# Time-Scale Version of Generalized Birkhoffian Mechanics and Its Symmetries and Conserved Quantities of Noether Type

Authors: Jin-Yue Chen; Yi Zhang
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

The time-scale version of Noether symmetry and conservation laws for three Birkhoffian mechanics, namely, nonshifted Birkhoffian systems, nonshifted generalized Birkhoffian systems, and nonshitfed constrained Birkhoffian systems, are studied. Firstly, on the basis of the nonshifted Pfaff-Birkhoff principle on time scales, Birkhoff’s equations for nonshifted variables are deduced; then, Noether’s quasi-symmetry for the nonshifted Birkhoffian system is proved and time-scale conserved quantity is presented. Secondly, the nonshifted generalized Pfaff-Birkhoff principle on time scales is proposed, the generalized Birkhoff’s equations for nonshifted variables are derived, and Noether’s symmetry for the nonshifted generalized Birkhoffian system is established. Finally, for the nonshifted constrained Birkhoffian system, Noether’s symmetry and time-scale conserved quantity are proposed and proved. The validity of the result is proved by examples.


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## Body

## 1. Introduction

Birkhoffian mechanics is a new stage in the development of analytical dynamics. It was first proposed by Birkhoff [1] and later developed by Santilli [2] and Mei et al. [3]. In literature [4], Mei proposed and studied in detail the dynamics of the generalized Birkhoffian systems. Since then, some scholars [5–10] have carried out a series of studies on this issue.The dynamics theory on a time scale unifies the dynamics of continuous systems, discrete systems, and quantum systems. The theory of time scale analysis can be traced back to Hilger [11], who first proposed the calculus theory on a measure chain. Time scale, as a special case of the measure chain, has strong representative, so it has attracted extensive attention. Bohner and Peterson [12] systematically studied time scale calculus and its dynamic equations. Agarwal and Bohner [13] began to study the time scale linear and nonlinear Hamiltonian systems and unify and extend the symplectic flow properties of continuous and discrete Hamiltonian system. In 2004, Bohner [14] studied the time scale variational problem for the first time. In 2008, Bartosiewicz and Torres [15] first carried out the researches about Noether’s theorem on time scales. They discovered that Noether’s conserved quantities can be derived without changing the time transformations. What is more, Bartosiewicz and his coworkers [16] also deduced the second Euler-Lagrange equation for variational problem on time scales. Based on the second Euler-Lagrange equations, they proposed another method to find the Noether conserved quantity. Afterwards, according to these two methods, many scholars have obtained some results have been obtained in the study of variational principle, dynamical equations, and Noether symmetries for the different mechanical systems, such as references [17–31].With the study on time scales, scholars began to study the time-scale version of the nonshifted variational problem. Bourdin [32] found that the Euler-Lagrange has greater convergence in the discrete case of the nonshifted variational problem. Anerot et al. [33] derived the Noether theorem for the shifted and nonshifted variational problems on time scales; they pointed out that the methods of deriving Noether conserved quantities on time scales by references [15, 16] were not correct. Song and Cheng [34] researched Noether symmetry on time scales for the nonshifted Birkhoffian systems, but the work was limited to free Birkhoffian systems and to Noether symmetries. Here, we will study the Noether symmetry for more general nonshifted Birkhoffian systems, including generalized Birkhoffian systems and constrained Birkhoffian systems, not only Noether symmetry but Noether quasi-symmetry. According to the study, it was found that the shifted variational problem are not suitable for the structure-preserving algorithm, while the nonshifted variational problem on time scales is suitable for the structure-preserving algorithm for discrete systems. Therefore, the research of the paper is of great significance.The structures of this article are as follows. In Section2, according to nonshifted Birkhoff’s equations, the Noether quasi-symmetry and time-scale conserved quantity are obtained. An example is given for discussion. In Section 3, about the nonshifted generalized systems on time scales, nonshifted generalized Pfaff-Birkhoff principle and equations are deduced. The Noether symmetries and time-scale conserved quantities are obtained. Then, an example is given for analysis. In Section 4, the equations for the nonshifted constrained Birkhoffian systems are deduced, and symmetries and time-scale conserved quantities are given. And an example is given. In Section 5, the conclusion is given.


## 2. Nonshifted Birkhoffian Systems on Time Scales

For the properties of calculus on a time scale, please refer to reference [12].


### 2.1. Nonshifted Birkhoff’s Equations

On a time scale, the nonshifted Pfaff action is(1)Saμ·=∫t1t2Rυt,aμaυΔ−Bt,aμΔt,where the endpoint conditions are aμt1=aμ1 and aμt2=aμ2. aμΔ is the delta derivative of aμ with respect to t. aμ∈Crd1,ΔT for t∈Tkk. The Birkhoffian B:T×ℝ2n⟶ℝ and Birkhoff’s functions Rμ:T×ℝ2n⟶ℝ are of Crd1,ΔT, where μ,υ=1,2,⋯,2n.The nonshifted Birkhoff’s equations on time scales are [34].
(2)Rμ∇=σ∇t∂Rυt,aρ∂aμaυΔ−∂Bt,aρ∂aμ,where σ=σt is nabla differentiable on T.


### 2.2. Quasi-symmetry and Conserved Quantity

Introduce infinitesimal transformations(3)t∗=t+εξ0t,aρ,aμ∗=aμ+εξμt,aρ,where ε is an infinitesimal parameter and ξ0 and ξμ are the generators.LetRυ1 and B1 be the other Birkhoffian and Birkhoff’s functions on time scales. If accurate to a small quantity of first order, this is true
(4)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫αt1αt2Rυ1t∗,aρ∗aυ∗Δ∗−B1t∗,aρ∗Δ∗t∗.Then, the nonshifted Pfaff action (1) is a quasi-invariant, where α:T⟶ℝ. Obviously, Rυ1,B1 and Rυ,B will satisfy the same equation, so we have
(5)Rυ1=Rυ+∂G∂aυ,B1=B−∂G∂t.Thus, equation (4) can be expressed as
(6)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫αt1αt2Rυt∗,aρ∗aυ∗Δ∗−Bt∗,aρ∗Δ∗t∗+∫t1t2ΔΔtGt,aρΔt,where G is a small quantity of first order.Definition 1.
If the nonshifted Pfaff action (1) is a quasi-invariant, in other words, for every infinitesimal transformations (3), the following relationship
(7)ΔS=−∫t1t2ΔΔtΔGΔt,always holds; the transformations (3) are referred to as Noether’s quasi-symmetric for the nonshifted Birkhoffian system (2).Criterion 2.
If the following equation(8)∂Rυ∂taυΔ−∂B∂tξ0+∂Rυ∂aμaυΔ−∂B∂aμξμ−Bξ0Δ+RυξυΔ+GΔ=0is satisfied, the transformations (3) are quasi-symmetric. Equation (8) is called the Noether identity.For equation (6),
(9)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫αt1αt2Rυt∗,aρ∗aυ∗Δ∗−Bt∗,aρ∗Δ∗t∗+∫t1t2ΔΔtGt,aρΔt=∫t1t2Rυαt,aρ∗∘αtaυ∗Δ∗αt−Bαt,aρ∗∘αtαΔtΔt+∫t1t2ΔΔtGt,aρΔt=∫t1t2Rυαt,aρ∗∘αtaρ∗∘αΔtαΔt−Bαt,aρ∗∘αtαΔtΔt+∫t1t2ΔΔtGt,aρΔt=∫t1t2Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔΔt+∫t1t2ΔΔtGt,aρΔt.We have(10)Rυt,aρaυΔ−Bt,aρ=Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔ+ΔΔtGt,aρTake the derivative ofε, on both sides of equation(10) and set ε=0, we get equation (8).Theorem 3.
If the transformations (3) satisfy Noether identity (8), then
(11)I=Rυξυσ+∫t1t∂Rν∂τaυΔ−∂B∂τξ0−Bξ0Δσ∇∇τ+Gσ=const.is the conserved quantity of this system (2).Proof.
From equation (8), we have
(12)∂Rυ∂tξ0+∂Rυ∂aμξμaυΔσ∇+RυξυΔσ∇−Bξ0Δσ∇−∂B∂tξ0σ∇−∂B∂aμξμσ∇+GΔσ∇=0.
Using equation (2), we get
(13)∂Rυ∂tξ0aυΔσ∇+ξ0B∇−ξ0σB∇−∂B∂tξ0σ∇+Rυξυσ∇+GΔσ∇=0.By nabla indefinite integral of equation (13), we can get conserved quantity (11).Example 4.
We can study the Hojman-Urrutia problem on time scales. This problem can be written to be a nonshifted Birkhoffian system on time scales. LetT=2n:n∈ℕ∪0, it is
(14)B=12a32+2a2a3−a42,R1=a2+a3,R2=0,R3=a4,R4=0.
From equation (8), we get
(15)a1Δ−a3ξ2+a1Δ−a3−a2ξ3+a3Δ+a4ξ4−Bξ0Δ+a3+a2ξ1Δ+a4ξ3Δ+GΔ=0.
It is easy to solve(16)ξ11=1,ξ01=ξ21=ξ31=ξ41=0,G1=0,(17)ξ02=0,ξ12=t,ξ22=0,ξ32=1,ξ42=0,G2=−a1.The generators (16) correspond to Noether symmetry, and generators (17) correspond to Noether quasi-symmetry.Based on Theorem3, we can get
(18)I1=a2+a3=const,(19)I2=a2+a3t+μt+a4−a1+μta1Δ=const.If we takeT=ℝ, we have
(20)I1=a2+a3=const,(21)I2=a2+a3t+a4−a1=const.


## 2.1. Nonshifted Birkhoff’s Equations

On a time scale, the nonshifted Pfaff action is(1)Saμ·=∫t1t2Rυt,aμaυΔ−Bt,aμΔt,where the endpoint conditions are aμt1=aμ1 and aμt2=aμ2. aμΔ is the delta derivative of aμ with respect to t. aμ∈Crd1,ΔT for t∈Tkk. The Birkhoffian B:T×ℝ2n⟶ℝ and Birkhoff’s functions Rμ:T×ℝ2n⟶ℝ are of Crd1,ΔT, where μ,υ=1,2,⋯,2n.The nonshifted Birkhoff’s equations on time scales are [34].
(2)Rμ∇=σ∇t∂Rυt,aρ∂aμaυΔ−∂Bt,aρ∂aμ,where σ=σt is nabla differentiable on T.


## 2.2. Quasi-symmetry and Conserved Quantity

Introduce infinitesimal transformations(3)t∗=t+εξ0t,aρ,aμ∗=aμ+εξμt,aρ,where ε is an infinitesimal parameter and ξ0 and ξμ are the generators.LetRυ1 and B1 be the other Birkhoffian and Birkhoff’s functions on time scales. If accurate to a small quantity of first order, this is true
(4)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫αt1αt2Rυ1t∗,aρ∗aυ∗Δ∗−B1t∗,aρ∗Δ∗t∗.Then, the nonshifted Pfaff action (1) is a quasi-invariant, where α:T⟶ℝ. Obviously, Rυ1,B1 and Rυ,B will satisfy the same equation, so we have
(5)Rυ1=Rυ+∂G∂aυ,B1=B−∂G∂t.Thus, equation (4) can be expressed as
(6)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫αt1αt2Rυt∗,aρ∗aυ∗Δ∗−Bt∗,aρ∗Δ∗t∗+∫t1t2ΔΔtGt,aρΔt,where G is a small quantity of first order.Definition 1.
If the nonshifted Pfaff action (1) is a quasi-invariant, in other words, for every infinitesimal transformations (3), the following relationship
(7)ΔS=−∫t1t2ΔΔtΔGΔt,always holds; the transformations (3) are referred to as Noether’s quasi-symmetric for the nonshifted Birkhoffian system (2).Criterion 2.
If the following equation(8)∂Rυ∂taυΔ−∂B∂tξ0+∂Rυ∂aμaυΔ−∂B∂aμξμ−Bξ0Δ+RυξυΔ+GΔ=0is satisfied, the transformations (3) are quasi-symmetric. Equation (8) is called the Noether identity.For equation (6),
(9)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫αt1αt2Rυt∗,aρ∗aυ∗Δ∗−Bt∗,aρ∗Δ∗t∗+∫t1t2ΔΔtGt,aρΔt=∫t1t2Rυαt,aρ∗∘αtaυ∗Δ∗αt−Bαt,aρ∗∘αtαΔtΔt+∫t1t2ΔΔtGt,aρΔt=∫t1t2Rυαt,aρ∗∘αtaρ∗∘αΔtαΔt−Bαt,aρ∗∘αtαΔtΔt+∫t1t2ΔΔtGt,aρΔt=∫t1t2Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔΔt+∫t1t2ΔΔtGt,aρΔt.We have(10)Rυt,aρaυΔ−Bt,aρ=Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔ+ΔΔtGt,aρTake the derivative ofε, on both sides of equation(10) and set ε=0, we get equation (8).Theorem 3.
If the transformations (3) satisfy Noether identity (8), then
(11)I=Rυξυσ+∫t1t∂Rν∂τaυΔ−∂B∂τξ0−Bξ0Δσ∇∇τ+Gσ=const.is the conserved quantity of this system (2).Proof.
From equation (8), we have
(12)∂Rυ∂tξ0+∂Rυ∂aμξμaυΔσ∇+RυξυΔσ∇−Bξ0Δσ∇−∂B∂tξ0σ∇−∂B∂aμξμσ∇+GΔσ∇=0.
Using equation (2), we get
(13)∂Rυ∂tξ0aυΔσ∇+ξ0B∇−ξ0σB∇−∂B∂tξ0σ∇+Rυξυσ∇+GΔσ∇=0.By nabla indefinite integral of equation (13), we can get conserved quantity (11).Example 4.
We can study the Hojman-Urrutia problem on time scales. This problem can be written to be a nonshifted Birkhoffian system on time scales. LetT=2n:n∈ℕ∪0, it is
(14)B=12a32+2a2a3−a42,R1=a2+a3,R2=0,R3=a4,R4=0.
From equation (8), we get
(15)a1Δ−a3ξ2+a1Δ−a3−a2ξ3+a3Δ+a4ξ4−Bξ0Δ+a3+a2ξ1Δ+a4ξ3Δ+GΔ=0.
It is easy to solve(16)ξ11=1,ξ01=ξ21=ξ31=ξ41=0,G1=0,(17)ξ02=0,ξ12=t,ξ22=0,ξ32=1,ξ42=0,G2=−a1.The generators (16) correspond to Noether symmetry, and generators (17) correspond to Noether quasi-symmetry.Based on Theorem3, we can get
(18)I1=a2+a3=const,(19)I2=a2+a3t+μt+a4−a1+μta1Δ=const.If we takeT=ℝ, we have
(20)I1=a2+a3=const,(21)I2=a2+a3t+a4−a1=const.


## 3. Nonshifted Generalized Birkhoffian Systems on Time Scales

### 3.1. Nonshifted Generalized Birkhoff’s Equations

The nonshifted generalized Pfaff-Birkhoff principle on time scales is [3].
(22)∫t1t2δRυaυΔ−B+Λυt,aρδaυΔt=0,where additional term Λυ:T×ℝ2n⟶ℝ, which is of Crd1,ΔT.From the principle (22), we have
(23)∫t1t2δRυt,aρaυΔ−Bt,aρ+Λυt,aρδaυΔt=∫t1t2∂Rυt,aρ∂aμaυΔδaμ+Rυt,aρδaυΔ−∂Bt,aρ∂aμδaμ+Λυt,aρδaυΔt=∫t1t2−∫t1σt∂Rυτ,aρ∂aμaυΔΔτ+Rμτ,aρ+∫t1σt∂Bτ,aρ∂aμΔτ−∫t1σtΛμτ,aρΔτδaμΔΔt=0We get(24)Rμ−∫t1σt∂Rυτ,aρ∂aμaυΔ−∂Bτ,aρ∂aμ+Λμτ,aρΔτ=const.Lethσt=∫t1σt∂Rυτ,aρ/∂aμaυΔ−∂Bτ,aρ/∂aμ+Λμτ,aρΔτ, we have
(25)Rμ−∫t1σt∂Rυτ,aρ∂aμaυΔ−∂Bτ,aρ∂aμ+Λμτ,aρΔτ∇=Rμ∇−hσ∇t=Rλ∇−σ∇thΔt=Rμ∇−σ∇t∫aσt∂Rυ∂aμaυΔ−∂B∂aμ+ΛμΔτΔ=Rμ∇−σ∇t∂Rυ∂aμaυΔ−∂B∂aμ+Λμ=0.Therefore, we get(26)σ∇t∂Rυ∂aμaυΔ−∂B∂aμ+Λμ=Rμ∇μ,υ=1,2,⋯,2n.Equation (26) is called nonshifted generalized Birkhoff’s equations. When Λμ=0, equation (26) becomes nonshifted Birkhoff’s equation (2).


### 3.2. Quasi-symmetry and Conserved Quantity

Definition 5.
If nonshifted Pfaff action (1) is a generalized quasi-invariant, that is, for every infinitesimal transformations (3), the following relationship
(27)ΔS=−∫t1t2ΔΔtΔG+ΛυδaυΔt,always holds, the transformations (3) are referred to as generalized quasi-symmetric for nonshifted generalized Birkhoffian system (26).Criterion 6.
If the following Noether identity(28)∂Rυ∂taυΔ−∂B∂tξ0+∂Rυ∂aμaυΔ−∂B∂aμξμ−Bξ0Δ+RυξυΔ+GΔ+Λυξυ−aυΔξ0=0,is satisfied, then transformations (3) are generalized quasi-symmetry.
For equation (27), we have
(29)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫αt1αt2Rυt∗,aρ∗aυ∗Δ∗−Bt∗,aρ∗Δ∗t∗+∫t1t2ΔΔtGt,aρ+Λυt,aρδaυΔt=∫t1t2Rυαt,aρ∗∘αtaυ∗Δ∗αt−Bαt,aρ∗∘αtαΔtΔt+∫t1t2ΔΔtGt,aρ+Λυt,aρδaυΔt=∫t1t2Rυαt,aρ∗∘αtaρ∗∘αΔtαΔt−Bαt,aρ∗∘αtαΔtΔt+∫t1t2ΔΔtGt,aρ+Λυt,aρδaυΔt=∫t1t2Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔΔt+∫t1t2ΔΔtGt,aρ+Λυt,aρδaυΔt.We have(30)Rυt,aρaυΔ−Bt,aρ=Rυt+εξ0t,aρ,aρ+εξρt,aρaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aθ,aρ+εξρt,aρt+εξ0t,aρΔ+ΔΔtGt,aρ+Λυt,aρδaυ.Take the derivative ofε on both sides of equation (30) and set ε=0, we obtain equation (28).Theorem 7.
If the transformations (3) satisfy Noether identity (28), then
(31)I=Rυξυσ+∫t1t∂Rυ∂τaυΔ−∂B∂τ−ΛυaυΔξ0−Bξ0Δσ∇∇τ+Gσ=const.is the conserved quantity for the system (26).Proof.
By equation (28), we have
(32)∂Rυ∂tξ0+∂Rυ∂aμξμaυΔσ∇+RυξυΔσ∇−Bξ0Δσ∇−∂B∂tξ0σ∇−∂B∂aμξμσ∇+GΔσ∇+Λυξυ−aυΔξ0σ∇=0.
Using equation (26), we have
(33)∂Rυ∂tξ0aυΔσ∇+ξ0B∇−ξ0σB∇−∂B∂tξ0σ∇+Rυξυσ+GΔσ∇+Λυξυ−aυΔξ0σ∇=0.
By nabla indefinite integral of equation (33), we can get conserved quantity (31).Example 8.
LetT=2n:n∈ℕ∪0, the nonshifted generalized Birkhoffian system on time scales is
(34)B=12a32+a2,R1=a3,R2=a4,R3=0,R4=0,Λ1=0,Λ2=0,Λ3=0,Λ4=−a4.
From equation (28), we get
(35)−ξ2−a3ξ3−Bξ0Δ+a3ξ1Δ+a4ξ2Δ−a4ξ4−a4Δξ0+GΔ=0.
Equation (35) has the following solution:
(36)ξ01=0,ξ11=0,ξ21=1,ξ31=0,ξ41=0,G1=t,(37)ξ02=0,ξ12=1,ξ22=0,ξ32=0,ξ42=0,G2=0.By Theorem7, the conserved quantities corresponding to the generators (36) and (37) are
(38)I1=a4+2t=const,(39)I2=a3=const.


## 3.1. Nonshifted Generalized Birkhoff’s Equations

The nonshifted generalized Pfaff-Birkhoff principle on time scales is [3].
(22)∫t1t2δRυaυΔ−B+Λυt,aρδaυΔt=0,where additional term Λυ:T×ℝ2n⟶ℝ, which is of Crd1,ΔT.From the principle (22), we have
(23)∫t1t2δRυt,aρaυΔ−Bt,aρ+Λυt,aρδaυΔt=∫t1t2∂Rυt,aρ∂aμaυΔδaμ+Rυt,aρδaυΔ−∂Bt,aρ∂aμδaμ+Λυt,aρδaυΔt=∫t1t2−∫t1σt∂Rυτ,aρ∂aμaυΔΔτ+Rμτ,aρ+∫t1σt∂Bτ,aρ∂aμΔτ−∫t1σtΛμτ,aρΔτδaμΔΔt=0We get(24)Rμ−∫t1σt∂Rυτ,aρ∂aμaυΔ−∂Bτ,aρ∂aμ+Λμτ,aρΔτ=const.Lethσt=∫t1σt∂Rυτ,aρ/∂aμaυΔ−∂Bτ,aρ/∂aμ+Λμτ,aρΔτ, we have
(25)Rμ−∫t1σt∂Rυτ,aρ∂aμaυΔ−∂Bτ,aρ∂aμ+Λμτ,aρΔτ∇=Rμ∇−hσ∇t=Rλ∇−σ∇thΔt=Rμ∇−σ∇t∫aσt∂Rυ∂aμaυΔ−∂B∂aμ+ΛμΔτΔ=Rμ∇−σ∇t∂Rυ∂aμaυΔ−∂B∂aμ+Λμ=0.Therefore, we get(26)σ∇t∂Rυ∂aμaυΔ−∂B∂aμ+Λμ=Rμ∇μ,υ=1,2,⋯,2n.Equation (26) is called nonshifted generalized Birkhoff’s equations. When Λμ=0, equation (26) becomes nonshifted Birkhoff’s equation (2).


## 3.2. Quasi-symmetry and Conserved Quantity

Definition 5.
If nonshifted Pfaff action (1) is a generalized quasi-invariant, that is, for every infinitesimal transformations (3), the following relationship
(27)ΔS=−∫t1t2ΔΔtΔG+ΛυδaυΔt,always holds, the transformations (3) are referred to as generalized quasi-symmetric for nonshifted generalized Birkhoffian system (26).Criterion 6.
If the following Noether identity(28)∂Rυ∂taυΔ−∂B∂tξ0+∂Rυ∂aμaυΔ−∂B∂aμξμ−Bξ0Δ+RυξυΔ+GΔ+Λυξυ−aυΔξ0=0,is satisfied, then transformations (3) are generalized quasi-symmetry.
For equation (27), we have
(29)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫αt1αt2Rυt∗,aρ∗aυ∗Δ∗−Bt∗,aρ∗Δ∗t∗+∫t1t2ΔΔtGt,aρ+Λυt,aρδaυΔt=∫t1t2Rυαt,aρ∗∘αtaυ∗Δ∗αt−Bαt,aρ∗∘αtαΔtΔt+∫t1t2ΔΔtGt,aρ+Λυt,aρδaυΔt=∫t1t2Rυαt,aρ∗∘αtaρ∗∘αΔtαΔt−Bαt,aρ∗∘αtαΔtΔt+∫t1t2ΔΔtGt,aρ+Λυt,aρδaυΔt=∫t1t2Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔΔt+∫t1t2ΔΔtGt,aρ+Λυt,aρδaυΔt.We have(30)Rυt,aρaυΔ−Bt,aρ=Rυt+εξ0t,aρ,aρ+εξρt,aρaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aθ,aρ+εξρt,aρt+εξ0t,aρΔ+ΔΔtGt,aρ+Λυt,aρδaυ.Take the derivative ofε on both sides of equation (30) and set ε=0, we obtain equation (28).Theorem 7.
If the transformations (3) satisfy Noether identity (28), then
(31)I=Rυξυσ+∫t1t∂Rυ∂τaυΔ−∂B∂τ−ΛυaυΔξ0−Bξ0Δσ∇∇τ+Gσ=const.is the conserved quantity for the system (26).Proof.
By equation (28), we have
(32)∂Rυ∂tξ0+∂Rυ∂aμξμaυΔσ∇+RυξυΔσ∇−Bξ0Δσ∇−∂B∂tξ0σ∇−∂B∂aμξμσ∇+GΔσ∇+Λυξυ−aυΔξ0σ∇=0.
Using equation (26), we have
(33)∂Rυ∂tξ0aυΔσ∇+ξ0B∇−ξ0σB∇−∂B∂tξ0σ∇+Rυξυσ+GΔσ∇+Λυξυ−aυΔξ0σ∇=0.
By nabla indefinite integral of equation (33), we can get conserved quantity (31).Example 8.
LetT=2n:n∈ℕ∪0, the nonshifted generalized Birkhoffian system on time scales is
(34)B=12a32+a2,R1=a3,R2=a4,R3=0,R4=0,Λ1=0,Λ2=0,Λ3=0,Λ4=−a4.
From equation (28), we get
(35)−ξ2−a3ξ3−Bξ0Δ+a3ξ1Δ+a4ξ2Δ−a4ξ4−a4Δξ0+GΔ=0.
Equation (35) has the following solution:
(36)ξ01=0,ξ11=0,ξ21=1,ξ31=0,ξ41=0,G1=t,(37)ξ02=0,ξ12=1,ξ22=0,ξ32=0,ξ42=0,G2=0.By Theorem7, the conserved quantities corresponding to the generators (36) and (37) are
(38)I1=a4+2t=const,(39)I2=a3=const.


## 4. Nonshifted Constrained Birkhoffian Systems on Time Scales

### 4.1. Nonshifted Constrained Birkhoff’s Equations

If the variables in nonshifted Birkhoffian system are not independent of each other on time scales, but subject to some constraints, these constraints are shown as(40)fβt,aμ=0β=1,2,⋯,2n.To calculate the isochronous variation of equation (40), we have
(41)∂fβ∂aμδaμ=0.From equation (41), we can get
(42)∫t1t2λβ∂fβ∂aμδaμΔt=0.By integrating by parts on time scales with equation (42), we get
(43)∫t1t2λβ∂fβ∂aμδaμΔt=δaμ∫t1tλβ∂fβ∂aμ∇τt1t2−∫t1t2∫t1σtλβ∂fβ∂aμ∇τδaμΔΔt=0,i.e.,
(44)∫t1t2∫t1σtλβ∂fβ∂aμ∇τδaμΔΔt=0.According to nonshifted Pfaff-Birkhoff principleδS=0,we get [34].
(45)∫t1t2−∫t1σt∂Rυτ,aρ∂aμaυΔΔτ+Rμτ,aρ+∫t1σt∂Bτ,aρ∂aμΔτδaμΔΔt=0.Add equations (44) to (45), we have
(46)∫t1t2−∫t1σt∂Rυτ,aρ∂aμaυΔΔτ+Rμτ,aρ+∫t1σt∂Bτ,aρ∂aμΔτ+∫t1σtλβ∂fβτ,aρ∂aμΔτδaμΔΔt=0We get(47)Rμ−∫t1σt∂Rυτ,aρ∂aμaυΔ−∂Bτ,aρ∂aμ−λβ∂fβτ,aρ∂aμΔτ=const.Lethσt=∫t1σt∂Rυτ,aρ/∂aμaυΔ−∂Bτ,aρ/∂aμ−λβ∂fβτ,aρ/∂aμΔτ, then
(48)Rμ−∫t1σt∂Rυτ,aρ∂aμaυΔ−∂Bτ,aρ∂aμ−λβ∂fβτ,aρ∂aμΔτ∇=Rμ∇−hσ∇t=Rλ∇−σ∇thΔt=Rμ∇−σ∇t∫aσt∂Rυ∂aμaυΔ−∂B∂aμ−λβ∂fβ∂aμΔτΔ=Rμ∇−σ∇t∂Rυ∂aμaυΔ−∂B∂aμ−λβ∂fβ∂aμ=0.Therefore, we get(49)σ∇t∂Rυ∂aμaυΔ−∂B∂aμ−λβ∂fβ∂aμ−Rμ∇=0μ,υ=1,2,⋯,2n;β=1,2,⋯,g.Equation (49) is called nonshifted constrained Birkhoff’s equations. If the system is nonsingular, by using equations (40) and (49), we can solve λβ=λβt,aρ. Equation (51) can be written as
(50)σ∇t∂Rυ∂aμaυΔ−∂B∂aμ−Pμ=Rμ∇,where
(51)Pμ=λβ∂fβ∂aμ.We call the system determined by equation (50) as the corresponding free Birkhoffian system.


### 4.2. Quasi-symmetry and Conserved Quantity

Definition 9.
If nonshifted Pfaff action (1) is a generalized quasi-invariant, in other words, for every infinitesimal transformations (3), the following relationship
(52)ΔS=−∫t1t2ΔΔtΔG+PυδaυΔt,always holds, the transformations (3) are referred to as generalized quasi-symmetric for the corresponding free Birkhoffian system (50).Criterion 10.
If the Noether identity(53)∂Rυ∂taυΔ−∂B∂tξ0+∂Rυ∂aμaυΔ−∂B∂aμξμ−Bξ0Δ+RυξυΔ+GΔ−Pυξυ−aυΔξ0=0,is satisfied, then transformations (3) are generalized quasi-symmetric for the corresponding free Birkhoffian system (50). If the restriction equation
(54)∂fβ∂aμξμ−aμΔξ0=0β=1,2,⋯,g,is also satisfied, then transformations (3) are generalized quasi-symmetric for the nonshifted constrained Birkhoffian system (40) and (49).Similar to the derivation of equation (29), from equation (52), we have
(55)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫t1t2Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔΔt+∫t1t2ΔΔtGt,aρ+Pυt,aρδaυΔtFrom equation (55), we have
(56)Rυt,aρaυΔ−Bt,aρ=Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔ+ΔΔtGt,aρ−Pυt,aρδaυ.Take the derivative ofε on both sides of equation (56) and set ε=0, we obtain equation (53).Theorem 11.
If the transformations (3) satisfy the Noether identity (53) and the restriction equation (54), then
(57)I=Rυξυσ+∫t1t∂Rυ∂τaυΔ−∂B∂τ+PυaυΔξ0−Bξ0Δσ∇∇τ+Gσ=const.is the conserved quantity of the system (40) and (49).Proof.
From equation (53), we have
(58)∂Rυ∂tξ0+∂Rυ∂aμξμaυΔσ∇+RυξυΔσ∇−Bξ0Δσ∇−∂B∂tξ0σ∇−∂B∂aμξμσ∇+GΔσ∇−Pυξυ−aυΔξ0σ∇=0.
Using equation (50), we have
(59)∂Rυ∂tξ0aυΔσ∇+ξ0B∇−ξ0σB∇−∂B∂tξ0σ∇+Rυξυσ∇+GΔσ∇−Pυξυ−aυΔξ0σ∇=0.By nabla indefinite integral of equation (59), we can get conserved quantity (57).Example 12.
LetT=2n:n∈ℕ∪0, the nonshifted constrained Birkhoffian system on time scales is
(60)B=12a12+a32+a42,R1=a3,R2=a4,R3=R4=0.The constraint equations are(61)f1=a1a3−c12=0,f2=a1+a4−c2=0.According to equation (49), we have
(62)a3∇=2−λ1a3−λ2−a1,a4∇=0,a1Δ−a3=λ1a1,a2Δ−a4=λ2.From equations (61) and (62), we have
(63)λ1=−a3a1,λ2=−a1+a32a1.Hence, we get(64)P1=−a1,P2=0,P3=−a3,P4=−a1+a32a1.According to equation (53), we have
(65)a1Δ−a3ξ3+a2Δ−a4ξ4−a1ξ1−Bξ0Δ+a3ξ1Δ+a4ξ2Δ−P1ξ1−a1Δξ0−P3ξ3−a3Δξ0−P4ξ4−a4Δξ0+GΔ=0.Equation (65) has the following solutions:
(66)ξ01=0,ξ11=0,ξ21=1,ξ31=0,ξ41=0,G1=0,(67)ξ02=0,ξ12=1,ξ22=0,ξ32=0,ξ42=0,G2=0.According to Theorem11, the conserved quantities corresponding to the generators (66) and (67) are
(68)I1=a4=const,(69)I2=a3=const.


## 4.1. Nonshifted Constrained Birkhoff’s Equations

If the variables in nonshifted Birkhoffian system are not independent of each other on time scales, but subject to some constraints, these constraints are shown as(40)fβt,aμ=0β=1,2,⋯,2n.To calculate the isochronous variation of equation (40), we have
(41)∂fβ∂aμδaμ=0.From equation (41), we can get
(42)∫t1t2λβ∂fβ∂aμδaμΔt=0.By integrating by parts on time scales with equation (42), we get
(43)∫t1t2λβ∂fβ∂aμδaμΔt=δaμ∫t1tλβ∂fβ∂aμ∇τt1t2−∫t1t2∫t1σtλβ∂fβ∂aμ∇τδaμΔΔt=0,i.e.,
(44)∫t1t2∫t1σtλβ∂fβ∂aμ∇τδaμΔΔt=0.According to nonshifted Pfaff-Birkhoff principleδS=0,we get [34].
(45)∫t1t2−∫t1σt∂Rυτ,aρ∂aμaυΔΔτ+Rμτ,aρ+∫t1σt∂Bτ,aρ∂aμΔτδaμΔΔt=0.Add equations (44) to (45), we have
(46)∫t1t2−∫t1σt∂Rυτ,aρ∂aμaυΔΔτ+Rμτ,aρ+∫t1σt∂Bτ,aρ∂aμΔτ+∫t1σtλβ∂fβτ,aρ∂aμΔτδaμΔΔt=0We get(47)Rμ−∫t1σt∂Rυτ,aρ∂aμaυΔ−∂Bτ,aρ∂aμ−λβ∂fβτ,aρ∂aμΔτ=const.Lethσt=∫t1σt∂Rυτ,aρ/∂aμaυΔ−∂Bτ,aρ/∂aμ−λβ∂fβτ,aρ/∂aμΔτ, then
(48)Rμ−∫t1σt∂Rυτ,aρ∂aμaυΔ−∂Bτ,aρ∂aμ−λβ∂fβτ,aρ∂aμΔτ∇=Rμ∇−hσ∇t=Rλ∇−σ∇thΔt=Rμ∇−σ∇t∫aσt∂Rυ∂aμaυΔ−∂B∂aμ−λβ∂fβ∂aμΔτΔ=Rμ∇−σ∇t∂Rυ∂aμaυΔ−∂B∂aμ−λβ∂fβ∂aμ=0.Therefore, we get(49)σ∇t∂Rυ∂aμaυΔ−∂B∂aμ−λβ∂fβ∂aμ−Rμ∇=0μ,υ=1,2,⋯,2n;β=1,2,⋯,g.Equation (49) is called nonshifted constrained Birkhoff’s equations. If the system is nonsingular, by using equations (40) and (49), we can solve λβ=λβt,aρ. Equation (51) can be written as
(50)σ∇t∂Rυ∂aμaυΔ−∂B∂aμ−Pμ=Rμ∇,where
(51)Pμ=λβ∂fβ∂aμ.We call the system determined by equation (50) as the corresponding free Birkhoffian system.


## 4.2. Quasi-symmetry and Conserved Quantity

Definition 9.
If nonshifted Pfaff action (1) is a generalized quasi-invariant, in other words, for every infinitesimal transformations (3), the following relationship
(52)ΔS=−∫t1t2ΔΔtΔG+PυδaυΔt,always holds, the transformations (3) are referred to as generalized quasi-symmetric for the corresponding free Birkhoffian system (50).Criterion 10.
If the Noether identity(53)∂Rυ∂taυΔ−∂B∂tξ0+∂Rυ∂aμaυΔ−∂B∂aμξμ−Bξ0Δ+RυξυΔ+GΔ−Pυξυ−aυΔξ0=0,is satisfied, then transformations (3) are generalized quasi-symmetric for the corresponding free Birkhoffian system (50). If the restriction equation
(54)∂fβ∂aμξμ−aμΔξ0=0β=1,2,⋯,g,is also satisfied, then transformations (3) are generalized quasi-symmetric for the nonshifted constrained Birkhoffian system (40) and (49).Similar to the derivation of equation (29), from equation (52), we have
(55)∫t1t2Rυt,aρaυΔ−Bt,aρΔt=∫t1t2Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔΔt+∫t1t2ΔΔtGt,aρ+Pυt,aρδaυΔtFrom equation (55), we have
(56)Rυt,aρaυΔ−Bt,aρ=Rυt+εξ0t,aρ,aρ+εξρt,aμaυ+εξυt,aρΔt+εξ0t,aρΔ−Bt+εξ0t,aρ,aρ+εξρt,aμt+εξ0t,aρΔ+ΔΔtGt,aρ−Pυt,aρδaυ.Take the derivative ofε on both sides of equation (56) and set ε=0, we obtain equation (53).Theorem 11.
If the transformations (3) satisfy the Noether identity (53) and the restriction equation (54), then
(57)I=Rυξυσ+∫t1t∂Rυ∂τaυΔ−∂B∂τ+PυaυΔξ0−Bξ0Δσ∇∇τ+Gσ=const.is the conserved quantity of the system (40) and (49).Proof.
From equation (53), we have
(58)∂Rυ∂tξ0+∂Rυ∂aμξμaυΔσ∇+RυξυΔσ∇−Bξ0Δσ∇−∂B∂tξ0σ∇−∂B∂aμξμσ∇+GΔσ∇−Pυξυ−aυΔξ0σ∇=0.
Using equation (50), we have
(59)∂Rυ∂tξ0aυΔσ∇+ξ0B∇−ξ0σB∇−∂B∂tξ0σ∇+Rυξυσ∇+GΔσ∇−Pυξυ−aυΔξ0σ∇=0.By nabla indefinite integral of equation (59), we can get conserved quantity (57).Example 12.
LetT=2n:n∈ℕ∪0, the nonshifted constrained Birkhoffian system on time scales is
(60)B=12a12+a32+a42,R1=a3,R2=a4,R3=R4=0.The constraint equations are(61)f1=a1a3−c12=0,f2=a1+a4−c2=0.According to equation (49), we have
(62)a3∇=2−λ1a3−λ2−a1,a4∇=0,a1Δ−a3=λ1a1,a2Δ−a4=λ2.From equations (61) and (62), we have
(63)λ1=−a3a1,λ2=−a1+a32a1.Hence, we get(64)P1=−a1,P2=0,P3=−a3,P4=−a1+a32a1.According to equation (53), we have
(65)a1Δ−a3ξ3+a2Δ−a4ξ4−a1ξ1−Bξ0Δ+a3ξ1Δ+a4ξ2Δ−P1ξ1−a1Δξ0−P3ξ3−a3Δξ0−P4ξ4−a4Δξ0+GΔ=0.Equation (65) has the following solutions:
(66)ξ01=0,ξ11=0,ξ21=1,ξ31=0,ξ41=0,G1=0,(67)ξ02=0,ξ12=1,ξ22=0,ξ32=0,ξ42=0,G2=0.According to Theorem11, the conserved quantities corresponding to the generators (66) and (67) are
(68)I1=a4=const,(69)I2=a3=const.


## 5. Conclusions

Time scale has been widely used in many fields. At present, most of the researches on time scales are about the shifted case. In this article, we studied the time-scale version of the nonshifted variational problem for three types of Birkhoffian systems. We proposed the nonshifted generalized Pfaff-Birkhoff principle, derived nonshifted generalized and constrained Birkhoff’s equation, studied Noether quasi-symmetries for these nonshifted Birkhoffian systems, and gave the condition of the symmetry resulting in conserved quantity and obtained conserved quantities for these nonshifted Birkhoffian systems on time scales. According to this passage, we also will research symmetries and time-scale conserved quantities for other nonshifted dynamical systems, including Lie and Mei symmetries.



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