# Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

Authors: C. J. Song
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Perturbation to Noether symmetry and adiabatic invariants are investigated for the generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative, respectively. Firstly, differential equations of motion for the generalized fractional Birkhoffian system are established. Secondly, Noether symmetry and conserved quantity are studied. Thirdly, perturbation to Noether symmetry and adiabatic invariants are presented for the generalized fractional Birkhoffian mechanics. And finally, several applications are discussed to illustrate the results and methods.


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## Body

## 1. Introduction

Compared with the integer order model, the fractional order model can describe the mechanical and physical behavior of the complex system more accurately. Miller and Ross [1] once pointed out that almost every field of science and engineering involves fractional calculus. Fractional calculus has been widely used in mathematics, physics, chemistry, signal processing, engineering, and so on [2]. Riemann-Liouville fractional derivative and Caputo fractional derivative are in common use. Cresson [3] presented two more general fractional derivatives in 2006, i.e., the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative.The study of the fractional calculus of variations was started in 1996, when Riewe [4, 5] was considering how to deal with the friction force and other forms of dissipative force in the classical mechanics and the quantum mechanics. Since then, Agrawal [6, 7], Atanacković [8], Almeida [9, 10], and some other scholars [11, 12] also studied fractional variational problems. However, all of the results were related to Lagrangian system or Hamiltonian system. As a matter of fact, there is a more general system called Birkhoffian system, which was introduced in 1927 [13]. Birkhoffian mechanics is a generalization of Hamiltonian mechanics, which is described in detail in [14]. Birkhoffian dynamics has gained significant headways [15, 16]. Recently, in [17], Luo first established Birkhoffian mechanics with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative.There is a set of unique integral theory in analytical mechanics, which is useful in solving differential equations of motion. Besides, symmetry and conserved quantity can help to reveal the intrinsic physical properties of the dynamic system. The commonly used symmetry methods [18] are Noether symmetry method, Lie symmetry method, and Mei symmetry method. Noether symmetry refers to the invariance of the Hamilton action under the infinitesimal transformations [19]. Lie symmetry refers to the invariance of the differential equations of motion under the infinitesimal transformations [20]. Mei symmetry means the invariance of the forms of the differential equations of motion when the dynamical functions, such as the kinetic energy, the potential energy, the generalized forces, the Lagrangian, the Hamiltonian, and the Birkhoffian, are replaced by the transformed functions under the infinitesimal transformations. Many important research results have been achieved in terms of symmetry and conserved quantity of the constrained mechanical systems [21].The fractional symmetry and conserved quantity were first studied by Frederico and Torres [22]. Based on the Riemann-Liouville fractional derivative, they established the fractional Noether theorem. Bourdin et al. [23] presented a new expression of the fractional conserved quantity and took the fractional harmonic oscillator as an example. Based on the Caputo fractional derivative, Muslih [24] extended the fractional Noether theorem of finite degree of freedom to the fractional field theory and presented the conserved quantity of the fractional Dirac field. In recent years, Zhang [25, 26], Jia [27], and Zhang [28] studied the Noether symmetry and conserved quantity of the fractional Birkhoffian system. Fu [29] investigated the Lie symmetry of the fractional nonholonomic Hamiltonian system and the corresponding inverse problem. Luo [30] presented the Mei symmetry and conserved quantity of the generalized fractional Hamiltonian system.When the system suffers small disturbance, the original conserved quantity will change. Perturbation to symmetry and adiabatic invariants are closely related to the integrability of the system, and they have been widely used in many fields, such as mathematics, physics, and mechanics. The classical adiabatic invariant refers to a slower physical quantity [31] when a parameter changes slowly in the system. In 1981, Djukić [32] discussed the disturbed Hamiltonian system and gave the corresponding adiabatic invariant. In 1996, Zhao and Mei [33] pointed out that adiabatic invariant was not just the product of the Hamiltonian system. Thereafter, perturbation to symmetry and adiabatic invariants were studied in many constrained mechanical systems such as the Lagrangian system [34], the Hamiltonian system [35], the nonholonomic system [36], and the Birkhoffian system [37], as well as the fractional constrained mechanical system [38, 39]. It is worth mentioning that [39] presented conserved quantity and adiabatic invariant for the fractional generalized Birkhoffian system on the basis of Riemann-Liouville fractional derivative, where Riemann-Liouville fractional derivative is one of the special cases of the combined Riemann-Liouville fractional derivative.Because the generalized fractional Birkhoffian system and the combined fractional derivatives are general, generalized fractional Birkhoff equation, Noether symmetry and conserved quantity, and perturbation to Noether symmetry and adiabatic invariant with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative will be studied here. Then several special cases of this paper will be discussed. This paper is organized as follows. In Section2, fractional derivatives and their properties are reviewed. In Section 3, generalized fractional Birkhoff equation is established through the generalized fractional Pfaff-Birkhoff-d’Alembert principle. Noether symmetry and conserved quantity for the generalized fractional Birkhoffian system are presented in Section 4. Perturbation to Noether symmetry and adiabatic invariant are investigated in Section 5. And in Section 6, several applications are discussed.


## 2. Some Preliminaries on Fractional Derivatives

Some fractional derivatives and their properties are listed in this section [17, 40].Letft be continuous and integrable, then the left and the right Riemann-Liouville fractional derivatives are(1)​t1RLDtαft=1Γn-αddtn∫t1tt-ξn-α-1fξdξ,(2)​tRLDt2βft=1Γn-β-ddtn∫tt2ξ-tn-β-1fξdξ,the left and the right Caputo fractional derivatives are(3)​t1CDtαft=1Γn-α∫t1tt-ξn-α-1ddξnfξdξ,(4)​tCDt2βft=1Γn-β∫tt2ξ-tn-β-1-ddξnfξdξ,and the Riesz-Riemann-Liouville fractional derivative and the Riesz-Caputo fractional derivative are(5)​t1RDt2αft=12Γn-αddtn∫t1t2t-ξn-α-1fξdξ,(6)​t1RCDt2αft=12Γn-α∫t1t2t-ξn-α-1ddξnfξdξ,where α and β are the orders of the fractional derivatives with n-1≤α,β<n.The combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative are defined as(7)​RLDγα,βft=γ​t1RLDtαft+-1n1-γ​tRLDt2βft,(8)​CDγα,βft=γ​t1CDtαft+-1n1-γ​tCDt2βft,where ​t1Dtα and ​tDt2β can help deal with dynamical systems exhibiting the arrow of time and γ determines the different quantity of information from the past and the future to keep track of the past and the future of the dynamics.Whenβ=α, γ=1/2, formulae (7) and (8) reduce to formulae (5) and (6), respectively. That is, the Riesz-Riemann-Liouville fractional derivative and the Riesz-Caputo fractional derivative are the special cases of the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative.Besides, whenα,β→1, we have(9)​t1Dt1=ddt,​tDt21=-ddt,Dγα,β=γ​t1Dt1+-1n1-γ​tDt21=ddt.The formulae for fractional integration by parts are(10)∫t1t2∗​t1RLDtαηdt=∫t1t2η​tCDt2α∗dt-∑j=0n-1-1n+j​t1RLDtα+j-nηtDn-1-j∗t1t2,(11)∫t1t2∗​tRLDt2βηdt=∫t1t2η​t1CDtβ∗dt-∑j=0n-1​tRLDt2β+j-n∗Dn-1-jηtt1t2,(12)∫t1t2∗​t1CDtαηdt=∫t1t2η​tRLDt2α∗dt+∑j=0n-1​tRLDt2α+j-n∗Dn-1-jηtt1t2,(13)∫t1t2∗​tCDt2βηdt=∫t1t2η​t1RLDtβ∗dt+∑j=0n-1-1n+j​t1RLDtβ+j-n∗Dn-1-jηtt1t2,(14)∫t1t2∗​t1RDt2αηdt=-1n∫t1t2η​t1RCDt2α∗dt+∑j=0n-1-1n+j​t1RDt2α+j-nηtDn-1-j∗t1t2,(15)∫t1t2∗​t1RLDt2αηdt=-1n∫t1t2η​t1RDt2α∗dt+∑j=0n-1-1j​t1RDt2α+j-n∗Dn-1-jηtt1t2.


## 3. Generalized Fractional Birkhoffian System

Generalized fractional Birkhoff equations in terms of the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative are established in this section.Case 1.
Based on the combined Riemann-Liouville fractional derivative, the fractional Pfaff action and the generalized fractional Pfaff-Birkhoff principle have their forms as follows:(16)SRLaν·=∫t1t2Rμt,aνt​RLDγα,βaμt-Bt,aνtdt,(17)∫t1t2δRμ​RLDγα,βaμ-B+δ′Wdt=0,​t1RLDtαδaμ=δ​t1RLDtαaμ,​tRLDt2βδaμ=δ​tRLDt2βaμ,δaμt=t1=δaμt=t2=0,where aμ=aμt is the Birkhoff’s variable, B=Bt,aν is the Birkhoffian, Rμ=Rμt,aν is the Birkhoff’s function, δ′W=Λμt,aνδaμ, μ,ν=1,2,⋯,2n, δ is the isochronous variation, n-1≤α,β<n, and Einstein summation convention is used in this text.
When0<α,β<1, it follows from(18)∫t1t2γRμ·δ​t1RLDtαaμdt=∫t1t2γRμ·​t1RLDtαδaμdt=∫t1t2γδaμ​tCDt2αRμdtand(19)∫t1t21-γRμ·δ​tRLDt2βaμdt=∫t1t21-γRμ·​tRLDt2βδaμdt=∫t1t21-γδaμ​t1CDtβRμdtthat(20)∫t1t2δRμ​RLDγα,βaμ-B+δ′Wdt=∫t1t2δRμ·​RLDγα,βaμ+Rμ·γδ​t1RLDtαaμ-1-γδ​tRLDt2βaμ-δB+δ′Wdt=∫t1t2∂Rν∂aμ​RLDγα,βaν+γ​tCDt2αRμ-1-γ​t1CDtβRμ-∂B∂aμ+Λμδaμdt=∫t1t2∂Rν∂aμ​RLDγα,βaν-​CD1-γβ,αRμ-∂B∂aμ+Λμδaμdt=0.
From the arbitrariness of the integration intervalt1,t2, we have(21)∂Rν∂aμ​RLDγα,βaν-​CD1-γβ,αRμ-∂B∂aμ+Λμδaμ=0.Formula (21) is called generalized fractional Pfaff-Birkhoff-d’Alembert principle with the combined Riemann-Liouville fractional derivative.
Due to the independence ofδaμ, we get(22)∂Rν∂aμ​RLDγα,βaν-​CD1-γβ,αRμ-∂B∂aμ+Λμ=0.Equation (22) is called generalized fractional Birkhoff equation with the combined Riemann-Liouville fractional derivative.Case 2.
Based on the combined Caputo fractional derivative, the fractional Pfaff action and the generalized fractional Pfaff-Birkhoff principle have their forms as follows:(23)SCaν·=∫t1t2Rμt,aνt​CDγα,βaμt-Bt,aνtdt,(24)∫t1t2δRμ​CDγα,βaμ-B+δ′Wdt=0,​t1CDtαδaμ=δ​t1CDtαaμ,​tCDt2βδaμ=δ​tCDt2βaμ,δaμt=t1=δaμt=t2=0.
Similarly, when0<α,β<1, we get(25)∂Rν∂aμ​CDγα,βaν-​RLD1-γβ,αRμ-∂B∂aμ+Λμ=0.Equation (25) is called generalized fractional Birkhoff equation with the combined Caputo fractional derivative.Remark 1.
Whenβ=α, γ=1/2, from (22) and (25), we have(26)∂Rν∂aμ​t1RDt2αaν-​t1RCDt2αRμ-∂B∂aμ+Λμ=0and(27)∂Rν∂aμ​t1RCDt2αaν-​t1RDt2αRμ-∂B∂aμ+Λμ=0.Equations (26) and (27) are called generalized fractional Birkhoff equations with the Riesz-Riemann-Liouville fractional derivative and the Riesz-Caputo fractional derivative, respectively.Remark 2.
WhenΛμ=0, μ=1,2,⋯,2n, from (22), (25), (26), and (27), we have(28)∂Rν∂aμ​RLDγα,βaν-​CD1-γβ,αRμ-∂B∂aμ=0,(29)∂Rν∂aμ​CDγα,βaν-​RLD1-γβ,αRμ-∂B∂aμ=0,(30)∂Rν∂aμ​t1RDt2αaν-​t1RCDt2αRμ-∂B∂aμ=0,and(31)∂Rν∂aμ​t1RCDt2αaν-​t1RDt2αRμ-∂B∂aμ=0.Equations (28)-(31) are consistent with the results in [17].Remark 3.
When0<α,β<1, the relationships between the Riemann-Liouville fractional derivative and the Caputo fractional derivative can be deduced from their definitions(32)
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.That is,(33)​t1RLDtαft=​t1CDtαft-1Γ1-αft1t-t1α.
Similarly, we have(34)​tRLDt2βft=​tCDt2βft+1Γ1-βft2t2-tβ.Substituting formulae (33) and (34) into formulae (18) and (19), we get(35)∫t1t2γRμ·δ​t1RLDtαaμdt=∫t1t2γRμ·​t1RLDtαδaμdt=∫t1t2γRμ·​t1CDtαδaμ-1Γ1-αδaμt1t-t1αdt=∫t1t2γRμ·​t1CDtαδaμdt=∫t1t2γδaμ·​tRLDt2αRμdtand(36)∫t1t21-γRμ·δ​tRLDt2βaμdt=∫t1t21-γδaμ​t1RLDtβRμdt.Using formulae (35) and (36), another form of the generalized fractional Birkhoff equation with the combined Riemann-Liouville fractional derivative can be achieved(37)∂Rν∂aμ​RLDγα,βaν-​RLD1-γβ,αRμ-∂B∂aμ+Λμ=0.
Similarly, another form of the generalized fractional Birkhoff equation with the combined Caputo fractional derivative can be achieved(38)∂Rν∂aμ​CDγα,βaν-​CD1-γβ,αRμ-∂B∂aμ+Λμ=0.Remark 4.
Whenβ=α, γ=1/2, from (37) and (38), we can get other forms of the generalized fractional Birkhoff equations with the Riesz-Riemann-Liouville fractional derivative and the Riesz-Caputo fractional derivative(39)∂Rν∂aμ​t1RDt2αaν-​t1RDt2αRμ-∂B∂aμ+Λμ=0,(40)∂Rν∂aμ​t1RCDt2αaν-​t1RCDt2αRμ-∂B∂aμ+Λμ=0.Remark 5.
Whenγ=1, it follows from (37) that(41)∂Rν∂aμ​t1RLDtαaν+​tRLDt2αRμ-∂B∂aμ+Λμ=0.Equation (41) is the generalized fractional Birkhoff equation with the Riemann-Liouville fractional derivative [39].Remark 6.
Whenγ=1, Λμ=0, μ=1,2,⋯,2n, it follows from (37) that(42)∂Rν∂aμ​t1RLDtαaν+​tRLDt2αRμ-∂B∂aμ=0.Equation (42) is the fractional Birkhoff equation with the Riemann-Liouville fractional derivative [41].Remark 7.
Whenα,β→1, (22), (25), (26), and (27) all reduce to the classical generalized Birkhoff equation [21](43)∂Rν∂aμ-∂Rμ∂aνa˙ν-∂B∂aμ-∂Rμ∂t+Λμ=0.The following discussions are all carried out under the condition0<α,β<1.


## 4. Noether Symmetry and Conserved Quantity

Assuming the infinitesimal transformations are(44)t-=t+Δt,a-νt-=aνt+Δaν,and their expansions are(45)t-=t+εξ00t,aμ+oε,a-νt-=aνt+εξν0t,aμ+oε,where ε is an infinitesimal parameter and ξ00 and ξν0 are infinitesimal generators, oε will be ignored in the following calculation.The fractional Pfaff actions (16) and (23) will change under the infinitesimal transformations (44); we denote them as ΔSRL=SRLa-νt--SRLaνt and ΔSC=SCa-νt--SCaνt, respectively.Definition 8.
The infinitesimal transformations (44) are called Noether symmetric transformations if and only if(46)ΔSRL=-∫t1t2ddtΔG+Λμδaμdtresp.ΔSC=-∫t1t2ddtΔG+Λμδaμdtholds, where ΔG=εG0t,aν, G0 is called a gauge function.
Noether symmetry can be verified from Noether symmetric transformations.
It follows from formula (46) and(47)ΔSRL=SRLa-νt--SRLaνt=∫t-1t-2Rμt-,a-νt-γ​t-1RLDt-α-1-γ​t-RLDt-2β​RLDγα,βa-μt--Bt-,a-νt-dt--∫t1t2Rμt,aνt​RLDγα,βaμt-Bt,aνtdt=∫t1t2Rν​RLDγα,βδaν+Rνddt​RLDγα,βaν+∂Rν∂t​RLDγα,βaν-∂B∂tΔt+∂Rν∂aμ​RLDγα,βaν-∂B∂aμΔaμ-γRν1Γ1-αddtt-t1-αaνt1Δt1+Rν​RLDγα,βaν-BddtΔt+1-γRν1Γ1-βddtt2-t-βaνt2Δt2dtthat(48)Rν​RLDγα,βξν0-a˙νξ00+Rνddt​RLDγα,βaν+∂Rν∂t​RLDγα,βaν-∂B∂tξ00+∂Rν∂aμ​RLDγα,βaν-∂B∂aμξμ0-γRνΓ1-αddtt-t1-αaνt1ξ00t1,aμt1+Rν​RLDγα,βaν-Bξ˙00+1-γRνΓ1-βddtt2-t-βaνt2ξ00t2,aμt2+G˙0+Λμξμ0-a˙μξ00=0.Formula (48) is called Noether identity with the combined Riemann-Liouville fractional derivative.
Similarly, we have(49)Rν​CDγα,βξν0-a˙νξ00+Rνddt​CDγα,βaν+∂Rν∂t​CDγα,βaν-∂B∂tξ00+∂Rν∂aμ​CDγα,βaν-∂B∂aμξμ0-RνγΓ1-αt-t1-αa˙νt1ξ00t1,aμt1+Rν​CDγα,βaν-Bξ˙00+1-γRνΓ1-βt2-t-βa˙νt2ξ00t2,aμt2+G˙0+Λμξμ0-a˙μξ00=0.Formula (49) is called Noether identity with the combined Caputo fractional derivative.Remark 9.
Whenβ=α, γ=1/2, from formulae (48) and (49), we get(50)Rν​t1RDt2αξν0-a˙νξ00+Rνddt​t1RDt2αaν+∂Rν∂t​t1RDt2αaν-∂B∂tξ00+∂Rν∂aμ​t1RDt2αaν-∂B∂aμξμ0+Rν2Γ1-αddtt-t2-αaνt2ξ00t2,aμt2+Rν​t1RDt2αaν-Bξ˙00-Rν2Γ1-αddtt-t1-αaνt1ξ00t1,aμt1+G˙0+Λμξμ0-a˙μξ00=0and (51)Rν​t1RCDt2αξν0-a˙νξ00+Rνddt​t1RCDt2αaν+∂Rν∂t​t1RCDt2αaν-∂B∂tξ00+Rν2Γ1-αt-t2-αa˙νt2ξ00t2,aμt2-t-t1-αa˙νt1ξ00t1,aμt1+∂Rν∂aμ​t1RCDt2αaν-∂B∂aμξμ0+Rν​t1RCDt2αaν-Bξ˙00+G˙0+Λμξμ0-a˙μξ00=0.Formula (50) and formula (51) are called Noether identities with the Riesz-Riemann-Liouville fractional derivative and the Riesz-Caputo fractional derivative, respectively.Definition 10.
A quantityI is called a conserved quantity if and only if d/dtI=0.Therefore, we have the following.Theorem 11.
If there exists a gauge functionG0 such that ξ00 and ξν0 satisfy the Noether identity (48) for the generalized fractional Birkhoffian system (22), then this system has a conserved quantity(52)IRL0=Rν​RLDγα,βaν-Bξ00+∫t1tRν​RLDγα,βξν0-a˙νξ00+ξν0-a˙νξ00​CD1-γβ,αRνdτ-∫t1tRνΓ1-αγddττ-t1-αaνt1ξ00t1,aμt1-RνΓ1-β1-γddτt2-τ-βaνt2ξ00t2,aμt2dτ+G0=const..Theorem 12.
If there exists a gauge functionG0 such that ξ00 and ξν0 satisfy the Noether identity (49) for the generalized fractional Birkhoffian system (25), then this system has a conserved quantity(53)IC0=Rν​CDγα,βaν-Bξ00+∫t1tRν​CDγα,βξν0-a˙νξ00+ξν0-a˙νξ00​RLD1-γβ,αRνdτ-∫t1tRνΓ1-αγτ-t1-αa˙νt1ξ00t1,aμt1-RνΓ1-β1-γt2-τ-βa˙νt2ξ00t2,aμt2dτ+G0=const..Theorem 13.
If there exists a gauge functionG0 such that ξ00 and ξν0 satisfy the Noether identity (50) for the generalized fractional Birkhoffian system (26), then this system has a conserved quantity(54)IR0=Rν​t1RDt2αaν-Bξ00+∫t1tRν​t1RDt2αξν0-a˙νξ00+ξν0-a˙νξ00​t1RCDt2αRνdτ+∫t1tRν2Γ1-αddττ-t2-αaνt2ξ00t2,aμt2-τ-t1-αaνt1ξ00t1,aμt1dτ+G0=const..Theorem 14.
If there exists a gauge functionG0 such that ξ00 and ξν0 satisfy the Noether identity (51) for the generalized fractional Birkhoffian system (27), then this system has a conserved quantity(55)IRC0=Rν​t1RCDt2αaν-Bξ00+∫t1tRν​t1RCDt2αξν0-a˙νξ00+ξν0-a˙νξ00​t1RDt2αRνdτ+∫t1tRν2Γ1-ατ-t2-αa˙νt2ξ00t2,aμt2-τ-t1-αa˙νt1ξ00t1,aμt1dτ+G0=const..
Whenα,β→1, we can get the classical conserved quantity [21].Theorem 15.
If there exists a gauge functionG0 such that ξ00 and ξν0 satisfy the Noether identity(56)Rνξ˙ν0-Bξ˙00+∂Rν∂ta˙ν-∂B∂tξ00+∂Rν∂aμa˙ν-∂B∂aμξμ0+G˙0+Λμξμ0-a˙μξ00=0for the classical generalized Birkhoffian system(57)∂Rν∂aμ-∂Rμ∂aνa˙ν-∂B∂aμ-∂Rμ∂t+Λμ=0,then this system has a conserved quantity(58)I=Rνξν0-Bξ00+G0.Remark 16.
LetG0=0, Λμ=0, μ=1,2,⋯,2n in Theorems 11–14; then we can get the conserved quantity for the fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative, the combined Caputo fractional derivative, the Riesz-Riemann-Liouville fractional derivative, and the Riesz-Caputo fractional derivative, respectively. And these results are consistent with the results in [26].


## 5. Perturbation to Noether Symmetry and Adiabatic Invariants

If a small disturbance is imposed on a mechanical system, then the conserved quantity for the original mechanical system may also change. In this section, we discuss the perturbation to Noether symmetry for the generalized fractional Birkhoffian system.Definition 17.
IfIz is a quantity with a small parameter ε whose highest power is z for a mechanical system, and dIz/dt is in direct proportion to εz+1, then Iz is called a z-th order adiabatic invariant for this system.
For the generalized fractional Birkhoffian system, assume it is disturbed by the small force ofεQμμ=1,2,⋯,2n. Under the small disturbance, suppose the gauge function G and the infinitesimal generators ξ0 and ξμ for the disturbed system are expressed as(59)G=G0+εG1+ε2G2+⋯,ξ0=ξ00+εξ01+ε2ξ02+⋯,ξμ=ξμ0+εξμ1+ε2ξμ2+⋯.Then we have the following.Theorem 18.
For the disturbed generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative(60)∂Rν∂aμ​RLDγα,βaν-​CD1-γβ,αRμ-∂B∂aμ+Λμ=εQμμ,ν=1,2,⋯,2n,if there exists a gauge function Gj (j=0,1,2,⋯,z, z is an integer) such that the infinitesimal generators ξ0j and ξνj satisfy (61)Rν​RLDγα,βξνj-a˙νξ0j+Rνddt​RLDγα,βaν+∂Rν∂t​RLDγα,βaν-∂B∂tξ0j+∂Rν∂aμ​RLDγα,βaν-∂B∂aμξμj-γRνΓ1-αddtt-t1-αaνt1ξ0jt1,aμt1+Rν​RLDγα,βaν-Bξ˙0j+1-γRνΓ1-βddtt2-t-βaνt2ξ0jt2,aμt2+G˙j+Λμξμj-a˙μξ0j-Qμξμj-1-a˙μξ0j-1=0,where ξμj-1=ξ0j-1=0 when j=0, then a z-th order adiabatic invariant exists as follows:(62)IRLz=∑j=0zεjRν​RLDγα,βaν-Bξ0j+∫t1tRν​RLDγα,βξνj-a˙νξ0j+ξνj-a˙νξ0j​CD1-γβ,αRνdτ-∫t1tRνΓ1-αγddττ-t1-αaνt1ξ0jt1,aμt1-RνΓ1-β1-γddτt2-τ-βaνt2ξ0jt2,aμt2dτ+Gj.Proof.
From formulae (60) and (61), we have(63)
d
d
t
I
R
L
z
=
∑
j
=
0
z
ε
j
R
ν
​
R
L
D
γ
α
,
β
a
ν
-
B
ξ
˙
0
j
+
∂
R
ν
∂
a
μ
a
˙
μ
+
∂
R
ν
∂
t
ξ
0
j
​
R
L
D
γ
α
,
β
a
ν
+
R
ν
ξ
0
j
d
d
t
​
R
L
D
γ
α
,
β
a
ν
-
∂
B
∂
a
μ
a
˙
μ
+
∂
B
∂
t
ξ
0
j
+
R
ν
​
R
L
D
γ
α
,
β
ξ
ν
j
-
a
˙
ν
ξ
0
j
+
ξ
ν
j
-
a
˙
ν
ξ
0
j
​
C
D
1
-
γ
β
,
α
R
ν
-
R
ν
Γ
1
-
α
γ
d
d
t
t
-
t
1
-
α
a
ν
t
1
ξ
0
j
t
1
,
a
μ
t
1
+
R
ν
Γ
1
-
β
1
-
γ
d
d
t
t
2
-
t
-
β
a
ν
t
2
ξ
0
j
t
2
,
a
μ
t
2
+
G
˙
j
=
∑
j
=
0
z
ε
j
-
∂
R
ν
∂
a
μ
​
R
L
D
γ
α
,
β
a
ν
-
​
C
D
1
-
γ
β
,
α
R
μ
-
∂
B
∂
a
μ
+
Λ
μ
ξ
μ
j
-
a
˙
μ
ξ
0
j
+
Q
μ
ξ
μ
j
-
1
-
a
˙
μ
ξ
0
j
-
1
=
∑
j
=
0
z
ε
j
-
ε
Q
μ
ξ
μ
j
-
a
˙
μ
ξ
0
j
+
Q
μ
ξ
μ
j
-
1
-
a
˙
μ
ξ
0
j
-
1
=
-
ε
z
+
1
Q
μ
ξ
μ
z
-
a
˙
μ
ξ
0
z
.
This proof is completed.Because the generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative has another form (37), we can present another form of the adiabatic invariant when the system is disturbed.Theorem 19.
For the disturbed generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative(64)∂Rν∂aμ​RLDγα,βaν-​RLD1-γβ,αRμ-∂B∂aμ+Λμ=εQμμ,ν=1,2,⋯,2n,if there exists a gauge function Gj (j=0,1,2,⋯,z, z is an integer) such that the infinitesimal generators ξ0j and ξνj satisfy(65)Rν​RLDγα,βξνj-a˙νξ0j+Rνddt​RLDγα,βaν+∂Rν∂t​RLDγα,βaν-∂B∂tξ0j+∂Rν∂aμ​RLDγα,βaν-∂B∂aμξμj-γRνΓ1-αddtt-t1-αaνt1ξ0jt1,aμt1+Rν​RLDγα,βaν-Bξ˙0j+1-γRνΓ1-βddtt2-t-βaνt2ξ0jt2,aμt2+G˙j+Λμξμj-a˙μξ0j-Qμξμj-1-a˙μξ0j-1=0,where ξμj-1=ξ0j-1=0 when j=0, then a z-th order adiabatic invariant exists as follows:(66)IRLz1=∑j=0zεjRν​RLDγα,βaν-Bξ0j+∫t1tRν​RLDγα,βξνj-a˙νξ0j+ξνj-a˙νξ0j​RLD1-γβ,αRνdτ-∫t1tRνΓ1-αγddττ-t1-αaνt1ξ0jt1,aμt1-RνΓ1-β1-γddτt2-τ-βaνt2ξ0jt2,aμt2dτ+Gj.Whenγ=1, we can get special theorem from Theorem 19 as follows.Theorem 20.
For the disturbed generalized fractional Birkhoffian system(67)∂Rν∂aμ​t1RLDtαaν+​tRLDt2αRμ-∂B∂aμ+Λμ=εQμμ,ν=1,2,⋯,2n,if there exists a gauge function Gj (j=0,1,2,⋯,z, z is an integer) such that the infinitesimal generators ξ0j and ξνj satisfy(68)Rν​t1RLDtαξνj-a˙νξ0j+Rνddt​t1RLDtαaν+∂Rν∂t​t1RLDtαaν-∂B∂tξ0j+∂Rν∂aμ​t1RLDtαaν-∂B∂aμξμj-RνΓ1-αddtt-t1-αaνt1ξ0jt1,aμt1+Rν​t1RLDtαaν-Bξ˙0j+G˙j+Λμξμj-a˙μξ0j-Qμξμj-1-a˙μξ0j-1=0,where ξμj-1=ξ0j-1=0 when j=0, then a z-th order adiabatic invariant exists as follows:(69)IRLz11=∑j=0zεjRν​t1RLDtαaν-Bξ0j+∫t1tRν​t1RLDταξνj-a˙νξ0j+ξνj-a˙νξ0j​τRLDt2αRνdτ-∫t1tRνΓ1-αddττ-t1-αaνt1ξ0jt1,aμt1dτ+Gj.Remark 21.
Whenaνt1=0ν=1,2,⋯,2n, Theorem 20 is consistent with the results in [39].Theorem 22.
For the disturbed generalized fractional Birkhoffian system with the combined Caputo fractional derivative(70)∂Rν∂aμ​CDγα,βaν-​RLD1-γβ,αRμ-∂B∂aμ+Λμ=εQμμ,ν=1,2,⋯,2n,if there exists a gauge function Gj (j=0,1,2,⋯,z, z is an integer) such that the infinitesimal generators ξ0j and ξνj satisfy(71)Rν​CDγα,βξνj-a˙νξ0j+Rνddt​CDγα,βaν+∂Rν∂t​CDγα,βaν-∂B∂tξ0j+∂Rν∂aμ​CDγα,βaν-∂B∂aμξμj-γRνΓ1-αt-t1-αa˙νt1ξ0jt1,aμt1+Rν​CDγα,βaν-Bξ˙0j+1-γRνΓ1-βt2-t-βa˙νt2ξ0jt2,aμt2+G˙j+Λμξμj-a˙μξ0j-Qμξμj-1-a˙μξ0j-1=0,where ξμj-1=ξ0j-1=0 when j=0, then a z-th order adiabatic invariant exists as follows:(72)ICz=∑j=0zεjRν​CDγα,βaν-Bξ0j+∫t1tRν​CDγα,βξνj-a˙νξ0j+ξνj-a˙νξ0j​RLD1-γβ,αRνdτ-∫t1tRνΓ1-αγτ-t1-αa˙νt1ξ0jt1,aμt1-RνΓ1-β1-γt2-τ-βa˙νt2ξ0jt2,aμt2dτ+Gj.Remark 23.
The generalized fractional Birkhoffian system with the combined Caputo fractional derivative has another form (38), so we can also present another form of the adiabatic invariant when the system is disturbed just like Theorem 19.
Whenβ=α, γ=1/2, we can get the adiabatic invariants for the disturbed generalized fractional Birkhoffian system with the Riesz-Riemann-Liouville fractional derivative and the Riesz-Caputo fractional derivative from Theorems 18 and 22, respectively.Theorem 24.
For the disturbed generalized fractional Birkhoffian system with the Riesz-Riemann-Liouville fractional derivative(73)∂Rν∂aμ​t1RDt2αaν-​t1RCDt2αRμ-∂B∂aμ+Λμ=εQμμ,ν=1,2,⋯,2n,if there exists a gauge function Gj (j=0,1,2,⋯,z, z is an integer) such that the infinitesimal generators ξ0j and ξνj satisfy(74)Rν​t1RDt2αξνj-a˙νξ0j+Rνddt​t1RDt2αaν+∂Rν∂t​t1RDt2αaν-∂B∂tξ0j+∂Rν∂aμ​t1RDt2αaν-∂B∂aμξμj+Rν2Γ1-αddtt-t2-αaνt2ξ0jt2,aμt2+Rν​t1RDt2αaν-Bξ˙0j+Λμξμj-a˙μξ0j-Rν2Γ1-αddtt-t1-αaνt1ξ0jt1,aμt1+G˙j-Qμξμj-1-a˙μξ0j-1=0,where ξμj-1=ξ0j-1=0 when j=0, then a z-th order adiabatic invariant exists as follows:(75)IRz=∑j=0zεjRν​t1RDt2αaν-Bξ0j+∫t1tRν​t1RDt2αξνj-a˙νξ0j+ξνj-a˙νξ0j​t1RCDt2αRνdτ+Gj+∫t1tRν2Γ1-αddττ-t2-αaνt2ξ0jt2,aμt2-τ-t1-αaνt1ξ0jt1,aμt1dτ.Theorem 25.
For the disturbed generalized fractional Birkhoffian system with the Riesz-Caputo fractional derivative(76)∂Rν∂aμ​t1RCDt2αaν-​t1RDt2αRμ-∂B∂aμ+Λμ=εQμμ,ν=1,2,⋯,2n,if there exists a gauge function Gj (j=0,1,2,⋯,z, z is an integer) such that the infinitesimal generators ξ0j and ξνj satisfy(77)
R
ν
​
t
1
R
C
D
t
2
α
ξ
ν
j
-
a
˙
ν
ξ
0
j
+
R
ν
d
d
t
​
t
1
R
C
D
t
2
α
a
ν
+
∂
R
ν
∂
t
​
t
1
R
C
D
t
2
α
a
ν
-
∂
B
∂
t
ξ
0
j
+
R
ν
​
t
1
R
C
D
t
2
α
a
ν
-
B
ξ
˙
0
j
+
R
ν
2
Γ
1
-
α
t
-
t
2
-
α
a
˙
ν
t
2
ξ
0
j
t
2
,
a
μ
t
2
-
t
-
t
1
-
α
a
˙
ν
t
1
ξ
0
j
t
1
,
a
μ
t
1
+
∂
R
ν
∂
a
μ
​
t
1
R
C
D
t
2
α
a
ν
-
∂
B
∂
a
μ
ξ
μ
j
+
Λ
μ
ξ
μ
j
-
a
˙
μ
ξ
0
j
-
Q
μ
ξ
μ
j
-
1
-
a
˙
μ
ξ
0
j
-
1
+
G
˙
j
=
0
,where ξμj-1=ξ0j-1=0 when j=0, then a z-th order adiabatic invariant exists as follows:(78)IRCz=∑j=0zεjRν​t1RCDt2αaν-Bξ0j+∫t1tRν​t1RCDt2αξνj-a˙νξ0j+ξνj-a˙νξ0j​t1RDt2αRνdτ+∫t1tRν2Γ1-ατ-t2-αa˙νt2ξ0jt2,aμt2-τ-t1-αa˙νt1ξ0jt1,aμt1dτ+Gj.Remark 26.
Whenα,β→1, since Γ0=∞, we can get the classical adiabatic invariant for the classical generalized Birkhoffian system [42].


## 6. Applications

Application 1.
The fractional Lotka biochemical oscillator model in terms of combined Riemann-Liouville fractional derivative has the form [17](79)12​RLDγα,βa2+12​CD1-γβ,αa2-α2-β2exp⁡a1=0,12​RLDγα,βa1+12​CD1-γβ,αa2-α2-β2exp⁡a1=0,where the Birkhoffian and Birkhoff functions are(80)B=α2a1-α1a2-β1exp⁡a2+β2exp⁡a1,R1=-12a2,R2=12a1.Formula (48) gives(81)-12a2​RLDγα,βξ10-a˙1ξ00+12a1​RLDγα,βξ20-a˙2ξ00+-12a2ddt​RLDγα,βa1+12a1×ddt​RLDγα,βa2ξ00-γ-12a21Γ1-αddtt-t1-αa1t1ξ00t1,aμt1-γ12a1×1Γ1-αddtt-t1-αa2t1ξ00t1,aμt1+1-γ-12a21Γ1-βddtt2-t-β×a1t2ξ00t2,aμt2+1-γ12a11Γ1-βddtt2-t-βa2t2ξ00t2,aμt2+-12a2​RLDγα,βa1+12a1​RLDγα,βa2-α2a1+α1a2+β1exp⁡a2-β2exp⁡a1ξ˙00+ξ1012​RLDγα,βa2-α2-β2exp⁡a1+ξ20-12​RLDγα,βa1+α1+β1exp⁡a2+G˙0=0,where(82)ddt​t1RLDtαaν=​t1RLDtαa˙ν+1Γ1-αddtt-t1-αaνt1,(83)ddt​tRLDt2βaν=​tRLDt2βa˙ν+1Γ1-βddtt2-t-βaνt2.Taking calculation, we have(84)ξ00=1,ξ10=ξ20=0,G0=0.It follows from Theorem 11 that(85)IRL0=-12a2​RLDγα,βa1+12a1​RLDγα,βa2-α2a1+α1a2+β1exp⁡a2-β2exp⁡a1+∫t1t12a2ddτ​RLDγα,βa1-12a1ddτ​RLDγα,βa2+12a˙1​CD1-γβ,αa2-12a˙2​CD1-γβ,αa1dτ.
Assume this system is disturbed byεQ1=εa2, εQ2=εa1; then from formula (61), we have(86)
-
1
2
a
2
​
R
L
D
γ
α
,
β
ξ
1
1
-
a
˙
1
ξ
0
1
+
1
2
a
1
​
R
L
D
γ
α
,
β
ξ
2
1
-
a
˙
2
ξ
0
1
+
-
1
2
a
2
d
d
t
​
R
L
D
γ
α
,
β
a
1
+
1
2
a
1
×
d
d
t
​
R
L
D
γ
α
,
β
a
2
ξ
0
1
-
γ
-
1
2
a
2
1
Γ
1
-
α
d
d
t
t
-
t
1
-
α
a
1
t
1
ξ
0
1
t
1
,
a
μ
t
1
-
γ
1
2
a
1
×
1
Γ
1
-
α
d
d
t
t
-
t
1
-
α
a
2
t
1
ξ
0
1
t
1
,
a
μ
t
1
+
1
-
γ
-
1
2
a
2
1
Γ
1
-
β
d
d
t
t
2
-
t
-
β
×
a
1
t
2
ξ
0
1
t
2
,
a
μ
t
2
+
1
-
γ
1
2
a
1
1
Γ
1
-
β
d
d
t
t
2
-
t
-
β
a
2
t
2
ξ
0
1
t
2
,
a
μ
t
2
+
-
1
2
a
2
​
R
L
D
γ
α
,
β
a
1
+
1
2
a
1
​
R
L
D
γ
α
,
β
a
2
-
α
2
a
1
+
α
1
a
2
+
β
1
exp
⁡
a
2
-
β
2
exp
⁡
a
1
ξ
˙
0
1
+
ξ
1
1
1
2
​
R
L
D
γ
α
,
β
a
2
-
α
2
-
β
2
exp
⁡
a
1
+
ξ
2
1
-
1
2
​
R
L
D
γ
α
,
β
a
1
+
α
1
+
β
1
exp
⁡
a
2
,
+
a
2
a
˙
1
+
a
1
a
˙
2
+
G
˙
1
=
0
.Solving formula (86), we get(87)ξ01=1,ξ11=ξ21=0,G1=-a1a2.From Theorem 18, we obtain(88)IRL1=-12a2​RLDγα,βa1+12a1​RLDγα,βa2-α2a1+α1a2+β1exp⁡a2-β2exp⁡a1+∫t1t12a2ddτ​RLDγα,βa1-12a1ddτ​RLDγα,βa2+12a˙1​CD1-γβ,αa2-12a˙2​CD1-γβ,αa1dτ+ε-12a2​RLDγα,βa1+12a1​RLDγα,βa2-α2a1+α1a2+β1exp⁡a2-β2exp⁡a1-a1a2+∫t1t12a2ddτ​RLDγα,βa1-12a1ddτ​RLDγα,βa2+12a˙1​CD1-γβ,αa2-12a˙2​CD1-γβ,αa1dτ.When α,β→1, we obtain the classical adiabatic invariant(89)I11=α2a1-α1a2-β1exp⁡a2+β2exp⁡a1+εα2a1-α1a2-β1exp⁡a2+β2exp⁡a1-a1a2.Application 2.
The fractional Whittaker model in terms of Riesz-Riemann-Liouville fractional derivative has the form [17](90)-12​t1RDt2αa1+12​t1RDt2αa4-12​t1RCDt2αa1-a4=0,-12​t1RDt2αa2+12​t1RDt2αa3-12​t1RCDt2αa2+12​t1RCDt2αa3-a1+a4=0,12​t1RDt2αa2+12​t1RCDt2αa2-a4=0,-12​t1RDt2αa4-12​t1RCDt2αa4+a3=0,where the Birkhoffian and Birkhoff functions are(91)B=122a1a4-a32-a42,R1=-12a2,R2=12a1-a4,R3=12a4,R4=12a2-a3.From formula (50), we have(92)
-
1
2
a
2
​
t
1
R
D
t
2
α
ξ
1
0
-
a
˙
1
ξ
0
0
+
1
2
a
1
-
a
4
​
t
1
R
D
t
2
α
ξ
2
0
-
a
˙
2
ξ
0
0
+
1
2
a
4
​
t
1
R
D
t
2
α
ξ
3
0
-
a
˙
3
ξ
0
0
+
1
2
a
2
-
a
3
​
t
1
R
D
t
2
α
ξ
4
0
-
a
˙
4
ξ
0
0
+
-
1
2
a
2
d
d
t
​
t
1
R
D
t
2
α
a
1
+
1
2
a
1
-
a
4
d
d
t
​
t
1
R
D
t
2
α
a
2
+
1
2
a
4
d
d
t
​
t
1
R
D
t
2
α
a
3
+
1
2
a
2
-
a
3
d
d
t
​
t
1
R
D
t
2
α
a
4
ξ
0
0
+
ξ
1
0
1
2
​
t
1
R
D
t
2
α
a
2
-
a
4
+
ξ
2
0
-
1
2
​
t
1
R
D
t
2
α
a
1
+
1
2
​
t
1
R
D
t
2
α
a
4
+
ξ
3
0
-
1
2
​
t
1
R
D
t
2
α
a
4
+
a
3
+
ξ
4
0
-
1
2
​
t
1
R
D
t
2
α
a
2
+
1
2
​
t
1
R
D
t
2
α
a
3
-
a
1
+
a
4
-
1
2
a
2
1
2
Γ
1
-
α
×
d
d
t
t
-
t
2
-
α
a
1
t
2
ξ
0
0
t
2
,
a
μ
t
2
+
1
2
a
1
-
a
4
1
2
Γ
1
-
α
d
d
t
t
-
t
2
-
α
a
2
t
2
×
ξ
0
0
t
2
,
a
μ
t
2
+
1
2
a
4
1
2
Γ
1
-
α
d
d
t
t
-
t
2
-
α
a
3
t
2
ξ
0
0
t
2
,
a
μ
t
2
+
1
2
a
2
-
a
3
×
1
2
Γ
1
-
α
d
d
t
t
-
t
2
-
α
a
4
t
2
ξ
0
0
t
2
,
a
μ
t
2
+
-
1
2
a
2
​
t
1
R
D
t
2
α
a
1
+
1
2
a
1
-
a
4
​
t
1
R
D
t
2
α
a
2
+
1
2
a
4
​
t
1
R
D
t
2
α
a
3
+
1
2
a
2
-
a
3
​
t
1
R
D
t
2
α
a
4
-
a
1
a
4
+
1
2
a
3
2
+
1
2
a
4
2
ξ
˙
0
0
+
1
2
a
2
1
2
Γ
1
-
α
×
d
d
t
t
-
t
1
-
α
a
1
t
1
ξ
0
0
t
1
,
a
μ
t
1
-
1
2
a
1
-
a
4
1
2
Γ
1
-
α
d
d
t
t
-
t
1
-
α
a
2
t
1
×
ξ
0
0
t
1
,
a
μ
t
1
-
1
2
a
4
1
2
Γ
1
-
α
d
d
t
t
-
t
1
-
α
a
ν
t
1
ξ
0
0
t
1
,
a
μ
t
1
-
1
2
a
2
-
a
3
×
1
2
Γ
1
-
α
d
d
t
t
-
t
1
-
α
a
ν
t
1
ξ
0
0
t
1
,
a
μ
t
1
+
G
˙
0
=
0
,where(93)ddt​t1RDt2αaν=​t1RDt2αa˙ν+12Γ1-αddtt-t1-αaνt1-t-t2-αaνt2.Taking calculation, we have(94)ξ00=1,ξ10=ξ20=ξ30=ξ40=0,G0=0.Theorem 13 gives(95)IR0=-12a2​t1RDt2αa1+12a1-a4​t1RDt2αa2+12a4​t1RDt2αa3+12a2-a3​t1RDt2αa4-122a1a4-a32-a42+∫t1t12a2ddτ​t1RDt2αa1-12a1-a4ddτ​t1RDt2αa2-12a4ddτ​t1RDt2αa3-12a2-a3ddτ​t1RDt2αa4+12a˙1​t1RCDt2αa2-12a˙2​t1RCDt2αa1-a4-12a˙3​t1RCDt2αa4-12a˙4​t1RCDt2αa2-a3dτ.
Assuming this system is disturbed byεQ1=0, εQ2=0, εQ3=εa4, εQ4=εa3, then formula (74) gives(96)
-
1
2
a
2
​
t
1
R
D
t
2
α
ξ
1
1
-
a
˙
1
ξ
0
1
+
1
2
a
1
-
a
4
​
t
1
R
D
t
2
α
ξ
2
1
-
a
˙
2
ξ
0
1
+
1
2
a
4
​
t
1
R
D
t
2
α
ξ
3
1
-
a
˙
3
ξ
0
1
+
1
2
a
2
-
a
3
​
t
1
R
D
t
2
α
ξ
4
1
-
a
˙
4
ξ
0
1
+
-
1
2
a
2
d
d
t
​
t
1
R
D
t
2
α
a
1
+
1
2
a
1
-
a
4
d
d
t
​
t
1
R
D
t
2
α
a
2
+
1
2
a
4
d
d
t
​
t
1
R
D
t
2
α
a
3
+
1
2
a
2
-
a
3
d
d
t
​
t
1
R
D
t
2
α
a
4
ξ
0
1
+
ξ
1
1
1
2
​
t
1
R
D
t
2
α
a
2
-
a
4
+
ξ
2
1
-
1
2
​
t
1
R
D
t
2
α
a
1
+
1
2
​
t
1
R
D
t
2
α
a
4
+
ξ
3
1
-
1
2
​
t
1
R
D
t
2
α
a
4
+
a
3
+
ξ
4
1
-
1
2
​
t
1
R
D
t
2
α
a
2
+
1
2
​
t
1
R
D
t
2
α
a
3
-
a
1
+
a
4
-
1
2
a
2
1
2
Γ
1
-
α
×
d
d
t
t
-
t
2
-
α
a
1
t
2
ξ
0
1
t
2
,
a
μ
t
2
+
1
2
a
1
-
a
4
1
2
Γ
1
-
α
d
d
t
t
-
t
2
-
α
a
2
t
2
×
ξ
0
1
t
2
,
a
μ
t
2
+
1
2
a
4
1
2
Γ
1
-
α
d
d
t
t
-
t
2
-
α
a
3
t
2
ξ
0
1
t
2
,
a
μ
t
2
+
1
2
a
2
-
a
3
×
1
2
Γ
1
-
α
d
d
t
t
-
t
2
-
α
a
4
t
2
ξ
0
1
t
2
,
a
μ
t
2
+
-
1
2
a
2
​
t
1
R
D
t
2
α
a
1
+
1
2
a
1
-
a
4
​
t
1
R
D
t
2
α
a
2
+
1
2
a
4
​
t
1
R
D
t
2
α
a
3
+
1
2
a
2
-
a
3
​
t
1
R
D
t
2
α
a
4
-
a
1
a
4
+
1
2
a
3
2
+
1
2
a
4
2
ξ
˙
0
1
+
1
2
a
2
1
2
Γ
1
-
α
×
d
d
t
t
-
t
1
-
α
a
1
t
1
ξ
0
1
t
1
,
a
μ
t
1
-
1
2
a
1
-
a
4
1
2
Γ
1
-
α
d
d
t
t
-
t
1
-
α
a
2
t
1
×
ξ
0
1
t
1
,
a
μ
t
1
-
1
2
a
4
1
2
Γ
1
-
α
d
d
t
t
-
t
1
-
α
a
ν
t
1
ξ
0
1
t
1
,
a
μ
t
1
-
1
2
a
2
-
a
3
×
1
2
Γ
1
-
α
d
d
t
t
-
t
1
-
α
a
ν
t
1
ξ
0
1
t
1
,
a
μ
t
1
+
a
4
a
˙
3
+
a
3
a
˙
4
+
G
˙
1
=
0
.Taking calculation, we have(97)ξ01=1,ξ11=ξ21=ξ31=ξ41=0,G1=-a3a4.From Theorem 24, we get(98)
I
R
1
=
-
1
2
a
2
​
t
1
R
D
t
2
α
a
1
+
1
2
a
1
-
a
4
t
1
R
D
t
2
α
a
2
+
1
2
a
4
​
t
1
R
D
t
2
α
a
3
+
1
2
a
2
-
a
3
​
t
1
R
D
t
2
α
a
4
-
1
2
2
a
1
a
4
-
a
3
2
-
a
4
2
+
∫
t
1
t
1
2
a
2
d
d
τ
​
t
1
R
D
t
2
α
a
1
-
1
2
a
1
-
a
4
d
d
τ
​
t
1
R
D
t
2
α
a
2
-
1
2
a
4
d
d
τ
​
t
1
R
D
t
2
α
a
3
-
1
2
×
a
2
-
a
3
d
d
τ
​
t
1
R
D
t
2
α
a
4
+
1
2
a
˙
1
​
t
1
R
C
D
t
2
α
a
2
-
1
2
a
˙
2
​
t
1
R
C
D
t
2
α
a
1
-
a
4
-
1
2
a
˙
3
​
t
1
R
C
D
t
2
α
a
4
-
1
2
a
˙
4
×
​
t
1
R
C
D
t
2
α
a
2
-
a
3
d
τ
+
ε
-
1
2
a
2
​
t
1
R
D
t
2
α
a
1
+
1
2
a
1
-
a
4
​
t
1
R
D
t
2
α
a
2
+
1
2
a
4
​
t
1
R
D
t
2
α
a
3
+
1
2
a
2
-
a
3
​
t
1
R
D
t
2
α
a
4
-
1
2
2
a
1
a
4
-
a
3
2
-
a
4
2
+
∫
t
1
t
1
2
a
2
d
d
τ
​
t
1
R
D
t
2
α
a
1
-
1
2
a
1
-
a
4
d
d
τ
​
t
1
R
D
t
2
α
a
2
-
1
2
a
4
d
d
τ
​
t
1
R
D
t
2
α
a
3
-
1
2
a
2
-
a
3
d
d
τ
​
t
1
R
D
t
2
α
a
4
+
1
2
a
˙
1
​
t
1
R
C
D
t
2
α
a
2
-
1
2
a
˙
2
​
t
1
R
C
D
t
2
α
a
1
-
a
4
-
1
2
a
˙
3
​
t
1
R
C
D
t
2
α
a
4
-
1
2
a
˙
4
​
t
1
R
C
D
t
2
α
a
2
-
a
3
d
τ
-
a
3
a
4
.
Whenα→1, we obtain the classical adiabatic invariant(99)I21=2a1a4-a32-a42+ε2a1a4-a32-a42-a3a4.Application 3.
The fractional Hojman–Urrutia model in terms of combined Caputo derivative has the form [17](100)​RLD1-γβ,αa2+​RLD1-γβ,αa3=0,​CDγα,βa1-a3=0,​CDγα,βa1-​RLD1-γβ,αa4-a3+a2=0,​CDγα,βa3+a4=0,where the Birkhoffian and Birkhoff functions are(101)B=12a32+2a2a3-a42,R1=a2+a3,R2=0,R3=a4,R4=0.From formula (49), we have(102)a2+a3​CDγα,βξ10-a˙1ξ00+a4​CDγα,βξ30-a˙3ξ00+a2+a3ddt​CDγα,βa1+a4ddt​CDγα,βa3ξ00+ξ20​CDγα,βa1-a3+ξ30​CDγα,βa1-a3-a2+ξ40×​CDγα,βa3+a4-a2+a3γΓ1-αt-t1-αa˙1t1ξ00t1,aμt1-a4×γΓ1-αt-t1-αa˙3t1ξ00t1,aμt1+a2+a3​CDγα,βa1+a4​CDγα,βa3-12a32-a2a3+12a42ξ˙00+a2+a31-γΓ1-βt2-t-βa˙1t2×ξ00t2,aμt2+a41-γΓ1-βt2-t-βa˙3t2ξ00t2,aμt2+G˙0=0,where(103)ddt​t1CDtαaν=​t1CDtαa˙ν+1Γ1-αt-t1-αa˙νt1,(104)ddt​tCDt2βaν=​tCDt2βa˙ν+1Γ1-βt2-t-βa˙νt2.Taking calculation, we have(105)ξ00=1,ξ10=0,ξ20=1,ξ30=0,ξ40=1,G0=0.It follows from Theorem 12 that(106)IC0=a2+a3​CDγα,βa1+a4​CDγα,βa3-12a32-a2a3+12a42-∫t1ta2+a3×ddτ​CDγα,βa1+a4ddτ​CDγα,βa3+a˙1​RLD1-γβ,αa2+a3+a˙3​RLD1-γβ,αa4dτ.
Assume this system is disturbed byεQ1=εa3, ,εQ2=0, εQ3=εa1, εQ4=0.
From formula (71), we have(107)a2+a3​CDγα,βξ11-a˙1ξ01+a4​CDγα,βξ31-a˙3ξ01+a2+a3ddt​CDγα,βa1+a4×ddt​CDγα,βa3ξ01+ξ21​CDγα,βa1-a3+ξ31​CDγα,βa1-a3-a2+ξ41​CDγα,βa3+a4-a2+a3γΓ1-αt-t1-αa˙1t1ξ01t1,aμt1-a4γΓ1-αt-t1-αa˙3t1×ξ01t1,aμt1+a2+a3​CDγα,βa1+a4​CDγα,βa3-12a32-a2a3+12a42ξ˙01+a2+a31-γΓ1-βt2-t-βa˙1t2ξ01t2,aμt2+a41-γΓ1-βt2-t-βa˙3t2×ξ01t2,aμt2+a3a˙1+a1a˙3+G˙1=0.From formula (107), we obtain(108)ξ01=1,ξ11=ξ21=ξ31=ξ41=0,G1=-a1a3.From Theorem 22, we achieve(109)IC1=a2+a3​CDγα,βa1+a4​CDγα,βa3-12a32-a2a3+12a42-∫t1ta2+a3×ddτ​CDγα,βa1+a4ddτ​CDγα,βa3+a˙1​RLD1-γβ,αa2+a3+a˙3​RLD1-γβ,αa4dτ+εa2+a3​CDγα,βa1+a4​CDγα,βa3-12a32-a2a3+12a42-∫t1ta2+a3ddτ​CDγα,βa1+a4ddτ​CDγα,βa3+a˙1​RLD1-γβ,αa2+a3+a˙3​RLD1-γβ,αa4dτ-a1a3.
Whenα,β→1, we obtain the classical adiabatic invariant(110)I31=12a32+a2a3-12a42+ε12a32+a2a3-12a42-a1a3.Application 4.
The fractional Hénon–Heiles model in terms of Riesz-Caputo derivative has the form [17](111)​t1RCDt2αa3+a1+2a1a2=0,​t1RCDt2αa4+a2-a22+a12=0,​t1RDt2αa1-a3=0,​t1RDt2αa2-a4=0,where the Birkhoffian and Birkhoff functions are(112)B=12a12+a22+a32+a42+2a2a12-23a23,R1=R2=0,R3=-a1,R4=-a2.Formula (51) gives(113)-a1​t1RCDt2αξ30-a˙3ξ00-a2​t1RCDt2αξ40-a˙4ξ00+-a1ddt​t1RCDt2αa3-a2ddt​t1RCDt2αa4ξ00-a1×12Γ1-αt-t2-αa˙3t2ξ00t2,aμt2-t-t1-αa˙3t1ξ00t1,aμt1-12Γ1-α×a2t-t2-αa˙4t2ξ00t2,aμt2-t-t1-αa˙4t1ξ00t1,aμt1-ξ10​t1RCDt2αa3+a1+2a1a2-ξ20​t1RCDt2αa4+a2+a12-a22-ξ30a3-ξ40a4+-a1​t1RCDt2αa3-a2​t1RCDt2αa4-12a12+a22+a32+a42+2a2a12-23a23ξ˙00+G˙0=0,where(114)ddt​t1RCDt2αaν=​t1RCDt2αa˙ν+12Γ1-αt-t1-αa˙νt1-t-t2-αa˙νt2.From formula (113), we have(115)ξ00=ξ10=ξ20=1,ξ30=ξ40=0,G0=0.From Theorem 14, we obtain(116)IRC0=-a1​t1RCDt2αa3-a2​t1RCDt2αa4-12a12+a22+a32+a42+2a2a12-23a23+∫t1ta1ddτ​t1RCDt2αa3+a2ddτ​t1RCDt2αa4+a˙3​t1RDt2αa1+a˙4​t1RDt2αa2dτ.
Assume this system is disturbed byεQ1=εQ2=0, εQ3=εa4, εQ4=εa3.
From formula (77), we have(117)-a1​t1RCDt2αξ31-a˙3ξ01-a2​t1RCDt2αξ41-a˙4ξ01+-a1ddt​t1RCDt2αa3-a2ddt​t1RCDt2αa4ξ01-a1×12Γ1-αt-t2-αa˙3t2ξ01t2,aμt2-t-t1-αa˙3t1ξ01t1,aμt1-12Γ1-α×a2t-t2-αa˙4t2ξ01t2,aμt2-t-t1-αa˙4t1ξ01t1,aμt1-ξ11​t1RCDt2αa3+a1+2a1a2-ξ21​t1RCDt2αa4+a2+a12-a22-ξ31a3-ξ41a4+-a1​t1RCDt2αa3-a2​t1RCDt2αa4-12a12+a22+a32+a42+2a2a12-23a23ξ˙01+a4a˙3+a3a˙4+G˙1=0.Calculating formula (117), we obtain(118)ξ01=1,ξ11=ξ21=ξ31=ξ41=0,G1=-a3a4.From Theorem 25, we get(119)IRC1=-a1​t1RCDt2αa3-a2​t1RCDt2αa4-12a12+a22+a32+a42+2a2a12-23a23+∫t1ta1ddτ​t1RCDt2αa3+a2ddτ​t1RCDt2αa4+a˙3​t1RDt2αa1+a˙4​t1RDt2αa2dτ+ε-a1​t1RCDt2αa3-a2×​t1RCDt2αa4-12a12+a22+a32+a42+2a2a12-23a23+∫t1ta1ddτ​t1RCDt2αa3+a2ddτ​t1RCDt2αa4+a˙3​t1RDt2αa1+a˙4​t1RDt2αa2dτ-a3a4.When α→1, we obtain the classical adiabatic invariant(120)I41=a12+a22+a32+a42+2a2a12-23a23+εa12+a22+a32+a42+2a2a12-23a23-a3a4.


## 7. Conclusions

Differential equations of motion, Noether symmetry, conserved quantity, perturbation to Noether symmetry, and adiabatic invariants are investigated for the generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative, the combined Caputo fractional derivative, the Riesz-Riemann-Liouville fractional derivative, and the Riesz-Caputo fractional derivative, respectively. The classical conserved quantity and the classical adiabatic invariant are all discussed as special cases.The fractional Lotka biochemical oscillator model with the combined Riemann-Liouville fractional derivative, the fractional Whittaker model with the Riesz-Riemann-Liouville fractional derivative, the fractional Hojman–Urrutia model with the combined Caputo fractional derivative, and the fractional Hénon–Heiles model with the Riesz-Caputo fractional derivative are discussed, and their first order adiabatic invariants are obtained.The main results obtained in this paper can help understand the internal properties and dynamical behaviors of the systems. Since generalized fractional Birkhoffian mechanics plays an important role in many fields of modern science and engineering, more research is worth doing.



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