JIA Yundie，ZHANG Yi

1.College of Mathematical Sciences，Suzhou University of Science and Technology，Suzhou 215009，P.R.China；2.College of Civil Engineering，Suzhou University of Science and Technology，Suzhou 215011，P.R.China

Abstract: In order to investigate the dynamic behavior of non-conservative systems，the Lie symmetries and conserved quantities of fractional Birkhoffian dynamics based on quasi-fractional dynamics model are proposed and studied.The quasi-fractional dynamics model here refers to the variational problem based on the definition of Riemann-Liouville fractional integral（RLFI），the variational problem based on the definition of extended exponentially fractional integral（EEFI），and the variational problem based on the definition of fractional integral extended by periodic laws（FIEPL）. First，the fractional Pfaff-Birkhoff principles based on quasi-fractional dynamics models are established，and the corresponding Birkhoff’s equations and the determining equations of Lie symmetry are obtained.Second，for fractional Birkhoffian systems based on quasi-fractional models，the conditions and forms of conserved quantities are given，and Lie symmetry theorems are proved. The Pfaff-Birkhoff principles，Birkhoff’s equations and Lie symmetry theorems of quasi-fractional Birkhoffian systems and classical Birkhoffian systems are special cases of this article.Finally，some examples are given.

Key words：quasi-fractional dynamics model；Lie symmetry；conserved quantity；fractional Birkhoffian system；Riemann-Liouville derivative

0 Introduction

Fractional calculus has been widely used in var⁃ious fields of engineering and science［1-3］in recent years，because it is more accurate to describe the dy⁃namic behavior and physical process of complex sys⁃tems than the integral order model. They are also used in many dynamical systems in interdisciplinary fields，such as electromechanical systems［4］，bio⁃medical systems［5］， mechanical systems［6-7］and mathematical systems［8］. In 1996—1997，Riewe［9-10］introduced fractional calculus into dynamics model⁃ing of non-conservative systems and established frac⁃tional Hamilton equations and fractional Lagrange equations. For fractional variational problems，Agrawal et al.，Atanacković et al.，and Torres et al.have made a more in-depth study［11-16］.

As a natural extension of Hamiltonian mechan⁃ics，Birkhoffian mechanics are more general than Hamiltonian mechanics. In 1927，Birkhoff intro⁃duced a more general equation than Hamilton equa⁃tion［17］，which was named Birkhoff’s equation by Santilli［18］in 1983. After that，Santilli studied the Birkhoff’s equation and its transformation theory. In 1996，Mei et al. set up the theoretical framework of Birkhoffian mechanics［19］and put forward its symme⁃try theory［20-22］. Due to the generality of Birkhoffian system，many scholars have introduced it into non⁃linear dynamical systems［23-24］，constrained dynami⁃cal systems［25］and quantum systems［26］. Moreover，fractional calculus has been introduced into Birkhof⁃fian system by many scholars and the corresponding symmetries have been obtained［27-33］.

In 1979，Lutzky［34］introduced the Lie method into the dynamical system，studied the invariant properties of second-order dynamical system under infinitesimal transformations of time，coordinates and velocity，and established the relation between Lie symmetry and Noether conserved quantities.Since then，Prince and Eliezer［35］extended Lutzky’s research to the classical Kepler problem，and ob⁃tained the corresponding Lie symmetry. In 1994，Zhao［36］extended Lie symmetry to non-conservative dynamical systems. Lie symmetries and conserved quantities have made important progress in con⁃strained mechanical systems，nonholonomic sys⁃tems and Birkhoffian system on time scales in recent years［37-45］.

In 2005，El-Nabulsi proposed a kind of nonconservative dynamics models based on the defini⁃tion of Riemann-Liouville fractional integration（RLFI）［46］，and further put forward the dynamics models based on the definition of extended exponen⁃tially fractional integral（EEFI）［47］and based on the definition of fractional integral extended by periodic laws（FIEPL）［48］. These three models are known as quasi-fractional dynamics models or El-Nabulsi models.The dynamic equations based on these mod⁃els are simple and similar to the classical conserva⁃tive system’s Lagrange equation. Its novelty lies in that the generalized fractional order external force corresponding to the dissipative force appears in the equations instead of the fractional order derivative，and the fractional order time integration only needs one parameter，while any number of fractional order parameters will appear in other models. After that，El-Nabulsi et al. and Frederico et al. established the equations of motion of the quasi-fractional dynamics models，and extended them to nonholonomic，holo⁃nomic and dissipative dynamical system［49-50］. In re⁃cent years，Zhang and his colleagues obtained the differential equations and symmetry theory of these models in Lagrangian system，Birkhoffian system and Hamiltonian system［51-57］. However，these stud⁃ies are limited to the integer order，and the fraction⁃al order is more accurate than the integer order in de⁃scribing the mechanical and physical behavior of complex systems. Therefore，in this article，we fur⁃ther consider the Lie symmetry for fractional Birk⁃hoffian system with Riemann-Liouville derivatives based on quasi-fractional dynamics models.

1 Preliminaries

We list some basic properties and definitions of fractional derivatives. For more detailed discussion and proof，please refer to Ref.［3］.

If functions f (t) and g(t) are integrable and continuous on interval[a，b]，the definitions of Rie⁃mann-Liouville derivative of order α is

where Γ(∗) is the Euler-Gamma function and 0 ≤α ＜1.

For 0 ＜α ＜1，β ＞0，we have

If f (t) vanishes at t=a and 0 ＜α ＜1，we have

2 Fractional Birkhoff’s Equations Under Quasi⁃fractional Dynam⁃ics Models

2.1 Fractional Birkhoff’s equations based on RLFI

According to Ref.［46］，the fractional Pfaff ac⁃tion based on RLFI can be expressed as

where aμare Birkhoff’s variables；B=B(τ，aν) is the Birkhoffian；Rμ=Rμ(τ，aν) are Birkhoff’s func⁃tions，0 ≤β ＜1；t is the observer time；τ is the in⁃trinsic time；τ ≠t，functions B and Rμare C2func⁃tions of their variables.

The fractional Pfaff-Birkhoff variational princi⁃ple based on RLFI can be expressed as

It is under the commutative conditions

and given terminal conditions

Using integration by parts to calculate the sec⁃ond term in Eq.（8）and considering Eq.（10），we have

Substituting Eq.（11）into principle in Eq.（8），we get

Since Eq.（12）is true for any integral interval[a，b] and the independence of δaμ，by using the fun⁃damental lemma［58］of the calculus of variations，we have

Eq.（13）is the bundle of fractional Birkhoff’s equations based on RLFI.

2.2 Fractional Birkhoff’s equations based on EEFI

According to Ref.［47］，the fractional Pfaff ac⁃tion based on EEFI can be expressed as

The fractional Pfaff-Birkhoff variational princi⁃ple based on EEFI can be expressed as

under the commutative conditions

and given terminal conditions

Similar to Eq.（12），from Eq.（17），we can easily get

On account of the independence of δaμand the lemma of the calculus of variations，from Eq.（18），we obtain

Eq.（19）is the bundle of fractional Birkhoff’s equations based on EEFI.

2.3 Fractional Birkhoff’s equations based on FIEPL

According to Ref.［48］，the fractional Pfaff ac⁃tion based on FIEPL can be expressed as

The fractional Pfaff-Birkhoff variational princi⁃ple based on FIEPL can be expressed as

under the commutative conditions

and given terminal conditions

Similar to Eq.（12），from Eq.（20），we can easily get

On account of the independence of δaμand the lemma of the calculus of variations，from Eq.（24），we obtain

Eq.（25）is the bundle of fractional Birkhoff’s equations based on FIEPL.

If β →1，Eqs.（13，19，25）become Birkhoff’s equations based on quasi-fractional models. If β →1，α →1，Eqs.（13，19，25）become the stan⁃dard Birkhoff’s equations［19］.

3 Lie Symmetry for Fractional Birkhoffian System

3.1 Lie symmetry based on RLFI

The infinitesimal transformation of the group is introduced as

and the extension formulas are

where ξ0and ξμare the infinitesimal generators；and ε is an infinitesimal parameter.

Under infinitesimal transformation in Eq.（26），there are［32］

The infinitesimal generator vector is introduced

and its β extension［32］is

Because of the invariance theory of differential equation under infinitesimal transformation，the in⁃variance of fractional Birkhoff’s equations in Eq.（13） under the infinitesimal transformation in Eq.（27）can lead to the following equations

By substituting operator in Eq.（30） into Eq.（31），we get

Eq.（32）is the determining equation.

Definition 1For the transformation in Eq.（27），if the determining equations in Eq.（32）are satisfied，the symmetry is the Lie symmetry of frac⁃tional Birkhoffian system Eq.（13）based on RLFI.

3.2 Lie symmetry based on EEFI

The invariance of Eq.（19）under the infinitesi⁃mal transformation in Eq.（27）can lead to the fol⁃lowing equations

By substituting operator Eq.（30）into Eq.（33），we get

Eq.（34）is the determining equation.

Definition 2For the transformation in Eq.（27），if the determining equations in Eq.（34）are satisfied，the symmetry is the Lie symmetry of fractional Birkhoffian system in Eq.（19）based on EEFI.

3.3 Lie symmetry based on FIEPL

The invariance of Eq.（25）under the infinitesi⁃mal transformation in Eq.（27）can obtian the follow⁃ing equations

By substituting operator Eq.（30） into Eq.（35），we get

Eq.（36）is the determining equation.

Definition 3For the transformation Eq.（27），if the determining equations in Eq.（36）are satis⁃fied，the symmetry is the Lie symmetry of fractional Birkhoffian system in Eq.（25）based on FIEPL.

4 Lie Symmetry Theorem of Frac⁃tional Birkhoffian System

Lie symmetry does not necessarily lead to con⁃served quantities. For the systems based on quasifractional models，the following theorems give the conditions and the conserved quantities led by Lie symmetry of this system.

4.1 Lie symmetry theorem based on RLFI

Theorem 1If the generators ξ0，ξμsatisfy the formula in Eq.（32）and there exists a gauge function G=G(τ，aν) that satisfies the following structural equation

then the fractional Birkhoffian system in Eq.（13）based on RLFI has the following conserved quantity

Proof

Substituting Eq.（37）into Eq.（39），we get

Due to Eq.（13），we obtain

Therefore，the theorem is proved.

We can describe Theorem 1 as Lie symmetry theorem of system（13）.

When β →1，the fractional Birkhoff’s equa⁃tions in Eq.（13）are reduced to

Eqs.（40） are Birkhoff’s equations based on quasi-fractional model given in Ref.［56］. And The⁃orem 1 is reduced to the following theorem.

Theorem 2If the generators ξ0and ξμsatisfy the following Lie symmetry equations

and there exists a gauge function G=G(τ，aν) which satisfies the following structural equation

then the system（40）has the following conserved quantity

When β →1 and α →1，Theorem 1 becomes the Lie symmetry theorem of classical Birkhoffian system［32］.

4.2 Lie symmetry theorem based on EEFI

Theorem 3If the generators ξ0，ξμsatisfy Eq.（34） and there exists a gauge function G=G(τ，aν) that satisfies the following structural equa⁃tion

then the fractional Birkhoffian system（19）based on EEFI has the following conserved quantity

Theorem 3Theorem 3 can be called Lie sym⁃metry theorem of fractional Birkhoffian system in Eq.（19）based on EEFI.

When β →1，the fractional Birkhoff’s equa⁃tions in Eq.（19）are reduced to

Eqs.（46） are Birkhoff’s equations based on quasi-fractional models given in Ref.［56］. And The⁃orem 3 is reduced to the following theorem.

Theorem 4If the generators ξ0，ξμsatisfy the following equations

and there exsits a gauge function G=G(τ，aν)which satisfies the following structural equation

then the system（46）contains the following con⁃served quantity

When β →1 and α →1，Theorem 3 becomes Lie symmetry theorem of classical Birkhoffian sys⁃tem［32］.

4.3 Lie symmetry theorem based on FIEPL

Theorem 5If the generators ξ0，ξμsatisfy Eq.（36） and there exists a gauge function G=G(τ，aν) that satisfies the following structural equa⁃tion

then the fractional Birkhoffian system in Eq.（25）based on FIEPL has the following conserved quanti⁃ty

Theorem 5Theorem 5 can be called Lie sym⁃metry theorem of fractional Birkhoffian system in Eq.（25）based on FIEPL.

When β →1，the fractional Birkhoff’s equa⁃tions in Eq.（25）become

Eqs.（52）are Birkhoff’s equations based on quasi-fractional models given in Ref.［56］. And The⁃orem 5 is reduced to the following theorem

Theorem 6If the infinitesimal generators ξ0，ξμsatisfy the following equations

and there exists a gauge function G=G(τ，aν)which satisfies the following structural equation

then the system（42）contains the following con⁃served quantity

When β →1 and α →1，Theorem 5 becomes Lie symmetry theorem of classical Birkhoffian sys⁃tem［32］.

5 Examples

（1）Example 1

A fractional Birkhoffian system is studied based on RLFI.Its Pfaff action is

According to Eq.（13），Birkhoff’s equations of the system can be written as

The structural Eq.（37）gives

Then

The generators in Eq.（59） satisfy Eq.（32）.Substituting the generators in Eq.（59）into Eq.（58），we have

According to Theorem 1，we obtain

Eq.（61）is the conserved quantity.

When β →1，Eq.（61）is reduced to

Eq.（62）is the conserved quantity of the quasifractional Birkhoffian system.

When β →1，α →1，Eq.（61）is reduced to

Eq.（63）is the conserved quantity of classical Birkhoffian system.

（2）Example 2

A fractional Birkhoffian system based on EEFI is elabrated.Its Pfaff action is

Birkhoff’s equations based on EEFI are

The structural Eq.（44）gives

Then

The generators in Eq.（67） satisfy Eq.（34）.Substituting the generators in Eq.（67） into Eq.（66），we have

According to Theorem 3，we obtain

Eq.（69）is the conserved quantity.

When β →1，from Eq.（69），we have

Eq.（70）is the conserved quantity of Birkhof⁃fian system based on EEFI.

When β →1，α →1，Eq.（69）becomes

Eq.（71）is the conserved quantity of classical Birkhoffian system.

（3）Example 3

Now we study a fractional Birkhoffian system based on FIEPL.Its Pfaff action is

Birkhoff’s equations under FIEPL are

The structural Eq.（50）gives

It is easy to verify that the system has the Lie symmetry

According to Theorem 5，corresponding to Lie symmetry in Eq.（75），the system has the following conserved quantity

If β →1，we get

Eq.（76）is the conserved quantity of Birkhof⁃fian system based on FIEPL.

If β →1，and α →1，we get

Eq.（78）is the conserved quantity of classical Birkhoffian system.

6 Conclusions

Since the fractional models can describe the dy⁃namics behavior of complex systems more accurate⁃ly，the research of fractional Birkhoffian system has always been a hot topic. The quasi-fractional dynam⁃ics models can simplify some complex problems in non-conservative dynamics，so it is quite valuable to study the symmetry of fractional Birkhoffian system based on quasi-fractional dynamics models. The Lie symmetry of fractional Birkhoffian system based on quasi-fractional dynamics models is proposed and studied in this paper. The determining equations for the systems based on RLFI，EEFI and FIEPL are established seperately. The corresponding structural Eqs.（44，50） and the conserved quantities of Eqs.（39，45，51）are obtained. The Lie symmetry theorems of quasi-fractional Birkhoffian systems and classical Birkhoffian systems are special cases of this paper.