# Mei Symmetry and New Conserved Quantities of Time-Scale Birkhoff’s Equations

Authors: Xiang-Hua Zhai; Yi Zhang
Journal: Complexity
 (2020)

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## Abstract

The time-scale dynamic equations play an important role in modeling complex dynamical processes. In this paper, the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations are studied. The definition and criterion of the Mei symmetry of the Birkhoffian system on time scales are given. The conditions and forms of new conserved quantities which are found from the Mei symmetry of the system are derived. As a special case, the Mei symmetry of time-scale Hamilton canonical equations is discussed and new conserved quantities for the Hamiltonian system on time scales are derived. Two examples are given to illustrate the application of results.


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## Body

## 1. Introduction

In 1988, Hilger [1] proposed the calculus on time scales to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. This theory is really important and plays a useful role in modeling complex dynamic processes and has a tremendous potential for applications [2–4].Symmetry is an important aspect in studying dynamic equations. From a symmetry, we can find a conserved quantity or the first integral of dynamic equations. The famous Noether symmetry theory which reveals the relationships between symmetries and conversation laws has been applied to a time-scale analogue of analytical mechanics, modern theoretical physics, and engineering [5–10]. Except for the Noether symmetry method, the other two popular symmetry methods, the Lie symmetry method [11] and the Mei symmetry method [12], are widely applied in studying dynamic systems. The Mei symmetry means that when the dynamic functions are replaced by the transformed functions under the infinitesimal transformations of group, the forms of the differential equations of motion keep invariant. Here, the dynamic functions are the Lagrangian, the Hamiltonian, the Birkhoffian, Birkhoff’s functions, the generalized forces, the generalized constraint reactions, etc. New kinds of conserved quantity can be led by Mei symmetry. This method has been successfully applied in equations of motion for Lagrangian systems, Hamiltonian systems, Birkhoffian systems, the motion of charged particles in an electromagnetic field, the equation of nonmaterial volumes, the equation of thin elastic rod, etc. [13–21].The Birkhoffian mechanics [22] represents a new stage in the development of analytical mechanics. It is a natural development of the Hamiltonian mechanics and has valuable applications in hadronic physics, spatial mechanics, statistical mechanics, biophysics, and engineering [23–26]. Over the past two decades, important achievements, including the symmetries and conserved quantities of Birkhoffian systems, the Birkhoffian dynamic inverse problems, the stability of motion of Birkhoff’s equations, the Birkhoffian systems with time delay, and the fractional Birkhoffian systems, have been made [27–38].The Pfaff–Birkhoff principle and Birkhoff’s equations were studied in a time-scale analogue, and their corresponding Noether symmetry and Lie symmetry as well as the conserved quantities have been studied [39,40] recently. However, the Mei symmetry of Birkhoffian systems on time scales has not been done. It is worth going a step further to find more new conserved quantities of Birkhoff’s equations by another symmetry method. In this paper, we will study the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations.This paper is organized as follows: in Section2, we first give the definition and criterion of Mei symmetry for the Birkhoffian system on time scales. Three kinds of conserved quantities led by Mei symmetry of the system on time scales are given in Section 3. As a special case, Section 4 deals with the Mei symmetry of time-scale Hamilton canonical equations and the corresponding new conserved quantities for the Hamiltonian system on time scales. Two examples are given in Section 5 to show the application of results. In the end, conclusions and future works are given.


## 2. Mei Symmetry of Time-Scale Birkhoff’s Equations

For basic knowledge about the calculus on time scales, the readers can refer to References [2, 3].The Pfaff–Birkhoff action on time scales is(1)S=∫t1t2Rωt,aυσtaωΔ−Bt,aυσtΔt.According to the Pfaff–Birkhoff principle on time scales,(2)δS=0,with boundary conditions(3)δaυt=t1=δaυt=t2=0and exchange relations(4)δaυΔ=δaυΔ,δaυσ=δaυσ,the time-scale Birkhoff’s equations can be derived, i.e.,(5)∂Rωt,aρσt∂aυσaωΔ−RυΔt,aρσt−∂Bt,aρσt∂aυσ=0,υ,ω,ρ=1,2,…,2n,where B:T×ℝn⟶ℝ is the Birkhoffian function, Rω:T×ℝn⟶ℝ are Birkhoff’s functions, aωΔ are the delta derivatives of the Birkhoffian variables aυt with respect to t, and aυσt=aυ∘σt. This result can be found in Ref. [39].Introduce the one-parameter infinitesimal transformations of time and variablesaυ as(6)t∗=t+εξ0t,aωt,aυ∗=aυt+εξυt,aωt,υ,ω=1,2,…,2n,where ξ0 and ξυ are infinitesimals and the infinitesimal parameter ε∈ℝ. The corresponding infinitesimal generator X0 is(7)X0=ξ0∂∂t+ξυ∂∂aυ,and the first extended infinitesimal generator is(8)X1=X0+ξυΔ−aυΔξ0Δ∂∂aυΔ.The BirkhoffianB and Birkhoff’s functions Rω become the new Birkhoffian B∗ and the new Birkhoff’s functions Rω∗ under transformations (6), that is,(9)B∗=Bt∗,aυ∗σ∗=Bt+εξ0,aυσ+εξυσ,Rω∗=Rωt∗,aυ∗σ∗=Rωt+εξ0,aυσ+εξυσ.By performing Taylor series expansion at the point ε=0, we have (10)B∗=Bt,aυσ+εX0B+Oε2,(11)Rω∗=Rωt,aυσ+εX0Rω+Oε2.Definition 1.
If the form of equation (5) keeps invariant when the Birkhoffian B and Birkhoff’s functions Rω are replaced by B∗ and Rω∗, respectively, i.e.,(12)∂Rω∗∂aυσaωΔ−Rυ∗Δ−∂B∗∂aυσ=0,υ,ω=1,2,…,2n,then this invariance is called the Mei symmetry of the Birkhoffian system on time scales.
From Definition 1 and formulae (10) and (11), we have the following.Criterion 1.
For the Birkhoffian system (5) on time scales, if the infinitesimals ξ0 and ξυ satisfy(13)∂X0Rω∂aυσaωΔ−X0RυΔ−∂X0B∂aυσ=0,υ,ω=1,2,…,2n,then the corresponding invariance is Mei symmetry.


## 3. New Conserved Quantities of Birkhoffian Systems on Time Scales

For the Birkhoffian system (5) on time scales, new conserved quantities can be found from Mei symmetry.Theorem 1.
If the infinitesimalsξ0 and ξυ of the Mei symmetry of the Birkhoffian system (5) and the gauge function G1t,aυσ satisfy the following structural equation:(14)X0RυaυΔ−X0Bξ0Δ+X1X0RυaυΔ−X0B+μt∂X0Rω∂aυσ−∂X0B∂taυΔξ0Δ+G1Δ=0,then the Mei symmetry of the system can lead to the new conserved quantity(15)I1=X0Rυξυ−X0B+μt∂X0Rω∂taωΔ−∂X0B∂tξ0+G1=const.Proof.
To prove that formula (15) is a conserved quantity, we need to prove(16)ΔΔtI=0.
Therefore, we have(17)ΔΔtI1=X0RυξυΔ+ΔΔtX0Rυξυσ−X0B+μt∂X0Rω∂taωΔ−∂X0B∂tξ0Δ−ΔΔtX0B+μt∂X0Rω∂taωΔ−∂X0B∂tξ0σ+G1Δ.
Here, for system (5), the following energy equation [40]:(18)ΔΔtB−μt∂B∂t−∂Rω∂taωΔ=∂B∂t−∂Rω∂taωΔ,holds. After the infinitesimal transformations, the new energy equation(19)ΔΔtB∗−μt∂B∗∂t−∂Rω∗∂taωΔ=∂B∗∂t−∂Rω∗∂taωΔ,holds. Noting formulae (10) and (11), we get(20)ΔΔtX0B−μt∂X0B∂t−∂X0Rω∂taωΔ=∂X0B∂t−∂X0Rω∂taωΔ.
Thus, combining equation (14) with (20) yields(21)ΔΔtI1=X0RυaυΔ−X0Bξ0Δ+X1X0RυaυΔ−X0B+μt∂X0Rω∂aυσ−∂X0B∂taυΔξ0Δ+G1Δ=0.
The proof is completed.
Theorem1 gives the Mei conserved quantity (15) on time scales led directly by Mei symmetry of system (5) with considering the structure equation (14). When T=ℝ, the conserved quantity (15) becomes the classical Mei conserved quantity.Remark 1.
IfT=ℝ, then σt=t and μt=0, and the conserved quantity (15) becomes the classical one [14](22)I1=X0Rυξυ−X0Bξ0+G1=const.Remark 2.
IfT=hℕ0=hi:i∈ℕ0,h>1, then σt=ht and μt=h−1t, and the conserved quantity (15) becomes the quantum one(23)I1=X0Rυξυ−X0B+h−1t∂X0Rω∂taωΔ−∂X0B∂tξ0+G1=const,where aωΔ=aωht−aωt/h−1t.Theorem 2.
If the infinitesimalsξ0 and ξυ of the Mei symmetry of the Birkhoffian system (5) and the gauge function G2t,aυσ satisfy the following condition:(24)aυσΔΔt∂X0Rω∂aυσaωΔ−∂X0B∂aυσ−aυΔΔΔtX0Rυ+G2Δ=0,then the Mei symmetry of the system can lead to the new conserved quantity(25)I2=∂X0Rω∂aυσaωΔ−∂X0B∂aυσaυσ−μtaυΔ+G2=const.With taking note of equation (13) and condition (24), we can derive that formula (25) is a conserved quantity of system (5).Theorem 3.
If the infinitesimalsξ0 and ξυ of the Mei symmetry of the Birkhoffian system (5) and the gauge function G3t,aυσ satisfy the following condition:(26)∂X0B∂t−∂X0Rω∂taωΔ+G3Δ=0,then the Mei symmetry of the system can lead to the new conserved quantity(27)I3=X0B−μt∂X0B∂t−∂X0Rω∂taωΔ+G3=const.With taking note of formulae (20) and (26), we can derive that formula (27) is a conserved quantity of system (5).Theorems2 and 3 give the other two kinds of conserved quantities (25) and (27) on time scales also led by Mei symmetry with considering conditions (24) and (26).


## 4. Mei Symmetry and Conserved Quantities of Time-Scale Hamilton Canonical Equations

The Birkhoffian mechanics is a natural development of the Hamiltonian mechanics; if we take(28)aυσ=qυσ,υ=1,2,…,n,pυ−n,υ=n+1,n+2,…,2n,Rυ=pυ,υ=1,2,…,n,0,υ=n+1,n+2,…,2n,B=H,then we can obtain the time-scale Hamiltonian canonical equations [8](29)qkΔ=∂H∂pk,pkΔ=−∂H∂qkσ,k=1,2,…,n.Now, we introduce the infinitesimal transformations(30)t∗=t+εξ0t,qst,pst,qk∗=qkt+εξkt,qst,pst,pk∗=pkt+εηkt,qst,pst,where ξ0, ξk, and ηk are infinitesimals and the infinitesimal parameter ε∈ℝ. Undergoing transformation (30), the Hamiltonian H becomes H∗. Then,(31)H∗=Ht,qkσ,pk+εX0H+Oε2,where(32)X0=ξ0∂∂t+ξk∂∂qk+ηk∂∂pk.Definition 2.
If the form of equations (29) keep invariant when the Hamiltonian H is replaced by H∗, i.e.,(33)qkΔ=∂H∗∂pk,pkΔ=−∂H∗∂qkσ,k=1,2,…,n,then this invariance is called the Mei symmetry of the Hamiltonian system on time scales.Criterion 2.
For the Hamiltonian system (29) on time scales, if the infinitesimals ξ0, ξk, and ηk satisfy(34)∂X0H∂qkσ=0,∂X0H∂pk=0,k=1,2,…,n,then the corresponding invariance is Mei symmetry.
Thus, we have the following three kinds of conserved quantities led by Mei symmetry of the Hamiltonian system.Theorem 4.
If the infinitesimalsξ0, ξk, and ηk of the Mei symmetry of the Hamiltonian system (29) and the gauge function G1t,qkσ,pk satisfy the following structural equation:(35)−X0Hξ0Δ+∂X0H∂pkηk+X0pkξkΔ−X1X0H−μt∂X0H∂qkσqkΔξ0Δ+G1Δ=0,where(36)X1=X0+ξkΔ−qkΔξ0Δ∂∂qkΔ+ηkΔ−pkΔξ0Δ∂∂pkΔ,then the Mei symmetry of the system can lead to the new conserved quantity(37)I1=X0pkξk+μt∂X0H∂t−X0Hξ0+G1=const.Remark 3.
IfT=ℝ, then σt=t and μt=0, and the conserved quantity (37) becomes the classical one [16](38)I1=X0pkξk−X0Hξ0+G1=const.Remark 4.
IfT=hℕ0=hi:i∈ℕ0,h>1, then σt=ht and μt=h−1t, and the conserved quantity (37) becomes the quantum one(39)I1=X0pkξk+h−1t∂X0H∂t−X0Hξ0+G1=const.Theorem 5.
If the infinitesimalsξ0, ξk, and ηk of the Mei symmetry of the Hamiltonian system (29) and the gauge function G2t,qkσ,pk satisfy the following condition:(40)−qkσΔΔt∂X0H∂qkσ−qkΔΔΔtX0pk+G2Δ=0,then the Mei symmetry of the system can lead to the new conserved quantity(41)I2=−∂X0H∂qkσqkσ−μtqkΔ+G2=const.Theorem 6.
If the infinitesimalsξ0, ξk, and ηk of the Mei symmetry of the Hamiltonian system (29) and the gauge function G3t,qkσ,pk satisfy the following condition:(42)∂X0H∂t+G3Δ=0,then the Mei symmetry of the system can lead to the new conserved quantity(43)I3=X0H−μt∂X0H∂t+G3=const.Theorems5 and 6 give the other two kinds of conserved quantities (41) and (43) on time scales also led by Mei symmetry with considering conditions (40) and (42).


## 5. Examples

Example 1.
Birkhoff’s functions and Birkhoffian of a fourth-order Birkhoffian system on time scales are(44)R1=a3σ,R2=a4σ,R3=0,R4=0,(45)B=12a3σ2+a4σ2+a2σ,which leads us to study the Mei symmetry and the conserved quantity of the system.
The equations of the system are(46)−a3σΔ=0,−a4σΔ−1=0,a1Δ−a3σ=0,a2Δ−a4σ=0.
If we choose the infinitesimals(47)ξ0=0,ξ1=0,ξ2=a4σ,ξ3=0,ξ4=−1,(48)ξ0=0,ξ1=−1,ξ2=−12a3σ2,ξ3=a3σ,ξ4=0,we can verify that infinitesimals (47) and (48) satisfy the criterion equation (13); therefore, the corresponding invariance is Mei symmetry.
Substituting infinitesimal (47) into the structure equation (14), we have the gauge function G1=−t. According to Theorem 1, we have(49)I1=−a4σ−t=const.
Substituting infinitesimal (48) into the structure equation (14), we have the gauge function G1=0. From Theorem 1, we have(50)I1=−a3σ=const.
The conserved quantities (49) and (50) show that different infinitesimals correspond to different forms of conserved quantities.
From formulae (47) and (24), we obtain(51)G2=0.
According to Theorem2, we have(52)I2=0.
This conserved quantity is a trivial one.
From formulae (48) and (26), we obtain(53)G3=0.
According to Theorem3, we have(54)I3=12a3σ2=const.
Conserved quantity (54) is led by the Mei symmetry of the system.Example 2.
The Hamiltonian of a system on time scales is(55)H=12p12+p22+p32+q1σ+q2σ,which leads us to study the Mei symmetry and the conserved quantity of the system.
The canonical equations of the system are(56)q1Δ=p1,q2Δ=p2,q3Δ=p3,p1Δ=−1,p2Δ=−1,p3Δ=0.
Taking the calculation, we have(57)X0H=ξ1+ξ2+η1p1+η2p2+η3p3.
If we choose the infinitesimals as(58)ξ0=0,ξ1=p1,ξ2=0,ξ3=0,η1=−1,η2=0,η3=0,(59)ξ0=0,ξ1=p2,ξ2=p1+p2,ξ3=0,η1=−1,η2=−2,η3=0,then(60)X0H=0.
Substituting (58) into (35), we obtain G1=−t. According to Theorem 4, we have(61)I1=−p1−t=const.
Substituting (59) into (35), we obtain G1=−5t. From Theorem 4, we have(62)I1=−2p1−3p2−5t=const.
The conserved quantities (61) and (62) show that different infinitesimals correspond to different forms of conserved quantities.
Substituting (58) and (59) into (40), respectively, we obtain G2=0. From Theorem 5, we get a trivial conserved quantity I2=0.
Taking note of formula (60), we obtain G3=0. From Theorem 6, we also get a trivial conserved quantity I3=0.


## 6. Conclusions

In recent years, the theory of time-scale calculus becomes widely available in describing plenty systems that contain both continuous-time and discrete-time domains. Symmetry theory is an important method to find a conserved quantity or the first integral of the equations of dynamical systems. This paper presented and studied the Mei symmetry of Birkhoffian systems on time scales. Three new forms of conserved quantities on time scales were derived from the Mei symmetry. In addition, as a special case, the Mei symmetry and conserved quantities for the Hamiltonian system are studied. The examples given illustrated the effectiveness of the results. The results of this paper show a method to find the conserved quantity or the first integral of dynamical equations on time scales. The method is universal. The continuous results and discrete results are special cases. By choosing different time scales, general results can be obtained directly without repeated proof.Further works about the Mei symmetry for general holonomic systems, nonholonomic systems, Hamiltonian systems on time scales, and Birkhoffian systems on time scales, as well as its relation to Lie symmetry on time scales, are still worth doing.



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