Chuan-Jing Song , Shi-Lei Shen

School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China

Keywords: Generalized fractional operator Birkhoffian system Noether symmetry Perturbation to Noether symmetry

ABSTRACT Compared with the Hamiltonian mechanics and the Lagrangian mechanics, the Birkhoffian mechanics is more general.The Birkhoffian mechanics is discussed on the basis of the generalized fractional operators, which are proposed recently.Therefore, differential equations of motion within generalized fractional op- erators are established.Then, in order to find the solutions to the differential equations, Noether sym- metry, conserved quantity, perturbation to Noether symmetry and adiabatic invariant are investigated.In the end, two applications are given to illustrate the methods and results.

Compared with the integer order model, the fractional order model can more accurately describe the dynamic behavior of the complex systems.Under the long-term exploration, many fractional operators have been put forward.The most widely used fractional operators are the Riemann-Liouville fractional operator, the Caputo fractional operator, the Riesz-Riemann-Liouville fractional operator and the Riesz-Caputo fractional operator.In 2010, Agrawal [1] pro- posed a new fractional operator, called generalized fractional oper- ator, which can be reduced to the above four operators in special cases.

Fractional calculus has practical applications in various fields [2–7] .In 1996, Riewe [8–9] first applied the fractional calculus to the study of the dynamics of the nonconservative mechanical systems, proposed and preliminarily studied the fractional varia- tional problems.This motivated several scholars to investigate frac- tional calculus of variations further, and those results include the fractional Lagrangian formulations and the fractional Hamiltonian formulations [10–17] .As one of the important research directions of the modern analytical mechanics, the Birkhoffian mechanics is more general than the Hamiltonian mechanics and the Lagrangian mechanics [18–19] .In recent years, scholars have also applied frac- tional calculus to the Birkhoffian system, and achieved a lot of re- search results [20–21] .Particularly, Zhang et al.[22] derived the generalized Birkhoffequation by using the generalized fractional operators proposed by Agrawal.

Noether symmetry is one of the useful methods to find solu- tions to the differential equations of motion, i.e., Noether sym- metry and conserved quantity are useful to reveal the inherent physical properties of the dynamic systems [23–24] .For example, for the fractional Birkhoffian mechanics, there are several achieve- ments on Noether theorem [25–30] .However, based on the def- inition of the classical conserved quantity, the study of Noether symmetry and conserved quantity of the Birkhoffian system within generalized fractional operators have not been involved.Therefore, this paper intends to study Noether symmetry for the Birkhoffian system in terms of generalized fractional operators and derive the corresponding conserved quantities.

When the system is disturbed by small disturbance, the sym- metry and conservation may change.The change of symmetry and its invariants are closely related to the integrability of the me- chanical system, so the research of this aspect is also of great sig- nificance.Some achievements have been made on the perturba- tion to Noether symmetry, and the corresponding adiabatic invari- ants have been obtained for fractional Birkhoffian systems [31–33] .Hence, the perturbation to Noether symmetry and adiabatic invari- ants of the Birkhoffian system within generalized fractional opera- tors are also studied here.

Generalized fractional operatorsK,AandBare introduced by Agrawal [1] .Here we only list their definitions and properties briefly.

The definition of the operatorsK,AandBare defined as

whereais a parameter set,mandωare two real numbers,κα(t,τ)is a kernel which may depend on a parameterα,nis an integer.

Remark 1 .Letκα(t,τ)=(t?τ)α?1/Γ(α), whenM=M1 =,M=M2 =andM=M3 =, the operatorAreduces to the left Riemann- Liouville fractional operator, the right Riemann-Liouville fractional operator and the Riesz-Riemann-Liouville fractional operator, respectively.Similarly, the operatorBreduces to the left Caputo fractional operator, the right Caputo fractional operator and the Riesz-Caputo fractional operator, respectively.The integration by parts formulae of operatorsK,AandBare

whereM*=,n?1<α

It is noted that in the following text we setn= 1 , i.e., 0<α<1 .

Differential equation of motion within operator A

Integration

is called Pfaff action within operatorA, whereaA=,BA(t,aA)is called Birkhoffian,RAν(t,aA)is called Birkhofffunctions,ν= 1,2,···,2n.

The isochronous variational principle

with the commutative relationship [22]

and the boundary condition

is called Pfaff-Birkhoffprinciple within operatorA, whereδmeans the isochronous variation.

From Eq.(8) , we have

whereμ,ν= 1,2,···,2n, and

Substituting Eq.(12) into Eq.(11) , we have

It follows from the independence ofδaμAand the arbitrariness of the interval [t1,t2 ] that

Equation (14) is called Birkhoffequation within operatorA.

Differential equation of motion within operator B

Integration

is called Pfaff action within operatorB, whereaB=,BB(t,aB)is called Birkhoffian,RBν(t,aB)are called Birkhofffunctions,ν= 1,2,···,2n.

The isochronous variational principle

with the commutative relationship [22]

and the boundary condition

is called Pfaff-Birkhoffprinciple within operatorB.

From Eqs.(6) , (16) , (17) , (18) , we have

Therefore,

Equation (20) is called Birkhoffequation within operatorB.

Remark 2.In fact, Birkhoffequation within operatorAand Birkhoffequation within operatorBhave been studied in Ref.[22] .Birkhoffequation within operatorB(Eq.(20)) obtained here is con- sistent with that in Ref.[22] .However, Birkhoffequation within operatorA(Eq.(14)) obtained here is different from that in Ref.[22] , becauseRAμ·in Eq.(12) is not equal to zero.

Remark 3.Letκα(t,τ)=(t?τ)α?1/Γ(α), from Eq.(14), Birkhoffequations within left Riemann-Liouville fractional opera- tor, right Riemann-Liouville fractional operator and Riesz-Riemann- Liouville fractional operator can be obtained by lettingM=M1,M=M2 andM=M3 , respectively.Similarly, from Eq.(20) , Birkhoff equations within left Caputo fractional operator, right Caputo frac- tional operator and Riesz-Caputo fractional operator can also be obtained.Furthermore, if letα→ 1 , then all of them reduce to the classical Birkhoffequation, which can be found in Ref.[24] .In ad- dition, Birkhoffequation obtained from Eq.(20) whenM=M3 is consistent with the result in Ref.[20] .

Noether symmetry means the invariance of the Pfaffaction.And conserved quantity can be deduced from Noether symmetry.Be- fore we present conserved quantities of the systems (Eqs.(14) and (20)), we first give its definition.

Definition 1.A quantityIis called a conserved quantity if and only if the condition dI/dt= 0 holds.

We begin with the change of the Pfaffaction within operatorA.

Noether symmetry within operator A

Assuming the Pfaffaction within operatorA(Eq.(7)) is changed under the following infinitesimal transformations

whose expansions are

whereθAis an infinitesimal parameter,andare called in- finitesimal generators within operatorA.

Then denoting the linear part ofasΔJAand neglecting the higher order ofθA, we have

where

It follows from Noether symmetry (ΔJA= 0) that

Equation (25) is called Noether identity within operatorA.

Equation (26) is called Noether-quasi identity within operatorA.

Therefore, we have:

Theorem 1For the Birkhoffian system within operatorA(E q.(14)) , if the infinitesimal generatorsandsatisfy the Noether identity (E q.(25)) , then there exists a conserved quan- tity

Proof .From Eqs.(14) and (25) , we have dIA0/dt= 0 .

Theorem 2For the Birkhoffian system with operatorA(Eq.(14)), if there exists a gauge functionsuch that the in- finitesimal generatorsandsatisfy the Noether-quasi iden- tity (Eq.(26)), then there exists a conserved quantity

Proof .From Eqs.(14) and (26) , we have dIAG0/dt= 0 .

Noether symmetry within operator B

Assuming the Pfaffaction within operatorB(Eq.(15)) is changed under the following infinitesimal transformations

whose expansions are

whereθBis an infinitesimal parameter,andare called in- finitesimal generators.

Then denoting the linear part ofasand neglecting the higher order ofθB, we have

where

It follows from Noether symmetry (ΔJB= 0) that

Equation (33) is called Noether identity within operatorB.

If letΔJB= ?is called gauge function, then from Eq.(31) , we get

Eq.(34) is called Noether-quasi identity within operatorB.

Therefore, we have

Theorem 3For the Birkhoffian system within operatorB(Eq.(20)), if the infinitesimal generatorsandsatisfy the Noether identity (Eq.(33)), then there exists a conserved quan- tity

Proof .From Eqs.(20) and (33) , we have dIB0/dt= 0 .

Theorem 4For the Birkhoffian system with operatorB(Eq.(20)), if there exists a gauge functionsuch that the in- finitesimal generatorsandsatisfy the Noether-quasi iden- tity (Eq.(34)), then there exists a conserved quantity

Proof .From Eqs.(20) and (34) , we have dIBG 0/dt= 0 .

Remark 4.Letκα(t,τ)=(t?τ)α?1/Γ(α), from Eqs.(25) , (26) and (28) , Noether identities, Noether-quasi identities and con- served quantities within left Riemann-Liouville fractional opera- tor, right Riemann-Liouville fractional operator and Riesz-Riemann- Liouville fractional operator can be obtained by lettingM=M1 ,M=M2 andM=M3 , respectively.Similarly, from Eqs.(33) , (34) and (36) ,Noether identities,Noether-quasi identities and con- served quantities within left Caputo fractional operator, right Ca- puto fractional operator and Riesz-Caputo fractional operator can also be obtained.Furthermore, if letα→ 1 , then all of them reduce to the classical Noether identity, Noether-quasi identity and conserved quantity, which can be found in Ref.[24] .In ad- dition, Noether identity and conserved quantity obtained from Eqs.(33) and (36) whenM=M3 are consistent with the results in Ref.[26] .

When the systems Eqs.(14) and ((20)) are disturbed by small forces, the conserved quantities may also change.Before we study the perturbation to Noether symmetry and adiabatic invariant, we first give the definition of the adiabatic invariant.

Definition 2.A quantityIzis called an adiabatic invariant ifIzcontains a parameterε, whose highest power isz, and also satisfies that dIz/dtis in proportion toεz+1 .

Supposing the Birkhoffian system with operatorA(Eq.(14)) is disturbed as

the gauge functionGA, the infinitesimal generatorsξA0andξAμare disturbed as

then we have:

Theorem 5For the disturbed Birkhoffian system within opera- torA(Eq.(37)), if there exists a gauge functionGsAsuch that the infinitesimal generatorsandsatisfy

Proof .From Eqs.(37) and (39) , we have=

Similarly, for the disturbed Birkhoffian system within operatorB

if the gauge functionGB, the infinitesimal generatorsξB0andξBμare disturbed as

then we have

Theorem 6For the disturbed Birkhoffian system with operatorB(Eq.(41)), if there exists a gauge functionGsBsuch that the in- finitesimal generatorsandsatisfy

Proof .From Eqs.(41) and (43) , we have

Remark 5.Letκα(t,τ)=(t?τ)α?1/Γ(α), from Theorem 5 and Theorem 6, adiabatic invariants within left Riemann-Liouville frac- tional operator, left Caputo fractional operator, right Riemann- Liouville fractional operator, right Caputo fractional operator, Riesz- Riemann-Liouville fractional operator and Riesz-Caputo fractional operator can be achieved by lettingM=M1 ,M=M2 andM=M3 , respectively.Furthermore, if letα→ 1 , then all of them reduce to the classical adiabatic invariant.

Remark 6.In Theorem 5 and Theorem 6, if letz= 0 , then con- served quantities in Theorem 2 and Theorem 4 can be obtained, respectively.

Remark 7.Because the Birkhoffian system is more general than the Hamiltonian system and the Lagrangian system, differential equations of motion, conserved quantities and adiabatic invariants for the Hamiltonian system and the Lagrangian system can be ob- tained through special transformations from Eq.(14) , Eq.(20) and Theorem 1 –Theorem 6.Particularly, differential equations of motion for the Lagrangian system obtained from Eq.(14) and Eq.(20) are consistent with the results in Ref.[1] .

Finally, we give two applications to illustrate the results and methods.

The differential equations of motion, Noether symmetry and conserved quantities, perturbation to Noether symmetry and adi- abatic invariants are investigated in terms of operator A and op- erator B, respectively.The Lotka biochemical oscillator model and the Hénon–Heiles model are presented below.Without loss of gen- erality, we study the former model using operator A and study the latter one using operator B.

Application 1 The Birkhoffian and Birkhofffunctions of the Lotka biochemical oscillator model have the forms

whereα1,α2,β1andβ2are constants.Try to find its conserved quantity and adiabatic invariant within operatorA.

Equation (14) gives the differential equation of motion within operatorAfor this model as

satisfy the Noether-quasi identity (Eq.(26)).Therefore, using The- orem 2, we get

Specially, letκα(t,τ)=M=M1 (orM=M2 orM=M3), andα→ 1 , we have

If the system is disturbed by ?εAWA1(t,aA)=and ?εAWA2(t,aA)=then we can verify that

is a solution to Eq.(39) .Therefore, using Theorem 5, we get

Furthermore, we can get higher order adiabatic invariant.

Application 2The Birkhoffian and Birkhoffequations of the Hénon–Heiles model are

Try to find its conserved quantity and adiabatic invariant within operatorB.

Firstly, from Eq.(20) , Birkhoffequation within operatorBcan be obtained as

is a solution to the Noether-quasi identity (Eq.(34)).Therefore, from Theorem 4, we get

Specially, letκα(t,τ)=M=M1 (orM=M2 orM=M3) andα→ 1 , we have

If the system is disturbed by ?εBWB1(t,aB)= 0 , ?εBWB2(t,aB)= 0 , ?εBWB3(t,aB)=?εBWB4(t,aB)=then we can verify that

satisfy Eq.(43) .Therefore, from Theorem 6, we have

Furthermore, we can get higher order adiabatic invariant.

Based on generalized fractional operators, Birkhoffequations, conserved quantities and adiabatic invariants are obtained.The Birkhoffequation within operatorA(Eq.(14)), the Birkhoffequa- tion within operatorB(Eq.(20)), the Noether theorems within op- eratorA(Theorem 1 –Theorem 2), the Noether theorems within operatorB(Theorem 3 –Theorem 4), the perturbation to Noether symmetry and adiabatic invariants within generalized fractional operators (Theorem 5 –Theorem 6), all of them are new work.

There are some other methods to find solutions to differential equations of motion except for Noether symmetry method, such as Lie symmetry method and Mei symmetry method.Besides, making use of MATLAB to simulate the applications to illustrate the results is also significant.Therefore, the discussions of different methods and the appliance of the MATLAB are likely to be the next work.

Declaration of Competing Interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foun- dation of China (Grants 11802193 and 11972241), the Natural Sci- ence Foundation of Jiangsu Province (Grant BK20191454) and the Young Scientific and Technological Talents Promotion Project of Suzhou Association for Science and Technology.