# A New Fractional Gradient Representation of Birkhoff Systems

Authors: Peng Wang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A fractional generalization of the gradient system to the Birkhoff mechanics, that is, a new fractional gradient representation of the Birkhoff system is investigated, in this paper. The definitions of the fractional gradient system are generalized to Birkhoff mechanics. Based on the definition, a general condition that a Birkhoff system can be a fractional gradient system is derived, the former studies for fractional gradient representation of the Birkhoff system are special cases of this paper. As applications of the results, the Birkhoff equations and fractional gradient expression of several classical nonlinear models are derived, such as the Hénon–Heiles equation, the Duffing oscillator model, and the Hojman–Urrutia equations. The results indicate, different from the former studies, that only gave the second-order gradient (integer order) expression for the Birkhoff system, an arbitrary fractional order gradient representation, exists for the Birkhoff system. The fractional potential function obtained from the general condition can determine the stationary states of these models in Birkhoffian expression.


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## Body

## 1. Introduction

Fractional integrals and derivatives are increasingly important in the modeling of science and engineering problems because they are more suitable for describing complex phenomena in science and engineering. The discussion of noninteger order derivatives dates back to Leibniz and L’Hopital in 1695, but until 1974 the first monography [1] written by Oldham and Spanier about fractional integrals and derivatives was published. Since then, fractional calculus has been applied in many fields such as physics, mechanics, etc. [2–15].A gradient system is an important dynamical system. The characteristics of a gradient system are suitable, especially to determine the stability of a system by using the Lyapunov function [16–20]. If a mechanical system can be transformed into a gradient system, the stability of the mechanical system can be studied through the characteristics of the gradient system. Many researchers have performed the gradient representations of constrained mechanical systems such as the Lagrange system, the Hamilton system, and the Birkhoff system and studied the stability of their solutions [21–27]. Tarasov [28–30] generalized the Hamilton system and the Lagrange system into fractional gradient expressions. Mei et al. [31, 32] studied the fractional gradient representation of the Birkhoff system, Chen et al. [33] gave a fractional gradient representation of Poincaré equations. However, their studies only provide a special condition with second order α=2 and only can provide a second order gradient system, which limits its application in constrained mechanical systems. In this paper, our goal is to generalize the fractional gradient representation of the Birkhoff system to any order.The Birkhoff system is an important dynamical system; it is an extension of the Hamilton system. Birkhoff mechanics has important applications in hadronic physics, statistic mechanics, space mechanics, biophysics, and engineering [34]. Birkhoff’s mechanical system is a more general, constrained mechanical system [35–37]. Conservative and nonconservative systems and holonomic and nonholonomic systems can all be expressed by Birkhoffian formulations, which have a generalized symplectic structure [38]. So, generalizing the fractional gradient system into the Birkhoff system has general significance. Since former studies about the fractional gradient representation of the Birkhoff system only gave its second-order expression. In this paper, we will show the order of the fractional gradient Birkhoff system is not only with α=2 but also can be other orders (integer or noninteger order), and give a general condition for the Birkhoff system transforming into a fractional gradient Birkhoff system.The structure of this paper is as follows: In Section2, we will fix the notations of fractional derivative and fractional exterior derivative and give a brief review of the definition of the fractional gradient system and its formulation. In Section 3, we will discuss under what conditions a Birkhoff system can transform into a fractional gradient system. In Section 4, we will apply the results that we got in Section 3 to several examples, such as the Hénon–Heiles equation, the Duffing oscillator model, and the Hojman–Urrutia equations. Section 5 is the conclusion.


## 2. Review of the Fractional Gradient System

Fractional derivatives have different definitions. In this paper, we will adopt the Caputo and Riemann–Liouville fractional derivative definitions. Based on the definition of the Riemann–Liouville fractional derivative, we have the definition ofDxα in the Riemann–Liouville fractional derivative sense.(1)DxaRLfx=1Γm−α∂m∂xm∫0xfydyx−yα−m+1m−1≤α<m,x>0,where m is the first whole number greater than α>0. The initial point of a fractional derivative is set to be zero. The Riemann–Liouville fractional derivative has some disadvantages in physical applications, such as the hyper-singular improper integral, where the order of singularity is higher than the dimension, and the nonzero of the fractional derivative of constants, which would make dissipation not vanish for a system in equilibrium. The Caputo fractional derivatives can choose the same initial value as the integer derivatives, which are more suitable for applying to physical problems. The Caputo fractional derivative has the following definition:(2)DxaCfx=1Γm−α∫axfymx−yα−m+1dym−1<α≤m,x>a,where(3)fym=dmdymfym=α+1.When0<α≤1,a=0, and m=1, we have(4)DxaCfx=1Γ1−α∫0xf′ydyx−yα.We can introduce fractional exterior derivatives as [29–31, 39](5)dα=dxiαDxiaRLordα=dxiαDxiaRL.For the convenience of expression, we useDxα to denote α order fractional derivative, and only point out whether it is the Riemann–Liouville fractional derivative or the Caputo fractional derivative as calculation.The differential equations of a gradient system have the form as(6)x˙i=−∂Vx∂xii=1,2,…,n,where x=x1,x2,…,xn, and V is a potential function.Following, we will review the definition and conditions of the fractional gradient system proposed by Tarasov [28–30].Definition 1.
For a dynamical system,(7)dxidt=Fixi=1,2,…,n
If a fractional differential 1-form,(8)ωα=Fixdxiα.
This is an exact form, that is,ωα=−dαV, where V=Vx is a continuous differential function, then system (7) is a fractional gradient system. Thus, we have Fix=−DxiαVx.It is known that to be exact, it is a sufficient condition to be closed; however, the converse statement is not necessarily true. From the Poincaré theorem that is verified to be true for fractional exterior derivatives, it can be deduced that any smooth 1-form (8) that is closed is also exact on a contractible open subset W of Rn [28]. We have the following proposition.Proposition 1.
If smooth functionsFi=Fix on a contractible open subset W of Rn satisfy the relations(9)DxjαFi−DxiαFj=0,i,j=1,2,…,n,then dynamical system (7) is a fractional gradient system with form(10)x˙i=−DxiαVxi=1,2,…,n,where V=Vx is a continuous differential function and DxiαVx=−Fix. Because the Riemann–Liouville fractional derivative of a constant is not necessary to be zero, so, Vx=C cannot define the stationary state of a fractional system (10), where C is constant. The stationary state of the fractional gradient system with the Riemann–Liouville fractional derivative can be defined by(11)Vx−∏i=0nxiα−m∑k1=0m−1⋯∑kn=0m−1Ck1⋯kn∏i=1nxiki=0,where Ck1⋯kn are constants and m is the first whole number greater than or equal to α.
If the fractional exact 1-form 8 equals to zero that isdαV=0 in the Caputo derivative, we can get Vx−C=0, which defines the stationary state of the fractional gradient system (10), where C is constant.


## 3. Fractional Gradient Representation of the Birkhoff System

An automatic Birkhoff system has form(12)Ωμνaa˙ν−∂Ba∂aμ=0,μ,ν=1,2,3,…,2n,where B=Ba is a Birkhoffian, and(13)Ωμν=∂Rν∂aμ−∂Rμ∂aν,is called the Birkhoffian tensor and Rμa are Birkhoff’s functions. If the system is nonsingularity, that is detΩμν≠0, the automatic Birkhoff system (12) can be expressed as(14)a˙μ=Ωμν∂B∂aνμ,ν=1,2,…,2n,where(15)ΩμνΩνρ=δρμ.Definition 2.
For a Birkhoff system (12) or (14), if the fractional differential 1-form(16)ωα=Ωμν∂B∂aνdaμα,is an exact form, that is ωα=−dαV, then it is a fractional gradient expression of the Birkhoff system.
Make exterior derivative to fractional differential 1-form (16), we have(17)dαωα=12DaραΩμν∂B∂aν−DaμαΩρν∂B∂aνdaρα∧daμα.
Ifdαωα=0 is closed, we have(18)DaραΩμν∂B∂aν−DaμαΩρν∂B∂aν=0.
So, we have the following proposition.Proposition 2.
If equation (14) on a contractible open subset W of Rn satisfies condition (18), it is an α order fractional gradient system.Remark 1.
Whenα=1, condition (18) degenerates to an integer derivative case, which gives the condition for a Birkhoff system transforming into a classical gradient expression [21].Whenα>1 (including integer and noninteger numbers), we call it an α order fractional gradient expression of the Birkhoff system.Remark 2.
Whenα=2, condition (18) gives the condition for a Birkhoff system transforming into a second-order fractional gradient system [31–33].We point out that0<α<1 cannot be a fractional gradient system for the Birkhoff system, it may be a fractional gradient system for the fractional Birkhoff system, which question we will discuss in future works. From (10), we have(19)Ωμν∂B∂aν=−DaμαVa,μ,ν=1,2,…,2n.According to Birkhoff 1-form (19), we can get the expression of the potential function Va.(20)Va=I0αΩμν∂B∂aν,where I0α is the fractional Riemann–Liouville integral symbol with definition(21)I0αfx=1Γα∫0xfydyx−y1−αx>0,α>0.Remark 3.
Hereα>1 can be any order, but we should point out that only (20) makes the Birkhoff 1-form is exact, and (19) is a fractional order gradient expression of the Birkhoff system.


## 4. Results

### 4.1. Application A: Hénon-Heiles Equation as a Birkhoff System and Fractional Gradient System

In this section, we will give the Birkhoff expression of the Hénon–Heiles equation and prove it can be a fractional gradient system with the Riemann–Liouville or the Caputo fractional derivatives, respectively, in the Birkhoff mechanics frame.The Hénon–Heiles equation is a sort of nonlinear, nonintegrable Hamilton system, which has broad applications in physics and applied mathematics [10, 40, 41]. The Hénon–Heiles equation’s form is(22)x¨=−x−2xy,y¨=−y+y2−x2.Let Birkhoff variables bea1=x,a2=y,a3=x˙,a4=y˙; a Birkhoff expression of H e´ non-Heiles problem is(23)B=12a12+a22+a32+a42+2a2a12−23a23,R1=R2=0R2=−a1R4=−a2.Therefore, we can get(24)Ωμν=00−10000−110000100, (25)Ωμν=Ωμν−1=00100001−10000−100.The Birkhoff equations can be expressed as(26)a˙1=a3,a˙2=a4,a˙3=−a1+2a1a2,a˙4=−a2+a12−a22.From (16), we can get a fractional differential 1-form of Birkhoff (26).(27)ωα=a3da1α+a4da2α−a1+2a1a2da3α−a2+a12−a22da4α.Its fractional exterior derivative is(28)dαωα=12−Da2αa3+Da1αa4da1α∧da2α+12−Da3αa3−Da1αa11+2a2da1α∧da3α+12−Da4αa3−Da1αa2+a12−a22da1α∧da4α+12−Da3αa4−Da2αa1+2a1a2da2α∧da3α+12−Da2αa2+a12−a22−Da4αa4da2α∧da4α12−Da3αa2+a12−a22+Da4αa1+2a1a2da3α∧da4α.Next, we will discuss its fractional gradient expression by the Caputo and Riemann–Liouville fractional derivatives, respectively.


### 4.2. Fractional Gradient Representation of the Birkhoff Equation (27) with the Caputo Fractional Derivative

Firstly, we study its fractional gradient system using the Caputo fractional derivative. As we take the Caputo fractional derivative, that is,Daρ=DaραC,ρ=1,2,3,4 in (28), when 0<α≤1, (29) becomes(29)dαωα=12Γ2−α−a31−α−1+2a2a11−αda1α∧da3α+12Γ2−α2a12−αda1α∧da4α+12Γ2−α2a1a21−αda2α∧da3αa21−α−2a22−α−a41−αda2α∧da4α.Sodαωα≠0, then (26) cannot be expressed as a fractional gradient system.When1<α≤2, we can get(30)dαωα=2Γ2−α−a12−αda1α∧da4α+a22−αda2α∧da4α,Sodαωα≠0, then (26) cannot be expressed a fractional gradient system.Whenm<α≤m+1 with m≥2, dαωα=0, so (26) can be expressed as an α order Caputo type fractional gradient system, where m<α≤m+1,m≥2. From (19), we can get(31)a3=Da1αCVa,a4=Da2αCVa,−a1−2a1a2=Da3αCVa,−a2−a12+a22=Da4αCVa.According to (20), we can get the potential function as(32)Va=1Γβ+1−a3a1β−a4a2β+a1+2a1a2a3β+a2+a12−a22a4β,where β>0. We can directly calculate that DaμαVa is not the Birkhoff system (27). In order to make the potential function V (33) is the potential function of the Birkhoff system (27), we have to modify (33) as(33)Va=Γβ+1−αΓ1−m+β−a3a1β−a4a2β+a1+2a1a2a3β+a2+a12−a22a4β.Here,m is the first integer greater than or equal to α>0. However, it does not guarantee the system is a fractional gradient system, so considering dαωα=0, we have m+1≥β=α>m≥2, the Birkhoff system (27) can be expressed as an α order Caputo fractional gradient system.The stationary states of the Birkhoff systems are defined by equation(34)Va−C=0,where C is constant.


### 4.3. Fractional Gradient Representation of the Birkhoff Equation (27) with the Riemann–Liouville Fractional Derivative

Secondly, we study the fractional gradient representation of the Birkhoff system (27) using the Riemann–Liouville fractional derivative. When we take the Riemann–Liouville fractional derivative that is Daρ=DaραRL,ρ=1,2,3,4, (29) becomes(35)dαωα=12Γ1−α−a3a2−α+a4a1−αda1α∧da2α−12Γ2−αa31−α+1+2a2a11−αda1α∧da3α+12−1Γ1−αa4−α+a2−a22a1−α−2Γ3−αa12−αda1α∧da4α+12−1Γ1−αa4a3−α+a1a2−α−2Γ2−αa1a21−αda2α∧da3α+12−1Γ2−αa21−α+a41−α−1Γ1−αa12a2−α+2Γ3−αa22−αda2α∧da4α−12Γ1−αa2+a12−a22a3−α−a1+2a1a2a4−αda3α∧da4α.Whenα=1, we have(36)d1ω1=−1+a2da1∧da3−a1da1∧da4−a1da2∧da3+a2−1da2∧da4≠0.So, Birkhoff system (27) is not a gradient system.Whenα=2, we have(37)d2ω2=da22∧da42−da12∧da42≠0So (27) is not an α=2 order fractional gradient system.Whenα=3, dαωα=0, so (27) is an α=3 order fractional gradient system.From (19), we can get(38)a3=Da13RLVa,a4=Da23RLVa,−a1−2a1a2=Da33RLVa,−a2−a12+a22=Da43RLVa.The potential function can be written as(39)Va=Γβ+1−αΓ1+β−a3a1β−a4a2β+a1+2a1a2a3β+a2+a12−a22a4β,with β=α.


### 4.4. Conclusion

In this section, we give the Birkhoff expression of the Hénon-Heiles equation and give its fractional gradient representation using the Caputo fractional derivative and the Riemann–Liouville fractional derivative, respectively. We can conclude that from this example, the Hénon–Heiles equation in the Birkhoff mechanics frame can be expressed as anα (noninteger) order fractional gradient system using the Caputo fractional derivative, or as a third-order fractional gradient system using the Riemann–Liouville fractional derivative. However, it cannot be a second-order gradient system, so our results are more general than those in [31–33]. We also conclude that a fractional gradient system is not a classical gradient system in general. From the calculus, we find that using the Riemann–Liouville fractional derivative can only give an integer order gradient system, but using the Caputo fractional derivative can give an integer or noninteger order gradient system, so the Caputo fractional derivative is more suitable to study fractional gradient systems than the Riemann–Lioville fractional derivative.


## 4.1. Application A: Hénon-Heiles Equation as a Birkhoff System and Fractional Gradient System

In this section, we will give the Birkhoff expression of the Hénon–Heiles equation and prove it can be a fractional gradient system with the Riemann–Liouville or the Caputo fractional derivatives, respectively, in the Birkhoff mechanics frame.The Hénon–Heiles equation is a sort of nonlinear, nonintegrable Hamilton system, which has broad applications in physics and applied mathematics [10, 40, 41]. The Hénon–Heiles equation’s form is(22)x¨=−x−2xy,y¨=−y+y2−x2.Let Birkhoff variables bea1=x,a2=y,a3=x˙,a4=y˙; a Birkhoff expression of H e´ non-Heiles problem is(23)B=12a12+a22+a32+a42+2a2a12−23a23,R1=R2=0R2=−a1R4=−a2.Therefore, we can get(24)Ωμν=00−10000−110000100, (25)Ωμν=Ωμν−1=00100001−10000−100.The Birkhoff equations can be expressed as(26)a˙1=a3,a˙2=a4,a˙3=−a1+2a1a2,a˙4=−a2+a12−a22.From (16), we can get a fractional differential 1-form of Birkhoff (26).(27)ωα=a3da1α+a4da2α−a1+2a1a2da3α−a2+a12−a22da4α.Its fractional exterior derivative is(28)dαωα=12−Da2αa3+Da1αa4da1α∧da2α+12−Da3αa3−Da1αa11+2a2da1α∧da3α+12−Da4αa3−Da1αa2+a12−a22da1α∧da4α+12−Da3αa4−Da2αa1+2a1a2da2α∧da3α+12−Da2αa2+a12−a22−Da4αa4da2α∧da4α12−Da3αa2+a12−a22+Da4αa1+2a1a2da3α∧da4α.Next, we will discuss its fractional gradient expression by the Caputo and Riemann–Liouville fractional derivatives, respectively.


## 4.2. Fractional Gradient Representation of the Birkhoff Equation (27) with the Caputo Fractional Derivative

Firstly, we study its fractional gradient system using the Caputo fractional derivative. As we take the Caputo fractional derivative, that is,Daρ=DaραC,ρ=1,2,3,4 in (28), when 0<α≤1, (29) becomes(29)dαωα=12Γ2−α−a31−α−1+2a2a11−αda1α∧da3α+12Γ2−α2a12−αda1α∧da4α+12Γ2−α2a1a21−αda2α∧da3αa21−α−2a22−α−a41−αda2α∧da4α.Sodαωα≠0, then (26) cannot be expressed as a fractional gradient system.When1<α≤2, we can get(30)dαωα=2Γ2−α−a12−αda1α∧da4α+a22−αda2α∧da4α,Sodαωα≠0, then (26) cannot be expressed a fractional gradient system.Whenm<α≤m+1 with m≥2, dαωα=0, so (26) can be expressed as an α order Caputo type fractional gradient system, where m<α≤m+1,m≥2. From (19), we can get(31)a3=Da1αCVa,a4=Da2αCVa,−a1−2a1a2=Da3αCVa,−a2−a12+a22=Da4αCVa.According to (20), we can get the potential function as(32)Va=1Γβ+1−a3a1β−a4a2β+a1+2a1a2a3β+a2+a12−a22a4β,where β>0. We can directly calculate that DaμαVa is not the Birkhoff system (27). In order to make the potential function V (33) is the potential function of the Birkhoff system (27), we have to modify (33) as(33)Va=Γβ+1−αΓ1−m+β−a3a1β−a4a2β+a1+2a1a2a3β+a2+a12−a22a4β.Here,m is the first integer greater than or equal to α>0. However, it does not guarantee the system is a fractional gradient system, so considering dαωα=0, we have m+1≥β=α>m≥2, the Birkhoff system (27) can be expressed as an α order Caputo fractional gradient system.The stationary states of the Birkhoff systems are defined by equation(34)Va−C=0,where C is constant.


## 4.3. Fractional Gradient Representation of the Birkhoff Equation (27) with the Riemann–Liouville Fractional Derivative

Secondly, we study the fractional gradient representation of the Birkhoff system (27) using the Riemann–Liouville fractional derivative. When we take the Riemann–Liouville fractional derivative that is Daρ=DaραRL,ρ=1,2,3,4, (29) becomes(35)dαωα=12Γ1−α−a3a2−α+a4a1−αda1α∧da2α−12Γ2−αa31−α+1+2a2a11−αda1α∧da3α+12−1Γ1−αa4−α+a2−a22a1−α−2Γ3−αa12−αda1α∧da4α+12−1Γ1−αa4a3−α+a1a2−α−2Γ2−αa1a21−αda2α∧da3α+12−1Γ2−αa21−α+a41−α−1Γ1−αa12a2−α+2Γ3−αa22−αda2α∧da4α−12Γ1−αa2+a12−a22a3−α−a1+2a1a2a4−αda3α∧da4α.Whenα=1, we have(36)d1ω1=−1+a2da1∧da3−a1da1∧da4−a1da2∧da3+a2−1da2∧da4≠0.So, Birkhoff system (27) is not a gradient system.Whenα=2, we have(37)d2ω2=da22∧da42−da12∧da42≠0So (27) is not an α=2 order fractional gradient system.Whenα=3, dαωα=0, so (27) is an α=3 order fractional gradient system.From (19), we can get(38)a3=Da13RLVa,a4=Da23RLVa,−a1−2a1a2=Da33RLVa,−a2−a12+a22=Da43RLVa.The potential function can be written as(39)Va=Γβ+1−αΓ1+β−a3a1β−a4a2β+a1+2a1a2a3β+a2+a12−a22a4β,with β=α.


## 4.4. Conclusion

In this section, we give the Birkhoff expression of the Hénon-Heiles equation and give its fractional gradient representation using the Caputo fractional derivative and the Riemann–Liouville fractional derivative, respectively. We can conclude that from this example, the Hénon–Heiles equation in the Birkhoff mechanics frame can be expressed as anα (noninteger) order fractional gradient system using the Caputo fractional derivative, or as a third-order fractional gradient system using the Riemann–Liouville fractional derivative. However, it cannot be a second-order gradient system, so our results are more general than those in [31–33]. We also conclude that a fractional gradient system is not a classical gradient system in general. From the calculus, we find that using the Riemann–Liouville fractional derivative can only give an integer order gradient system, but using the Caputo fractional derivative can give an integer or noninteger order gradient system, so the Caputo fractional derivative is more suitable to study fractional gradient systems than the Riemann–Lioville fractional derivative.


## 5. Application B: The Birkhoff Equation of Duffing Oscillator Models and Its Fractional Gradient System

In this section, we will give the Birkhoff equation expression of the Duffing oscillator models and prove it can be a fractional gradient system with the Riemann–Liouville or the Caputo fractional derivatives, respectively, in the Birkhoff mechanics frame.Duffing oscillator models and their equation are widely discussed in nonlinear science:(40)y¨+y+ϵy3=0,where ϵ is the nonlinear stiffness coefficient of the Duffing oscillator. Take Birkhoff’s variables as(41)a1=y,a2=y˙.We can get Birkhoff expression of the Duffing oscillator equation(42)R1=−a2,R2=0,B=12a12+12a22+14ϵa14,with Birkhoff equations(43)a˙1=a2,a˙2=−a1−ϵa13.From (16), we can get a fractional differential 1-form of Birkhoff equation (43).(44)ωα=a2da1α+−a1−ϵa13da2α.Its fractional exterior derivative is(45)12Da2α−a2+Da1α−a1−ϵa13da1α∧da2α.Next, we will discuss its fractional gradient expression using the Caputo and the Riemann–Liouville fractional derivatives, respectively.


### 5.1. Fractional Gradient Representation of Birkhoff System (43) with the Caputo Fractional Derivative

Firstly, we study its fractional gradient system using the Caputo fractional derivative. As we take the Caputo fractional derivative, that is,Daρ=DaραC,ρ=1,2 in (43), When 0<α≤1, (45) becomes(46)dαωα=12−1Γ1−α1−αa21−α+a11−α−6εΓ4−αa13−αda1α∧da2α.Thendαωα≠0, so the Duffing oscillator system expressed by the Birkhoff equation (43) is not a fractional gradient system.When1<α≤2, we can get(47)dαωα=−6ϵΓ2−α2−α3−αa13−αda1α∧da2α.Thendαωα≠0, so the Duffing oscillator system expressed by the Birkhoff equation (43) is not a fractional gradient system.When2<α≤3, we can get(48)dαωα=−6ϵΓ3−α3−αa13−αda1α∧da2α.Thendαωα≠0, so the Duffing oscillator system expressed by the Birkhoff equation (43) is not a fractional gradient system.Whenm<α≤m+1 with m≥3, dαωα=0, so equation (43) can be expressed as an α order Caputo type fractional gradient system, where m<α≤m+1,m≥3. From (19), we can get(49)−a1−ϵa1=Da2αCVa,a2=Da1αCVa.According to (20) and condition (18), we can get the potential function as(50)Va=1Γ1−m+α−a2a1α+a1a2α+ϵa13a2α.The stationary state of system 43 in the Caputo derivative is defined by equation (35).


### 5.2. Fractional Gradient Representation of Birkhoff System (43) with the Riemann–Liouville Fractional Derivative

Secondly, we study the fractional gradient representation of the Birkhoff system (43) using the Riemann–Liouville fractional derivative. When we take the Riemann–Liouville fractional derivative, that isDaρ=DaραRL,ρ=1,2, (45) becomes(51)dαωα=12−1Γ2−αa21−α+a11−α−6ϵΓ4−αa13−αda1α∧da2α.Whenα=1, we have(52)d1ω1=12−2−3ϵa12da11∧da21.Thendαωα≠0, so (43) is not a fractional gradient system.Whenα=2, we have(53)d1ω1=−3ϵa1da12∧da22.Thendαωα≠0, so (43) is not a fractional gradient system.Whenα=3, we have(54)d1ω1=−3ϵda13∧da23.Thendαωα≠0, so (43) is not a fractional gradient system.Whenα=4, dαωα=0, so (43) is an α order fractional gradient system. From equation (19), we can get(55)−a1−ϵa1=Da24RLVa,−a2=Da14RLVa.The potential function is(56)Va=1Γ1+α−a2a1α+a1a2α+ϵa13a2α.


### 5.3. Conclusion

In this section, we give the Birkhoff equation expression of the Duffing oscillator problem and give its fractional gradient representation using the Caputo fractional derivative and the Riemann–Liouville fractional derivative, respectively. We conclude that, from this example, the Duffing oscillator model in the Birkhoff mechanics frame can be expressed as aα (noninteger) order fractional gradient system using the Caputo fractional derivative or a fourth order fractional gradient system using the Riemann–Liouville fractional derivative. However, it cannot be a second-order gradient system, so our results are more general than those in [31–33]. We also conclude that a fractional gradient system is not a classical gradient system in general. From the calculus, we find that using the Riemann–Liouville fractional derivative can only give an integer order gradient system, but using the Caputo fractional derivative can give an integer or noninteger order gradient system, so the Caputo fractional derivative is more suitable to study fractional gradient systems than the Riemann–Liouville fractional derivative.


## 5.1. Fractional Gradient Representation of Birkhoff System (43) with the Caputo Fractional Derivative

Firstly, we study its fractional gradient system using the Caputo fractional derivative. As we take the Caputo fractional derivative, that is,Daρ=DaραC,ρ=1,2 in (43), When 0<α≤1, (45) becomes(46)dαωα=12−1Γ1−α1−αa21−α+a11−α−6εΓ4−αa13−αda1α∧da2α.Thendαωα≠0, so the Duffing oscillator system expressed by the Birkhoff equation (43) is not a fractional gradient system.When1<α≤2, we can get(47)dαωα=−6ϵΓ2−α2−α3−αa13−αda1α∧da2α.Thendαωα≠0, so the Duffing oscillator system expressed by the Birkhoff equation (43) is not a fractional gradient system.When2<α≤3, we can get(48)dαωα=−6ϵΓ3−α3−αa13−αda1α∧da2α.Thendαωα≠0, so the Duffing oscillator system expressed by the Birkhoff equation (43) is not a fractional gradient system.Whenm<α≤m+1 with m≥3, dαωα=0, so equation (43) can be expressed as an α order Caputo type fractional gradient system, where m<α≤m+1,m≥3. From (19), we can get(49)−a1−ϵa1=Da2αCVa,a2=Da1αCVa.According to (20) and condition (18), we can get the potential function as(50)Va=1Γ1−m+α−a2a1α+a1a2α+ϵa13a2α.The stationary state of system 43 in the Caputo derivative is defined by equation (35).


## 5.2. Fractional Gradient Representation of Birkhoff System (43) with the Riemann–Liouville Fractional Derivative

Secondly, we study the fractional gradient representation of the Birkhoff system (43) using the Riemann–Liouville fractional derivative. When we take the Riemann–Liouville fractional derivative, that isDaρ=DaραRL,ρ=1,2, (45) becomes(51)dαωα=12−1Γ2−αa21−α+a11−α−6ϵΓ4−αa13−αda1α∧da2α.Whenα=1, we have(52)d1ω1=12−2−3ϵa12da11∧da21.Thendαωα≠0, so (43) is not a fractional gradient system.Whenα=2, we have(53)d1ω1=−3ϵa1da12∧da22.Thendαωα≠0, so (43) is not a fractional gradient system.Whenα=3, we have(54)d1ω1=−3ϵda13∧da23.Thendαωα≠0, so (43) is not a fractional gradient system.Whenα=4, dαωα=0, so (43) is an α order fractional gradient system. From equation (19), we can get(55)−a1−ϵa1=Da24RLVa,−a2=Da14RLVa.The potential function is(56)Va=1Γ1+α−a2a1α+a1a2α+ϵa13a2α.


## 5.3. Conclusion

In this section, we give the Birkhoff equation expression of the Duffing oscillator problem and give its fractional gradient representation using the Caputo fractional derivative and the Riemann–Liouville fractional derivative, respectively. We conclude that, from this example, the Duffing oscillator model in the Birkhoff mechanics frame can be expressed as aα (noninteger) order fractional gradient system using the Caputo fractional derivative or a fourth order fractional gradient system using the Riemann–Liouville fractional derivative. However, it cannot be a second-order gradient system, so our results are more general than those in [31–33]. We also conclude that a fractional gradient system is not a classical gradient system in general. From the calculus, we find that using the Riemann–Liouville fractional derivative can only give an integer order gradient system, but using the Caputo fractional derivative can give an integer or noninteger order gradient system, so the Caputo fractional derivative is more suitable to study fractional gradient systems than the Riemann–Liouville fractional derivative.


## 6. Application C: Hojman-Urrutia Equations as a Birkhoff System and Fractional Gradient System

In this section, we will give the Birkhoff equation expression of the Hojman–Urrutia equations, and prove it can be a fractional gradient system with the Riemann–Liouville or Caputo fractional derivatives, respectively, in the Birkhoff mechanics frame.The Hojman–Urrutia equations have an important status in the development of inverse problems in Lagrangian mechanics and Birkhoffian mechanics [36]. Their equations are written in the following form:(57)x¨+y˙=0,y¨+y=0.Leta1=x,a2=y,a3=a˙1,a4=a˙2; we can get one types of its Birkhoff expression.(58)R1=a2+a3,R2=0,R3=a4,R4=0,B=12a32+2a2a3−a42.So we can get(59)Ωμν=0−1−101000100−10010,and(60)Ωμν=Ωμν−1=0100−100−1000101−10.with Birkhoff equations(61)a˙1=a3,a˙2=a4,a˙3=−a4,a˙4=−a2,From (16), we can get a fractional differential 1-form of Birkhoff (57).(62)ωα=a3da1α+a4da2α−a4da3α−a2da4α.Its fractional exterior derivative is(63)dαωα=12Da1αa4−Da2αa3da1α∧da2α−12Da1αa4+Da3αa3da1α∧da3α−12Da1αa2+Da4αa3da1α∧da4α−12Da2αa4+Da3αa4da2α∧da3α−12Da2αa2+Da4αa4da2α∧da4α+12−Da3αa2+Da4αa4da3α∧da4α.


### 6.1. Fractional Gradient Representation of Birkhoff Systme (60) with the Caputo Fractional Derivative

Next, we will discuss its fractional gradient expression by the Caputo and the Riemann–Liouville fractional derivative, respectively.Firstly, we study its fractional gradient system using the Caputo fractional derivative. As we take the Caputo fractional derivative, that is,Daρ=DaραC,ρ=1,2,3,4 in (59), When 0<α≤1, (62) becomes(64)dαωα=−12Γ2−αa31−αda1α∧da3α−12Γ2−αa21−α+a41−αda2α∧da4α+12Γ2−αa41−αda3α∧da4α.Then,dαωα≠0, so the Hojman–Urrutia equation expressed by Birkhoff (57) is not a fractional gradient system.Whenm−1<α≤m with m≥2, we can get dαωα=0, so (57) can be expressed as an α order Caputo type fractional gradient system, where m−1<α≤m,m≥2. From (19), we can get(65)a3=Da1αCVa,a4=Da2αCVa,−a4=Da3αCVa,−a2=Da4αCVa.According to (20) and condition (18), we can get the potential function as(66)Va=1Γ1−n+α−a3a1α−a4a2α+a4a3α+a2a4α.Which can define the stationary state of system (60) in the sense of the Caputo fractional derivatives in equation (35).


### 6.2. Fractional Gradient Representation of Birkhoff System (60) with the Riemann–Liouville Fractional Derivative

Secondly, we study fractional gradient representation of the Birkhoff system (60) using the Riemann–Liouville fractional derivative. When we take the Riemann–Liouville fractional derivative that is Daρ=DaραRL,ρ=1,2,3,4, (62) becomes(67)dαωα=12Γ1−αa4a1−α−a3a2−αda1α∧da2α−121Γ1−αa4a1−α+1Γ2−αa31−αda1α∧da3α−12Γ1−αa2a1−α+a3a4−αda1α∧da4α−12Γ1−αa4a2−α+a4a3−αda2α∧da3α−12Γ2−αa21−α+a41−αda2α∧da4α+12−1Γ1−αa2a3−α+1Γ2−αa41−αda3α∧da4α.Whenα=1, we get(68)d1ω1=−12da11∧da31−da21∧da41−12da31∧da41.Thend1ω1≠0, so (60) is not a fractional gradient system.Whenα=2, d2ω2=0, so (60) is an α order fractional gradient system with potential function with α=2. From equation (19), we can get(69)a3=Da11CVa,a4=Da22CVa,−a4=Da33CVa,−a2=Da44CVa.The potential function is(70)Va=Γ1Γ1+α−a3a1α−a4a2α+a4a3α+a2a4α.


### 6.3. Conclusion

In this section, we give the Birkhoff expression of the Hojman–Urrutia equation and give its fractional gradient representation using the Caputo fractional derivative and the Riemann–Liouville fractional derivative, respectively. We can conclude that from this example, the Hojman–Urrutia equation in the Birkhoff mechanics frame can be expressed as anα (noninteger) order fractional gradient system using the Caputo fractional derivative or a second-order fractional gradient system using the Riemann–Liouville fractional derivative, so our results are more general than those in [31–33]. We also conclude that a fractional gradient system is not a classical gradient system in general. From the calculus, we find that using the Riemann–Liouville fractional derivative can only give an integer order gradient system, but using the Caputo fractional derivative can give an integer or noninteger order gradient system, so the Caputo fractional derivative is more suitable to study fractional gradient systems than the Riemann–Liouville fractional derivative.


## 6.1. Fractional Gradient Representation of Birkhoff Systme (60) with the Caputo Fractional Derivative

Next, we will discuss its fractional gradient expression by the Caputo and the Riemann–Liouville fractional derivative, respectively.Firstly, we study its fractional gradient system using the Caputo fractional derivative. As we take the Caputo fractional derivative, that is,Daρ=DaραC,ρ=1,2,3,4 in (59), When 0<α≤1, (62) becomes(64)dαωα=−12Γ2−αa31−αda1α∧da3α−12Γ2−αa21−α+a41−αda2α∧da4α+12Γ2−αa41−αda3α∧da4α.Then,dαωα≠0, so the Hojman–Urrutia equation expressed by Birkhoff (57) is not a fractional gradient system.Whenm−1<α≤m with m≥2, we can get dαωα=0, so (57) can be expressed as an α order Caputo type fractional gradient system, where m−1<α≤m,m≥2. From (19), we can get(65)a3=Da1αCVa,a4=Da2αCVa,−a4=Da3αCVa,−a2=Da4αCVa.According to (20) and condition (18), we can get the potential function as(66)Va=1Γ1−n+α−a3a1α−a4a2α+a4a3α+a2a4α.Which can define the stationary state of system (60) in the sense of the Caputo fractional derivatives in equation (35).


## 6.2. Fractional Gradient Representation of Birkhoff System (60) with the Riemann–Liouville Fractional Derivative

Secondly, we study fractional gradient representation of the Birkhoff system (60) using the Riemann–Liouville fractional derivative. When we take the Riemann–Liouville fractional derivative that is Daρ=DaραRL,ρ=1,2,3,4, (62) becomes(67)dαωα=12Γ1−αa4a1−α−a3a2−αda1α∧da2α−121Γ1−αa4a1−α+1Γ2−αa31−αda1α∧da3α−12Γ1−αa2a1−α+a3a4−αda1α∧da4α−12Γ1−αa4a2−α+a4a3−αda2α∧da3α−12Γ2−αa21−α+a41−αda2α∧da4α+12−1Γ1−αa2a3−α+1Γ2−αa41−αda3α∧da4α.Whenα=1, we get(68)d1ω1=−12da11∧da31−da21∧da41−12da31∧da41.Thend1ω1≠0, so (60) is not a fractional gradient system.Whenα=2, d2ω2=0, so (60) is an α order fractional gradient system with potential function with α=2. From equation (19), we can get(69)a3=Da11CVa,a4=Da22CVa,−a4=Da33CVa,−a2=Da44CVa.The potential function is(70)Va=Γ1Γ1+α−a3a1α−a4a2α+a4a3α+a2a4α.


## 6.3. Conclusion

In this section, we give the Birkhoff expression of the Hojman–Urrutia equation and give its fractional gradient representation using the Caputo fractional derivative and the Riemann–Liouville fractional derivative, respectively. We can conclude that from this example, the Hojman–Urrutia equation in the Birkhoff mechanics frame can be expressed as anα (noninteger) order fractional gradient system using the Caputo fractional derivative or a second-order fractional gradient system using the Riemann–Liouville fractional derivative, so our results are more general than those in [31–33]. We also conclude that a fractional gradient system is not a classical gradient system in general. From the calculus, we find that using the Riemann–Liouville fractional derivative can only give an integer order gradient system, but using the Caputo fractional derivative can give an integer or noninteger order gradient system, so the Caputo fractional derivative is more suitable to study fractional gradient systems than the Riemann–Liouville fractional derivative.


## 7. Conclusion

In this paper, we give a definition and a general condition of a Birkhoff system to be a fractional gradient system with the Caputo fractional derivative and the Riemann–Liouville fractional derivative, respectively. Whenα=1, the definition and condition degenerate into those of the classical gradient system, and when α=2, our results goes back to those of the second order gradient representation [31–33], so the classical gradient system and a second-order gradient system can be considered as special cases of the fractional gradient representation of the Birkhoff system obtained in this paper.As applications, we apply the results we get in this paper to the Hénon–Heiles equation, the Duffing oscillator model, and the Hojman–Urrutia equations. The Birkhoff equation expression and fractional gradient expression of these classical equations are given. It shows that the orders of the fractional gradient Birkhoff system are not only withα=2 but can also be other integers or noninteger orders in this paper, so our results are more general.The limitation of this paper is that we only discuss the fractional gradient representation of an autonomous Birkhoff system. The more general cases of the fractional gradient representation of nonautonomous Birkhoff systems and the fractional gradient representation of the fractional Birkhoff system [42] will be considered in future work.



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