College of Civil Engineering，Suzhou University of Science and Technology，Suzhou 215011，P.R.China

Abstract: This paper presents fractional generalized canonical transformations for fractional Birkhoffian systems within Caputo derivatives.Firstly，based on fractional Pfaff-Birkhoff principle within Caputo derivatives，fractional Birkhoff’s equations are derived and the basic identity of constructing generalized canonical transformations is proposed.Secondly，according to the fact that the generating functions contain new and old variables，four kinds of generating functions of the fractional Birkhoffian system are proposed，and four basic forms of fractional generalized canonical transformations are deduced.Then，fractional canonical transformations for fractional Hamiltonian system are given.Some interesting examples are finally listed.

Key words：fractional Birkhoffian system；generalized canonical transformation；fractional Pfaff-Birkhoff principle；generating function

0 Introduction

As is known to all，the transformation of vari?ables is an important means used by analytical me?chanics to study problems.It is often very difficult to solve the general dynamical equation，so it is a very important research topic to use the method of variable transformation to make the differential equa?tion to be easier to solve［1］.The transformation that keeps the form of Hamilton canonical equations un?changed is called canonical transformation.The pur?pose of canonical transformation is to find new Ham?iltonian function through transformation，so that it has more concise forms and more cyclic coordi?nates，so as to simplify the solution of the problem.Hamilton canonical transformation is the basis of Hamilton-Jacobi equation and perturbation theory，and has a wide range of applications in celestial me?chanics and other fields［2］.Under certain conditions，canonical transformation can be extended to non?holonomic systems［3-4］and weakly nonholonomic systems［5］.The transformation theory of Birkhoff’s equations was first introduced by Santilli［6］.Wu and Mei［7］extended the transformation theory to the gen?eralized Birkhoffian system.For Birkhoffian sys?tems，we studied their generalized canonical trans?formations，and gave six kinds of transformation for?mulas［8-9］.The generalized canonical transforma?tions were extended to second-order time-scale Birk?hoffian systems［10］.

In 1996，Riewe introduced fractional deriva?tives in his study of modeling of nonconservative mechanics［11］.In recent decades，fractional models have been widely used in various fields of mechanics and engineering due to their historical memory and spatial nonlocality，which can more succinctly and accurately describe complex dynamic behavior，ma?terial constitutive relations and physical proper?ties［12-22］.However，the transformation theory based on fractional model is still an open subject.In Ref.［23］，we presented fractional canonical transforma?tions for fractional Hamiltonian systems.Here we will work on generalized canonical transformations of fractional Birkhoffian systems.We will set up the basic identity of constructing generalized canonical transformations.According to different cases of gen?erating functions containing new and old variables，we will give four kinds of basic forms of generating functions and their corresponding generalized canoni?cal transformation formulae.

1 Fractional Calculus

The fractional left derivative of Riemann-Liou?ville type is defined as［24］

The right derivative is

The fractional left derivative of Caputo type is defined as

The right derivative is

The fractional-order integration by parts formu?lae are［15］

2 Fractional Birkhoffian Mechanics

The fractional Pfaff action can be written as

whereRβ=Rβ(t，aγ) (β=1，2，…，2n) are Birk?hoff’s functions，B=B(t，aγ) is the Birkhoffian，andaγ(γ=1，2，…，2n)are Birkhoff’s variables.

The isochronous variational principle

with commutative relation

and the endpoint condition

is called the fractional Pfaff-Birkhoff principle within Caputo derivatives.

Expanding Principle（9）yields

Integrating by parts，and using Eqs.（6）and（11），we get

Substituting Eq.（13）into Eq.（12），we get

Since the interval[t1，t2] is arbitrary，andδaβis independent，we get

Eq.（15）can be called fractional Birkhoff’s equations.

If takeα→1，then Eq.（15）gives

Eq.（16）is Birkhoff’s equation given in Ref.［6］.

Let

Then Principle（9）and Eq.（15）become

Eq.（18）is the fractional Hamilton principle and Eq.（19）is fractional Hamilton equations.

3 Fractional Generalized Canoni?cal Transformations

The isochronous transformations from the old variableaβto the new variableare

Let the transformed Birkhoffian and Birkhoff’s functions be

If Eq.（15）is still valid under the new variablesi.e.

then Eq.（20）is then called generalized canonical transformations of fractional Birkhoffian system（15）.Obviously，if both the old and new variables satisfy

then Eq.（20）is the generalized canonical transfor?mations.Since the starting and ending positions of the comparable motions of the system are defined，there are

Based on Eqs.（23）and（24），considering Eq.（25），if the relationship between the old and new variables

is satisfied，the transformations are generalized ca?nonical transformations of fractional Birkhoffian sys?tems and vice versa.Eq.（26）is called the basic identity for constructing generalized canonical trans?formations.Because generalized canonical transfor?mations depend entirely on the choice of any func?tionF，it is called the generating function.

4 Generating Function and Trans?formations

For convenience，Birkhoff’s variables are ex?pressed asa={as，as}，and Birkhoff’s functions are expressed asR={Rs，Rs}，wheres=1，2，…，n.Thus，Eq.（26）can be expressed as

whereRsandRsare functions oft，ajandaj(s，j=1，2，…，n)，andfunctions oft，andAccording to the fact that the generating function contains new and old variables，the following frac?tional generalized canonical transformations are pre?sented.

4.1 Generalized canonical transformations based on generating functions of the first kind

Let the generating function be

Substituting Eq.（29）into Eq.（27），we get

4.2 Generalized canonical transformations based on generating functions of the second kind

Let the generating function be

Then we have

From Eq.（34），we have

4.3 Generalized canonical transformations based on generating functions of the third kind

Let the generating function be

Then we have

Substituting Eq.（37）into Eq.（27），we get

From Eq.（38），we have

4.4 Generalized canonical transformations based on generating functions of the fourth kind

Let the generating function be

Substituting Eq.（41）into Eq.（27），we get

It should be pointed out that the four fractional generalized canonical transformations determined by the four kinds of generating functions are only part of the transformations.Of course，only these four fractional generalized canonical transformations are quite extensive.

5 Canonical Transformations of Fractional Hamiltonian Systems

Let

and

whereqsare the generalized coordinates，psthe gen?eralized momenta，andHis the Hamiltonian.Then Eq.（27）becomes

This is the basic identity for constructing canon?ical transformations of fractional Hamiltonian sys?tem.Thus，the results of generating functions and generalized canonical transformations of fractional Birkhoffian systems are reduced to generating func?tions and fractional canonical transformations of frac?tional Hamiltonian systems.The results are as fol?lows：

（1）The first kind of generating function and corresponding fractional canonical transformation are

（2）The second kind of generating function and corresponding fractional canonical transformation are

（3）The third kind of generating function and corresponding fractional canonical transformation are

（4）The fourth kind of generating function and corresponding fractional canonical transformation are

Whenα→1，the results above are reduced to the classical integer-order generating functions and canonical transformations for Hamiltonian sys?tems［1-2］.

6 Examples

In the following，some simple but important examples are given to illustrate the effects of gener?ating functions and fractional generalized canonical transformations.

Example 1If the generating functionF1is

then Eq.（31）gives

The transformation（56）shows that the new Birkhoff’s functionsdepend on the old variablesa={as，as}，and the old Birkhoff’s func?tionsR={Rs，Rs} are associated with the new vari?ables

Accordingly，for the fractional Hamiltonian system（19），let’s take the generating function as

then the transformations are

Example 2If the generating functionF2is

then Eq.（35）gives

Accordingly，for the fractional Hamiltonian system（19），let us take the generating function as

then the transformations are

This is an identity transformation.

Example 3If the generating functionF3is

then Eq.（39）gives

Wherein，it is assumed thatR={Rs，Rs} does not explicitly containt.

Accordingly，for the fractional Hamiltonian system（19），let us take the generating function as

then the transformations are

Example 4If the generating functionF4is

then Eq.（43）gives

Wherein，it is assumed thatR={Rs，Rs} and=do not explicitly containt.The transforma?tions（68）are the same as transformations（56）.Therefore，the selection of different generating func?tions may correspond to the same generalized canon?ical transformations.

Accordingly，for the fractional Hamiltonian system（19），let us take the generating function as

then the transformations are

7 Conclusions

In this paper，the generalized canonical trans?formations of fractional Birkhoffian systems are studied.Four basic forms of generalized canonical transformations are established by different choices of generating functions.The canonical transforma?tions of fractional Hamiltonian systems are the spe?cial cases.As a novel mathematical tool，fractional calculus has been widely used in engineering，me?chanics，materials and other research fields in recent years because it can more accurately describe com?plex dynamics problems with spatial nonlocality and historical memory.Birkhoffian mechanics is a new development of Hamiltonian mechanics，and canoni?cal transformation is an important means of analyti?cal mechanics，so the research on this topic is of great significance.