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

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.


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][2][3][4][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][12][13][14][15][16][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 nonstan-dard 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][25][26][27][28][29][30]. Bartosiewicz and Torres presented Noether's theorem [31] and the second Euler-Lagrange equations [32] for the timescale 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][35][36][37][38][39][40][41][42][43][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.

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

Hamilton Principle and Its Extension. The time-scale nonshifted Hamilton action is
where H : T × ℝ n × ℝ n ⟶ ℝ is the Hamiltonian, q Δ i ðtÞ is the generalized velocity, i.e., delta derivative of generalized coordinate q i ðtÞ with respect to time t, and p i ðtÞ is the generalized momentum, i = 1, 2, ⋯, n. All functions are of C 1,Δ rd ðT Þ. We refer to equation (1) as nonshifted Hamilton action in which the variables are q i and p i (instead of q σ i and p σ i or q ρ i and p ρ i ) [45], where σ is the forward jump operator and ρ is the backward jump operator.
The variational principle with the endpoint conditions and the commutative relation is called the time-scale nonshifted Hamilton principle.
Principle (2) can be generalized as where Q i = Q i ðt, q j , p j Þ are nonpotential generalized forces.
Principle (5) is the nonshifted Hamilton principle for timescale general holonomic systems.

Hamiltonian
System. From principle (2), it is easy to derive Equation (6) contains nonshifted Hamilton canonical equations for the time-scale Hamiltonian system.
If we take T = ℝ, then the equations in (6) are reduced to the classical Hamilton canonical equations.
If we take T = ℤ, then the equations in (6) become where Δ is the forward difference and ∇ is the backward difference.
2.3. General Holonomic System. From principle (5), it is easy to derive Equation (8) contains nonshifted Hamilton equations for the time-scale general holonomic system.

Nonholonomic System. The nonholonomic constraints are
The restriction applied by constraints in (9) on virtual displacements δq i is
Assume that the Lagrangian is L = Lðt, q i , q Δ i Þ, and let p i = ∂L/∂q Δ i , and then q Δ i = q Δ i ðt, q j , p j Þ. Thus, equations (9) and (10) can be written as The nonshifted Hamilton equations can be expressed as If the system is nondegenerate, the constraint multiplier λ α 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 where Π i = λ α Φ * αi 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).

Mei Symmetry
where υ ∈ ℝ is a small parameter. Under the transformations in (15), the Hamiltonian H is transformed into H, and then where hold, then the transformations in (15) are said to be Mei symmetric.

Criterion 2. If the transformations in (15) satisfy the following criterion equations
then they correspond to the Mei symmetry of the time-scale Hamiltonian system (6).

General Holonomic System. Assume that the time-scale
Hamiltonian H and the time-scale generalized forces Q i undergo transformations in (15) to become H and and then we have the following.
hold, then the transformations in (15) are said to be Mei symmetric.

Criterion 4. If the transformations in (15) satisfy the following criterion equations
then they correspond to the Mei symmetry of the time-scale general holonomic system (8).

Nonholonomic
System. Assume that the Hamiltonian H, the generalized forces Q i , the constraining forces Π i , and the constraints ϕ * α on time scales undergo transformations in (15) and then we have the following.

Definition 5.
For the time-scale corresponding holonomic system (14), if hold, then the transformations in (15) are said to be Mei symmetric.
Criterion 6. If the transformations in (15) satisfy the following criterion equations 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 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 then they correspond to the Mei symmetry of the time-scale nonholonomic system determined by (11) and (13).

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 where G M is the gauge function that satisfies Proof.
Theorem 9 is the Mei symmetry theorem for the time-scale nonshifted Hamiltonian system (6), and equation (27) is called a Mei conservation law.

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

Advances in Mathematical Physics
where G M is the gauge function that satisfies Proof.
Theorem 10 is the Mei symmetry theorem for the time-scale nonshifted general holonomic system (8).

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 where G M is the gauge function that satisfies 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 G M satisfies equation (35).
We call Theorem 12 the Mei symmetry theorem for timescale 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).

Examples
Example 1. The time scale is T = f2 m : m ∈ ℕ 0 g, and the Hamiltonian is According to equation (6), we get To do the calculation, we have From equation (18), we obtain the following criterion equations: Substituting (40) and (41) into equation (28) and using equation (37), we can get