inconsistence in [18], regarding a sign of (36).

Conclusion
In this article we unify, subsume and significantly extend the necessary optimality conditions available in the literature of the fractional calculus of variations. It should be mentioned, however, that since fractional operators are nonlocal, it can be extremely challenging to find analytical solutions to fractional problems of the calculus of variations and, in many cases, solutions may not exist. In our paper we give two examples with analytic solutions, and many more can be found borrowing different kernels from the book [47]. On the other hand, one can easily choose examples for which the fractional Euler-Lagrange differential equations are hard to solve, and in that case one needs to use numerical methods [2,6,48,49]. The question of existence of solutions to fractional variational problems is a complete open area of research. This needs attention. Indeed, in the absence of existence, the necessary conditions for extremality are vacuous: one cannot characterize an entity that does not exist in the first place. For solving a problem of the fractional calculus of variations one should proceed along the following three steps: (i) first, prove that a solution to the problem exists; (ii) second, verify the applicability of necessary optimality conditions; (iii) finally, apply the necessary conditions which identify the extremals (the candidates). Further elimination, if necessary, identifies the minimizer(s) of the problem. All three steps in the above procedure are crucial. As mentioned by Young in [59], the calculus of variations has born from the study of necessary optimality conditions, but any such theory is "naive" until the existence of minimizers is verified. The process leading to the existence theorems was introduced by Leonida Tonelli in 1915 by the so-called direct method [56]. During two centuries, mathematicians were developing "the naive approach to the calculus of variations". There was, of course, good reasons why the existence problem was only solved in the beginning of XX century, two hundred years after necessary optimality conditions began to be studied: see [13,57] and references therein. Similar situation happens now with the fractional calculus of variations: the subject is only fifteen years old, and is still in the "naive period". We believe time has come to address the existence question, and this will be considered in a forthcoming paper.