Noether symmetries and conserved quantities for fractional forced Birkhoffian systems

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Noether symmetries and conserved quantities for fractional forced Birkhoffian systems Qiuli Jia a,b,∗ , Huibin Wu a,c , Fengxiang Mei d a

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China c Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China d School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China b

a r t i c l e

i n f o

Article history: Received 9 September 2014 Available online xxxx Submitted by P.G. Lemarie-Rieusset Keywords: Fractional Pfaff–Birkhoff variational principle Fractional forced Birkhoffian system Fractional Noether theorem Fractional conserved quantity

a b s t r a c t In this paper Noether symmetries and conserved quantities for fractional forced Birkhoffian systems are studied. Firstly, a new fractional Pfaff–Birkhoff variational principle with Riemann–Liouville derivatives and generalized force is established. And from which the fractional forced Birkhoff equations are derived. Secondly, the definitions and the criteria of the invariance of a functional under an ε-parameter group of infinitesimal transformations are given. Finally, the fractional Noether theorems for forced Birkhoffian systems are obtained. And an illustrative example is presented. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Fractional Calculus (FC) is a subfield of mathematics which deals with derivatives and integrals of non-integer order. Its beginning dates back to 1695 when L’Hopital asked Leibniz the meaning of a fractional derivative of order n = 1/2 in his letter. After that, many well-known mathematicians, such as Euler, Fourier, Abel, Laplace, Liouville, Riemann, etc. contributed to its development [39,54,58]. FC is an important mathematical subject with different notions of fractional-order derivatives available. The most popular ones are those of Riemann–Liouville, Caputo and Grünwald–Letnikov. FC has been widely used to understand complex dynamical behaviors in classical and quantum systems, also describe and discuss the physics process like randomness and chaotic non-diffusion of complicated chaotic dynamical systems. At present, FC plays an important role in various widespread fields of science and engineering, such as mechanics (classic and

* Corresponding author at: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China. E-mail address: [email protected] (Q. Jia). http://dx.doi.org/10.1016/j.jmaa.2016.04.067 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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quantum), thermodynamics, optics, control theory, chemistry, biology, economics, geology, astrophysics, probability and statistics, signal and image processing, and so on [5,39,41,42]. The Fractional Calculus of Variations (FCV) unifies the calculus of variations and the fractional calculus by inserting fractional derivatives into the variational functionals, which occurs naturally in many problems of mechanics, physics and engineering, in order to provide more accurate mathematical models to describe the physical phenomenon [49]. The research of the FCV began with Riewe’s work [56,57] which discussed the dynamical modeling problems of a nonconservative system. The further study on the FCV can be found in Refs. [1–4,6] which extended the fractional variational problems to the fractional optimal control problems. Furtherly, Ref. [8] considered a natural generalization of the FCV, and along the thought of Ref. [8], Ref. [7] discussed the necessary and sufficient condition of the FCV. Afterwards, Refs. [10–12, 36,37,40,47,48,55,63,66,69] applied the FCV to the study of fractional Lagrangian systems and fractional Hamiltonian systems. In 2005, Ref. [16] presented a new method for nonconservative dynamical models that is called the El-Nabulsi’s fractional model, i.e., fractional action-like variational approach, which was generalized in recent years [17–21,26,38]. The conservative physical system implies frictionless motion and is a kind of simplification of the real dynamical world. Almost all systems contain internal damping and are subject to external forces. For nonconservative dynamical systems, the conservation laws are broken so that the standard Lagrangian or Hamiltonian formalism is no longer fit to describe the behavior of the system. Therefore, the Newtonian dissipative dynamical system is a complement to the conservative system since not only energy, but also other physical quantities, such as linear or angular momentums, are not conserved. In this case, the classical Noether theorem is no longer valid. However, it is still possible to achieve the validity of the Noether principle by using the FCV. Accordingly, the FCV provides a more realistic approach for physics, mechanics and engineering, which permits the consideration of non-conservative systems in a natural way [49]. The fractional Noether theorem plays the central role in the FCV. The study of symmetries and conserved quantities for the FCV is an important aspect of the fractional order dynamics. The fractional variational symmetry is defined by the parameter transformation which keeps a problem of the calculus of fractional variations or fractional optimal control invariant. Their importance is reflected in the existence of conservation laws. Refs. [27,28,30,31] first studied the invariance of the FCV, obtained the fractional Noether theorem, and extended it to the fractional optimal control problem, as well as the cases of Caputo derivatives and Riesz–Caputo derivatives. Ref. [9] further discussed the invariance conditions of the FCV and gave the fractional Noether theorem based on the classical concept of a conserved quantity. Ref. [15] defined a class of fractional operators, and received the Noether theorem for this operator. Ref. [32] considered the Noether theorem of the nonconservative system under the fractional action-like framework, and Ref. [29] further generalized the situation to the Lagrangian that contains higher order derivatives. Nowadays, the Noether theorem has been recognized as one of the most important results of the FCV and the fractional optimal control. Birkhoffian mechanics is a kind of natural generalization of the Hamiltonian mechanics [13,59]. Its core is the Pfaff–Birkhoff principle and the Birkhoff equations. The Hamilton principle is a special case of the Pfaff–Birkhoff principle. And Hamilton canonical equations are the special case of Birkhoff equations, which remain the same under a canonical transformation, and become the Birkhoff equations under a general noncanonical transformation. So, the theory of Birkhoffian mechanics is applicable to Hamiltonian mechanics, Lagrangian mechanics and Newtonian mechanics, and also applicable to general holonomic and nonholonomic mechanics [52,53]. Simultaneously Birkhoffian mechanics can still be applied to quantum mechanics, statistical mechanics, atomic and molecular physics, hadron physics, biological physics and engineering, and so on [59]. In 1989, Ref. [33] pointed out that the study of Birkhoffian mechanics is an important direction of modern analytical mechanics. In 1996, Ref. [53] established the basic theoretical framework for the Birkhoffian dynamics. In 1997, Ref. [34] indicated that the symmetry theory of Birkhoffian system is an

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important aspect for the study of Birkhoffian dynamics, and presented the symmetry of Birkhoffian system. Over the past 20 years, great progress has been achieved in the study of Birkhoffian dynamics [35,43,44,50, 51,61,62,64,67]. In 2013, Zhang and Zhou introduced the idea of El-Nabulsi’s fractional model to Birkhoffian mechanics known as El-Nabulsi’s fractional Birkhoffian system, presented and discussed the fractional Pfaffian variational problem, and derived the fractional Birkhoff equations and the fractional Noether theorem [65]. The novelty of the fractional Pfaffian variational problem is that the fractional integral over time only needs one parameter, and the derived fractional Birkhoff equations are similar to the classical ones, in absence of fractional derivatives, but in the presence of the fractional generalized external force acting on the system [45]. Recently, the fractional conserved quantities and fractional variational symmetries based on El-Nabulsi’s fractional model were furtherly studied and some important results have been found [14,45,46,60]. Ref. [45] studied the Noether symmetries and the conserved quantities for a non-conservative Hamiltonian system with holonomic or nonholonomic constraints under the El-Nabulsi dynamical model which was based on a fractional integral extended by periodic laws. Ref. [14] studied the problem of perturbation to Noether symmetries and adiabatic invariants for a Birkhoffian system under small disturbance based on the ElNabulsi dynamical model. Ref. [60] studied Lie symmetry, conserved quantities and adiabatic invariants for El-Nabulsi’s fractional Birkhoffian system. Ref. [46] focused on studying Noether’s theorem in phase space for fractional variational problems from extended exponentially fractional integral introduced by ElNabulsi. As a continuation, Zhou and Zhang studied the Noether symmetry and the conserved quantity of a fractional Birkhoffian system within the Riemann–Liouville fractional derivatives [68]. While, most of the studies were related to the fractional Hamiltonian systems and the fractional Birkhoffian systems, rather than the fractional forced Birkhoffian systems. In the sense that forced Birkhoffian equations correspond to the forced Birkhoffian representation of original mechanical systems, fractional forced Birkhoffian equations correspond to the fractional forced Birkhoffian representation of original mechanical systems. For the original mechanical systems with some unpleasant terms, such as coupling and dissipative terms, we can give them easily a fractional forced Birkhoffian representation but difficultly a fractional Birkhoffian representation. Thus theoretically speaking, the fractional forced Birkhoffian systems are more universal in comparison with fractional Birkhoffian systems and easier to be constructed from the practical point. In addition, the fractional forced Birkhoffian systems are closely associated with control problems for the term of generalized force. Therefore, it seems to be very important to investigate fractional forced Birkhoffian systems. In this paper, we focus on studying Noether symmetries and conserved quantities for the fractional forced Birkhoffian systems. We will give a new fractional Pfaff–Birkhoff variational principle, from which derive the fractional forced Birkhoff equations, and obtain the Noether theorems for the fractional forced Birkhoffian systems. Our fractional Birkhoffian model in this paper is based on the Riemann–Liouville derivatives under the framework of the fractional calculus. The novelty of the fractional Pfaff–Birkhoff variational principle in this paper is that it contains fractional derivatives and generalized force, and the derived El-Nabulsi–Birkhoff fractional equations are fractional differential equations. Our results center on fractional forced Birkhoffian systems which contain fractional Birkhoffian systems as a particular case (when generalized force equals to zero). The paper is organized as follows. In Section 2, we will recall the definitions of the right and left Riemann–Liouville fractional derivatives and their main properties. In Section 3, we will introduce the idea of fractional model to Birkhoffian mechanics, present a new fractional Pfaff–Birkhoff variational principle, and from which derive the fractional forced Birkhoff Equations. Section 4 is devoted to the study of the fractional Noether theorems. Section 5 is an illustrative example. Section 6 presents conclusions of this research. Up to now, using the methods of the FCV to study the Birkhoffian mechanics is just at the beginning, its deepness and breadth are limited and much work is still required.

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2. Preliminaries In this section, we recall the definitions of the right and left Riemann–Liouville fractional derivatives and their main properties. Definition 1. [28] Let f be a continuous and integrable function in the interval [C1, C2 ]. For all t ∈ [C1 , C2 ], the left Riemann–Liouville fractional derivative C1 Dtα f (t), and the right Riemann–Liouville fractional α derivative t DC f (t), of order α, are defined in the following way: 2 α C1 Dt f (t)

α t DC2 f (t)

1 = Γ(n − α)

1 = Γ(n − α)



d dt

n  t (t − θ)n−α−1 f (θ)dθ,

(1)

n C2 (θ − t)n−α−1 f (θ)dθ,

(2)

C1



d − dt

t

where n ∈ N, n − 1 ≤ α < n, and Γ is the Euler gamma function. Remark 1. If α is an integer, then from Definition 1 one can obtain the standard derivative, which is  α C1 Dt f (t) =

d dt

α

 f (t),

α t DC2 f (t) =

−

d dt

α f (t).

Theorem 1. [28] Let f and g be two continuous functions on [C1 , C2 ]. We are assuming that C1 Dtp f (t) and p t DC2 g(t), p > 0, exist at every point t ∈ [C1 , C2 ] and are continuous. Then, for all t ∈ [C1 , C2 ], the following properties hold: (1) for p > 0, C1 Dtp (f (t) + g(t)) = C1 Dtp f (t) + C1 Dtp g(t); (2) for p ≥ q ≥ 0, C1 Dtp (C1 Dt−q f (t)) = C1 Dtp−q f (t); (3) for p > 0, C1 Dtp (C1 Dt−p f (t)) = f (t) (fundamental property of the Riemann–Liouville fractional derivatives); (4) if f (t) or g(t) vanish at t = C1 and t = C2 together with their derivatives up to order k, where k is the C C p biggest integer smaller than p, C12 (C1 Dtp f (t))g(t)dt = C12 f (t) t DC g(t)dt. 2 For convenience, we now introduce the notions for partial derivative which will be used throughout this paper: the partial derivative of a function K : RM → R with respect to its ith argument xi will be denoted ∂K by ∂x or ∂i K, i = 1, 2, · · · , M (M ∈ N). i 3. Fractional Pfaff–Birkhoff principle and fractional forced Birkhoff equations Find an extreme value of the integral functional

S(a, F ) =

t2  2n t1

Rvα (t, a) t1 Dtα av

− B (t, a) +

v=1

2n 

Rvβ (t, a) t Dtβ2 av

 + F (t) · a(t) dt

(3)

v=1

under the commutation relations α v t1 Dt (δa )

= δ(t1 Dtα av ),

β v t Dt2 (δa )

= δ(t Dtβ2 av ),

(4)

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and the boundary conditions δav |t=t1 = δav |t=t2 = 0,

(5)

where B (t, a) is the Birkhoffian, Rvα (t, a), Rvβ (t, a), v = 1, 2, · · · , 2n, are Birkhoff’s functions, a = (a1 , a2 , · · · , a2n ) is Birkhoffian variable, Rvα (t, a), Rvβ (t, a) and B (t, a) are C 2 functions, F (t) is the generalized force, δ() is the contemporaneous variation, and 0 < α ≤ 1, 0 < β ≤ 1. Remark 2. One can consider more general problem with the case α > 1, β > 1 in the integral functional (3). And our results can be easily formulated in that case. So we will consider only the case 0 < α ≤ 1, 0 < β ≤ 1 in the integral functional (3). Similar to the Pfaff–Birkhoff variational principle [52,53], we give the following fractional Pfaff–Birkhoff variational principle. Definition 2. Under the commutation relations (4) and the boundary conditions (5), the contemporaneous variation principle: δS = 0, is called fractional Pfaff–Birkhoff variational principle. Taking the contemporaneous variation of the functional (3), i.e., δ{S(a, F )} = 0, we have t2  2n   2n  μ=1

t1

+

v=1

∂Rvα ∂aμ

α v t1 Dt a

∂Rvβ β v + t Dt2 a ∂aμ



 ∂B − μ + Fμ δaμ ∂a



2n   Rvα δ(t1 Dtα av ) + Rvβ δ(t Dtβ2 av ) dt = 0.

(6)

v=1

Using the commutation relations (4) and the formula of integration by parts, we get t2

t2 Rvα (t, a)δ(t1 Dtα av )dt

t1

Rvα (t, a) t1 Dtα (δav )dt

= t1

t2 = (δav ) t Dtα2 Rvα dt,

(7)

t1

and by the same reasoning, t2 Rvβ (t, a)δ(t Dtβ2 av )dt t1

t2 = (δav )t1 Dtβ Rvβ dt.

(8)

t1

Substituting (7), (8) into (6), we obtain t2  2n  2n  t1

μ=1 v=1

∂Rvα ∂aμ

α v t1 Dt a +

∂Rvβ β v t Dt2 a ∂aμ

 −



∂B β β α α μ δa dt = 0. + F + D R + D R μ t t2 μ t1 t μ ∂aμ

Considering the arbitrariness of the integral interval [t1 , t2 ], we have 2n   2n   ∂Rα μ=1

v=1

v α v t D a ∂aμ 1 t

+

∂Rvβ β v t Dt2 a ∂aμ

 −

And considering the independence of δaμ , we obtain

 ∂B β β α α μ + F + D R + D R μ t t2 μ t1 t μ δa = 0. ∂aμ

(9)

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6 2n   ∂Rα

v α v t D a ∂aμ 1 t

v=1

∂Rvβ β v + t Dt2 a ∂aμ

 −

∂B + Fμ + t Dtα2 Rμα + t1 Dtβ Rμβ = 0, ∂aμ

(10)

where μ = 1, 2, · · · , 2n. Formula (10) can be called the fractional forced Birkhoff equations in the Riemann– Liouville sense. We can obtain the following result. Proposition 1. If a makes functional (3) take its minimum, then a satisfies the fractional forced Birkhoff equations (10). Remark 3. If the fractional integral functional (3) only contains the left fractional derivative in the Riemann– Liouville sense, then the equations (10) become 2n  ∂Rα v=1

v α v t D a ∂aμ 1 t

−

∂B + Fμ + t Dtα2 Rμα = 0, ∂aμ

(11)

where μ = 1, 2, · · · , 2n. Let α → 1, then the equations (11) become 2n   ∂R v=1

v ∂aμ

−

∂Rμ ∂av



 a˙ v −

∂Rμ ∂B + μ ∂a ∂t

 + Fμ = 0,

(12)

where μ = 1, 2, · · · , 2n. The equations (12) are the standard forced Birkhoff equations which are the special case of equations (10) [52]. 4. Noether theorems Definition 3. Functional (3) is said to be invariant under an ε-parameter group of infinitesimal transformations ¯ (t) = a(t) + εξ(t, a) + o(ε) a

(13)

if T2   2n T1

T1

+

μ=1

T2   2n

=

Rμα (t, a) t1 Dtα aμ

2n 

Rμβ (t, a) t Dtβ2 aμ

 − B(t, a) + F (t) · a(t) dt

μ=1

¯) Rμα (t, a

α μ ¯ + t1 Dt a

μ=1

2n 

 ¯ ) t Dtβ2 a ¯ ) + F (t) · a ¯ (t) dt Rμβ (t, a ¯μ − B(t, a

(14)

μ=1

for any subinterval [T1 , T2 ] ⊆ [t1 , t2 ]. Proposition 2. If functional (3) is invariant under Definition 3, then 2n   2n  ∂Rμα μ=1

−

v=1

2n   v=1

∂av

 ξv

α μ t1 Dt a

+

2n   2n  ∂Rμβ μ=1

v=1

∂av

 ξv

β μ t Dt2 a

+

2n  μ=1

Rμα

α t1 Dt ξμ

+

2n 

Rμβ t Dtβ2 ξμ

μ=1



∂B − Fv ξv = 0. ∂av

(15)

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Proof. Having in mind that condition (14) is valid for any subinterval [T1 , T2 ] ⊆ [t1 , t2 ], we can get rid off the integral signs in (14): 2n 

Rμα (t, a)

α μ t1 Dt a +

μ=1

=

2n 

2n 

Rμβ (t, a) t Dtβ2 aμ − B(t, a) + F (t) · a(t)

μ=1

¯) Rμα (t, a

α μ ¯ + t1 Dt a

μ=1

2n 

¯ ) t Dtβ2 a ¯ ) + F (t) · a ¯ (t). Rμβ (t, a ¯μ − B(t, a

(16)

μ=1

Differentiating both sides of equation (16) with respect to ε, and substituting ε = 0, we obtain 2n   2n  ∂Rμα μ=1

+

v=1

2n  μ=1

Rμα

∂av

 ξv

α μ t1 Dt a

+

2n   2n  ∂Rμβ μ=1

d 1 Γ(1 − α) dt

t

−α

(t − τ )

v=1

∂av

ξ μ dτ +

 ξv

2n 

−

2n   ∂B v=1

Rμβ

μ=1

t1

β μ t Dt2 a

1 Γ(1 − β)

∂av

 − Fv ξv

  t2 d − (t − τ )−β ξμ dτ dt t

= 0.

(17)

By (1), (2) and (17), we obtain (15). This ends the proof. 2 Definition 4. [49] A quantity C(t, a, t1 Dtα a, t Dtβ2 a) is said to be fractional conserved if and only if it is possible to write C in the form C=

m 

Ci1 (t, a, t1 Dtα a, t Dtβ2 a) · Ci2 (t, a, t1 Dtα a, t Dtβ2 a),

(18)

i=1

for some m ∈ N and some functions Ci1 , Ci2 , i = 1, · · · , m, where each pair Ci1 , Ci2 , i = 1, · · · , m, satisfies j1

j2

Dtγi (Ci i (t, a, t1 Dtα a, t Dtβ2 a), Ci i (t, a, t1 Dtα a, t Dtβ2 a)) = 0,

(19)

where γi ∈ {α, β}, ji1 = 1 and ji2 = 2 (or ji1 = 2 and ji2 = 1), along all the solutions of the fractional forced Birkhoff equations (10). Remark 4. In (19), Dtγ (f, g) is defined by Dtγ (f, g) = −g t Dtγ2 f + f t1 Dtγ g, where functions f and g are C 1 functions on the interval [t1 , t2 ], and t ∈ [t1 , t2 ], γ ∈ R0+ . When γ = 1, we have d Dt1 (f, g) = −g t Dt12 f + f t1 Dt1 g = f˙g + f g˙ = (f g), dt and Dt1 (f, g) = Dt1 (g, f ), but generally Dtγ (f, g) = Dtγ (g, f ). Proposition 3. If functional (3) is invariant under Definition 3, then C(t, a) =

2n   α Rμ (t, a) − Rμβ (t, a) ξμ (t, a) μ=1

is a fractional conserved quantity of the fractional forced Birkhoffian system (10).

(20)

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Proof. The fractional forced Birkhoff equations (10) can be rewritten as  ∂B − Fμ = μ ∂a v=1 2n



∂Rvα ∂aμ

α v t1 Dt a +

∂Rvβ β v t Dt2 a ∂aμ

 + t Dtα2 Rμα + t1 Dtβ Rμβ .

(21)

Since functional (3) is invariant under Definition 3, substituting (21) into (15), we obtain 2n 

Rμα

α t1 Dt ξμ +

μ=1

=

2n 

2n 

Rμβ t Dtβ2 ξμ −

μ=1

α α t Dt2 Rμ

 + t1 Dtβ Rμβ ξμ

μ=1

2n  



 Dtα Rμα , ξμ − Dtβ ξμ , Rμβ = 0.

μ=1

The proof is complete. 2 Definition 5. Functional (3) is said to be invariant under an ε-parameter group of infinitesimal transformations ¯ (t) = a(t) + εξ(t, a) + o(ε) a t¯ = t + εζ(t, a) + o(ε)

(22)

if T2   2n T1

=

α μ t1 Dt a +

μ=1

T¯2  2n T¯1

Rμα (t, a)

2n 

 Rμβ (t, a) t Dtβ2 aμ − B(t, a) + F (t) · a(t) dt

μ=1

Rμα (t¯,

¯) a

α μ ¯ t1 Dt a

μ=1

+

2n 

Rμβ (t¯,

¯) a

β μ ¯ t Dt2 a

 ¯ ¯ ¯ ) + F (t) · a ¯ (t) dt − B(t, a

(23)

μ=1

for any subinterval [T1 , T2 ] ⊆ [t1 , t2 ]. Proposition 4. If functional (3) is invariant under Definition 5, then  2n 2n  

α  β  α β C t, a, t1 Dt a, t Dt2 a = Rμ − Rμ ξμ + (1 − α) Rμα μ=1

+ (1 − β)

2n 

Rμβ t Dtβ2 aμ − B +

μ=1

α μ t1 Dt a

μ=1 2n 

 Fμ aμ ζ

(24)

μ=1

is fractional conserved. Proof. We reparameterize the time (the independent variable t) by the Lipschitzian transformation t ∈ [t1 , t2 ] → σf (λ) ∈ [σ1 , σ2 ] which satisfies tσ = f (λ) = 1 if λ = 0. Functional (3) can be reduced, in this way, to an autonomous functional: ¯ S(t(·), a(·), F (·)) =

σ2   2n σ1

(Rμα (t(σ), a(t(σ)))

μ=1

α μ σ1 Dt(σ) a (t(σ))

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+

Rμβ (t(σ), a(t(σ))) t(σ) Dσβ2 aμ (t(σ))

9

 − B (t(σ), a(t(σ))) + F (t(σ)) · a(t(σ)) tσ dσ,

μ=1

where t(σ1 ) = t1 , t(σ2 ) = t2 , and

α σ1 Dt(σ) a(t(σ))

d (tσ )−α = Γ(1 − α) dt(σ)

and by the same reasoning,

d 1 = Γ(1 − α) dt(σ) σ

σf  (λ)

σf (λ) − τ

−α −1  a τ f (λ) dτ

t1 f (λ)

(σ − s)−α a(s)ds = (tσ )−α

t1 (tσ )2

Dσα a(σ),

t1 (tσ )2

β t(σ) Dσ2 a(t(σ))

= (tσ )−β σ Dβ t2 a(σ). (tσ )2

So, we have ¯ S(t(·), a(·), F (·)) =

σ2   2n σ1

+

Rμα (t(σ), a(t(σ)))(tσ )−α

μ=1 2n 

t1 (tσ )2

Rμβ (t(σ), a(t(σ)))(tσ )−β σ Dβ t2

(tσ )2

μ=1

=

σ2   2n σ1

+

Rμα (t(σ), a(t(σ)))(tσ )−α+1

μ=1 2n 

+

¯ α (t(σ), a(t(σ)), t ) R μ σ

μ=1 2n 

t1 (tσ )2

t2   2n t1

¯ Dσα aμ (σ) − B(t(σ), a(t(σ)), tσ )

  ¯ (t(σ), t ) · a ¯ β (t(σ), a(t(σ)), t ) σ Dβ t aμ (σ) + F ¯ R (t(σ), t ) μ σ σ σ dσ 2 (tσ )2

μ=1

=

Dσα aμ (σ) − B(t(σ), a(t(σ)))tσ

(tσ )2

σ2   2n σ1

t1 (tσ )2

 a (σ) + F (t(σ)) · a(t(σ)) tσ dσ μ

 Rμβ (t(σ), a(t(σ)))(tσ )−β+1 σ Dβ t2 aμ (σ) + F (t(σ)) · a(t(σ))tσ dσ

μ=1

=

Dσα aμ (σ) − B(t(σ), a(t(σ)))

Rμα (t, a) t1 Dtα aμ

μ=1

+

2n 

Rμβ (t, a) t Dtβ2 aμ

 − B(t, a) + F (t) · a(t) dt

μ=1

= S(a(·), F (·)). ¯ By hypothesis, functional S(t(·), a(·), F (·)) is invariant under Definition 5, then S(a(·), F (·)) is invariant under Definition 3.

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Let   L = L t, a(t), t1 Dtα a, t Dtβ2 a =

2n 

Rμα (t, a)

α μ t1 Dt a +

μ=1

2n 

Rμβ (t, a) t Dtβ2 aμ − B(t, a) + F (t) · a(t),

μ=1

we have  ¯=L ¯ t(σ), a(t(σ)), t , L σ =

2n 

 t1 (tσ )2

Dσα a(σ), σ Dβ t2 a(σ) (tσ )2

 Rμα t(σ), a(t(σ)) (tσ )−α+1

μ=1

+

2n 

t1 (tσ )2

 Dσα aμ (σ) − B t(σ), a(t(σ)) tσ

 Rμβ t(σ), a(t(σ)) (tσ )−β+1 σ Dβ t2 aμ (σ) + F (t(σ)) · a(t(σ))tσ (tσ )2

μ=1

=

2n 

 ¯ α t(σ), a(t(σ)), t R μ σ

μ=1

+

2n 

t1 (tσ )2

 ¯ t(σ), a(t(σ)), t Dσα aμ (σ) − B σ

 

 ¯ t(σ), t · a ¯ β t(σ), a(t(σ)), t σ Dβ t aμ (σ) + F ¯ t(σ), tσ . R μ σ σ 2 (tσ )2

μ=1

Then applying Proposition 3 to

 σ2 σ1

 C¯ t(σ), a(σ), tσ ,

¯ Ldσ, we get that 

t1 (t σ )2

Dσα a(σ), σ Dβ t2 a(σ) (tσ )2

¯ − ∂5 L) ¯ · ξ + ∂ Lζ ¯ = (∂4 L ∂tσ

(25)

is fractional conserved. For λ = 0, t1 (tσ )2

Dσα a(σ) = t1 Dtα a(t),

σD

β t2 (tσ )2

a(σ) = t Dtβ2 a(t),

and we can obtain ¯ − ∂5 L ¯ = ∂3 L − ∂4 L, ∂4 L

(26)

and   σ  −α−1 d ∂ ¯ ) −α(t σ −α ¯ L = ∂ L · (σ − τ ) a(τ )dτ 4 ∂tσ Γ(1 − α) dσ t1 (tσ )2

 ¯· + ∂5 L = −α∂3 L ·

−β(tσ )−β−1 Γ(1 − β)

α t1 Dt a

(−

− β∂4 L ·

d ) dσ

t2 (tσ )2



 (τ − σ)−β a(τ )dτ + L

σ β t Dt2 a

+ L.

(27)

Substituting (26) and (27) into (25), we receive the fractional conserved quantity (24). This completes the proof. 2

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Propositions 3 and 4 are called Noether theorems for fractional forced Birkhoffian systems. The results above are of universal meaning which take the theory of Noether symmetries and conserved quantities for a standard forced Birkhoffian system as a special case. For example, if taking α = β = 1, then Propositions 3 and 4 give Noether theory of a standard forced Birkhoffian system [52]. Recently the fractional action-like variational approach has gained increasing importance both in mathematical and in physical theories. The results obtained in this paper can be extended to the case of El-Nabulsi–Pfaff–Birkhoff–d’Alembert fractional principle formulated in Refs. [14,65]. In this paper we mainly discuss the Birkhoffian aspect. Similarly, we can consider the Hamiltonian aspect. Assume that the configuration of a mechanical system is determined by n generalized coordinates qk (k = 1, 2, · · · , n), the Lagrangian of the system is L(t, qk , t1 Dtα qk , t Dtβ2 qk ). Introduce the generalized momentum and Hamiltonian pαk =

∂

∂L ∂L , H = H(t, qk , pαk , pβk ). α q , pβk = D t1 t k ∂ t Dtβ2 qk

Find an extreme value of the Hamiltonian integral functional

S=

t2   n t1

pαk t1 Dtα qk

+

k=1

n 

 pβk t Dtβ2 qk

− H + F · q dt

k=1

under the commutation relations δ(t1 Dtα qk ) = t1 Dtα (δqk ), δ(t Dtβ2 qk ) = t Dtβ2 (δqk ), and the boundary conditions δqk |t=t1 = δqk |t=t2 = 0, where 0 < α ≤ 1, 0 < β ≤ 1, F is generalized force. And our results can be similarly formulated in the Hamiltonian aspect. If taking F = 0, we can obtain the Noether theory of a fractional Hamiltonian system [70]. 5. Illustrative example In the above, the Noether theorems of the forced Birkhoffian systems, which are about the relationship between the symmetry and the conserved quantity, are presented. The behavior of a mechanical system or the state of a physical system can be described by the above theorems when they can be converted to the Birkhoffian systems. In the following, we give an illustrative example. Example. Under the conditions (4) and (5), consider the fractional integral functional t2 S(a, F ) =

 3 a

α 1 t1 Dt a

+ a4 t Dtβ2 a2 − B + F · a dt,

t1

where the Birkhoffian and Birkhoff’s functions are 2 1  2 1 3 1 1 a − arctan bt + a4 − ln(1 + b2 t2 ) , 2 b 2 2b β α 3 α α α R1 = a , R2 = R3 = R4 = 0, R1 = R3β = R4β = 0, R2β = a4 , B=

respectively, a = (a1 , a2 , a3 , a4 ) and 0 < α ≤ 1, 0 < β ≤ 1.

(28)

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Taking the contemporaneous variation of the functional (28), we have 

  1 + F1 δa1 + t1 Dtα a4 + F2 δa2 + t1 Dtα a1 − a3 + arctan bt + F3 δa3 b  β 2 1 ln(1 + b2 t2 ) + F4 δa4 = 0. + t Dt2 a − a4 + 2b α 3 t Dt2 a

(29)

By the independence of δa1 , δa2 , δa3 and δa4 , we obtain α 3 t Dt2 a

+ F1 = 0,

β 2 t Dt2 a

− a4 +

α 4 t1 Dt a

+ F2 = 0,

α 1 t1 Dt a

− a3 +

1 arctan bt + F3 = 0, b

1 ln(1 + b2 t2 ) + F4 = 0, 2b

(30)

which are the corresponding fractional forced Birkhoff equations. If choosing the transformation (ζ, ξ1 , ξ2 , ξ3 , ξ4 ) = (t, 1, 1, 0, 0) to make the fractional integral functional invariant under Definition 5, by Proposition 4, we get a fractional conserved quantity

  C t, a, t1 Dtα a, t Dtβ2 a = a3 − a4 + (1 − α)a3

α 1 t1 Dt a

+ (1 − β)a4 t Dtβ2 a2 − B + F · a t.

Remark 5. Similarly, if choosing the transformation (ζ, ξ1 , ξ2 , ξ3 , ξ4 ) = (0, 1, 0, 0, 0) or (0, 0, −1, 0, 0) to make the fractional integral functional invariant under Definition 5, by Proposition 4, we can obtain the corresponding fractional conserved quantity C = a3 or a4 , which is a generalized momentum of the system [52]. Remark 6. In our illustrations most of the Lagrangians belong to the class of non-standard Lagrangians. Non-standard Lagrangians were introduced by Arnold and discussed largely in literature [22–25]. In nonstandard Lagrangians, no obvious identification of the kinetic and potential energy terms can be made. These Lagrangians play an important role in nonlinear differential equations, dynamical systems and theoretical physics. In fact non-standard Lagrangians are introduced in the literature through different forms depending on the dynamical problem under study. This type of non-standard Lagrangians in our paper plays a crucial role in the study of Noether symmetries and conserved quantities of fractional forced Birkhoffian systems and deserve attention. For α = β = 1, F = 0, it is just Pfaff type. More features connected to these non-standard Lagrangians will be investigated carefully in a future work. This is an important issue that deserves special attention and more studies. 6. Conclusions Based on the fractional calculus of variations, we have discussed the fractional variational problem of forced Birkhoffian systems, presented a new fractional Pfaff–Birkhoff variational principle, from which derived the fractional forced Birkhoff equations, and established its fractional Noether theory in the Riemann–Liouville sense which describes the connection of the fractional symmetry and the fractional conserved quantity. The results of this paper are of universal significance. The corresponding ones of standard Birkhoffian system are special cases of this paper. Further work could include the extension of symmetries and conserved quantities for fractional variational problems to various constrained fractional mechanical systems. The method of this paper is complete. However, in the real-world for a complicated example finding conserved quantities is not so easy, which deserves numerical techniques. Developing numerical methods will be a big step forward. Extending our work numerically will be addressed carefully in a forthcoming work. The fractional variational theory of Birkhoffian system is still in its childhood so that there is much remains to be done.

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