# Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

Authors: Yi Zhang
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

The Mei symmetry and conservation laws for time-scale nonshifted Hamilton equations are explored, and the Mei symmetry theorem is presented and proved. Firstly, the time-scale Hamilton principle is established and extended to the nonconservative case. Based on the Hamilton principles, the dynamic equations of time-scale nonshifted constrained mechanical systems are derived. Secondly, for the time-scale nonshifted Hamilton equations, the definitions of Mei symmetry and their criterion equations are given. Thirdly, Mei symmetry theorems are proved, and the Mei-type conservation laws in time-scale phase space are driven. Two examples show the validity of the results.


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## Body

## 1. Introduction

As we all know, the symmetry property of dynamic systems is the invariance of some physical quantity to the infinitesimal transformations of a group in mathematical form and can be expressed as a conservation law in physics [1–5]. The classical symmetries we are familiar with include Noether symmetry and Lie symmetry. Mathematically, the Noether symmetry is the invariance of the action functional under the infinitesimal transformations of a group, and physically, it is the Noether conservation laws [4, 5]. The Lie symmetry is the invariance of the differential equation under the infinitesimal transformations of a group [2, 6, 7]. Correspondingly, the Hojman conservation laws can be derived [5, 8, 9]. The Mei symmetry we are going to study was first proposed by Professor Mei in 2000 [10] and later popularized by many scholars [11–17]. Compared with Noether or Lie symmetry, Mei symmetry is a new kind of symmetry, and it refers to the invariance that the dynamic functions after infinitesimal transformation still make the dynamic equations hold. From Mei symmetry, a new kind of conservation law can be brought about, which is different from the Noether or Hojman one and called the Mei conserved quantity. The Mei symmetry theorem has been extended to fractional-order mechanical systems [18, 19] and nonstandard Lagrangian dynamics [20]. Recently, we studied Mei symmetries on time scales [21, 22], but the research is preliminary and limited to Lagrange equations and Birkhoff equations.The time scale, namely, any nonempty closed subset of the real number set, was first introduced by Dr. Hilger [23]. Since the real numbers and the integers themselves are special time scales, it is possible to deal with not only the continuous system and the discrete system uniformly but also the complex dynamic processes that have both continuity and discretization in time scales. In the past three decades, the time-scale analysis has been continuously improved and its application has been expanded [24–30]. Bartosiewicz and Torres presented Noether’s theorem [31] and the second Euler-Lagrange equations [32] for the time-scale shifted Lagrange systems. Anerot and his collaborators [33] proved Noether’s theorem in the time-scale version of Lagrange systems under the framework of shifted and nonshifted delta calculus of variation, and the results are also a correction of References [31, 32]. In the past decade, the study of time-scale dynamics and its symmetry has attracted extensive attention and made important progress, as shown in References [34–44]. However, the research is mainly limited to the following: (1) conservative system, (2) Noether symmetry, and (3) Noether-type conservation laws. Furthermore, according to Bourdin’s study [45], the results of the nonshifted case at the discrete level maintain the variational structure and related properties, although so far there have been few studies on the time-scale nonshifted variational problem.This paper will focus on exploring Mei symmetry for time-scale nonshifted general holonomic systems and nonholonomic systems under the Hamiltonian framework, prove Mei symmetry theorems, and derive Mei conservation laws. The time-scale Mei symmetry is different from Noether or Lie symmetry on time scales. Mathematically, it is a new symmetry under the infinitesimal transformations of a group. Physically, it leads to a new kind of conservation law. The time-scale Mei symmetry not only unifies the Mei symmetry of the continuous case and the discrete case but also can obtain various discrete models due to the arbitrariness of time scales, which provides a new perspective for the structure-preserving algorithm of mechanical systems. Up to now, although there have been many research studies on Noether or Lie symmetry on time scales, the research on time-scale nonshifted Mei symmetry has not been reported.


## 2. Time-Scale Nonshifted Dynamic Equations

For time-scale calculus and its basic properties, the reader is recommended to refer to [24, 25].


### 2.1. Hamilton Principle and Its Extension

The time-scale nonshifted Hamilton action is(1)A=∫t1t2piqiΔ−Ht,qit,pitΔt,where H:T×ℝn×ℝn⟶ℝ is the Hamiltonian, qiΔt is the generalized velocity, i.e., delta derivative of generalized coordinate qit with respect to time t, and pit is the generalized momentum, i=1,2,⋯,n. All functions are of Crd1,ΔT. We refer to equation (1) as nonshifted Hamilton action in which the variables are qi and pi (instead of qiσ and piσ or qiρ and piρ) [45], where σ is the forward jump operator and ρ is the backward jump operator.The variational principle(2)δA=0,with the endpoint conditions
(3)δqit=t1=δqit=t2=0,and the commutative relation
(4)δqiΔ=δqiΔ,δpiΔ=δpiΔ,is called the time-scale nonshifted Hamilton principle.Principle (2) can be generalized as
(5)∫t1t2δpiqiΔ−H+QiδqiΔt=0,where Qi=Qit,qj,pj are nonpotential generalized forces. Principle (5) is the nonshifted Hamilton principle for time-scale general holonomic systems.


### 2.2. Hamiltonian System

From principle (2), it is easy to derive
(6)qiΔ=∂H∂pi,pi∇=−σ∇∂H∂qi.Equation (6) contains nonshifted Hamilton canonical equations for the time-scale Hamiltonian system.If we takeT=ℝ, then the equations in (6) are reduced to the classical Hamilton canonical equations.If we takeT=ℤ, then the equations in (6) become
(7)Δqik=∂Htk,qjk,pjk∂pik,∇pik=−∂Htk,qjk,pjk∂qik,where Δ is the forward difference and ∇ is the backward difference.


### 2.3. General Holonomic System

From principle (5), it is easy to derive
(8)qiΔ=∂H∂pi,pi∇=−σ∇∂H∂qi+σ∇Qi.Equation (8) contains nonshifted Hamilton equations for the time-scale general holonomic system.


### 2.4. Nonholonomic System

The nonholonomic constraints are(9)ϕαt,qi,qiΔ=0α=1,2,⋯,m.The restriction applied by constraints in (9) on virtual displacements δqi is
(10)Φαit,qj,qjΔδqi=0.IfΦαi is independent of ∂ϕα/∂qiΔ, then formula (9) represents non-Chetaev constraints, and if Φαi=∂ϕα/∂qiΔ, then formula (9) represents Chetaev constraints.Assume that the Lagrangian isL=Lt,qi,qiΔ, and let pi=∂L/∂qiΔ, and then qiΔ=qiΔt,qj,pj. Thus, equations (9) and (10) can be written as
(11)ϕα∗t,qi,pi=0,(12)Φαi∗t,qj,pjδqi=0.The nonshifted Hamilton equations can be expressed as(13)qiΔ=∂H∂pi,pi∇=−σ∇∂H∂qi+σ∇Qi+σ∇λαΦαi∗.If the system is nondegenerate, the constraint multiplierλα can be solved by equations (11) and (13) as a function of generalized coordinates, generalized momentum, and time. Thus, equation (13) can be written as
(14)qiΔ=∂H∂pi,pi∇=−σ∇∂H∂qi+σ∇Qi+Πi,where Πi=λαΦαi∗ represent the constraining forces corresponding to the nonholonomic constraints. Equation (14) can be treated as a holonomic system corresponding to the nonholonomic system determined by (11) and (13). If constraint equation (11) is initially satisfied, then the motion of the nonholonomic system determined by (11) and (13) is solved from equation (14).


## 2.1. Hamilton Principle and Its Extension

The time-scale nonshifted Hamilton action is(1)A=∫t1t2piqiΔ−Ht,qit,pitΔt,where H:T×ℝn×ℝn⟶ℝ is the Hamiltonian, qiΔt is the generalized velocity, i.e., delta derivative of generalized coordinate qit with respect to time t, and pit is the generalized momentum, i=1,2,⋯,n. All functions are of Crd1,ΔT. We refer to equation (1) as nonshifted Hamilton action in which the variables are qi and pi (instead of qiσ and piσ or qiρ and piρ) [45], where σ is the forward jump operator and ρ is the backward jump operator.The variational principle(2)δA=0,with the endpoint conditions
(3)δqit=t1=δqit=t2=0,and the commutative relation
(4)δqiΔ=δqiΔ,δpiΔ=δpiΔ,is called the time-scale nonshifted Hamilton principle.Principle (2) can be generalized as
(5)∫t1t2δpiqiΔ−H+QiδqiΔt=0,where Qi=Qit,qj,pj are nonpotential generalized forces. Principle (5) is the nonshifted Hamilton principle for time-scale general holonomic systems.


## 2.2. Hamiltonian System

From principle (2), it is easy to derive
(6)qiΔ=∂H∂pi,pi∇=−σ∇∂H∂qi.Equation (6) contains nonshifted Hamilton canonical equations for the time-scale Hamiltonian system.If we takeT=ℝ, then the equations in (6) are reduced to the classical Hamilton canonical equations.If we takeT=ℤ, then the equations in (6) become
(7)Δqik=∂Htk,qjk,pjk∂pik,∇pik=−∂Htk,qjk,pjk∂qik,where Δ is the forward difference and ∇ is the backward difference.


## 2.3. General Holonomic System

From principle (5), it is easy to derive
(8)qiΔ=∂H∂pi,pi∇=−σ∇∂H∂qi+σ∇Qi.Equation (8) contains nonshifted Hamilton equations for the time-scale general holonomic system.


## 2.4. Nonholonomic System

The nonholonomic constraints are(9)ϕαt,qi,qiΔ=0α=1,2,⋯,m.The restriction applied by constraints in (9) on virtual displacements δqi is
(10)Φαit,qj,qjΔδqi=0.IfΦαi is independent of ∂ϕα/∂qiΔ, then formula (9) represents non-Chetaev constraints, and if Φαi=∂ϕα/∂qiΔ, then formula (9) represents Chetaev constraints.Assume that the Lagrangian isL=Lt,qi,qiΔ, and let pi=∂L/∂qiΔ, and then qiΔ=qiΔt,qj,pj. Thus, equations (9) and (10) can be written as
(11)ϕα∗t,qi,pi=0,(12)Φαi∗t,qj,pjδqi=0.The nonshifted Hamilton equations can be expressed as(13)qiΔ=∂H∂pi,pi∇=−σ∇∂H∂qi+σ∇Qi+σ∇λαΦαi∗.If the system is nondegenerate, the constraint multiplierλα can be solved by equations (11) and (13) as a function of generalized coordinates, generalized momentum, and time. Thus, equation (13) can be written as
(14)qiΔ=∂H∂pi,pi∇=−σ∇∂H∂qi+σ∇Qi+Πi,where Πi=λαΦαi∗ represent the constraining forces corresponding to the nonholonomic constraints. Equation (14) can be treated as a holonomic system corresponding to the nonholonomic system determined by (11) and (13). If constraint equation (11) is initially satisfied, then the motion of the nonholonomic system determined by (11) and (13) is solved from equation (14).


## 3. Mei Symmetry

### 3.1. Hamiltonian System

Introduce the infinitesimal transformations of the group:(15)t¯=t+υξ0t,qi,pi,q¯it¯=qit+υξit,qj,pj,p¯it¯=pit+υζit,qj,pji,j=1,2,⋯,n,where υ∈ℝ is a small parameter. Under the transformations in (15), the Hamiltonian H is transformed into H¯, and then
(16)H¯=Ht¯,q¯it¯,p¯it¯=Ht,qit,pit+υY0H+Oυ2,where Y0=ξ0∂/∂t+ξi∂/∂qi+ζi∂/∂pi.Definition 1.
For the time-scale nonshifted Hamiltonian system (6), if
(17)qiΔ=∂H¯∂pi,pi∇=−σ∇∂H¯∂qi,hold, then the transformations in (15) are said to be Mei symmetric.Criterion 2.
If the transformations in (15) satisfy the following criterion equations
(18)∂Y0H∂pi=0,−σ∇∂Y0H∂qi=0,then they correspond to the Mei symmetry of the time-scale Hamiltonian system (6).


### 3.2. General Holonomic System

Assume that the time-scale HamiltonianH and the time-scale generalized forces Qi undergo transformations in (15) to become H¯ and Q¯i, where
(19)H¯=Ht,qi,pi+υY0H+Oυ2,Q¯i=Qit,qj,pj+υY0Qi+Oυ2,and then we have the following.Definition 3.
For the time-scale nonshifted general holonomic system (8), if
(20)qiΔ=∂H¯∂pi,pi∇=−σ∇∂H¯∂qi+σ∇Q¯i,hold, then the transformations in (15) are said to be Mei symmetric.Criterion 4.
If the transformations in (15) satisfy the following criterion equations
(21)∂Y0H∂pi=0,−σ∇∂Y0H∂qi+σ∇Y0Qi=0,then they correspond to the Mei symmetry of the time-scale general holonomic system (8).


### 3.3. Nonholonomic System

Assume that the HamiltonianH, the generalized forces Qi, the constraining forces Πi, and the constraints ϕα∗ on time scales undergo transformations in (15) to become
(22)H¯=Ht,qi,pi+υY0H+Oυ2,Q¯i=Qit,qj,pj+υY0Qi+Oυ2,Π¯i=Πit,qj,pj+υY0Πi+Oυ2,ϕ¯α∗=ϕα∗t,qj,pj+υY0ϕα∗+Oυ2,and then we have the following.Definition 5.
For the time-scale corresponding holonomic system (14), if
(23)qiΔ=∂H¯∂pi,pi∇=−σ∇∂H¯∂qi+σ∇Q¯i+Π¯i,hold, then the transformations in (15) are said to be Mei symmetric.Criterion 6.
If the transformations in (15) satisfy the following criterion equations
(24)∂Y0H∂pi=0,−σ∇∂Y0H∂qi+σ∇Y0Qi+Y0Πi=0,then they correspond to the Mei symmetry of the time-scale corresponding holonomic system (14).Definition 7.
For the time-scale nonholonomic system determined by (11) and (13), if equation (23) and the following equation
(25)ϕ¯α∗=ϕα∗t¯,q¯it¯,p¯it¯=0,hold, then the transformations in (15) are said to be Mei symmetric.Criterion 8.
If the transformations in (15) satisfy criterion equation (24) and the following restriction equation
(26)Y0ϕα∗=0,then they correspond to the Mei symmetry of the time-scale nonholonomic system determined by (11) and (13).


## 3.1. Hamiltonian System

Introduce the infinitesimal transformations of the group:(15)t¯=t+υξ0t,qi,pi,q¯it¯=qit+υξit,qj,pj,p¯it¯=pit+υζit,qj,pji,j=1,2,⋯,n,where υ∈ℝ is a small parameter. Under the transformations in (15), the Hamiltonian H is transformed into H¯, and then
(16)H¯=Ht¯,q¯it¯,p¯it¯=Ht,qit,pit+υY0H+Oυ2,where Y0=ξ0∂/∂t+ξi∂/∂qi+ζi∂/∂pi.Definition 1.
For the time-scale nonshifted Hamiltonian system (6), if
(17)qiΔ=∂H¯∂pi,pi∇=−σ∇∂H¯∂qi,hold, then the transformations in (15) are said to be Mei symmetric.Criterion 2.
If the transformations in (15) satisfy the following criterion equations
(18)∂Y0H∂pi=0,−σ∇∂Y0H∂qi=0,then they correspond to the Mei symmetry of the time-scale Hamiltonian system (6).


## 3.2. General Holonomic System

Assume that the time-scale HamiltonianH and the time-scale generalized forces Qi undergo transformations in (15) to become H¯ and Q¯i, where
(19)H¯=Ht,qi,pi+υY0H+Oυ2,Q¯i=Qit,qj,pj+υY0Qi+Oυ2,and then we have the following.Definition 3.
For the time-scale nonshifted general holonomic system (8), if
(20)qiΔ=∂H¯∂pi,pi∇=−σ∇∂H¯∂qi+σ∇Q¯i,hold, then the transformations in (15) are said to be Mei symmetric.Criterion 4.
If the transformations in (15) satisfy the following criterion equations
(21)∂Y0H∂pi=0,−σ∇∂Y0H∂qi+σ∇Y0Qi=0,then they correspond to the Mei symmetry of the time-scale general holonomic system (8).


## 3.3. Nonholonomic System

Assume that the HamiltonianH, the generalized forces Qi, the constraining forces Πi, and the constraints ϕα∗ on time scales undergo transformations in (15) to become
(22)H¯=Ht,qi,pi+υY0H+Oυ2,Q¯i=Qit,qj,pj+υY0Qi+Oυ2,Π¯i=Πit,qj,pj+υY0Πi+Oυ2,ϕ¯α∗=ϕα∗t,qj,pj+υY0ϕα∗+Oυ2,and then we have the following.Definition 5.
For the time-scale corresponding holonomic system (14), if
(23)qiΔ=∂H¯∂pi,pi∇=−σ∇∂H¯∂qi+σ∇Q¯i+Π¯i,hold, then the transformations in (15) are said to be Mei symmetric.Criterion 6.
If the transformations in (15) satisfy the following criterion equations
(24)∂Y0H∂pi=0,−σ∇∂Y0H∂qi+σ∇Y0Qi+Y0Πi=0,then they correspond to the Mei symmetry of the time-scale corresponding holonomic system (14).Definition 7.
For the time-scale nonholonomic system determined by (11) and (13), if equation (23) and the following equation
(25)ϕ¯α∗=ϕα∗t¯,q¯it¯,p¯it¯=0,hold, then the transformations in (15) are said to be Mei symmetric.Criterion 8.
If the transformations in (15) satisfy criterion equation (24) and the following restriction equation
(26)Y0ϕα∗=0,then they correspond to the Mei symmetry of the time-scale nonholonomic system determined by (11) and (13).


## 4. Mei Symmetry Theorems

### 4.1. Hamiltonian System

Theorem 9.
Assuming that the transformations in (15) satisfy criterion equation (18), then the time-scale nonshifted Hamiltonian system (6) has a new conservation law, such as
(27)IM=∫t1tξ0−σ∇∂Y0H∂t+∇∇tY0H∇t+Y0piξiσ−Y0Hξ0σ+GMσ,where GM is the gauge function that satisfies
(28)ΔΔtY0piρξi+Y0piξiΔ−Y0Y0H−Y0Hξ0Δ+GMΔ=0.Proof.
(29)∇∇tIM=−ξ0σ∇∂Y0H∂t+ξ0∇∇tY0H+ξi∇∇tY0pi+σ∇ξiΔY0pi−ξ0∇∇tY0H−σ∇ξ0ΔY0H+∇∇tGMσ=σ∇Y0piξiΔ+ΔΔtY0piρξi−Y0Y0H−Y0Hξ0Δ+GMΔ+ξiσ∇∂Y0H∂qi+ζiσ∇∂Y0H∂pi.
Substituting equations (18) and (28) into (29), we get
(30)∇∇tIM=0.
Therefore, equation (27) is a conservation law.Theorem9 is the Mei symmetry theorem for the time-scale nonshifted Hamiltonian system (6), and equation (27) is called a Mei conservation law.


### 4.2. General Holonomic System

Theorem 10.
Assuming that the transformations in (15) satisfy criterion equation (21), then the time-scale nonshifted general holonomic system (8) has a new conservation law, such as
(31)IM=∫t1tξ0−σ∇Y0QiqiΔ−σ∇∂Y0H∂t+∇∇tY0H∇t+Y0piξiσ−Y0Hξ0σ+GMσ,where GM is the gauge function that satisfies
(32)ΔΔtY0piρξi+Y0piξiΔ−Y0Y0H−Y0Hξ0Δ+Y0Qiξi−qiΔξ0+GMΔ=0.Proof.
(33)∇∇tIM=−σ∇Y0QiqiΔ−ξ0σ∇∂Y0H∂t+ξ0∇∇tY0H+ξi∇∇tY0pi+σ∇ξiΔY0pi−ξ0∇∇tY0H−σ∇ξ0ΔY0H+∇∇tGMσ=σ∇Y0piξiΔ+ΔΔtY0piρξi−Y0Y0H−Y0Hξ0Δ+GMΔ+Y0Qiξi−qiΔξ0+ξiσ∇∂Y0H∂qi−Y0Qi+ζiσ∇∂Y0H∂pi.
Substituting equations (21) and (32) into (33), we obtain ∇/∇tIM=0. So the theorem is proved.Theorem10 is the Mei symmetry theorem for the time-scale nonshifted general holonomic system (8).


### 4.3. Nonholonomic System

Theorem 11.
Assuming that the transformations in (15) satisfy criterion equation (24), then the time-scale corresponding holonomic system (14) has a new conservation law, such as
(34)IM=∫t1tξ0−σ∇Y0Qi+Y0ΠiqiΔ−σ∇∂Y0H∂t+∇∇tY0H∇t+Y0piξiσ−Y0Hξ0σ+GMσ,where GM is the gauge function that satisfies
(35)ΔΔtY0piρξi+Y0piξiΔ−Y0Y0H−Y0Hξ0Δ+Y0Qi+Y0Πiξi−qiΔξ0+GMΔ=0.Theorem 12.
Assuming that the transformations in (15) satisfy equations (24) and (26), then the time-scale nonholonomic system determined by (11) and (13) has a new conservation law (34), where the gauge function GM satisfies equation (35).We call Theorem12 the Mei symmetry theorem for time-scale nonshifted nonholonomic systems determined by (11) and (13). Here, Theorem 11 establishes the relationship between the Mei symmetry and the conservation law of time-scale Hamilton equations in (14).


## 4.1. Hamiltonian System

Theorem 9.
Assuming that the transformations in (15) satisfy criterion equation (18), then the time-scale nonshifted Hamiltonian system (6) has a new conservation law, such as
(27)IM=∫t1tξ0−σ∇∂Y0H∂t+∇∇tY0H∇t+Y0piξiσ−Y0Hξ0σ+GMσ,where GM is the gauge function that satisfies
(28)ΔΔtY0piρξi+Y0piξiΔ−Y0Y0H−Y0Hξ0Δ+GMΔ=0.Proof.
(29)∇∇tIM=−ξ0σ∇∂Y0H∂t+ξ0∇∇tY0H+ξi∇∇tY0pi+σ∇ξiΔY0pi−ξ0∇∇tY0H−σ∇ξ0ΔY0H+∇∇tGMσ=σ∇Y0piξiΔ+ΔΔtY0piρξi−Y0Y0H−Y0Hξ0Δ+GMΔ+ξiσ∇∂Y0H∂qi+ζiσ∇∂Y0H∂pi.
Substituting equations (18) and (28) into (29), we get
(30)∇∇tIM=0.
Therefore, equation (27) is a conservation law.Theorem9 is the Mei symmetry theorem for the time-scale nonshifted Hamiltonian system (6), and equation (27) is called a Mei conservation law.


## 4.2. General Holonomic System

Theorem 10.
Assuming that the transformations in (15) satisfy criterion equation (21), then the time-scale nonshifted general holonomic system (8) has a new conservation law, such as
(31)IM=∫t1tξ0−σ∇Y0QiqiΔ−σ∇∂Y0H∂t+∇∇tY0H∇t+Y0piξiσ−Y0Hξ0σ+GMσ,where GM is the gauge function that satisfies
(32)ΔΔtY0piρξi+Y0piξiΔ−Y0Y0H−Y0Hξ0Δ+Y0Qiξi−qiΔξ0+GMΔ=0.Proof.
(33)∇∇tIM=−σ∇Y0QiqiΔ−ξ0σ∇∂Y0H∂t+ξ0∇∇tY0H+ξi∇∇tY0pi+σ∇ξiΔY0pi−ξ0∇∇tY0H−σ∇ξ0ΔY0H+∇∇tGMσ=σ∇Y0piξiΔ+ΔΔtY0piρξi−Y0Y0H−Y0Hξ0Δ+GMΔ+Y0Qiξi−qiΔξ0+ξiσ∇∂Y0H∂qi−Y0Qi+ζiσ∇∂Y0H∂pi.
Substituting equations (21) and (32) into (33), we obtain ∇/∇tIM=0. So the theorem is proved.Theorem10 is the Mei symmetry theorem for the time-scale nonshifted general holonomic system (8).


## 4.3. Nonholonomic System

Theorem 11.
Assuming that the transformations in (15) satisfy criterion equation (24), then the time-scale corresponding holonomic system (14) has a new conservation law, such as
(34)IM=∫t1tξ0−σ∇Y0Qi+Y0ΠiqiΔ−σ∇∂Y0H∂t+∇∇tY0H∇t+Y0piξiσ−Y0Hξ0σ+GMσ,where GM is the gauge function that satisfies
(35)ΔΔtY0piρξi+Y0piξiΔ−Y0Y0H−Y0Hξ0Δ+Y0Qi+Y0Πiξi−qiΔξ0+GMΔ=0.Theorem 12.
Assuming that the transformations in (15) satisfy equations (24) and (26), then the time-scale nonholonomic system determined by (11) and (13) has a new conservation law (34), where the gauge function GM satisfies equation (35).We call Theorem12 the Mei symmetry theorem for time-scale nonshifted nonholonomic systems determined by (11) and (13). Here, Theorem 11 establishes the relationship between the Mei symmetry and the conservation law of time-scale Hamilton equations in (14).


## 5. Examples

Example 1.
The time scale isT=2m:m∈ℕ0, and the Hamiltonian is
(36)H=12p12+p22+q1.
According to equation (6), we get
(37)q1Δ=p1,q2Δ=p2,p1∇=−σ∇,p2∇=0.
To do the calculation, we have(38)Y0H=ξ1+ζ1p1+ζ2p2.
From equation (18), we obtain the following criterion equations:
(39)∂ξ1∂p1+∂ζ1∂p1p1+ζ1+∂ζ2∂p1p2=0,∂ξ1∂p2+∂ζ1∂p2p1+∂ζ2∂p2p2+ζ2=0,∂ξ1∂q1+∂ζ1∂q1p1+∂ζ2∂q1p2=0,∂ξ1∂q2+∂ζ1∂q2p1+∂ζ2∂q2p2=0.
From equation (39), we have
(40)ξ01=0,ξ11=−p2,ξ21=p1ρ,ζ11=0,ζ21=1,(41)ξ02=0,ξ12=−p1−p2,ξ22=p1ρ+p2ρ,ζ12=1,ζ22=1.
Substituting (40) and (41) into equation (28) and using equation (37), we can get
(42)GM1=t,GM2=p1+p2+t.
According to Theorem9, we have
(43)IM1=p1+σt=const.,IM2=p1+p2+σt=const.,where σt=2t for T=2ℕ0. These are the two Mei conserved quantities of the system.Example 2.
We investigate a time-scale nonholonomic system whose Lagrangian function is(44)L=12q1Δ2+q2Δ2+q3Δ2−q1,and the constraint equation is
(45)ϕ=q2Δ−tq1Δ=0,and the virtual displacements of the system satisfy
(46)δq1−δq2=0.
The generalized momentum and the Hamiltonian are(47)p1=∂L∂q1Δ=q1Δ,p2=∂L∂q2Δ=q2Δ,p3=∂L∂q3Δ=q3Δ,H=12p12+p22+p32+q1.
In canonical coordinates, constraint equation (45) becomes
(48)ϕ∗=p2−tp1=0.
The time-scale dynamic equations are(49)q1Δ=p1,q2Δ=p2,q3Δ=p3,p1∇=−σ∇+σ∇λ,p2∇=−σ∇λ,p3∇=0.
From equations (48) and (49), we can get
(50)λ=tρσ∇−p1σ∇1+tρ,and therefore, the nonholonomic constraining forces are
(51)Π1=tρσ∇−p1σ∇1+tρ,Π2=−tρσ∇−p1σ∇1+tρ,Π3=0.
Thus, equations in (49) become
(52)q1Δ=p1,q2Δ=p2,q3Δ=p3,p1∇=−σ∇+tρσ∇−p11+tρ,p2∇=−tρσ∇−p11+tρ,p3∇=0.
To do the calculation, we obtain(53)Y0H=ξ1+ζ1p1+ζ2p2+ζ3p3,Y0Π1=−Y0Π2=ξ0∂∂ttρσ∇−p1σ∇1+tρ−ζ1σ∇1+tρ,Y0Π3=0,Y0ϕ∗=ζ2−ξ0p1−tζ1.
Take the generators as(54)ξ0=0,ξ1=−p2+p3,ξ2=p1ρ+p2ρ,ξ3=p3ρ,ζ1=0,ζ2=1,ζ3=−1.
The generators in (54) satisfy equation (24), according to Criterion 6, which correspond to the Mei symmetry of the corresponding holonomic system (52). However, the generators in (54) do not satisfy Y0ϕ∗=0, so they are not the Mei symmetry of the nonholonomic system determined by (48) and (49). Substituting equation (54) into equation (35), we solve
(55)GM=t.
So from Theorem11, we have
(56)IM=p1+p2−p3+σt=const.
This is the Mei conserved quantity of the corresponding holonomic system (52).
If we take the generators as(57)ξ0=0,ξ1=−p3,ξ2=0,ξ3=p3ρ+t,ζ1=0,ζ2=0,ζ3=1.
According to Criterion8, the generators in (57) correspond to the Mei symmetry of the nonholonomic system determined by (48) and (49). Substituting equation (57) into equation (35), we get
(58)GM=−t.
From Theorem12, we have
(59)IM=p3=const.


## 6. Conclusions

The study of symmetry and conservation laws has always been an important aspect of analytical mechanics. Noether symmetry and Lie symmetry are two different classical symmetries. Mei symmetry is essentially different from these two, and it can directly yield a Mei conservation law. In this paper, we proved Mei symmetry theorems for time-scale nonshifted mechanical systems under the Hamiltonian framework. The main work of this paper is to extend Mei’s concept of form invariance to arbitrary time-scale frameworks and obtain the new conservation laws for time-scale nonshifted mechanical systems. The details are as follows.Firstly, we proposed the time-scale Hamilton principles (2) and (5) and derived the time-scale dynamic equations (6), (8), and (13). Secondly, we defined Mei symmetry for time-scale mechanical systems under the Hamiltonian framework and derived its criterion equations. Thirdly, we proposed and proved Mei symmetry theorems for time-scale mechanical systems under the Hamiltonian framework and obtained the Mei conservation laws (27), (31), and (34).If we takeT=ℝ, then we have σt=t and μt=0, so the above results are reduced to the continuous versions of the variational principle, dynamic equations, and Mei symmetry theorems under the Hamiltonian framework.If we takeT=hℤ, then we have σt=t+h and μt=h, so the above results are reduced to the discrete versions of the variational principle, dynamic equations, and Mei symmetry theorems with step size h.Since there are many alternatives to real and integer numbers on the time scale, our results are general.



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*Source: 7329399-2021-11-17.xml*