Issue 
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 2, April 2023



Page(s)  106  116  
DOI  https://doi.org/10.1051/wujns/2023282106  
Published online  23 May 2023 
Mathematics
CLC number: O 316
A Study on Time Scale NonShifted Hamiltonian Dynamics and Noether's Theorems
College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China
Received:
8
May
2022
The timescale nonshifted Hamiltonian dynamics are investigated, including both general holonomic systems and nonholonomic systems. The timescale nonshifted Hamilton principle is presented and extended to nonconservative system, and the dynamic equations in Hamiltonian framework are deduced. The Noether symmetry, including its definition and criteria, for timescale nonshifted Hamiltonian dynamics is put forward, and Noether's theorems for both holonomic and nonholonomic systems are presented and proved. The nonshifted Noether conservation laws are given. The validity of the theorems is verified by two examples.
Key words: nonshifted Hamiltonian dynamics / Noether symmetry / nonshifted Noether conservation laws / time scales
Biography: ZHANG Yi, male, Ph.D., Professor, research direction: analytical mechanics. Email: zhy@mail.usts.edu.cn
Fundation item: Supported by the National Natural Science Foundation of China (11972241, 12272248, 11572212), the Natural Science Foundation of Jiangsu Province (BK20191454)
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Timescale analysis originated from the work of Hilger^{ [1]}, which unifies and generalizes the differential equation and difference equation theory within the framework of timescale calculus, thus avoiding the repeated solution of the two different kinds of equations and revealing the essential differences between continuous and discrete systems ^{[24]}. Time scale theory has been widely applied and achieved many achievements in various fields^{ [410]}. In last two decades, some new advances have emerged on the study of timescale dynamics and its symmetries, such as kinetic equations^{ [1113]}, optimal control problems^{ [14,15]}, fractional variational problems^{ [1618]}, Noether theorems^{ [1822]}, Lie symmetries^{ [2325]}, Mei symmetries^{ [26,27]}, canonical transformation and HamiltonJacobi method^{ [28,29]}, timedelay dynamics^{ [30]}, Herglotz variational problems^{ [31]}, higherorder delta derivatives^{ [32]}, etc. However, Ref. [33] indicated that, possibly affected by Bohner's original paper^{ [11]}, most studies on timescale variational problems were only shifted variational problems. In fact, the numerical calculation scheme for variational problems based on nonshifted action functionals on time scales has good properties^{ [33,34]}. Recently, three symmetries and conservation laws for nonshifted timescales dynamic systems were studied in Refs. [25, 27, 3537]. The study of nonshifted variational problems on time scales is a new but important research direction of analytical mechanics.
Conservation law plays an irreplaceable role in the solution and reduction of differential equations and stability analysis, so it has always been a research hotspot in the field of analytical mechanics^{ [3844]}. Using symmetry to find conservation laws is an effective method. However, the exploration of symmetry and conservation laws theory on time scales is still very preliminary. Moreover, the studies are mainly limited to Lagrangian systems or Birkhoffian systems so far, and nonconservative forces or nonholonomic constraints are not considered. Since the phase space of mechanical system has natural symplectic structure, it is easier to describe mathematically than Lagrangian mechanics. For some cases, the symmetry that is difficult to find in configuration space can be found in phase space, and the corresponding conserved quantity can be derived by using Noether's theorem in the canonical form^{ [45]}. Here we will explore the nonshifted Hamiltonian dynamics, Noether symmetry and nonshifted conservation laws on time scales under Hamiltonian framework.
1 NonShifted TimeScale Hamiltonian Dynamics
For time scale calculus and its basic operations, please refer to Refs. [2, 3, 12].
The timescale nonshifted Hamilton action reads
where is a certain curve, T is the Hamiltonian on time scales, and are generalized coordinate and generalized momentum, is the delta derivative of with respect to , where . All functions belong to T.
1.1 Hamilton System
The isochronous variational principle
with endpoint conditions
and commutative relations
is called the timescale nonshifted 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 , , according to DuboisReymond lemma ^{[11]}, we get
where and are some constants. Taking the nabla derivative of (7), we have
So there are
Eqs.(9) are the Hamilton canonical equations for the nonshifted Hamilton system on time scales.
If take T, then , Eqs.(9) become
Eqs.(10) are the classical Hamilton canonical equations.
1.2 General Holonomic Mechanical System in Phase Space
For a general holonomic mechanical system, we extend principle (2) as follows
where are nonpotential generalized forces.
Similar to the derivation of Eq. (6), from principle (11), we get
From the independence of , according to DuboisReymond lemma ^{[11]}, we get
where and are some constants. Hence, we have
Eqs.(14) are the timescale dynamic equations of the general holonomic system. When , Eqs. (14) are reduced to Eqs. (9), which are the nonshifted Hamilton canonical equations.
1.3 Nonholonomic Mechanical System in Phase Space
Consider the system is subject to bilateral ideal nonholonomic constraints
and the virtual displacements need to meet the conditions
where . In general, is independent of , and the constraints are of nonChetaev. If , then the constraints are of Chetaev.
If the nonshifted Lagrangian is , then
According to Eq. (17), 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 multiplied by each of Eqs. (19) and summing over , and integrating the equation on the interval , and by integration by parts, we get
Taking into account conditions (3), we get
Add Eq. (21) to Eq. (12), and we get
According to the Lagrange multiplier method, without loss of generality, choose the multiplier such that , and using DuboisReymond lemma ^{[11]}, from Eq. (22), we get
where are some constants. So there are
Assuming that the system is nonsingular, by using Eqs. (24) and (18), can be solved as the function of , and . Therefore, Eqs.(24) can be expressed as
where 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 and satisfy Eq. (18), namely
then the solution of (25) is the desired solution of timescale nonholonomic systems (18) and (24).
2 Noether Symmetry for TimeScale Hamiltonian Dynamics
2.1 Noether Symmetry for Hamilton System
The infinitesimal transformations are
where , and are the generating functions, is the infinitesimal parameter, and . Let the map be a strictly increasing T function, whose image is denoted as T, delta derivative is , forward jump operator , and .
Under the transformation (27), the Hamilton action (1) reads
Therefore, the nonisochronous variation , namely the mainline part of difference relative to , is
By straightforward calculation, formula (29) can also be expressed as
Formulas (29) and (30) are two mutually equivalent nonisochronous variational formulas of nonshifted Hamilton action on time scales.
Definition 1 For the timescale nonshifted Hamilton system (9), the transformation (27) is said to be Noether symmetric, if and only if .
By using Eqs. (29) and (30), we obtain:
Criterion 1 For the timescale nonshifted Hamilton system (9), if the Noether identity
holds, then the transformation (27) is Noether symmetric.
Criterion 2 If the generating functions , and solve the equation
then the transformation (27) is Noether symmetric for the timescale Hamilton system (9).
Assume that is another Hamiltonian on time scales, if, considering the firstorder approximation, the transformation (27) satisfies the following relation
then the action (1) is called quasiinvariant. Both and satisfy the same timescale Hamilton equations. So there is
where is the gauge function.
Definition 2 For the timescale nonshifted Hamilton system (9), the transformation (27) is said to be Noether quasisymmetric, if and only if
By using Eqs.(29) and (30), we obtain:
Criterion 3 For the timescale nonshifted Hamilton system (9), if the generalized Noether identity
holds, then the transformation (27) is Noether quasisymmetric.
Criterion 4 If the generating functions , and solve the equation
then the transformation (27) is Noether quasisymmetric for the timescale Hamilton system (9).
2.2 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 quasiinvariant.
Definition 3 For the timescale nonshifted general holonomic system (14), the transformation (27) is said to be generalized Noether quasisymmetric, if and only if
Criterion 5 For the timescale nonshifted general holonomic system (14), if the generalized Noether identity
holds, then the transformation (27) is generalized Noether quasisymmetric.
Criterion 6 If the generating functions , and solve the equation
then the transformation (27) is generalized Noether quasisymmetric for the timescale general holonomic system (14).
2.3 Noether Symmetry for Nonholonomic Mechanical System in Phase Space
For the nonholonomic system, we have
Definition 4 For the corresponding holonomic system (25), the transformation (27) is said to be generalized Noether quasisymmetric, if and only if
Criterion 7 For the corresponding holonomic system (25), if the generalized Noether identity
holds, then the transformation (27) is generalized Noether quasisymmetric.
Criterion 8 If the generating functions , and solve the equation
then the transformation (27) is generalized Noether quasisymmetric for the corresponding holonomic system (25).
The restriction conditions of Eqs. (19) on the generating functions are
Definition 5 For the timescale nonshifted 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 quasisymmetric.
Criterion 9 For the timescale nonshifted 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 , and solve the equation (44) and the restriction conditions (45), then the transformation (27) is generalized Noether quasisymmetric for the timescale nonholonomic system determined by (18) and (24).
3 Noether Theorems for TimeScales Hamiltonian Dynamics
Noether symmetry is closely related to conservation laws. Here we establish and prove Noether's theorems for timescale nonshifted holonomic and nonholonomic Hamiltonian dynamics.
3.1 Noether Theorems for Hamilton System
Theorem 1 For the timescale nonshifted Hamilton system (9), if the transformation (27) is Noether symmetric, then
is a nonshifted Noether conserved quantity.
Proof Due to
Substituting the nonshifted Hamilton equations (9) and the Noether identity (31) into (47), we get
Therefore, formula (46) is a nonshifted Noether conserved quantity.
Theorem 2 For the timescale nonshifted Hamilton system (9), if the transformation (27) is Noether quasisymmetric, then
is a nonshifted 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 timescale nonshifted Hamilton system.
3.2 Noether Theorems for General Holonomic Mechanical System in Phase Space
For the general holonomic mechanical system, we have
Theorem 3 For the timescale nonshifted general holonomic system (14), if the transformation (27) is generalized Noether quasisymmetric, then
is a nonshifted Noether conserved quantity.
Proof Due to
This completes the proof.
Theorem 3 is Noether's theorem for timescale nonshifted general holonomic system under Hamiltonian framework.
3.3 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 quasisymmetric, then
is a nonshifted Noether conserved quantity.
Proof Taking the nabla derivative of (52), and using Eqs.(25) and (43), we get the results.
Theorem 5 For the timescale nonshifted nonholonomic system determined by (18) and (24), if the transformation (27) is generalized Noether quasisymmetric, then formula (52) is a nonshifted Noether conserved quantity.
Theorem 5 and Theorem 4 are Noether's theorems for timescale nonshifted nonholonomic system and its corresponding holonomic system under Hamiltonian framework.
4 Examples
Example 1 Consider a nonshifted holonomic nonconservative system on time scale T, 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
This is the classic HojmanUrrutia problem^{ [46]}.
The generalized Noether identity (40) reads
Since , Eq.(58) has a solution
By Theorem 3, we obtain
The conserved quantity (61) and (62) correspond to the quasisymmetry (59) and (60). Assume the initial conditions , , , . Let . The numerical simulation results of conserved quantities (61) and (62) on are shown in Fig. 1.
Fig. 1 The values of conserved quantities (61) and (62) on 
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 AppellHamel problem ^{[47]} on time scales. The nonshifted 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 timescale dynamical equations can be expressed as
Taking the nabla derivative of (67), we get
Substituting (68) into (69), we can get
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) have a solution
From Theorem 5, we get the Noether conserved quantity
If T, then , and formula (75) becomes
This is the classical conservation law of energy, which has been given in Ref.[47].
If T, then , , , and formula (75) becomes
This is the Noether conserved quantity of the discrete version with unit step size.
5 Conclusion
The timescale 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 timescale nonshifted Hamilton principle and extended it to nonconservative systems, and derived the dynamic equations for nonshifted Hamilton systems, nonshifted general holonomic systems and nonshifted nonholonomic systems. We defined Noether symmetries and gave their criteria. We proved Noether's theorems for nonshifted Hamilton systems, nonshifted general holonomic systems and nonshifted nonholonomic systems, and obtained the nonshifted Noether conserved quantities. The ideas presented here can be applied to solving the symmetries of complex dynamics under timescale framework, such as nonlinear problems.
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All Figures
Fig. 1 The values of conserved quantities (61) and (62) on 

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