A Study on Time Scale Non-Shifted Hamiltonian Dynamics and Noether 􀆳 s Theorems

: The time-scale non-shifted Hamiltonian dynamics are investigated, including both general holonomic systems and nonholo‐ nomic systems. The time-scale non-shifted Hamilton principle is presented and extended to nonconservative system, and the dynamic equa‐ tions in Hamiltonian framework are deduced. The Noether symmetry, including its definition and criteria, for time-scale non-shifted Hamil‐ tonian dynamics is put forward, and Noether 􀆳 s theorems for both holonomic and nonholonomic systems are presented and proved. The non-shifted Noether conservation laws are given. The validity of the theorems is verified by two examples.


Non-Shifted Time-Scale Hamiltonian Dynamics
For time scale calculus and its basic operations, please refer to Refs. [2,3,12]. The time-scale non-shifted Hamilton action reads where γ is a certain curve, H:T k´Rn´Rn ® R is the Hamiltonian on time scales, q s and p s are generalized coordinate and generalized momentum, q D s is the delta derivative of q s with respect to t, where s = 12n. All functions belong to C 1D rd (T).

Hamilton System
The isochronous variational principle and commutative relations is called the time-scale non-shifted Hamilton principle. By carrying out the variational operation of Eq. (2) and using the relations (4), we can get Substituting the endpoint conditions (3) into Eq. (5), we get From the independence of δq D s , δp D s ( s = 12n ) , according to Dubois-Reymond lemma [11] , we get where C s and D s are some constants. Taking the nabla derivative of (7), we have So there are Eqs. (9) are the Hamilton canonical equations for the non-shifted Hamilton system on time scales.

General Holonomic Mechanical System in Phase Space
For a general holonomic mechanical system, we extend principle (2) as follows where Q″ s = Q″ s ( tq k (t) p k (t) ) are non-potential generalized forces.
Similar to the derivation of Eq. (6), from principle (11), we get From the independence of δq D s  δp D s ( s = 12n ) , according to Dubois-Reymond lemma [11] , we get where C′ s and D′ s are some constants. Hence, we have Eqs. (14) are the time-scale dynamic equations of the general holonomic system. When Q″ s = 0, Eqs. (14) are reduced to Eqs. (9), which are the non-shifted Hamilton canonical equations.

Nonholonomic Mechanical System in Phase Space
Consider the system is subject to g bilateral ideal nonholonomic constraints and the virtual displacements δq s need to meet the conditions where β = 12g. In general, F βs is independent of ¶f β ¶q D s , and the constraints are of non-Chetaev. If F βs = ¶f β ¶q D s , then the constraints are of Chetaev.
If the non-shifted Lagrangian is L = L ( tq s q D s ) , then According to Eq. (17), q D s = q D s ( tq j p j ) can be solved, and then substituted into Eqs. (15) and (16), thus the constraints (15) and restriction conditions (16) can be written as By introducing the constraint multiplier λ β multiplied by each of Eqs. (19) and summing over β, and integrating the equation on the interval [t 1 t 2 ], and by integration by parts, we get Taking into account conditions (3), we get According to the Lagrange multiplier method, without loss of generality, choose the multiplier λ β such that C″ β = 0 ( β = 12g ) , and using Dubois-Reymond lemma [11] , from Eq. (22), we get where C″ g + 1 C″ n  D″ s are some constants. So there are Assuming that the system is non-singular, by using Eqs. (24) and (18), λ β can be solved as the function of q s , p s and t. Therefore, Eqs. (24) can be expressed as where Λ s = λ β F ͂ βs are the constraint forces corresponding to the nonholonomic constraints (18). Eqs. (25) can be regarded as a holonomic system corresponding to the nonholonomic system determined by Eqs. (18) and (24). If the initial values of q s and p s satisfy Eq. (18), namely then the solution of (25) is the desired solution of time-scale nonholonomic systems (18) and (24).

Noether Symmetry for Hamilton System
The infinitesimal transformations are where ξ 0 , ξ s and η s are the generating functions, ε is the infinitesimal parameter, and sj = 12n. Let the map t  α (t) = t + εξ 0 + o (ε) be a strictly increasing C 1D rd (T) function, whose image is denoted as T, delta derivative is D, forward jump operator σ, and σ  α = α  σ.
Under the transformation (27), the Hamilton action (1) reads Therefore, the nonisochronous variation D * S, namely the main-line part of difference By straightforward calculation, formula (29) can also be expressed as Formulas (29) and (30) are two mutually equivalent nonisochronous variational formulas of non-shifted Hamilton action on time scales.
Definition 1 For the time-scale non-shifted Hamilton system (9), the transformation (27) is said to be Noether symmetric, if and only if D * S = 0. By using Eqs. (29) and (30), we obtain: Criterion 1 For the time-scale non-shifted Hamilton system (9), if the Noether identity holds, then the transformation (27) is Noether symmetric. Criterion 2 If the generating functions ξ 0 , ξ s and η s solve the equation Ñ then the transformation (27) is Noether symmetric for the time-scale Hamilton system (9). Assume that H 1 is another Hamiltonian on time scales, if, considering the first-order approximation, the transformation (27) satisfies the following relation then the action (1) where G = G ( tq s p s ) is the gauge function.

Definition 2
For the time-scale non-shifted Hamilton system (9), the transformation (27) is said to be Noether quasi-symmetric, if and only if By using Eqs. (29) and (30), we obtain: Criterion 3 For the time-scale non-shifted Hamilton system (9), if the generalized Noether identity holds, then the transformation (27) is Noether quasi-symmetric.
then the transformation (27) is Noether quasi-symmetric for the time-scale Hamilton system (9).

Noether Symmetry for General Holonomic Mechanical System in Phase Space
For the general holonomic system, if the following relation is satisfied, then the action (1) is called generalized quasi-invariant. Definition 3 For the time-scale non-shifted general holonomic system (14), the transformation (27) is said to be generalized Noether quasi-symmetric, if and only if Criterion 5 For the time-scale non-shifted general holonomic system (14), if the generalized Noether identity then the transformation (27) is generalized Noether quasi-symmetric for the time-scale general holonomic system (14).

Noether Symmetry for Nonholonomic Mechanical System in Phase Space
For the nonholonomic system, we have Criterion 7 For the corresponding holonomic system (25), if the generalized Noether identity holds, then the transformation (27) (44) then the transformation (27) is generalized Noether quasi-symmetric for the corresponding holonomic system (25). The restriction conditions of Eqs. (19) on the generating functions are

Definition 5
For the time-scale non-shifted nonholonomic system determined by (18) and (24), if and only if the formula (42) and restriction conditions (45) hold, then the transformation (27) is said to be generalized Noether quasi-symmetric.
Criterion 9 For the time-scale non-shifted nonholonomic system determined by (18) and (24), if the generalized Noether identity (43) and the restriction conditions (45) hold, then the transformation (27) is generalized Noether quasisymmetric.
Criterion 10 If the generating functions ξ 0 , ξ s and η s solve the equation (44) and the restriction conditions (45), then the transformation (27) is generalized Noether quasi-symmetric for the time-scale nonholonomic system determined by (18) and (24).

Noether Theorems for Time-Scales Hamiltonian Dynamics
Noether symmetry is closely related to conservation laws. Here we establish and prove Noether s theorems for time-scale non-shifted holonomic and nonholonomic Hamiltonian dynamics.

Noether Theorems for Hamilton System
Theorem 1 For the time-scale non-shifted Hamilton system (9), if the transformation (27) is Noether symmetric, then is a non-shifted Noether conserved quantity. Proof Due to Ñ Ñt Substituting the non-shifted Hamilton equations (9) and the Noether identity (31) into (47), we get Ñ Ñt Therefore, formula (46) is a non-shifted Noether conserved quantity. Theorem 2 For the time-scale non-shifted Hamilton system (9), if the transformation (27) is Noether quasisymmetric, then is a non-shifted Noether conserved quantity. Proof Taking the nabla derivative of (49), and using Eqs. (9) and (36), we get the result immediately. Theorem 1 and 2 are Noether s theorems for time-scale non-shifted Hamilton system.

Noether Theorems for General Holonomic Mechanical System in Phase Space
For the general holonomic mechanical system, we have Theorem 3 For the time-scale non-shifted general holonomic system (14), if the transformation (27) is generalized Noether quasi-symmetric, then is a non-shifted Noether conserved quantity. Proof Due to Ñ Ñt This completes the proof. Theorem 3 is Noether s theorem for time-scale non-shifted general holonomic system under Hamiltonian framework.

Noether Theorems for Nonholonomic Mechanical System in Phase Space
For the nonholonomic mechanical system, we have Theorem 4 For the corresponding holonomic system (25), if the transformation (27) is generalized Noether quasi-symmetric, then is a non-shifted Noether conserved quantity. Proof Taking the nabla derivative of (52), and using Eqs. (25) and (43), we get the results. Theorem 5 For the time-scale non-shifted nonholonomic system determined by (18) and (24), if the transformation (27) is generalized Noether quasi-symmetric, then formula (52) is a non-shifted Noether conserved quantity.
Theorem 5 and Theorem 4 are Noether s theorems for time-scale non-shifted nonholonomic system and its corresponding holonomic system under Hamiltonian framework.

Examples
Example 1 Consider a non-shifted holonomic non-conservative system on time scale T= {2 m :m Î N 0 }, and let the Lagrangian function be The generalized forces are The generalized momenta and the Hamiltonian are According to equation (14), the Hamilton equations are If we take T= , then Eqs.(56) are reduced to q̇1 = p 1 q̇2 = p 2 ṗ1 + p 2 = 0ṗ2 + q 2 = 0 (57) This is the classic Hojman-Urrutia problem [46] . The generalized Noether identity (40) reads Since σ (t) = 2t, Eq.(58) has a solution By Theorem 3, we obtain The conserved quantity (61) and (62)  From Fig. 1, it can be seen intuitively that conserved quantities obtained from formulae (61) and (62) are both constants, which shows the correctness of Theorem 3.
Example 2 Let us study Appell-Hamel problem [47] on time scales. The non-shifted Lagrangian and nonholonomic constraint are respectively The constraint (64) is of Chetaev type. The generalized momenta and the Hamiltonian are In canonical coordinates, the constraint (64) can be shown as The time-scale dynamical equations can be expressed as Taking the nabla derivative of (67), we get Substituting (68) into (69), we can get λ = 2mg ( ) Therefore, the nonholonomic constraint forces are From (43), the generalized Noether identity for the system is The restriction condition (45) reads The equations (72) and (73) This is the classical conservation law of energy, which has been given in Ref. [47].

Conclusion
The time-scale calculus provides an excellent mathematical tool for exploring the dynamics of continuous and discrete systems or their mixtures, and has attracted extensive attentions. In this paper, we proposed the time-scale nonshifted Hamilton principle and extended it to non-conservative systems, and derived the dynamic equations for nonshifted Hamilton systems, non-shifted general holonomic systems and non-shifted nonholonomic systems. We defined Noether symmetries and gave their criteria. We proved Noether s theorems for non-shifted Hamilton systems, nonshifted general holonomic systems and non-shifted nonholonomic systems, and obtained the non-shifted Noether conserved quantities. The ideas presented here can be applied to solving the symmetries of complex dynamics under timescale framework, such as nonlinear problems.