Noether's theorem of fractional Birkhoffian systems

Accepted Manuscript Noether’s Theorem of Fractional Birkhoffian systems

Hong-Bin Zhang, Hai-Bo Chen

PII: DOI: Reference:

S0022-247X(17)30719-9 http://dx.doi.org/10.1016/j.jmaa.2017.07.056 YJMAA 21587

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Journal of Mathematical Analysis and Applications

Received date:

23 July 2016

Please cite this article in press as: H.-B. Zhang, H.-B. Chen, Noether’s Theorem of Fractional Birkhoffian systems, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2017.07.056

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Noether’s Theorem of Fractional Birkhoffian systems Hong-Bin Zhang

∗

and Hai-Bo Chen

College of Mechanical and Electronic Engineering , Chaohu University, Hefei 238000, People’s Republic of China

Abstract In this paper, we study Noether type symmetry theorem to fractional Birkhoffian systems with Riemann-Liouville derivatives. This theorem provides an explicit algorithmic way to compute a constant for any Birkhoffian systems admitting a symmetry. Finally, we extend our Noether’s theorem to fractional Birkhoffian systems base on Caputo or Riesz derivatives. Key words: Birkhoffian systems. Fractional Calculus. Noether’s theorem. Symmetries

1

Introduction The concept of symmetry plays an important role in science and engineering. Symmetries are

described by transformations, which result in the same object after the transformations are carried out. They are described mathematically by parameter groups of transformations [1-4]. Their importance, as recognized by Noether in 1918 [5], is connected with the existence of conservation laws that can be used to reduce the order of the Euler-Lagrange differential equations [6]. Noether’s symmetry theorem is nowadays recognized as one of the most beautiful results of the calculus of variations [7]. The fractional calculus is an area of current strong research with many different and important applications [8-11]. In the last two decades, its importance in the calculus of variations has been perceived, and a fractional variational theory began to be developed by several different authors [12-20]. Most part of the results in this direction make use of fractional derivatives in the sense of Riemann- Liouville, Caputo or Riesz [21-24]. In 1927, the American mathematician Birkhoff [25] presented a new form integral variational principle and give a new equations of motion in his famous works. In 1978, the American physicist Santilli studied the Birkhoffian equations and its transformation theory [26,27]. In 1989, A.S. Galiullan point out [28] that it is an important developmental direction of modern analytical mechanics to study the Birkhoffian dynamics. In 1992, Mei and his co-workers constructed the theoretical framework of Birkhoffian dynamics [29]. Since then, The Significant progress was made in the study of Birkhoffian dynamics [30-37]. It can be applied to many fields, such as quantum mechanics, atomic and molecular physics, hadron physics, biological physics and so on. Recently, Fractional Birkhoffian dynamics has called the attention of some researchers, S.K. Luo [38] presented a unified fractional Pfaff-Birkhoffian principle by using the definitions ∗

Corresponding author. E-mail:[email protected]

1

of combined fractional derivative, and deduced Birkhoffian equations in terms of Riemann-Liouville, Caputo, Riesz and Riesz-Caputo fractional derivatives, respectively. Using Agrawal’s new operators, author present a generalized fractional Birkhoffian equations [39]. Y. Zhang [40] presented the variational problems for fractional Birkhoffian systems and given a Noether theorem in terms of conservation laws as it is done in the classical theory.This conserved quantities to be constant in time. However, This result is unsatisfactory, because this conserved quantity is defined by an integral relation and it is not explicit. In this paper, we will borrow a recent “transfer formula” from [41] that allows to study the existence of explicit conservation laws for fractional Birkhoffian systems introduced in [38]. The article is organized as follows: In Sect.2, a brief summery of some definitions of fractional integrals and derivatives, and their basic properties. In Sect.3, we review of the classical Noether’s theorem of Birkhoffian systems. The main contributions of the paper appear in Sect.4. We establish two forms Noether’s theorem of fractional Birkhoffian systems based on Riemann-Liouville derivatives and new differential operators, respectively. Finally, a conclusion is given in Sect.5.

2

Preliminaries of fractional calculus In this section we fix notations by collecting the definitions and properties of fractional integrals and

derivatives needed in the sequel [9-12]. Definition 1 ( Riemann-Liouville fractional integrals ). Let f be a continuous function in the interval [t1 , t2 ]. For t ∈ [t1 , t2 ], the left Rieminn-Liouville fractional integral Riemann-Liouville fractional integral

α t1 It f (t)

and the right

α t It2 f (t)

of order α, are defined by  t 1 α I f (t) = (t − θ)α−1 f (θ)dθ t1 t Γ(α) t1  t2 1 α (θ − t)α−1 f (θ)dθ t It2 f (t) = Γ(α) t Where Γ is the Euler gamma function and 0 < α < 1 .

(1) (2)

Definition 2 (Fractional derivatives in the sense of Riemann-Liouville ). Let f be a continuous function in the interval [t1 , t2 ]. For t ∈ [t1 , t2 ], the left Rieminn-Liouville fractional derivative and the right Riemann-Liouville fractional derivative t Dtα2 f (t) of order α , are defined by  d t d 1 1−α α f (t) = (t − θ)−α f (θ)dθ t1 Dt f (t) = t1 I t dt Γ(1 − α) dt t1  d t2 d 1−α −1 α f (t) = (θ − t)−α f (θ)dθ t Dt2 f (t) = − tI dt t2 Γ(1 − α) dt t

α t1 Dt f (t)

(3) (4)

Theorem 3. Let f and g be two continuous functions in the interval [t1 , t2 ]. For all t ∈ [t1 , t2 ], the following property holds: α t1 Dt (f (t)

+ g(t)) = t1 Dtα f (t) + t1 Dtα g(t)

Theorem 4. If f , g and the fractional derivatives t ∈ [t1 , t2 ], then



t2 t1

 f (t)t1 Dtα g(t)dt =

α t1 D t g t2

t1

(5)

and t Dtα2 f are continuous at every point

g(t)t Dtα2 f (t)dt

(6)

for any 0 < α < 1. Moreover, formula (6) is still valid for α = 1 provided f or g are zero at t = t1 and t = t2 .

2

3

Review of the classical Noether’s theorem of Birkhoff systems The classical variational problems of Birkhoffian systems as following [29,44]  t2 (Rμ (t, a)a˙ μ − B(t, a))dt → min S(a) =

(7)

t1

subject to the terminal conditions aμ |t=t1 = aμ1 ,

aμ |t=t2 = aμ2

(μ = 1, 2, · · · , 2n)

(8)

where a˙ μ = daμ /dt, the Birkhoffian B : [t1 , t2 ]×R2n → R, and Birkhoff’s functions Rμ : [t1 , t2 ]×R2n → R are assumed C 2 -functions with respect to all its arguments. Let P = P (t, aμ , a˙ μ ) = Rμ (t, a)a˙ μ − B(t, a). We denote by ∂i P the partial derivative of P with respect to its ith argument, i = 1, · · · , 4. Definition 5

(Invariance without transforming the time). Functional (7) is said to be invariant

under an -parameter group of infinitesimal transformations a ¯μ (t) = aμ (t) + ξμ (t, a) + o() If, and only if,



tb



tb

(Rμ (t, a)a˙ μ − B(t, a))dt =

ta

(9)

μ

(Rμ (t, a ¯ )a ¯˙ − B(t, a ¯))dt

(10)

ta

for any subinterval [ta , tb ] ⊆ [t1 , t2 ]. Theorem 6

(Necessary condition of invariance). If functional (7) is invariant under transformation

(9), then Rμ ξ˙μ + ( Proof :

∂Rμ μ ∂B a˙ − ν )ξν = 0 ∂aν ∂a

(μ, ν = 1, 2, · · · , 2n)

(11)

Equation (10) is equivalent to

(Rμ (t, aν )a˙ μ − B(t, aν )) = (Rμ (t, aν + ξν + o())(a˙ μ + ξ˙μ + o()) − B(t, aν + ξν + o())

(12)

Differentiating both sides of Eq.(12) with respect to , then substituting  = 0, we obtain equality (11). Definition 7

(Conserved quantity).

Quantity I(t, a) is said to be conserved if, and only if,

dI(t, a)/dt = 0 along all the solutions of the Birkhoffian equations. 2n  ∂Rν ∂Rμ ν ∂B ∂Rμ =0 ( μ − )a˙ − μ − ν ∂a ∂a ∂a ∂t μ=1

Theorem 8

(μ, ν = 1, 2, · · · , 2n)

(13)

(Noether’s theorem without transforming time). If functional (7) is invariant under

the one-parameter group of transformations (9), then I(t, a) =

∂P ξμ = Rμ (t, a)ξμ ∂ a˙ μ

(14)

is conserved. Proof : Using the Birkhoffian equations (13) and the necessary condition of invariance (11), we obtain d d (Rμ (t, a)ξμ (t, a)) = (Rμ (t, a))ξμ (t, a) + Rμ (t, a)ξ˙μ (t, a) dt dt =(

∂Rμ ν ∂B ∂Rν ν ∂Rμ + a˙ )ξμ (t, a) + ( μ − a˙ )ξμ (t, a) ∂t ∂aν ∂a ∂aμ

3

= (( Definition 9

∂Rν ∂Rμ ν ∂B ∂Rμ )ξμ (t, a) = 0 − )a˙ − μ − ∂aμ ∂aν ∂a ∂t

Functional (7) is said to be invariant under the one-parameter group of infinitesimal

transformations

If, and only if



tb

t¯ = t + τ (t, a) + o()

(15)

a ¯μ (t) = aμ (t) + ξμ (t, a) + o()

(16)

 (Rμ (t, a)a˙ μ − B(t, a))dt =

ta

t¯(tb ) t¯(ta )

μ

(Rμ (t¯, a ¯ )a ¯˙ − B(t¯, a ¯))dt¯

(17)

for any subinterval [ta , tb ] ⊆ [t1 , t2 ]. Theorem 10

(Noether’s theorem). If functional (7) is invariant, in the sense of definition 9, then I(t, a) = Rμ (t, a)ξμ (t, a) − B(t, a)τ (t, a)

(18)

is conserved. Proof : In order to prove Noether’s theorem in its general form, we adopt a technique used in [42]. Every non-autonomous problem (7) is equivalent to an autonomous one, considering t as a dependent variable. For that we consider a Lipschitzian one-to-one transformation [t1 , t2 ]  t −→ σ ∈ [σt1 , σt2 ] Such that



t2

S(a(·)) =  =

(Rμ (t, a)a˙ μ − B(t, a))dt

t1 σt2 σt1

(Rμ (t(σ), a(t(σ)))



σt2

=

(Rμ (t(σ), a(t(σ)))

σt1



σt2

=

σt1

daμ (a(t(σ))) dσ dt(σ) dσ

− B(t(σ), a(t(σ))))

dt(σ) dσ dσ

(aμσ ) − B(t(σ), a(t(σ))))tσ dσ tσ

(Rμ (t(σ), a(t(σ)))(aμσ ) − B(t(σ), a(t(σ)))tσ )dσ  =

σt2 σt1

P¯ (t(σ), aμ (t(σ)), tσ , (aμσ ) )dσ

Where t(σt1 ) = t1 , t(σt2 ) = t2 , tσ =

dt(σ) dσ , B(t(σ), a(t(σ)))tσ . If

(aμσ ) =

daμ (t(σ)) dσ

and P¯ = P¯ (t(σ), aμ (t(σ)), tσ , (aμσ ) ) =

Rμ (t(σ), a(t(σ)))(aμσ ) − functional S(a(·)) is invariant in the sense of definition 9, ¯ then functional S(t(·), a(t(·))) is invariant in the sense of definition 5. Applying Theorem 8, we obtain that C(t, aμ , tσ , (aμσ ) ) =

∂ P¯ ∂ P¯ τ μ  ξμ + ∂(aσ ) ∂tσ

(19)

is a conserved quantity. Since

∂3 P¯ = −∂3 P

∂4 P¯ = ∂3 P

(20)

(aμσ ) tσ

(21)

+ P = P − ∂3 P · a˙ μ

Substituting (20) and (21) into (19), we arrive to the intended conclusion (18).

4

4

Main results In this section, we study two forms Noether’s theorem of fractional Birkhoffian systems. (1) Noether’s theorem of fractional Birkhoffian systems with Riemann-Liouville derivatives The variational problems of fractional Birkhoffian systems with Riemann-Liouville derivatives as

following [38]



t2

Sf [a(·)] = t1

(Rμ (t, a)t1 Dtα aμ (t) − B(t, a))dt → min

(22)

subject to the terminal conditions aμ |t=t1 = aμ1 ,

aμ |t=t2 = aμ2 ,

(μ = 1, 2, · · · , 2n)

where 0 < α ≤ 1, the Birkhoffian B : [t1 , t2 ] × R2n → R, and Birkhoff’s functions Rμ : [t1 , t2 ] × R2n → R are assumed C 2 -functions with respect to all its arguments. Let Pf = Pf (t, aμ (t), t1 Dtα aμ (t)) = Rμ (t, a)t1 Dtα aμ (t) − B(t, a). If aμ (t) is a minimizer of problem (22), then it satisfies the fractional Birkhoffian

Theorem 11 equations [40]

∂Rν (t, a) ∂B(t, a) α ν α =0 t1 Dt a + t Dt2 Rμ (t, a) − ∂aμ ∂aμ Definition 12 if,

(23)

We say that functional (22) is invariant under the transformations (9) if, and only



tb ta

 (Rμ (t, a)t1 Dtα aμ − B(t, a))dt =

tb ta

(Rμ (t, a ¯)t1 Dtα a ¯μ − B(t, a ¯))dt

(24)

for any subinterval [ta , tb ] ⊆ [t1 , t2 ]. The next theorem establishes a necessary condition of invariance, of extreme importance for our objectives. Theorem 13

If functional (22) is invariant under transformations (9), then

∂2 Pf (t, aμ (t), t1 Dtα aμ (t))ξν (t, a) + ∂3 Pf (t, aμ (t), t1 Dtα aμ (t))t1 Dtα ξμ (t, a) = 0 Proof :

(25)

Having in mind that condition (24) is valid for any subinterval [ta , tb ] ⊆ [t1 , t2 ], we can get

rid of the integral signs in (24). Differentiating this condition with respect to , substituting  = 0, and using the definitions and properties of the Riemann-Liouville fractional derivatives given we arrive to 0 = ∂2 Pf (t, aμ (t), t1 Dtα aμ (t)) · ξν (t, a) + ∂3 Pf (t, aμ (t), t1 Dtα aμ (t))· 1 d d [ d Γ(1 − α) dt



t

(t − θ)

t1

d  a (θ)dθ + Γ(1 − α) dt

−α μ



t

(t − θ)−α ξμ (θ, a)dθ]=0

t1

above equation is equivalent to (25). Remark : Using the Birkhoffian equation(23), the necessary condition of invariance (25) is equivalent to Rμ (t, a)t1 Dtα ξμ (t, a) − ξμ (t, a)t Dtα2 Rμ (t, a) = 0 In [41], the following theorem is proved.

5

(26)

( Transfer formula [41]). Consider functions f, g ∈ C ∞ ([t1 , t2 ]; R) and assume the

Theorem 14

following condition (C ): the sequences (g (k) · t1 Itk−α f )k∈N\0 and (f (k) · t Itk−α g)k∈N\0 converge uniformly 2 to on [t1 , t2 ]. Then, the following equality holds: g · t1 Dtα f − f · t Dtα2 g =

∞ d  [ ((−1)r g (r) · t1 Itr+1−α f + f (r) · t Itr+1−α g)] 2 dt r=0

Theorem 15 (Fractional Noether’s theorem without transformation of time ).If functional (22) is invariant, in the sense of definition (12), Let aμ (t) be a solution of Birkhoffian equations (23), functions ξμ and Rμ satisfy condition (C ) of Theorem 14, then the following equality holds: ∞ d  [ ((−1)r Rμ(r) · t1 Itr+1−α ξμ + ξμ(r) · t Itr+1−α Rμ )] = 0 2 dt r=0

(27)

Proof : We combine equation (26) and theorem 14. This theorem provides an explicit algorithmic way to compute a constant for any fractional Birkhoffian systems admitting a symmetry. An arbitrary closed approximation of this quantity can be obtained with a truncature of the infinite sum. The next definition gives a more general notion of invariance for integral functional (22). The main result of this section, the Theorem 17, is formulated with the help of this definition. Definition 16 Functional (22) is said to be invariant under the  - parameter group of infinitesimal transformations

If



tb ta

t¯ = t + τ (t, a) + o()

(28)

a ¯μ (t) = aμ (t) + ξμ (t, a) + o()

(29)

 (Rμ (t, a)t1 Dtα aμ

− B(t, a))dt =

t¯(tb ) t¯(ta )

(Rμ (t¯, a ¯μ )t¯1 Dt¯α a ¯μ − B(t¯, a ¯μ ))dt¯

(30)

For any subinterval [ta , tb ] ⊆ [t1 , t2 ]. Our next result gives a general from of Noether’s theorem for fractional Birkhoffian systems. Theorem 17

(Fractional Noether’s theorem). If functional (22) is invariant, in the sense of defini-

μ

tion 16, Let a (t) be a solution of Birkhoffian equations (23), functions ξμ and Rμ satisfy condition (C ) of Theorem 14, then the following equality holds: ∞ d  [ ((−1)r Rμ(r) · t1 Itr+1−α ξμ + ξμ(r) · t Itr+1−α Rμ ) + τ (Pf − αRμ · t1 Dtα aμ )] = 0 2 dt r=0

(31)

Proof : According to the strategy used by Frederico and Torres in [22], we reparametrize the time (the independent variable) by the Lipschitz transformation [t1 , t2 ]  t → σf (λ) ∈ [σt1 , σt2 ] That satisfies tσ =

dt(σ) = f (λ) = 1 if λ = 0 dσ

Functional (22) is reduced, in this way, to an autonomous functional:  σt2 α S¯f [t(·), aμ (t(·))] = Pf (t(σ), aμ (t(σ)), σt1 Dσ(t) aμ (t(σ)))tσ dσ σt1

6

(32)

(33)

Where t(σt1 ) = t1 and t(σt2 ) = t2 . Using the definitions and properties of fractional derivatives given in section 2, we get successively that α μ σt1 Dσ(t) a (t(σ))

d 1 Γ(1 − α) dt(σ)

=

=

(tσ )−α d Γ(1 − α) dσ



σ t1 (tσ )2

= (tσ )−α We then have



σt2

S¯f [t(·), aμ (t(·))] = . =





σf (λ)

(σ − s)−α aμ (s)ds

t1 (tσ )2

Dσα aμ (σ)

Pf (t(σ), aμ (t(σ)),

σt1 σt2

P¯f (t(σ), aμ (t(σ)), tσ ,

σt1



t2

= t1

(σf (λ) − θ)aμ (θf −1 (λ))dθ

σt1

t1 (tσ )2

t1 (tσ )2

α Dσ(t) aμ (t(σ)))tσ dσ

α Dσ(t) aμ (t(σ)))dσ

Pf (t, aμ (t), t1 Dtα aμ (t))dt = Sf [aμ (·)]

If the integral functional (22) is invariant in the sense of definition (16), then the integral functional (33) is invariant in the sense of definition (12). It follows from Theorem 17 that ∞

 d ∂ (r) [τ  P¯f + ((−1)r ∂4 P¯f · t1 Itr+1−α ξμ + ξμ(r) t Itr+1−α ∂4 P¯f )] = 0 2 dt ∂tσ r=0

(34)

For λ = 0, the condition (32) allow us to write that t1 (tσ )2

α Dσ(t) aμ (t(σ)) = t1 Dtα aμ (t)

And, therefore, we get ∂4 P¯f = ∂3 Pf and

∂ ¯ ∂ (t )−α d P f = P f + ∂ 3 Pf  [ σ  ∂tσ ∂tσ Γ(1 − α) dσ



σ t1  )2 (tσ

(35)

(σ − s)−α aμ (s)ds]tσ = Pf − α∂3 Pf · t1 Dtα aμ

(36)

We obtain (31) substituting (35) and (36) into equaqtion (34). (2) Noether’s theorem of fractional Birkhoffian systems with new operators α The following three new operators be introduced by Agrawal [43]. The integral operator KM of order

α, which is defined as follow  α Kt 1 ,t,t2 ,p,q



t

=p

b

kα (t, τ )f (τ )dτ + q a

t

α kα (t, τ )f (τ )dτ = KM f (t)

(37)

α Two differential operators Aα M and BM , which are defined as follow n n−α α Aα t1 ,t,t2 ,p,q f (t) = D KM f (t) = AM f (t)

(38)

n−α n α α f (t) = KM D f (t) = BM f (t) Bt 1 ,t,t2 ,p,q

(39)

α α The new Operators KM , Aα M and BM satisfy the following formulas  t2  t2 α α g(t)KM f (t)dt = f (t)KM ∗ g(t)dt t1

t1

7

(40)



t2 t1



b a

 n g(t)Aα M f (t)dt = (−1)

 α g(t)BM f (t)dt = (−1)n

t2 t1

t2 t1

n−1 

α−1−j (−D)j g(t)AM f (t) |tt21

(41)

α+j−n (−1)j AM g(t)Dn−1−j f (t) |tt21 ∗

(42)

α f (t)BM ∗ g(t)dt +

j=0

f (t)Aα M ∗ g(t)dt +

n−1  j=0

where t1 < t < t2 , M = t1 , t, t2 , p, q is a parameter set, and M ∗ = t1 , t, t2 , q, p . D is the classical derivative operator, kα (τ, t) is a kernel which may depend on a parameter α, f (t) and g(t) are sufficiently smooth functions, the parameters p and q are two real numbers, n − 1 < α < n. Firstly, we consider the variational problems of the fractional Birkhoffian systems with fractional differential operator Aα M [39] 

t2

SN [a(·)] = t1

μ (Rμ (t, a)Aα M a (t) − B(t, a))dt → min

(43)

subject to the terminal conditions (aμ )k |t=t1 = aμ1,k ,

(aμ )k |t=t2 = aμ2,k ,

k ∈ {0, 1, · · · , n − 1},

(μ = 1, 2, · · · , 2n)

where n − 1 < α < n, the Birkhoffian B : [t1 , t2 ] × R2n → R, and Birkhoff’s functions Rμ : [t1 , t2 ] × μ R2n → R are assumed C 2 -functions with respect to all its arguments. Let PA = PA (t, aμ (t), Aα M a (t)) = μ Rμ (t, a)Aα M a (t) − B(t, a).

If aμ (t) is a minimizer of problem (43), then it satisfies the fractional Birkhoffian

Theorem 18 equations [39]

∂B(t, a) ∂Rν (t, a) α ν α AM a + (−1)n BM =0 ∗ Rμ (t, a) − ∂aμ ∂aμ Definition 19 if,

(44)

We say that functional (43) is invariant under the transformations (9) if, and only 

tb ta

 μ (Rμ (t, a)Aα Ma

− B(t, a))dt =

tb ta

(Rμ (t, a ¯)Aα ¯μ − B(t, a ¯))dt Ma

(45)

for any subinterval [ta , tb ] ⊆ [t1 , t2 ]. The next theorem establishes a necessary condition of invariance. Theorem 20

If functional (43) is invariant under transformations (9), then

μ μ α μ α ∂2 PA (t, aμ (t), Aα M a (t))ξν (t, a) + ∂3 PA (t, a (t), AM a (t))AM ξμ (t, a) = 0

Proof :

(46)

Having in mind that condition (45) is valid for any subinterval [ta , tb ] ⊆ [t1 , t2 ], we can get

rid of the integral signs in (45). Differentiating this condition with respect to , substituting  = 0, and using the definitions and properties of the operator Aα M given we arrive to μ μ α μ α 0 = ∂2 PA (t, aμ (t), Aα M a (t)) · ξν (t, a) + ∂3 PA (t, a (t), AM a (t)) · AM ξμ (t, a)

Remark : Using the Birkhoffian equation (44), the necessary condition of invariance (46) is equivalent to μ α n α μ α μ ∂3 PA (t, aμ (t), Aα M a (t)) · AM ξμ (t, a) − (−1) ξμ (t, a) · BP ∗ ∂3 PA (t, a (t), AM a (t)) = 0

Imitating the proof of literature [41], we can obtain the following theorem

8

(47)

( Transfer formula ). Consider functions f, g ∈ C ∞ ([t1 , t2 ]; R) and assume the

Theorem 21

k−α k−α following condition (C): the sequences (g (k) · KM f )k∈N\0 and (f (k) · KM ∗ g)k∈N\0 converge uniformly

to on [t1 , t2 ]. Then, the following equality holds: n α g · Aα M f − (−1) f · BM ∗ g =

∞ d  r+1−α r+1−α [ ((−1)r g (r) · KM f + (−1)n f (r) · KM g)] ∗ dt r=0

If functional (43) is invariant, in the sense of definition 19, Let aμ (t) be a solution of

Theorem 22

Birkhoffian equations (44), functions ξμ and Rμ satisfy condition (C ) of Theorem 21, then the following equality holds:

Proof

∞ d  r+1−α r+1−α [ ((−1)r Rμ(r) · KM ξμ + (−1)n ξμ(r) · KM Rμ )] = 0 ∗ dt r=0

(48)

We combine equation (46) and theorem 21.

The next definition gives a more general notion of invariance for integral functional (43). Definition 23

Functional (43) is said to be invariant under the parameter group of infinitesimal

transformations t¯ = t + τ (t, a) + o() a ¯μ (t) = aμ (t) + ξμ (t, a) + o() If



tb ta

 μ (Rμ (t, a)Aα M a − B(t, a))dt =

t¯(tb ) t¯(ta )

(Rμ (t¯, a ¯μ )Aα ¯μ − B(t¯, a ¯μ ))dt¯ ¯a M

(49)

¯ = t¯1 , t¯, t¯2 , p, q . For any subinterval [ta , tb ] ⊆ [t1 , t2 ]. Where M Our next result gives a general from of Noether’s theorem for fractional Birkhoffian systems with the operator Aα M. Theorem 24

If functional (43) is invariant, in the sense of definition 23. Let aμ (t) be a solution of

Birkhoffian equations (44), functions ξμ and Rμ satisfy condition (C ) of Theorem 21, then the following equality holds: ∞ d  r+1−α r+1−α μ [ ((−1)r Rμ(r) · KM ξμ + (−1)n ξμ(r) · KM Rμ ) + τ (PA − α∂3 PA · Aα ∗ M a )] = 0 dt r=0

(50)

Proof : Every non-autonomous problem (43) is equivalent to an autonomous one, considering t as a dependent variable. For that we reparametrize the time (the independent variable t) by the Lipschitzian transformation [t1 , t2 ]  t → σf (λ) ∈ [σt1 , σt2 ] That satisfies tσ =

dt(σ) = f (λ) = 1 if λ = 0 dσ

Functional (43) is reduced, in this way, to an autonomous functional:  σt2 μ  S¯A [t(·), aμ (t(·))] = PA (t(σ), aμ (t(σ)), Aα M a (t(σ)))tσ dσ

(51)

(52)

σt1

Where t(σt1 ) = t1 and t(σt2 ) = t2 . Using the definitions and properties of fractional derivatives given in [39], we have

9

Case 1 If M = M1 = t1 , t, t2 , 1, 0 , μ α μ Aα M1 a (t(σ)) = σ(t1 ) Dσ(t) a (t(σ)) =

=

d n 1 ( ) Γ(n − α) dt(σ)

(tσ )−α d n ( ) Γ(n − α) dσ



σ t1 (tσ )2

= (tσ )−α



σf (λ) t1 f (λ)

(σf (λ) − θ)n−α−1 aμ (θf −1 (λ))dθ

(σ − s)n−α−1 aμ (s)ds

t1 (tσ )2

Dσα aμ (σ)

and, using the same reasoning Case 2 If M = M2 = t1 , t, t2 , 0, 1 μ  −α α μ Aα σ D t2 a (σ) M2 a (t(σ)) = (tσ ) (tσ )2

Case 3 If M = M3 = t1 , t, t2 , 12 , 12 , μ  −αR α μ Aα t1 D t2 a (σ) M3 a (t(σ)) = (tσ ) (tσ )2

(tσ )2

Introducing the operator Aα Mσ , we have μ  −α α AMσ aμ (σ) Aα M a (t(σ)) = (tσ )

where Mσ = σ(t1 ), σ(t), σ(t2 ), p, q . We then have 

σt2

S¯A [t(·), aμ (t(·))] = . =



σt1 σt2 σt1

μ  PA (t(σ), aμ (t(σ)), Aα M a (t(σ)))tσ dσ

μ P¯A (t(σ), aμ (t(σ)), tσ , Aα Mσ a (t(σ)))dσ



t2

= t1

μ PA (t, aμ (t), Aα M a (t))dt

= SA [aμ (·)] If the integral functional (43) is invariant in the sense of definition (23), then the integral functional (52) is invariant in the sense of definition (19). It follows from Theorem 24 that ∞  d ∂ (r) r+1−α r+1−α [τ  P¯A + ((−1)r ∂4 P¯A · KM ξμ + (−1)n ξμ(r) KM ∂4 P¯A )] = 0 ∗ dt ∂tσ r=0

(53)

For λ = 0, the condition (51) allow us to write that μ α μ Aα Mσ a (t(σ)) = AM a (t)

And, therefore, we get ∂4 P¯A = ∂3 PA

(54)

∂ ¯ μ PA = PA − α∂3 PA · Aα Ma ∂tσ

(55)

and

10

We obtain (50) substituting (54) and (55) into equaqtion (53). Second, we consider the variational problems of the fractional Birkhoffian systems with fractional α differential operator BM [39]



t2

SB [a(·)] = t1

α μ (Rμ (t, a)BM a (t) − B(t, a))dt → min

(56)

subject to the terminal conditions (aμ )k |t=t1 = aμ1,k ,

(aμ )k |t=t2 = aμ2,k ,

k ∈ {0, 1, · · · , n − 1},

(μ = 1, 2, · · · , 2n)

where n − 1 < α < n, the Birkhoffian B : [t1 , t2 ] × R2n → R, and Birkhoff’s functions Rμ : [t1 , t2 ] × α μ R2n → R are assumed C 2 -functions with respect to all its arguments. Let PB = PB (t, aμ (t), BM a (t)) = α μ Rμ (t, a)BM a (t) − B(t, a).

Using the same reasoning, we can obtain following results: Theorem 25

If is a minimizer of problem (56), then it satisfies the fractional Birkhoffian equations

[39] ∂Rν (t, a) α ν ∂B(t, a) BM a + (−1)n Aα =0 M ∗ Rμ (t, a) − ∂aμ ∂aμ Definition 26 if,

(57)

We say that functional (56) is invariant under the transformations (9) if, and only 

tb ta

 α μ (Rμ (t, a)BM a − B(t, a))dt =

tb ta

α μ (Rμ (t, a ¯)BM a ¯ − B(t, a ¯))dt

(58)

for any subinterval [ta , tb ] ⊆ [t1 , t2 ]. Theorem 27

If functional (56) is invariant under transformations (9), then

α μ μ α ∂2 PB (t, aμ (t), BM a (t))ξν (t, a) + ∂3 PB (t, aμ (t), Aα M a (t))BM ξμ (t, a) = 0

(59)

Remark: Using the Birkhoffian equation (57), the necessary condition of invariance (59) is equivalent to α μ α μ α μ ∂3 PB (t, aμ (t), BM a (t)) · BM ξμ (t, a) − (−1)n ξμ (t, a) · Aα P ∗ ∂3 PB (t, a (t), BM a (t)) = 0

(60)

If functional (56) is invariant, in the sense of definition 26, Let aμ (t) be a solution of

Theorem 28

Birkhoffian equations (57), functions ξμ and Rμ satisfy condition (C ) of Theorem 21, then the following equality holds:

∞ d  r+1−α r+1−α [ ((−1)r Rμ(r) · KM ξμ + (−1)n ξμ(r) · KM Rμ )] = 0 ∗ dt r=0

Definition 29

(61)

Functional (56) is said to be invariant under the parameter group of infinitesimal

transformations t¯ = t + τ (t, a) + o() a ¯μ (t) = aμ (t) + ξμ (t, a) + o() If



tb ta

 α μ (Rμ (t, a)BM a − B(t, a))dt =

t¯(tb ) t¯(ta )

α μ (Rμ (t¯, a ¯μ )BM ¯ − B(t¯, a ¯μ ))dt¯ ¯a

¯ = t¯1 , t¯, t¯2 , p, q . For any subinterval [ta , tb ] ⊆ [t1 , t2 ]. Where M

11

(62)

Theorem 30

If functional (56) is invariant, in the sense of definition 29. Let aμ (t) be a solution of

Birkhoffian equations (57), functions ξμ and Rμ satisfy condition (C ) of Theorem 21, then the following equality holds: ∞ d  r+1−α r+1−α α μ [ ((−1)r Rμ(r) · KM ξμ + (−1)n ξμ(r) · KM Rμ ) + τ (PB − α∂3 PB · BM a )] = 0 ∗ dt r=0

(63)

Obviously, when the parameter p and q take different values, above results can give the Noether theorem of fractional Birkhoffian systems with Riemann-Liouvill, caputo, Riesz derivatives, respectively.

5

Conclusion In this paper, we borrow a recent “transfer formula”, proved a Noether’s theorem for fractional

Birkhoffian systems with Riemann-Liouvill derivatives invariant under a symmetry group. In contrary to previous results in this direction, it provides an explicit conservation law. The given formula is algorithmic ,it can be used to computer conserved quantity to arbitrary high order approximations. Then we further α extend Noether’s theorem to fractional Birkhoffian systems base on the new operators Aα M and BM .

The results not only adapted to the fractional Birkhoffian systems with Riemann-Liouvill derivatives and adapted to the fractional Birkhoffian systems with Caputo or Riesz derivatives. Acknowledgments We express our sincere thinks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos.10872037,11472063 and the Natural Science Foundation of Anhui Province under Grant No. 070416226.

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