Noether symmetry for fractional Hamiltonian system

Physics Letters A 383 (2019) 125914

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Physics Letters A www.elsevier.com/locate/pla

Noether symmetry for fractional Hamiltonian system C.J. Song School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, PR China

a r t i c l e

i n f o

Article history: Received 11 June 2019 Received in revised form 13 August 2019 Accepted 20 August 2019 Available online 29 August 2019 Communicated by B. Malomed Keywords: Fractional Hamilton equation Noether quasi-symmetry Perturbation to Noether quasi-symmetry

a b s t r a c t Fractional Hamiltonian systems within combined Riemann-Liouville fractional order derivative and combined Caputo fractional order derivative are established. Then Noether quasi-symmetry and conserved quantity for the fractional Hamiltonian systems are presented. Thirdly, perturbation to Noether quasisymmetry and adiabatic invariant are studied. Several special cases are discussed in each section. And finally, two applications, i.e., the fractional Lotka biochemical oscillator model and the fractional isotropic harmonic oscillator model, are discussed to illustrate the results and methods. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Fractional order calculus almost has been involved in every field of science and engineering [1]. Riemann-Liouville fractional order derivative (RL) and Caputo fractional order derivative (C) are most popular among various kinds of definitions of fractional order derivative. RL is introduced earlier than C. Mathematical calculations will be simplified when RL is used, which is the main reason for RL’s popularity. However, RL has its own two drawbacks at that time. One is that the Riemann-Liouville fractional order derivative of a constant is not zero and the other is that RL requires fractional order initial conditions in many applications. But those fractional order initial conditions are generally not specified, and many people believe that they are not physical. In contrast, Caputo fractional order derivative of a constant is zero, and a fractional differential equation defined in terms of C requires standard boundary conditions. So C becomes popular among engineers and scientists. However, after C was introduced, Agrawal [2] proved that fractional order initial conditions may have physical meaning, and he illustrated this by an example. He also argued that fractional terminal conditions may be needed even when a problem is described in terms of C. That means RL and C have their own advantages and disadvantages. Calculus of variations with fractional order derivative was started in 1996 in Refs. [3,4], where Riewe stated the history of fractional order derivative, established the fractional order Lagrangian system and the fractional order Hamiltonian system. Agrawal [2,5] presented Lagrange equations in terms of RL as well as C. Because the classical equivalent Lagrangian is very useful to deal with the Hamilton-Jacobi equation, Baleanu [6] extended the notion of equivalent Lagrangian to the fractional case, constructing the fractional equivalent Lagrangian and the fractional Hamiltonian within RL. In Ref. [7], combined Caputo fractional order derivative (CC) was introduced, by which EulerLagrange equation, isoperimetric problem and transversality condition were proved. And the result in Ref. [8] is only a special case of that in Ref. [7]. In 2012, Euler-Lagrange equation, isoperimetric problem and transversality condition in terms of classical and combined Caputo fractional order derivative were obtained [9]. Those results are important because the Lagrangian within both classical and combined Caputo fractional order derivative enables scholars to make use of all the mathematical machinery of classical mechanics to investigate non-conservative systems. Recently, combined Riemann-Liouville fractional order derivative (CRL) and CC were used to study Birkhoffian system, and Birkhoffian mechanics was established in Ref. [10]. Other important fractional order equations can be found in Refs. [11–16], which will not be described here in detail. Based on Ref. [5], Frederico and Torres [17] studied fractional Noether symmetry, and gave a definition of fractional conserved quantity. However, that definition was analyzed to be wrong in Ref. [18]. And many believe that the definition of fractional conserved quantity ´ definition, Frederico and Lazo [20] presented constants of motion for gave by Atanackovic´ [19] is more reasonable. Using Atanackovic’s dissipative system and discussed Noether’s conditions for the fractional optimal control problem on the basis of Ref. [9]. Fu [21] studied

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C.J. Song / Physics Letters A 383 (2019) 125914

fractional Lie symmetry for the nonholonomic Hamiltonian system in terms of RL, giving the corresponding conserved quantity. Jin [22] presented Noether theorem for the non-conservative systems with time delay in phase space. Zhang [23,24] investigated Noether symmetry and conserved quantity for the classical fractional Birkhoffian system and the fractional Birkhoffian system with time delay. Based on Ref. [10], Song and Zhang [25,26] further studied Noether theorem for fractional Birkhoffian system and generalized fractional Birkhoffian system in terms of four kinds of fractional order derivatives, and several applications were discussed. In this letter, differential equation of motion, Noether quasi-symmetry and perturbation to Noether quasi-symmetry will be discussed on the basis of four kinds of fractional order derivative, i.e., CRL, CC, Riesz-Riemann-Liouville fractional order derivative (RRL) and RieszCaputo fractional order derivative (RC), for the fractional Hamiltonian system. The remainder of this letter is organized as follows. Preliminaries of fractional order derivatives are listed simply in section 2. Fractional Hamilton equations with CRL and CC are established in section 3. In section 4, Noether quasi-symmetry and conserved quantity for fractional Hamiltonian system and fractional Lagrangian system are presented. In section 5, several results of perturbation to Noether quasi-symmetry and adiabatic invariant are obtained. Fractional Lotka biochemical oscillator model and fractional isotropic harmonic oscillator model are discussed as two applications in section 6. 2. Preliminaries CRL, CC, RRL, RC and the formulae for fractional integration by parts are listed as follows [10,15]. For the continuous and integrable function f (t ), the left RL, the right RL, the left C, the right C, the RRL and the RC are defined as

RL α t1 D t

RL β t D t2

C α t1 D t

C β t D t2

R α t1 D t2

 (n − α ) 1

f (t ) =

f (t ) =

f (t ) =

f (t ) =

RC α t1 D t2



1

f (t ) =

 (n − β)

 (n − α )

n t2  d − (ξ − t )n−β−1 f (ξ ) dξ,

(2)

dt

t

(t − ξ )n−α −1



n

d

f (ξ ) dξ,

dξ

(3)

t1

 (n − β)

n  d f (ξ ) dξ, (ξ − t )n−β−1 − dξ

(4)

t



1 2 (n − α )

f (t ) =

(1)

t1

t2

1

(t − ξ )n−α −1 f (ξ ) dξ,

dt

t

1

n  t

d

n t2

d dt

2 (n − α )

(5)

t1

t2

1

|t − ξ |n−α −1 f (ξ ) dξ,

n −α −1



|t − ξ |

d

n

dξ

f (ξ ) dξ,

(6)

t1

where α and β are the orders of the fractional derivatives, n − 1 ≤ α , β < n. Furthermore, the CRL and the CC are defined as RL C

α ,β

Dγ

f (t ) = γ

RL α t1 D t

β

f (t ) + (−1)n (1 − γ ) RtL D t2 f (t ) ,

(7)

α ,β β D γ f (t ) = γ tC1 D tα f (t ) + (−1)n (1 − γ ) Ct D t2 f (t ) ,

(8)

β

where t1 D tα and t D t2 show the arrow of time, γ is a parameter, which determines the different quantity of information from the past and the future. Particularly, substituting β = α , γ = 12 into formulae (7) and (8), RRL and RC can be obtained. Assuming α , β → 1, we get 1 t1 D t

= d/dt ,

1 t D t2

= −d/dt ,

α ,β

Dγ

=γ

1 t1 D t

+ (−1)n (1 − γ ) t D t12 = d/dt .

(9)

The formulae for fractional integration by parts are [15]

t2

t2 [∗] RtL1 D tα

t1

t2 t1

η Ct D tα2 [∗] dt −

ηdt = t2

ηdt = t1

(−1)n+ j

R L α + j −n t1 D t



η (t ) D n−1− j [∗] tt21 ,

(10)

j =0

t1

β [∗] RtL D t2

n −1 

η tC1 D tβ [∗] dt −

n −1  j =0

R L β+ j −n [∗] D n−1− j t D t2



η (t ) tt21 ,

(11)

C.J. Song / Physics Letters A 383 (2019) 125914

t2

t2 [∗] tC1 D tα

t1

n −1 

η RtL D tα2 [∗] dt +

ηdt =

β [∗] Ct D t2

t2

η Rt1L D tβ [∗] dt +

ηdt =

t1

n −1 

t2

(−1)n+ j

t2 α η RC t 1 D t 2 [∗] dt +

n

ηdt = (−1)

t1

R L β+ j −n [∗] D n−1− j t1 D t

 

η (t ) tt21 ,

(13)

t2

η tR1 D tα2 [∗] dt +

n

ηdt = (−1)

t1

n −1 

(−1)n+ j

R α + j −n t1 D t2



η (t ) D n−1− j [∗] tt21 ,

(14)

j =0

t1

t2 [∗] Rt1L D tα2

(12)

j =0

t1

[∗] tR1 D tα2

 

η (t ) tt21 ,

j =0

t1

t2

R L α + j −n [∗] D n−1− j t D t2

3

n −1 

(−1) j

R α + j −n [∗] D n−1− j t1 D t2



η (t ) tt21 .

(15)

j =0

t1

3. Fractional Hamilton equation Fractional Hamilton equations with CRL and CC are established as follows. It is noted that Einstein summation convention is used, and 0 < α , β < 1 is assumed in the whole text. Supposing a mechanical system is determined by the generalized coordinate q j , j = 1, 2, · · · , n, then from the fractional Lagrangian





α ,β

L t , q j , D γ q j , the generalized momentum and the Hamiltonian can be obtained

pi =

  α ,β ∂ L t, q j , Dγ q j



α ,β

α ,β



H = p i · D γ qi − L t , q j , D γ q j ,

,

α ,β ∂ D γ qi

i , j = 1, 2, · · · , n .

(16)

Then a unified fractional Hamilton action can be defined as

t2

SH =



α ,β

p i · D γ qi − H t , q j , p j



dt ,

(17)

t1

where q j , p j ∈ C1 , the Hamiltonian H is a C1 function with respect to all its arguments. The simultaneous variational principle

δSH = δ

t2



α ,β

p i · D γ qi − H t , q j , p j



dt = 0

(18)

t1

with α ,β

  δqi t =t1 = δqi t =t2 = 0

α ,β

δ D γ qi = D γ δqi ,

(19)

is called unified fractional Hamilton principle. In the following text, two sections will be given to establish fractional Hamilton equations, one with CRL, and the other with CC. 3.1. Fractional Hamilton equation with CRL



α ,β



Based on CRL, the fractional Lagrangian is L R L t , q j , R L D γ q j , the generalized momentum and the Hamiltonian are

p R Li =

  α ,β ∂ L R L t, q j , R L Dγ q j

H R L = p R Li ·

,

α ,β ∂ R L D γ qi

RL

α ,β



α ,β



D γ qi − L R L t , q j , R L D γ q j ,

i , j = 1, 2, · · · , n .

(20)

Then the unified fractional Hamilton action changes to

S H RL =

t2

p R Li ·

RL

α ,β



D γ qi − H R L t , q j , p R L j



dt .

(21)

t1

The simultaneous variational principle

δ S H RL = δ

t2 t1

with

α ,β



p R Li · R L D γ qi − H R L t , q j , p R L j



dt = 0

(22)

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C.J. Song / Physics Letters A 383 (2019) 125914

RL α t1 D t δqi

RL β t D t2 δqi

= δ Rt1L D tα qi ,

  δqi t =t1 = δqi t =t2 = 0

β

= δ RtL D t2 qi ,

(23)

is called fractional Hamilton principle with CRL. From formula (22), we have

t2

δ S H RL = δ



α ,β



p R Li ·

RL

D γ qi − H R L t , q j , p R L j

δ p R Li ·

RL

D γ qi + p R Li · δ R L D γ qi − δ H R L t , q j , p R L j

dt

t1

=

t2

α ,β



α ,β



dt

t1

t2  =

δ p R Li ·

RL

∂ H RL ∂ H RL α ,β β,α D γ qi − δqi · C D 1−γ p R Li − δqi − δ p R Li ∂ qi

t1

t2  =

 RL

δ p R Li ·

D γ qi −

∂ H RL ∂ p R Li

t2



α ,β

t1



 C

− δqi ·

∂ p R Li

β,α

D 1−γ p R Li +

∂ H RL ∂ qi

 dt

 dt = 0,

(24)

where

t2 p R Li · δ

RL

α ,β D γ qi dt =

t1

p R Li · δ

γ

β − (1 − γ ) RtL D t2 qi dt

RL α t1 D t qi

t1

=

t2

γ p R Li ·

RL α t1 D t δqi

− (1 − γ ) p R Li ·

RL β t D t2 δqi

dt

t1

t2 t2 β,α C α C β = δqi · γ t D t2 p R Li − δqi · (1 − γ ) t1 D t p R Li dt = − δqi · C D 1−γ p R Li dt . t1

(25)

t1

From formula (20), we get RL

α ,β

D γ qi =

∂ H RL . ∂ p R Li

(26)

Substituting formula (26) into formula (24), we have

t2

 δqi ·

C

β,α

D 1−γ p R Li +

t1

∂ H RL ∂ qi

 dt = 0.

(27)

It is followed from the arbitrariness of [t 1 , t 2 ] and the independence of δq i that C

β,α

D 1−γ p R Li = −

∂ H RL . ∂ qi

(28)

Formula (26) and formula (28) together are called fractional Hamilton equation with CRL. Remark 1. The result of fractional Hamilton equation here is as the same as that in Ref. [10] although they adopt different methods. 3.2. Fractional Hamilton equation with CC



α ,β



Based on CC, the fractional Lagrangian is L C t , q j , C D γ q j , the generalized momentum and the Hamiltonian are

pC i =

  α ,β ∂ LC t, q j , C Dγ q j α ,β ∂ C D γ qi

H C = pC i ·

,

C

α ,β



α ,β



D γ qi − L C t , q j , C D γ q j ,

i , j = 1, 2, · · · , n .

(29)

Then the unified fractional Hamilton action changes to

S HC =

t2 t1

pC i ·

C

α ,β



D γ qi − H C t , q j , p C j



dt .

(30)

C.J. Song / Physics Letters A 383 (2019) 125914

5

The simultaneous variational principle

δ S HC = δ

t2



α ,β

p C i · C D γ qi − H C t , q j , p C j



dt = 0

(31)

t1

with C α t1 D t δqi

C β t D t2 δqi

= δ tC1 D tα qi ,

  δqi t =t1 = δqi t =t2 = 0

β

= δ Ct D t2 qi ,

(32)

is called fractional Hamilton principle with CC. From formula (31), we have

δ S HC = δ

t2



α ,β

p C i · C D γ qi − H C t , q j , p C j



dt

t1

t2

 α ,β α ,β = δ p C i · C D γ qi + p C i · δ C D γ qi − δ H C t , q j , p C j dt t1

=

t2  α ,β δ p C i · C D γ qi − δqi ·

RL

β,α

D 1 −γ p C i −

t1

 ∂ HC ∂ HC δqi − δ p C i dt ∂ qi ∂ pC i

    t2  ∂ HC ∂ HC C α ,β R L β,α δ p C i · D γ qi − − δqi · = D 1 −γ p C i + dt = 0, ∂ pC i ∂ qi

(33)

t1

where

t2

α ,β p C i · δ D γ qi dt =

t2

C

t1

pC i · δ





γ tC1 D tα qi − (1 − γ ) Ct D tβ2 qi dt

t1

=

t2



γ p C i · tC1 D tα δqi − (1 − γ ) p C i · Ct D tβ2 δqi dt

t1

t2 = δqi · γ

RL α t D t2 p C i

− δ q i · (1 − γ )

RL β t1 D t p C i



t2 dt = − δqi ·

t1

RL

β,α



D 1−γ p C i dt .

(34)

t1

From formula (29), we get C

α ,β

D γ qi =

∂ HC . ∂ pC i

(35)

Substituting formula (35) into formula (33), we have

t2

 RL

δqi ·

β,α D 1 −γ p C i

t1

∂ HC + ∂ qi

 dt = 0.

(36)

It is followed from the arbitrariness of [t 1 , t 2 ] and the independence of δq i that RL

β,α

D 1 −γ p C i = −

∂ HC . ∂ qi

(37)

Formula (35) and formula (37) together are called fractional Hamilton equation with CC. Remark 2. When β = α , R α t1 D t2 qi

=

∂HR , ∂ p Ri

γ = 12 , from formulae (26), (28) and formulae (35), (37), we have RC α t 1 D t 2 p Ri

=−

∂HR , ∂ qi

(38)

and RC α t1 D t2 qi

=

∂ H RC , ∂ p RC i

R α t 1 D t 2 p RC i

=−

∂ H RC . ∂ qi

Equation (38) and equation (39) are called fractional Hamilton equations with RRL and RC, respectively.

(39)

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C.J. Song / Physics Letters A 383 (2019) 125914

Remark 3. Equation (38) and equation (39) can also be obtained through fractional Hamilton principles with RRL and RC, and the methods are similar to Sec. 3.1 as well as Sec. 3.2. Remark 4. Equation (39) is consistent with the result in Ref. [15]. Remark 5. Because the relationships [26]

1 f (t 1 ) ,  (1 − α ) (t − t 1 )α 1 f (t 2 ) RL β C β t D t 2 f (t ) = t D t 2 f (t ) +  (1 − β) (t 2 − t )β RL α t1 D t

f (t ) = tC1 D tα f (t ) +

(40) (41)

exist between RL and C, we can get another forms of fractional Hamilton equations with CRL and CC from formulae (24), (25), (33) and (34) as follows:

∂ H RL , ∂ p R Li ∂ HC C α ,β D γ qi = , ∂ pC i RL

α ,β

RL

D γ qi =

Remark 6. When β = α ,

∂HR , ∂ p Ri ∂ H RC RC α , t1 D t2 qi = ∂ p RC i

R α t1 D t2 qi

=

C

β,α

D 1−γ p R Li = −

β,α

D 1 −γ p C i = −

∂ H RL , ∂ qi

(42)

∂ HC . ∂ qi

(43)

γ = 12 , from formulae (42) and (43), we have ∂HR , ∂ qi ∂ H RC RC α . t 1 D t 2 p RC i = − ∂ qi

R α t 1 D t 2 p Ri

=−

(44) (45)

Equation (44) and equation (45) are another forms of fractional Hamilton equations with RRL and RC, respectively. Remark 7. The following fractional Lagrange equations with CRL and CC

∂ L R L C β,α ∂ LRL − D 1 −γ α ,β = 0, ∂ qi ∂ R L D γ qi

(46)

∂ L C R L β,α ∂ LC − D 1 −γ α ,β = 0 C ∂ qi ∂ D γ qi

(47)

can be obtained through

t2



α ,β



L R L t , q j , R L D γ q j dt = 0,

δ SLRL = δ

RL α t1 D t δqi

= δ RtL1 D tα qi ,

RL β t D t2 δqi

β

= δ RtL D t2 qi ,

t1

  δqi t =t1 = δqi t =t2 = 0

(48)

and

t2



α ,β



L C t , q j , C D γ q j dt = 0,

δ S LC = δ

C α t1 D t δqi

= δ tC1 D tα qi ,

C β t D t2 δqi

β

= δ Ct D t2 qi ,

t1

  δqi t =t1 = δqi t =t2 = 0.

(49)

Remark 8. Equation (46) is as the same as that in Ref. [10] although they adopt different methods.

α , β → 1, from formulae (26), (28), formulae (35), (37), and formulae (46), (47), we have 

 ∂ H t, q j , p H j ∂ H t, q j , p H j q˙ i = , p˙ H i = − ∂ pHi ∂ qi

Remark 9. When



and



 ∂ L t , q j , q˙ j d ∂ L t , q j , q˙ j − = 0. ∂ qi dt ∂ q˙ i

Equations (50) and (51) are the classical Hamilton equation and the classical Lagrange equation [27].

(50)

(51)

C.J. Song / Physics Letters A 383 (2019) 125914

7

4. Noether quasi-symmetry and conserved quantity For the fractional Hamiltonian system, if the time t, the generalized coordinate q i and the generalized momentum p i change under infinitesimal transformations, then the unified fractional Hamilton action will become

S¯ H =

t¯2



α ,β

p¯ i · D γ q¯ i − H t¯, q¯ j , p¯ j



dt¯.

(52)

t¯1

 t2

0

G dt, then the infinitesimal transformations are called Noether



 symmetric transformations, which determine Noether quasi-symmetry of the system, where G 0 = θ G 0 t , q j , p j , G 0 t , q j , p j is called a gauge function, θ is an infinitesimal parameter. The linear part of S¯ H − S H is denoted as S H , if S H = −

d t 1 dt

Actually, Noether quasi-symmetry can lead to conserved quantity, which helps find the solutions to the differential equation of motion, d and reduce the freedom of the mechanical system. A quantity I is called a conserved quantity if and only if dt I = 0. Supposing the infinitesimal transformations are



t¯ = t + t ,



q¯ i t¯ = qi (t ) + qi ,

p¯ i t¯ = p i (t ) + p i ,

(53)

and the linear part of their expansions are







t¯ = t + θξ00 t , q j , p j ,







q¯ i t¯ = qi (t ) + θξi0 t , q j , p j ,

ξ00 ,

ξi0





p¯ i t¯ = p i (t ) + θ ηi0 t , q j , p j ,

(54)

0 i

where θ is an infinitesimal parameter, and η are infinitesimal generators. In the following text, two sections will be presented to discuss the fractional conserved quantity, one with CRL, and the other with CC. 4.1. Conserved quantity with CRL Based on CRL, the fractional Hamilton action changes to

S¯ H R L =

t¯2

p¯ R Li ·

RL



α ,β

D γ q¯ i − H R L t¯, q¯ j , p¯ R L j



dt¯.

(55)

t¯1

Then we have

S H R L = S¯ H R L − S H R L =

t¯2

p¯ R Li ·

RL



 α ,β D γ q¯ i − H R L t¯, q¯ j , p¯ R L j dt¯ −

t¯1

t2

p R Li ·

RL



α ,β

D γ qi − H R L t , q j , p R L j



dt

t1

t2 ( p R Li + p R Li ) ·

=

RL

α ,β

t1

d

1−γ

+

d

α ,β

D γ qi + R L D γ δqi + t

(t 2 − t )−β qi (t 2 ) t 2





RL

dt

d

γ

α ,β

D γ qi −

 (1 − α ) dt

(t − t 1 )−α qi (t 1 ) t 1

 − H R L t + t , q j + q j , p R L j + p R L j

 (1 − β) dt t2

 α ,β − p R Li · R L D γ qi − H R L t , q j , p R L j dt

 1+

d dt





t dt

t1

t2 =

p R Li ·

RL



α ,β

D γ δqi +

t1

p R Li ·

d

RL

dt

α ,β

D γ qi −

∂ H RL ∂t



t −

 (1 − α ) dt

  ∂ H RL

qi + p R Li · (t 2 − t )−β qi (t 2 ) t 2 − dt ∂ qi

p R Li · (1 − γ ) d

+

 (1 − β)

d

p R Li · γ

RL

(t − t 1 )−α qi (t 1 ) t 1

α ,β

 d

D γ qi − H R L ·

dt





t dt ,

where RL

α ,β

α ,β

α ,β

D γ q¯ i = R L D γ qi + R L D γ δqi + t

+ Let S H R L = −

p R Li ·

RL

d

 (1 − β) dt

 t2

d t 1 dt

α ,β

Dγ

1−γ



d

RL

dt

α ,β

D γ qi −

 (t 2 − t )−β qi (t 2 ) t 2 .

dt

RL

α ,β

D γ qi −

(t − t 1 )−α qi (t 1 ) t 1

 (56)

0 

G R L dt, we have

  d ξ R0Li − q˙ i ξ R0L0 + p R Li ·

d

γ

 (1 − α ) dt

∂ H RL ∂t



 ξ R0L0 + p R Li ·

RL



D γ qi − H R L · ξ˙ R0L0 α ,β

8

C.J. Song / Physics Letters A 383 (2019) 125914

d 

 ∂ H R L 0 ξ (t − t 1 )−α qi (t 1 ) ξ R0L0 t 1 , q j (t 1 ) , p R L j (t 1 ) −  (1 − α ) dt ∂ qi R Li



  p R Li · (1 − γ ) d  + (t 2 − t )−β qi (t 2 ) ξ R0L0 t 2 , q j (t 2 ) , p R L j (t 2 ) + G˙ 0R L t , q j , p R L j = 0.  (1 − β) dt −

p R Li · γ

(57)

Formula (57) is called Noether identity with CRL. Theorem 1. For the fractional Hamiltonian system (26) and (28), if the gauge function G 0R L , the infinitesimal generators ξ R0 L0 and ξ R0 Li satisfy formula (57), then there exists a conserved quantity

I H R L0 =

t

RL

p R Li ·

α ,β

    β,α ξ R0Li − q˙ i ξ R0L0 + ξ R0Li − q˙ i ξ R0L0 · C D 1−γ p R Li dτ + p R Li ·



Dγ

RL



α ,β

D γ qi − H R L · ξ R0L0

t1

t −

p R Li

 (1 − α ) dτ

t1

−

 (τ − t 1 )−α qi (t 1 ) ξ R0L0 t 1 , q j (t 1 ) , p R L j (t 1 )

d 

γ



 (1 − γ ) d  dτ + G 0R L . (t 2 − τ )−β qi (t 2 ) ξ R0L0 t 2 , q j (t 2 ) , p R L j (t 2 )  (1 − β) dτ

Proof. Using formulae (26), (28) and (57), we have

d dt



p˙ R Li ·

∂ H RL ∂ H RL ∂ H RL d R L α ,β α ,β D γ qi + p R Li · D γ qi − − q˙ i − p˙ R Li



· ξ R0L0 ∂ qi ∂ p R Li       α ,β α ,β β,α + p R Li · R L D γ qi − H R L · ξ˙ R0L0 + p R Li · R L D γ ξ R0Li − q˙ i ξ R0L0 + ξ R0Li − q˙ i ξ R0L0 · C D 1−γ p R Li

 p R Li γ d  − (t − t 1 )−α qi (t 1 ) ξ R0L0 t 1 , q j (t 1 ) , p R L j (t 1 ) + G˙ 0R L  (1 − α ) dt

 p R Li (1 − γ ) d  + (t 2 − t )−β qi (t 2 ) ξ R0L0 t 2 , q j (t 2 ) , p R L j (t 2 )  (1 − β) dt    ∂ H RL  0 β,α ξ R Li − q˙ i · ξ R0L0 + ξ R0Li − q˙ i · ξ R0L0 · C D 1−γ p R Li = 0. = ∂ qi

I H R L0 =

This proof is completed.

RL

(58)

dt

∂t

2

4.2. Conserved quantity with CC Based on CC, the fractional Hamilton action changes to

S¯ H C =

t¯2

p¯ C i ·

C



α ,β

D γ q¯ i − H C t¯, q¯ j , p¯ C j



dt¯.

(59)

t¯1

Then we have

S HC

= S¯ H C − S H C =

t¯2

p¯ C i ·

C

 α ,β D γ q¯ i − H C t¯, q¯ j , p¯ C j dt¯ −

t¯1

t2

pC i ·

C

α ,β



dt

t1

t2 d α ,β α ,β α ,β = ( p C i + p C i ) · C D γ qi + C D γ δqi + t C D γ qi − dt

t1

+

−



D γ qi − H C t , q j , p C j



1−γ

 (1 − β)

t2

pC i ·

C

−β

(t 2 − t )

q˙ i (t 2 ) t 2



α ,β

 

D γ qi − H C t , q j , p C j

γ  (1 − α )



(t − t 1 )−α q˙ i (t 1 ) t 1 



− H C t + t , q j + q j , p C j + p C j





1+

d dt



t dt

dt

t1

t2

=





 pC i · γ

(t − t 1 )−α q˙ i (t 1 ) t 1  (1 − α ) t1   d   ∂ HC p C i · (1 − γ )

α ,β

qi + p C i · C D γ qi − H C · t dt , + (t 2 − t )−β q˙ i (t 2 ) t 2 −  (1 − β) ∂ qi dt pC i ·

C

α ,β D γ δqi +

pC i ·

d

dt

C

∂ HC α ,β D γ qi − ∂t

t −



C.J. Song / Physics Letters A 383 (2019) 125914

9

where C

α ,β

α ,β

α ,β

D γ q¯ i = C D γ qi + C D γ δqi + t

Let S H C = −

 t2

d t 1 dt

0

G C dt, we have

d dt

C

γ

α ,β

D γ qi −

 (1 − α )

 (t − t 1 )−α q˙ i (t 1 ) t 1 +



1−γ

 (t 2 − t )−β q˙ i (t 2 ) t 2 .



 (1 − β)

     ∂ HC d α ,β α ,β ξC00 + p C i · C D γ qi − H C · ξ˙C00 ξC0i − q˙ i ξC00 + p C i · C D γ qi − dt ∂t

 ∂ H C 0 pC i · γ  − ξ (t − t 1 )−α q˙ i (t 1 ) ξC00 t 1 , q j (t 1 ) , p C j (t 1 ) −  (1 − α ) ∂ qi C i



  p C i · (1 − γ )  + (t 2 − t )−β q˙ i (t 2 ) ξC00 t 2 , q j (t 2 ) , p C j (t 2 ) + G˙ 0C t , q j , p C j = 0.  (1 − β)

pC i ·

C

α ,β

(60)



Dγ

(61)

Formula (61) is called Noether identity with CC. Theorem 2. For the fractional Hamiltonian system (35) and (37), if the gauge function G 0C , the infinitesimal generators ξC00 and ξC0i satisfy formula (61), then there exists a conserved quantity



I H C 0 = pC i ·

C



α ,β

D γ qi − H C · ξC00 +

t

pC i ·

C

α ,β



Dγ

   ξC0i − q˙ i ξC00 + ξC0i − q˙ i ξC00 ·

RL

β,α



D 1−γ p C i dτ

t1

t + G 0C −

pC i

 (1 − α )

t1

−

γ

 (τ − t 1 )−α q˙ i (t 1 ) ξC00 t 1 , q j (t 1 ) , p C j (t 1 )





 (1 − γ )  dτ . (t 2 − τ )−β q˙ i (t 2 ) ξC00 t 2 , q j (t 2 ) , p C j (t 2 )  (1 − β)

(62)

Proof. Using formulae (35), (37) and (61), we have

 ∂ HC ∂ HC ∂ HC d C α ,β C α ,β ˙ ˙ ˙ D γ qi − − qi − p C i · ξC00 I H C 0 = p C i · D γ qi + p C i · dt dt ∂t ∂ qi ∂ pC i       α ,β α ,β + p C i · C D γ qi − H C · ξ˙C00 + p C i · C D γ ξC0i − q˙ i ξC00 + ξC0i − q˙ i ξC00 ·

 pC i γ  − (t − t 1 )−α q˙ i (t 1 ) ξC00 t 1 , q j (t 1 ) , p C j (t 1 ) + G˙ 0C  (1 − α )

 p C i (1 − γ )  + (t 2 − t )−β q˙ i (t 2 ) ξC00 t 2 , q j (t 2 ) , p C j (t 2 )  (1 − β)    ∂ HC  0 β,α = ξC i − q˙ i · ξC00 + ξC0i − q˙ i · ξC00 · R L D 1−γ p C i = 0. ∂ qi 

d

This proof is completed. Remark 10. When β = α , Theorem 2 as follows.

RL

β,α

D 1 −γ p C i

2

γ = 12 , we can get conserved quantities for fractional Hamiltonian system with RRL and RC from Theorem 1 and

0 0 Theorem 3. For the fractional Hamiltonian system (38), if the gauge function G 0R , the infinitesimal generators ξ R0 and ξ Ri satisfy

p Ri ·

R α t1 D t2

  d 0 0 − q˙ i ξ R0 ξ Ri + p Ri ·



dt

R α t1 D t2 qi

−

∂HR ∂t



 0 ξ R0 + p Ri ·

R α t1 D t2 qi

 0 − H R · ξ˙ R0

 ∂ H R 0 0 t 1 , q j (t 1 ) , p R j (t 1 ) − ξ (t − t 1 )−α qi (t 1 ) ξ R0  (1 − α ) dt ∂ qi Ri



  p Ri · (1 − γ ) d  0 + t 2 , q j (t 2 ) , p R j (t 2 ) + G˙ 0R t , q j , p R j = 0, (t 2 − t )−β qi (t 2 ) ξ R0  (1 − β) dt −

d 

p Ri · γ

then there exists a conserved quantity

I H R0 =

t









0 0 0 0 p Ri · tR1 D tα2 ξ Ri − q˙ i ξ R0 − q˙ i ξ R0 + ξ Ri ·

RC α t 1 D t 2 p Ri





dτ + p Ri ·

R α t1 D t2 qi

t1

t +

G 0R

−

p Ri t1

γ

 0 t 1 , q j (t 1 ) , p R j (t 1 ) (τ − t 1 )−α qi (t 1 ) ξ R0

d 

 (1 − α ) dτ

 0 − H R · ξ R0

(63)

10

C.J. Song / Physics Letters A 383 (2019) 125914



 (1 − γ ) d  0 t 2 , q j (t 2 ) , p R j (t 2 ) dτ . (t 2 − τ )−β qi (t 2 ) ξ R0  (1 − β) dτ

−

(64)

0 0 Theorem 4. For the fractional Hamiltonian system (39), if the gauge function G 0RC , the infinitesimal generators ξ RC 0 and ξ RC i satisfy

   ∂ H RC 0 RC α ˙0 ξ RC 0 + p RC i · t 1 D t 2 q i − H RC · ξ RC 0 dt ∂t

 ∂ H RC 0 p RC i · γ  0 − − ξ (t − t 1 )−α q˙ i (t 1 ) ξ RC 0 t 1 , q j (t 1 ) , p RC j (t 1 )  (1 − α ) ∂ qi RC i  



  p RC i · (1 − γ ) 0 + G˙ 0RC t , q j , p RC j = 0, + (t 2 − t )−β q˙ i (t 2 ) ξ RC 0 t 2 , q j (t 2 ) , p RC j (t 2 )  (1 − β)

p RC i ·

RC α t1 D t2

  d 0 0 ˙ − q ξ ξ RC i RC 0 + p RC i · i



RC α t1 D t2 qi

−

(65)

then there exists a conserved quantity

I H RC 0 =

t

p RC i ·



   0 0 ˙ 0 ˙ 0 ξ RC i − q i ξ RC 0 + ξ RC i − q i ξ RC 0 ·

RC α t1 D t2

R α t 1 D t 2 p RC i





dτ + p RC i ·

RC α t1 D t2 qi

 0 0 − H RC · ξ RC 0 + G RC

t1

t −

p RC i t1

 0 (τ − t 1 )−α q˙ i (t 1 ) ξ RC 0 t 1 , q j (t 1 ) , p RC j (t 1 )



γ  (1 − α )



 (1 − γ )  −β ˙ 0 − dτ . (t 2 − τ ) qi (t 2 ) ξ RC 0 t 2 , q j (t 2 ) , p RC j (t 2 )  (1 − β)

(66)

Remark 11. Conserved quantities for fractional Lagrangian system with CRL and CC can be obtained as follows. Theorem 5. For the fractional Lagrangian system (46), if the gauge function G 0L R L , the infinitesimal generators ξ L0R L0 and ξ L0R Li satisfy

  α ,β ∂ L R L t, q j , R L Dγ q j α ,β

∂ R L D γ qi

·

RL





α ,β Dγ ξ L0R Li − q˙ i ξ L0R L0 +



∂ LRL

d

RL

∂ LRL α ,β D γ qi +



ξ L0R L0 α ,β · ∂t ∂ R L D γ qi dt

 d  (t − t 1 )−α qi (t 1 ) ξ L0R L0 t 1 , q j (t 1 )

∂ LRL 0 ∂ LRL γ ξ L R Li + L R L · ξ˙L0R L0 − α ,β ∂ qi ∂ R L D γ qi  (1 − α ) dt   

 1−γ d  α ,β −β 0 − + G˙ 0L R L t , q j , R L D γ q j = 0, (t 2 − t ) qi (t 2 ) ξ L R L0 t 2 , q j (t 2 )  (1 − β) dt

+

(67)

then there exists a conserved quantity

⎡

t

⎣

I L R L0 =

  α ,β ∂ L R L t, q j , R L Dγ q j α ,β

∂ R L D γ qi

t1

t − t1

∂ LRL α ,β ∂ R L D γ qi

·

RL







⎤



α ,β β,α Dγ ξ L0R Li − q˙ i ξ L0R L0 + ξ L0R Li − q˙ i ξ L0R L0 · C D 1−γ

∂ LRL α ,β

∂ R L D γ qi

⎦ dτ

 (τ − t 1 )−α qi (t 1 ) ξ L0R L0 t 1 , q j (t 1 )

d 

γ

 (1 − α ) dτ

  

 (1 − γ ) d  α ,β −β 0 − dτ + G 0L R L t , q j , R L D γ q j + L R L · ξ L0R L0 . (t 2 − τ ) qi (t 2 ) ξ L R L0 t 2 , q j (t 2 )  (1 − β) dτ Proof. From formulae (46) and (67), we have

d dt

I L R L0 =

  α ,β ∂ L R L t, q j , R L Dγ q j 

∂

R L D α ,β q

γ

i

·

RL

α ,β

Dγ



   β,α ξ L0R Li − q˙ i ξ L0R L0 + ξ L0R Li − q˙ i ξ L0R L0 · C D 1−γ 

∂ LRL ∂ LRL ∂ LRL d R L α ,β + q˙ i + D γ qi · ξ L0R L0 + L R L · ξ˙L0R L0 α ,β R L ∂t ∂ qi ∂ D γ qi dt

 ∂ LRL γ d  − (t − t 1 )−α qi (t 1 ) ξ L0R L0 t 1 , q j (t 1 ) α ,β ∂ R L D γ qi  (1 − α ) dt 

 (1 − γ ) d  + G˙ 0L R L − (t 2 − t )−β qi (t 2 ) ξ L0R L0 t 2 , q j (t 2 )  (1 − β) dt   ∂ LRL ∂ LRL 0 ∂ LRL β,α 0 ˙ = ξ L0R Li − q˙ i ξ L0R L0 · C D 1−γ α ,β + ∂ q qi · ξ L R L0 − ∂ q · ξ L R Li = 0. i i ∂ R L D γ qi +

∂ LRL ∂

R L D α ,β q

γ

i

(68)

C.J. Song / Physics Letters A 383 (2019) 125914

11

2

This proof is completed.

0 0 Theorem 6. For the fractional Lagrangian system (47), if the gauge function G 0LC , the infinitesimal generators ξ LC 0 and ξ LC i satisfy

  α ,β ∂ LC t, q j , C Dγ q j

·

α ,β

∂ C D γ qi

C







α ,β 0 ˙ 0 Dγ ξ LC i − q i ξ LC 0 +

∂ LC

d

C

∂ LC α ,β D γ qi +



0 ξ LC 0 α ,β · ∂t ∂ C D γ qi dt 

 0 (t − t 1 )−α q˙ i (t 1 ) ξ LC 0 t 1 , q j (t 1 )

∂ LC 0 ∂ LC γ 0 ˙ + ξ + L C · ξ LC 0 − α ,β C ∂ qi LC i  1 − α) ( ∂ D γ qi    

 1−γ 0 ˙ 0LC t , q j , C D αγ ,β q j = 0, + t , q t G − ( ) (t 2 − t )−β q˙ i (t 2 ) ξ LC 2 2 j 0  (1 − β)

(69)

then there exists a conserved quantity

⎡

t

⎣

I LC 0 = t1

t − t1

−

  α ,β ∂ LC t, q j , C Dγ q j α ,β

∂ C D γ qi

∂ LC α ,β ∂ C D γ qi

1−γ

γ  (1 − α )





⎤



∂ LC α ,β

∂ C D γ qi

⎦ dτ

 0 (τ − t 1 )−α q˙ i (t 1 ) ξ LC 0 t 1 , q j (t 1 )



  

 0 0 C α ,β 0 t , q t d τ + G t , q , D q + L C · ξ LC ( ) (t 2 − τ )−β q˙ i (t 2 ) ξ LC 2 2 j j j γ LC 0 0.



 (1 − β)

Remark 12. When can be deduced.

·



α ,β 0 0 R L β,α ˙ 0 ˙ 0 Dγ D 1 −γ ξ LC i − q i ξ LC 0 + ξ LC i − q i ξ LC 0 ·

C

(70)

α , β → 1, the classical conserved quantities for the classical Hamiltonian system and the classical Lagrangian system

0 and ξ H0 i satisfy Theorem 7. For the classical Hamilton equation (50), if the gauge function G 0H , the infinitesimal generators ξ H0

0 p H i · ξ˙ H0 i − H · ξ˙ H0 −

 ∂H 0 ∂H 0 ξ − ξ + G˙ 0H t , q j , p H j = 0, ∂ t H0 ∂ qi H i

(71)

then there exists a conserved quantity 0 I H0 = p H i ξ H0 i − H ξ H0 + G 0H .

(72)





0 0 Theorem 8. For the classical Lagrange equation (51), if the gauge function G 0L t , q j , q˙ j , the infinitesimal generators ξ L0 and ξ Li satisfy

   ∂ L t , q j , q˙ j ∂L 0 ∂L 0 ∂L 0 0 ˙ ˙ · ξ Li − qi − L · ξ˙L0 + ξ + ξ + G˙ 0L = 0, ∂ q˙ i ∂ q˙ i ∂ t L0 ∂ qi Li

(73)

then there exists a conserved quantity

I L0 =

  ∂L 0 ∂L 0 ξ Li − q˙ i − L ξ L0 + G 0L . ∂ q˙ i ∂ q˙ i

(74)

Remark 13. The results of Theorem 7 and Theorem 8 are consistent with the results in Ref. [27]. 5. Perturbation to Noether quasi-symmetry and adiabatic invariant When the mechanical system is disturbed by small forces, the original conserved quantity may also change. In this section, we will study perturbation to Noether quasi-symmetry and adiabatic invariant. There are many ways to impose the disturbance ε W i on the mechanical system, and here we choose

G = G 0 + ε G 1 + ε2 G 2 + · · · ,

ξ0 = ξ00 + ε ξ01 + ε 2 ξ02 + · · · ,

ξi = ξi0 + ε ξi1 + ε 2 ξi2 + · · · ,

(75)

where ε is a small parameter, G is the gauge function of the disturbed system, ξ0 and ξi are the infinitesimal generators of the disturbed system. A quantity I z , which has a parameter ε and the highest power of ε is z, is called an adiabatic invariant if dI z /dt is in direct proportion to ε z+1 . Several theorems will be presented. Theorem 9. For the disturbed fractional Hamiltonian system RL

α ,β

D γ qi =

∂ H RL , ∂ p R Li

C

β,α

D 1−γ p R Li = −

 ∂ H RL − ε W R Li t , q j , p R L j , ∂ qi

(76)

12

C.J. Song / Physics Letters A 383 (2019) 125914

m m if the gauge function G m R L , the infinitesimal generators ξ R L0 and ξ R Li satisfy

p R Li ·

RL

 α ,β m ξ R Li − q˙ i ξ RmL0 +



p R Li ·

Dγ

d

RL

dt

α ,β

D γ qi −

∂ H RL ∂t



 ξ RmL0 + p R Li ·

RL



D γ qi − H R L · ξ˙ RmL0 α ,β

d



  ∂ H R L m ξ + G˙ m (t − t 1 )−α qi (t 1 ) ξ RmL0 t 1 , q j (t 1 ) , p R L j (t 1 ) − R L t, q j , p R L j  (1 − α ) dt ∂ qi R Li  

 p R Li · (1 − γ ) d

−1 + = 0, (t 2 − t )−β qi (t 2 ) ξ RmL0 t 2 , q j (t 2 ) , p R L j (t 2 ) − W R Li ξ RmLi−1 − q˙ i ξ RmL0  (1 − β) dt p R Li · γ

−

(77)

then there exists an adiabatic invariant

I H R Lz =

z 

ε

m

m =0

⎧ t ⎨ ⎩

p R Li ·

RL





 α ,β m β,α ξ R Li − q˙ i ξ RmL0 + ξ RmLi − q˙ i ξ RmL0 · C D 1−γ p R Li dτ

Dγ

t1



+ p R Li ·

RL



α ,β D γ qi − H R L · ξ RmL0 −

t p R Li t1

γ

d

 (1 − α ) dτ

 (τ − t 1 )−α qi (t 1 ) ξ RmL0 t 1 , q j (t 1 ) , p R L j (t 1 )

⎫  ⎬

 (1 − γ ) d

− (t 2 − τ )−β qi (t 2 ) ξ RmL0 t 2 , q j (t 2 ) , p R L j (t 2 ) dτ + G m RL , ⎭  (1 − β) dτ

(78)

−1 where ξ RmLi−1 = ξ RmL0 = 0 when m = 0.

Proof. Using formulae (76) and (77), we have

d dt

I H R Lz =

 ∂ H RL ∂ H RL ∂ H RL − q˙ i − p˙ R Li · ξ RmL0 dt ∂t ∂ qi ∂ p R Li m =0   

 α ,β α ,β

β,α + p R Li · R L D γ qi − H R L · ξ˙ RmL0 + p R Li · R L D γ ξ RmLi − q˙ i ξ RmL0 + ξ RmLi − q˙ i ξ RmL0 · C D 1−γ p R Li z 



p˙ R Li ·

εm

RL

α ,β

D γ qi + p R Li ·

d

RL

α ,β

D γ qi −

 p R Li γ d

(t − t 1 )−α qi (t 1 ) ξ RmL0 t 1 , q j (t 1 ) , p R L j (t 1 ) + G˙ m RL  (1 − α ) dt 

 p R Li (1 − γ ) d

+ (t 2 − t )−β qi (t 2 ) ξ RmL0 t 2 , q j (t 2 ) , p R L j (t 2 )  (1 − β) dt

z   

m  m  C β,α m −1 m −1 m ∂ H RL m m ξ R Li − q˙ i · ξ R L0 + ξ R Li − q˙ i · ξ R L0 · D 1−γ p R Li + W R Li ξ R Li − q˙ i ξ R L0 = ε ∂ qi −

m =0 z 

=









−1 εm −ε W R Li ξ RmLi − q˙ i · ξ RmL0 + W R Li ξ RmLi−1 − q˙ i ξ RmL0



 = −ε z+1 W R Li ξ Rz Li − q˙ i · ξ Rz L0 .

m =0

This proof is completed.

2

Theorem 10. For the disturbed fractional Hamiltonian system C

α ,β

D γ qi =

∂ HC , ∂ pC i

RL

β,α

D 1 −γ p C i = −

 ∂ HC − ε W C i t, q j , pC j , ∂ qi

(79)

m m if the gauge function G m C , the infinitesimal generators ξC 0 and ξC i satisfy

pC i ·

C

 α ,β m ξC i − q˙ i ξCm0 +



pC i ·

Dγ

d dt

C

α ,β

D γ qi −

∂ HC ∂t



  α ,β ξCm0 + p C i · C D γ qi − H C · ξ˙Cm0



  ∂ H C m ξC i + G˙ m (t − t 1 )−α q˙ i (t 1 ) ξCm0 t 1 , q j (t 1 ) , p C j (t 1 ) − C t, q j , pC j  (1 − α ) ∂ qi  

 p C i · (1 − γ )

−β ˙ m + (t 2 − t ) qi (t 2 ) ξC 0 t 2 , q j (t 2 ) , p C j (t 2 ) − W C i ξCmi−1 − q˙ i ξCm0−1 = 0,  (1 − β)

−

pC i · γ



then there exists an adiabatic invariant

I HC z =

z  m =0

εm

⎧ t ⎨ ⎩

t1

pC i ·

C





 α ,β m β,α ξC i − q˙ i ξCm0 + ξCmi − q˙ i ξCm0 · R L D 1−γ p C i dτ + G m C

Dγ

(80)

C.J. Song / Physics Letters A 383 (2019) 125914



C

+ pC i ·



α ,β D γ qi − H C

t · ξCm0

−

pC i t1

γ  (1 − α )

13



 (τ − t 1 )−α q˙ i (t 1 ) ξCm0 t 1 , q j (t 1 ) , p C j (t 1 )

⎫  ⎬

 (1 − γ )

− (t 2 − τ )−β q˙ i (t 2 ) ξCm0 t 2 , q j (t 2 ) , p C j (t 2 ) dτ , ⎭  (1 − β)

(81)

where ξCmi−1 = ξCm0−1 = 0 when m = 0. Theorem 11. For the disturbed fractional Hamiltonian system R α t1 D t2 qi

∂HR , ∂ p Ri

=

RC α t 1 D t 2 p Ri

=−

 ∂HR − ε W Ri t , q j , p R j , ∂ qi

(82)

m m if the gauge function G m R , the infinitesimal generators ξ R0 and ξ Ri satisfy

p Ri ·

R α t1 D t2



m ξ Ri

m − q˙ i ξ R0





+ p Ri ·

d dt

R α t1 D t2 qi

∂HR − ∂t



 m + p Ri · ξ R0

R α t1 D t2 qi

 m − H R · ξ˙ R0



  ∂ H R m m t 1 , q j (t 1 ) , p R j (t 1 ) − ξ Ri + G˙ m (t − t 1 )−α qi (t 1 ) ξ R0 R t, q j , p R j  (1 − α ) dt ∂ qi  

 p Ri · (1 − γ ) d

m −1 m −1 m = 0, t 2 , q j (t 2 ) , p R j (t 2 ) − W Ri ξ Ri − q˙ i ξ R0 + (t 2 − t )−β qi (t 2 ) ξ R0  (1 − β) dt

−

d

p Ri · γ

(83)

then there exists an adiabatic invariant z 

I H Rz =

εm

m =0

⎧ t ⎨ ⎩

R α t1 D t2



 m  m m m ξ Ri + ξ Ri − q˙ i ξ R0 · − q˙ i ξ R0

RC α t 1 D t 2 p Ri





dτ + G m R + p Ri ·

R α t1 D t2 qi

 m − H R · ξ R0

t1

t −

p Ri ·

p Ri

d

γ

 (1 − α ) dτ

t1

 m t 1 , q j (t 1 ) , p R j (t 1 ) (τ − t 1 )−α qi (t 1 ) ξ R0

⎫  ⎬ 

(1 − γ ) d

m t 2 , q j (t 2 ) , p R j (t 2 ) dτ , − (t 2 − τ )−β qi (t 2 ) ξ R0 ⎭  (1 − β) dτ

(84)

m−1 m−1 where ξ Ri = ξ R0 = 0 when m = 0.

Theorem 12. For the disturbed fractional Hamiltonian system RC α t1 D t2 qi

=

∂ H RC , ∂ p RC i

R α t 1 D t 2 p RC i

=−

 ∂ H RC − ε W RC i t , q j , p RC j , ∂ qi

(85)

m m if the gauge function G m RC , the infinitesimal generators ξ RC 0 and ξ RC i satisfy

p RC i ·

RC α t1 D t2



  d m m ˙ ξ RC + − q ξ p RC i · i RC 0 i

dt

−



∂ H RC ∂t

 m ξ RC + p RC i · 0

RC α t1 D t2 qi

 m − H RC · ξ˙ RC 0

 ∂ H RC m ξ RC i + G˙ m RC t , q j , p RC j  (1 − α ) ∂ qi  

 p RC i · (1 − γ )

m −1 m −1 m ˙ i ξ RC + = 0, (t 2 − t )−β q˙ i (t 2 ) ξ RC 0 t 2 , q j (t 2 ) , p RC j (t 2 ) − W RC i ξ RC i − q 0  (1 − β)

−

p RC i · γ

−α

(t − t 1 )

m q˙ i (t 1 ) ξ RC 0



RC α t1 D t2 qi



t 1 , q j (t 1 ) , p RC j (t 1 )

−

(86)

then there exists an adiabatic invariant

I H RC z =

z  m =0

εm

⎧ t ⎨ ⎩

p RC i

t1

RC α t1 D t2



 m  m ˙ m ˙ m ξ RC i − q i ξ RC 0 + ξ RC i − q i ξ RC 0 ·

R α t 1 D t 2 p RC i





dτ + p RC i ·

RC α t1 D t2 qi

 m m − H RC · ξ RC 0 + G RC

t1

t −

p RC i ·

γ  (1 − α )





m (τ − t 1 )−α q˙ i (t 1 ) ξ RC 0 t 1 , q j (t 1 ) , p RC j (t 1 )

⎫  ⎬

 (1 − γ )

m − dτ , (t 2 − τ )−β q˙ i (t 2 ) ξ RC 0 t 2 , q j (t 2 ) , p RC j (t 2 ) ⎭  (1 − β) m−1 m−1 where ξ RC = ξ RC 0 = 0 when m = 0. i

(87)

14

C.J. Song / Physics Letters A 383 (2019) 125914

Theorem 13. For the disturbed fractional Lagrangian system

  ∂ L R L C β,α ∂ LRL R L α ,β − D 1 −γ = ε W t , q , D q , L R Li j j γ α ,β ∂ qi ∂ R L D γ qi

(88)

m m if the gauge function G m L R L , the infinitesimal generators ξ L R L0 and ξ L R Li satisfy

  α ,β ∂ L R L t, q j , R L Dγ q j α ,β

∂ R L D γ qi

·

RL

 α ,β m ξ L R Li − q˙ i ξ LmR L0 + Dγ



∂ LRL α ,β

∂ R L D γ qi

·

d

RL

dt

∂ LRL α ,β D γ qi +



∂t

ξ LmR L0

 

 ∂ LRL m ∂ LRL γ d

R L α ,β ξ L R Li + G˙ m Dγ q j − (t − t 1 )−α qi (t 1 ) ξ LmR L0 t 1 , q j (t 1 ) L R L t, q j , α ,β ∂ qi ∂ R L D γ qi  (1 − α ) dt   

 1−γ d

−β m + L R L · ξ˙LmR L0 − W L R Li ξ LmR−Li1 − q˙ i ξ LmR−L01 = 0, − (t 2 − t ) qi (t 2 ) ξ L R L0 t 2 , q j (t 2 )  (1 − β) dt

+

then there exists an adiabatic invariant

I L R Lz =

z 

⎧ ⎨

t

εm L R L · ξLmR L0 + ⎩

m =0

t

t1

∂ LRL

−

α ,β ∂ R L D γ qi

t1

  ⎡ α ,β ∂ L R L t, q j , R L Dγ q j ⎣ · α ,β ∂ R L D γ qi

RL

α ,β

Dγ





(89)

⎤



β,α

ξ LmR Li − q˙ i ξ LmR L0 + ξ LmR Li − q˙ i ξ LmR L0 · C D 1−γ

∂ LRL α ,β

∂ R L D γ qi

⎦ dτ

d

γ

 (1 − α ) dτ

 (τ − t 1 )−α qi (t 1 ) ξ LmR L0 t 1 , q j (t 1 )

⎫   ⎬

 (1 − γ ) d

R L α ,β Dγ q j , − (t 2 − τ )−β qi (t 2 ) ξ LmR L0 t 2 , q j (t 2 ) dτ + G m L R L t, q j , ⎭  (1 − β) dτ

(90)

where ξ LmR−Li1 = ξ LmR−L01 = 0 when m = 0. Proof. From formulae (88) and (89), we have

d dt

I L R Lz =

z 

εm

m =0

  ⎧ ⎨ ∂ L R L t , q j , R L D αγ ,β q j ⎩

∂



R L D α ,β q

γ

·

RL

α ,β m ξ

Dγ

L R Li

i



 β,α − q˙ i ξ LmR L0 + ξ LmR Li − q˙ i ξ LmR L0 · C D 1−γ



∂ LRL ∂

R L D α ,β q

γ

i

∂ LRL ∂ LRL ∂ LRL d R L α ,β + q˙ i + D γ qi · ξ LmR L0 + L R L · ξ˙LmR L0 α ,β ∂t ∂ qi ∂ R L D γ qi dt

 ∂ LRL γ d

− (t − t 1 )−α qi (t 1 ) ξ LmR L0 t 1 , q j (t 1 ) α ,β R L  1 − α dt ( ) ∂ D γ qi ⎫  ⎬

 (1 − γ ) d

+ G˙ m − (t 2 − t )−β qi (t 2 ) ξ LmR L0 t 2 , q j (t 2 ) LRL ⎭  (1 − β) dt !   z    

m ∂ LRL ∂ LRL m −1 m −1 m C β,α m = ε D 1 −γ · ξ L R Li − q˙ i · ξ L R L0 + W L R Li ξ L R Li − q˙ i ξ L R L0 α ,β − ∂ q i ∂ R L D γ qi +

m =0

=

z 









εm −ε W L R Li ξLmR Li − q˙ i ξLmR L0 + W L R Li ξLmR−Li1 − q˙ i ξLmR−L01



 = −ε z+1 W L R Li ξ LzR Li − q˙ i ξ LzR L0 .

m =0

2

This proof is completed.

Theorem 14. For the disturbed fractional Lagrangian system

  ∂ L C R L β,α ∂ LC α ,β − D 1 −γ = ε W LC i t , q j , C D γ q j , α ,β ∂ qi ∂ C D γ qi

(91)

m m if the gauge function G m LC , the infinitesimal generators ξ LC 0 and ξ LC i satisfy

  α ,β ∂ LC t, q j , C Dγ q j α ,β ∂ C D γ qi

−

∂ LC α ,β

∂ C D γ qi

·

C

 α ,β m m ξ LC i − q˙ i ξ LC Dγ 0 +

γ  (1 − α )



(t − t 1 )

−α



∂ LC

d

α ,β · ∂ C D γ qi dt

C

 m q˙ i (t 1 ) ξ LC 0 t 1 , q j (t 1 ) −

∂ LC α ,β D γ qi + ∂t

1−γ

 (1 − β)





  m C α ,β ˙m ˙m ξ LC 0 + G LC t , q j , D γ q j + L C · ξ LC 0 −β

(t 2 − t )



 m ˙qi (t 2 ) ξ LC 0 t 2 , q j (t 2 )

C.J. Song / Physics Letters A 383 (2019) 125914

+

15

  ∂ LC m m −1 m −1 ˙ ξ LC i − W LC i ξ LC − q ξ = 0, i LC 0 i ∂ qi

then there exists an adiabatic invariant

I LCm =

z 

ε

⎧ ⎨

m

⎩

m =0

t −

t m L C · ξ LC 0

⎣

  α ,β ∂ LC t, q j , C Dγ q j α ,β

∂ C D γ qi

t1

∂ LC α ,β

t1

+

⎡

∂ C D γ qi

(92)



γ  (1 − α )

−α

(τ − t 1 )

·

C

 m  R L β,α α ,β m m m ˙ i ξ LC ξ LC i − q˙ i ξ LC Dγ D 1 −γ 0 + ξ LC i − q 0 ·

 m q˙ i (t 1 ) ξ LC 0 t 1 , q j (t 1 ) −

1−γ



 (1 − β)

(t 2 − τ )

−β

⎤ ∂ LC α ,β

∂ C D γ qi

⎦ dτ



 m ˙qi (t 2 ) ξ LC dτ 0 t 2 , q j (t 2 )

⎫ ⎬



C α ,β + Gm , LC t , q j , D γ q j ⎭

(93)

m−1 m−1 where ξ LC i = ξ LC 0 = 0 when m = 0.

Theorem 15. For the disturbed classical Hamiltonian system

 ∂ H t, q j , p H j q˙ i = , ∂ pHi



 ∂ H t, q j , p H j p˙ H i = − − εW Hi t, q j , p H j , ∂ qi

(94)

m m if the gauge function G m H , the infinitesimal generators ξ H0 and ξ H i satisfy

m p H i · ξ˙ Hmi − H · ξ˙ H0 −

 

 ∂H m ∂H m m −1 m −1 ξ H0 − ξ H i + G˙ m − q˙ i ξ H0 = 0, H t, q j , p H j − W H i ξH i ∂t ∂ qi

(95)

then there exists an adiabatic invariant

IHz =

z 

"

#

m εm p H i ξ Hmi − H ξ H0 + Gm H ,

(96)

m =0 m−1 where ξ Hmi−1 = ξ H0 = 0 when m = 0.

Theorem 16. For the disturbed classical Lagrangian system





 ∂ L t , q j , q˙ j d ∂ L t , q j , q˙ j − = ε W Li t , q j , q˙ j , ∂ qi dt ∂ q˙ i

(97)



m m ˙ if the gauge function G m L t , q j , q j , the infinitesimal generators ξ L0 and ξ Li satisfy

     ∂ L t , q j , q˙ j ∂L m ∂L m ˙m ∂L m −1 m −1 m m ˙ · ξ Li − q˙ i − L · ξ˙L0 + ξ L0 + ξ Li + G L − W Li ξ Li − q˙ i ξ L0 = 0, ∂ q˙ i ∂ q˙ i ∂t ∂ qi

(98)

then there exists an adiabatic invariant

I Lz =

z  m =0

εm

   ∂L m ∂L m ξ Li − q˙ i − L ξ L0 + Gm L , ∂ q˙ i ∂ q˙ i

(99)

m−1 m−1 where ξ Li = ξ L0 = 0 when m = 0.

Remark 14. Theorems 9–16 reduce to Theorems 1–8 when z = 0 respectively, i.e., when the Hamiltonian system is undisturbed, we can get a conserved quantity. When the Hamiltonian system is disturbed, we can get a z-th order adiabatic invariant. 6. Applications Two applications of the fractional Hamiltonian system in terms of CRL and CC are given. Application 1. The Lotka biochemical oscillator, which can be used to describe the competition between different species in the field of ecology, is an important class of biological model [28]. Now, the fractional Lotka biochemical oscillator model with CRL will be presented through the fractional Hamiltonian method. The fractional differential equation of the Lotka biochemical oscillator is RL

where

α ,β

D γ x1 = α1 + β1 exp x2 ,

α1 , α2 , β1 , β2 are constants.

C

β,α

D 1−γ x2 = α2 + β2 exp x1 ,

(100)

16

C.J. Song / Physics Letters A 383 (2019) 125914

Let x1 = q, x2 = p R L , we can get the Hamiltonian

H R L = α1 p R L − α2 q + β1 exp p R L − β2 exp q,

(101)

and the fractional Lotka biochemical oscillator model with CRL is RL

α ,β

C

D γ q = α1 + β1 exp p R L ,

When

β,α

D 1−γ p R L = α2 + β2 exp q.

(102)

α , β → 1, the classical differential equation of the Lotka biochemical oscillator and the classical Hamiltonian can be obtained x˙ 2 = α2 + β2 exp x1 ,

x˙ 1 = α1 + β1 exp x2 ,

H = α1 p H − α2 q + β1 exp p H − β2 exp q.

(103)

It is followed from formula (57) that

p R L · ξ R0L0 ·

−

d

RL

dt pRL · γ



α ,β

Dγ q + pRL ·

RL



D γ q − α1 p R L + α2 q − β1 exp p R L + β2 exp q · ξ˙ R0L0 α ,β

 (t − t 1 )−α q (t 1 ) ξ R0L0 t 1 , q j (t 1 ) , p R L j (t 1 ) + p R L ·

d 

 (1 − α ) dt

+ (α2 + β2 exp q) ξ R0L +

RL

α ,β

Dγ



 ξ R0L − q˙ ξ R0L0 + G˙ 0R L

p R L · (1 − γ ) d 

 (1 − β)

 (t 2 − t )−β q (t 2 ) ξ R0L0 t 2 , q j (t 2 ) , p R L j (t 2 ) = 0.

dt

(104)

Taking calculation, we have

ξ R0L0 = −1,

ξ R0L = 0,

G 0R L = 0, 

α ,β γ d R L α ,β d where dt D γ q = R L D γ q˙ + (1−α ) dt (t − t 1 )−α q (t 1 ) −

t  I H R L0 =

pRL ·

d

RL

dτ

α ,β



β,α

D γ q + q˙ C D 1−γ p R L

(105) 1−γ d (1−β) dt



dτ − p R L ·





RL

(t 2 − t )−β q (t 2 ) . From Theorem 1, we can get a conserved quantity α ,β



Dγ q − H RL .

(106)

t1

When the fractional Lotka biochemical oscillator model is disturbed as RL

α ,β

C

D γ q = α1 + β1 exp p R L ,

β,α

D 1−γ p R L = α2 + β2 exp q − ε (2q + 1) ,

(107)

using formula (77), we have

ξ R1L0 · p R L ·

d

RL

dt



α ,β

Dγ q + pRL ·

RL



D γ q − α1 p R L + α2 q − β1 exp p R L + β2 exp q · ξ˙ R1L0 α ,β

 pRL · γ d

(t − t 1 )−α q (t 1 ) ξ R1L0 t 1 , q j (t 1 ) , p R L j (t 1 )  (1 − α ) dt  

 p R L · (1 − γ ) d

+ (α2 + β2 exp q) · ξ R1L + (t 2 − t )−β q (t 2 ) ξ R1L0 t 2 , q j (t 2 ) , p R L j (t 2 ) − (2q + 1) ξ R0L − q˙ ξ R0L0 = 0.  (1 − β) dt + G˙ 1R L + p R L ·

RL

 α ,β 1 ξ R L − q˙ ξ R1L0 −

Dγ

(108)

Solving formula (108), we obtain

ξ R1L0 = 1,

ξ R1L = 0,

G 1R L = q2 + q.

(109)

From Theorem 9, we can get the first order adiabatic invariant

t  I H R L1 =

pRL ·

d

RL

dτ

α ,β



β,α

D γ q + q˙ C D 1−γ p R L



dτ − p R L ·

RL

α ,β

D γ q − α1 p R L + α2 q − β1 exp p R L

t1

 + β2 exp q) + ε q2 + q + p R L · t  pRL ·

−

d dτ

RL

α ,β

D γ q + q˙ ·

C

RL

α ,β

D γ q − α1 p R L + α2 q − β1 exp p R L + β2 exp q



β,α

D 1 −γ p R L



⎤

dτ ⎦ .

(110)

t1

When

α , β → 1, from formulae (106) and (110), we have

I H0 = α1 p H − α2 q + β1 exp p H − β2 exp q,



2



I H1 = α1 p H − α2 q + β1 exp p H − β2 exp q + ε q + q − α1 p H + α2 q − β1 exp p H + β2 exp q , where formula (111) is consistent with the result in Ref. [27].

(111) (112)

C.J. Song / Physics Letters A 383 (2019) 125914

17

Application 2. The fractional Lagrangian of the two dimensional isotropic harmonic oscillator is

1

LC =

2

 m

α ,β

C

D γ q1

2

+



α ,β

C

D γ q2

2 

1 − k (q1 )2 + (q2 )2 ,

(113)

2

where k, m are constants. Then from formula (113), we can introduce the generalized momentum and the Hamiltonian

pC 1 =

  α ,β ∂ LC t, q j , C Dγ q j ∂

C D α ,β q

γ

α ,β

= mC D γ q1 ,

1

  α ,β ∂ LC t, q j , C Dγ q j

pC 2 =



α ,β α ,β α ,β H C = p C 1 · C D γ q1 + p C 2 · C D γ q2 − L C t , q j , C D γ q j

= pC 1 · =

pC 1 m

( pC 1)

2

2m

+

+ pC 2 · (pC 2)

2

2m

pC 2 m

1

− m

 p

2

C1

2

+

m

1

p

C2

2 

∂ 

C D α ,β q

γ

α ,β

= mC D γ q2 ,

2

1 + k (q1 )2 + (q2 )2

m

2



+ k (q1 )2 + (q2 )2 .

(114)

2

When α , β → 1, the classical Lagrangian and the classical Hamiltonian of the two dimensional isotropic harmonic oscillator can be obtained as

L=

1 2





m q˙ 21 + q˙ 22 −

 1  2 k q1 + q22 , 2

H=

( p H1 )2 2m

+

1 + k (q1 )2 + (q2 )2 .

( p H2 )2 2m

(115)

2

Substituting formula (114) into formulae (35) and (37), we get C

α ,β

D γ q1 =

pC 1 m

C

,

α ,β

D γ q2 =

pC 2 m

RL

,

β,α

RL

D 1−γ p C 1 = −kq1 ,

β,α

D 1−γ p C 2 = −kq2 .

(116)

Formula (116) is the fractional isotropic harmonic oscillator model with CC. It is followed from formula (61) that

   d d α ,β α ,β α ,β ξC01 − q˙ 1 ξC00 + p C 2 · C D γ ξC02 − q˙ 2 ξC00 + ξC00 · p C 1 · C D γ q1 + ξC00 · p C 2 · C D γ q2 dt dt !  ( p C 1 )2 ( p C 2 )2 k  C α ,β C α ,β 2 2 + p C 1 · D γ q1 + p C 2 · D γ q2 − − − · ξ˙C00 (q1 ) + (q2 )

pC 1 ·

C

α ,β



Dγ

2m

2m

2



 pC 1 · γ  pC 2 · γ d  − (t − t 1 )−α q˙ 1 (t 1 ) ξC00 − (t − t 1 )−α q2 (t 1 ) ξC00 t 1 , q j (t 1 ) , p C j (t 1 )  (1 − α )  (1 − α ) dt 

 p C 1 · (1 − γ ) − kq1 ξC01 − kq2 ξC02 + (t 2 − t )−β q˙ 1 (t 2 ) ξC00 t 2 , q j (t 2 ) , p C j (t 2 )  (1 − β)  p C 2 · (1 − γ ) d  + (t 2 − t )−β q2 (t 2 ) ξC00 + G˙ 0C = 0.  (1 − β) dt

(117)

Taking calculation, we have

ξC00 = −1,

ξC01 = ξC02 = 0,

G 0C = 0,

(118)

  α ,β γ 1−γ

d C α ,β where dt D γ q1 = C D γ q˙ 1 + (1−α ) (t − t 1 )−α q˙ 1 (t 1 ) − (1−β) (t 2 − t )−β q˙ 1 (t 2 ) ,  1−γ

−β ˙ q2 (t 2 ) . From Theorem 2, we can get a conserved quantity (1−β) (t 2 − t )

d C α ,β D γ q2 dt

α ,β = C D γ q˙ 2 +

γ

(1−α )

 (t − t 1 )−α q˙ 2 (t 1 ) −



!

I H C 0 = − pC 1 ·

C

 ( p C 1 )2 ( p C 2 )2 1  α ,β α ,β D γ q1 + p C 2 · C D γ q2 − − − k (q1 )2 + (q2 )2

t  +

pC 1 ·

2m

d dτ

C

α ,β

D γ q1 + p C 2 ·

d dτ

C

α ,β

2m

D γ q2 + q˙ 1 ·

RL

2

β,α

D 1−γ p C 1 + q˙ 2 ·

RL

β,α



D 1−γ p C 2 dτ .

(119)

t1

When the fractional isotropic harmonic oscillator model is disturbed as C

α ,β

D γ q1 =

pC 1 m

,

using formula (80), we have

C

α ,β

D γ q2 =

pC 2 m

,

RL

β,α

D 1−γ p C 1 = −kq1 − εq2 ,

RL

β,α

D 1−γ p C 2 = −kq2 − εq1 ,

(120)

18

C.J. Song / Physics Letters A 383 (2019) 125914

pC 1 ·

C

  α ,β 1 α ,β

ξC 1 − q˙ 1 ξC10 + p C 2 · C D γ ξC12 − q˙ 2 ξC10 + α ,β

2

( pC 1)

α ,β

+ p C 1 · C D γ q1 + p C 2 · C D γ q2 − −

 pC 1 ·

Dγ

2m

−

( pC 2)

d

C

dt

!

1 

2

− k (q1 )2 + (q2 )2

2m

d

α ,β

D γ q1 + p C 2 ·



2

C

dt



α ,β

D γ q2 ξC10

· ξ˙C10 − kq1 ξC11 − kq2 ξC12

 (t − t 1 )−α q˙ 1 (t 1 ) ξC10 t 1 , q j (t 1 ) , p C j (t 1 ) + G˙ 1C − W C 1 q˙ 1 − W C 2 q˙ 2



pC 1 · γ

 (1 − α )

 p C 1 · (1 − γ )

+ (t 2 − t )−β q˙ 1 (t 2 ) ξC10 t 2 , q j (t 2 ) , p C j (t 2 )  (1 − β)

 pC 2 · γ

− (t − t 1 )−α q˙ 2 (t 1 ) ξC10 t 1 , q j (t 1 ) , p C j (t 1 )  (1 − α )

 p C 2 · (1 − γ )

+ (t 2 − t )−β q˙ 2 (t 2 ) ξC10 t 2 , q j (t 2 ) , p C j (t 2 ) = 0.  (1 − β)

(121)

Solving formula (121), we obtain

ξC10 = 1,

ξC11 = ξC12 = 0,

G 1C = q1 q2

(122)

From Theorem 10, we can get the first order adiabatic invariant

!

I H C 1 = − pC 1 ·

C

 ( p C 1 )2 ( p C 2 )2 1  α ,β α ,β D γ q1 + p C 2 · C D γ q2 − − − k (q1 )2 + (q2 )2 2m

t  pC 1 ·

+ t1

−ε

⎧ t ⎨  ⎩

d

C

dτ

pC 1 ·

α ,β

D γ q1 + p C 2 ·

d

C

dτ

d

C

dτ

2m

α ,β

D γ q2 + q˙ 1 ·

RL

2

β,α

D 1−γ p C 1 + q˙ 2 ·

RL



β,α

D 1−γ p C 2 dτ

d C α ,β α ,β β,α β,α D γ q1 + p C 2 · D γ q2 + q˙ 1 · R L D 1−γ p C 1 + q˙ 2 · R L D 1−γ p C 2 dτ

t1

α ,β

α ,β

− p C 1 · C D γ q1 + p C 2 · C D γ q2 − When

( p C 1 )2 2m

−

( p C 2 )2 2m

 dτ

⎫

!

⎬  1  − k (q1 )2 + (q2 )2 − q1 q2 . ⎭ 2

(123)

α , β → 1, from formulae (119) and (123), we have

I H0 = I H1 =

( p H1 )2 2m

( p H1 )2 2m

+ +

( p H2 )2 2m

( p H2 )2 2m

+ +

k 2

 (q1 )2 + (q2 )2 ,

k 2

2

(q1 ) + (q2 )

2



−ε

(124)

( p H1 )2 2m

+

( p H2 )2 2m

+

k 2

2

2

(q1 ) + (q2 )



! − q1 q2 ,

(125)

where formula (124) is consistent with the result in Ref. [27]. 7. Results and discussion Fractional Hamiltonian mechanics with CRL and CC are investigated, including fractional differential equation of motion, Noether quasisymmetry, fractional conserved quantity, perturbation to Noether quasi-symmetry and fractional adiabatic invariant. In addition, fractional Lagrangian mechanics with CRL and CC, the classical conserved quantity and the classical adiabatic invariant for the classical Hamiltonian system are also discussed as special cases. The fractional isotropic harmonic oscillator model in terms of CC and the fractional Lotka biochemical oscillator model in terms of CRL are discussed, and the conserved quantities as well as their first order adiabatic invariants are presented. The results of Noether symmetry, conserved quantity, perturbation to Noether symmetry and adiabatic invariant can be obtained by letting the gauge function disappear in this text. However, those results do not be listed in detail. Acknowledgements This work was supported by the National Natural Science Foundation of China (grant numbers 11802193, 11572212), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (grant number 18KJB130005), the Jiangsu Government Scholarship for Overseas Studies, the Science Research Foundation of Suzhou University of Science and Technology (grant number 331812137) and the Natural Science Foundation of Suzhou University of Science and Technology.

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