Xue Tian, Yi Zhang

a School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, China

b School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, 232001, Anhui, China

c College of Civil Engineering, Suzhou University of Science and Technology, Suzhou, 215011, Jiangsu, China

Keywords:Time-scale non-shifted system Hamiltonian system Structure-preserving algorithm Noether conserved quantity

ABSTRACT The variational calculus of time-scale non-shifted systems includes both the traditional continuous and traditional significant discrete variational calculus. Not only can the combination of Δ and ?derivatives be beneficial to obtaining higher convergence order in numerical analysis, but also it prompts the timescale numerical computational scheme to have good properties, for instance, structure-preserving. In this letter, a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is proposed. By using the time-scale discrete variational method and calculus theory, and taking a discrete time scale in the variational principle of non-shifted Hamiltonian systems, the corresponding discrete Hamiltonian principle can be obtained. Furthermore, the time-scale discrete Hamilton difference equations, Noether theorem, and the symplectic scheme of discrete Hamiltonian systems are obtained. Finally, taking the Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems as examples, they show that the time-scale discrete variational method is a structure-preserving algorithm. The new algorithm not only provides a numerical method for solving time-scale non-shifted dynamic equations but can be calculated with variable step sizes to improve the computational speed.

In classical mechanics, such as Newtonian mechanics and Lagrangian mechanics, Hamiltonian formalism has a salient symmetric form, and regularity of motion is most apparent in Hamiltonian formalism [1]. All actual physical processes whose dissipative effect is negligible can express in Hamiltonian formalism [2]. No matter the process is classical, quantum or relativistic, or no matter its degree of freedom is finite or infinite, the physical process can always express in proper Hamiltonian formalism [3]. Mathematicians and physicists have paid much attention to Hamiltonian systems on account of symmetry and simplicity. The basic theory of continuous Hamiltonian systems has been investigated for a long time. At present, with the development of information technology and the wide application of digital computers, many mathematical models supported by discrete Hamiltonian systems have appeared,and the study of discrete systems theory has attracted more and more people’s attention [4–11]. However, although there are some differences between a discrete system and its corresponding continuous system, there are striking similarities and duality [1].

As a mathematical tool for establishing a unified framework for continuous and discrete systems, the time-scale theory was proposed by Hilger in his doctoral thesis in 1988 [12]. Subsequently,Bohner and Peterson [13], Agarwal et al. [14], Bohner and Peterson[15] realized that time-scale calculus is bridging the gap between the continuous and discrete aspects. The unified approach means that sophisticated new models can take into account dozens of variables. Besides, it is worth noting that the time-scale calculus of dynamic equations can also representq-difference equations when the time scale is T=qN0(q ＞1)or T=qZ∪{0}, which have crucial applications in quantum theory [14]. Thus, the time-scale dynamical equations can provide models for continuous systems, discrete systems and quantum systems simultaneously. The complexity of time-scale theory enriches the research content of dynamic systems, and time-scale calculus provides a new method for solving complex dynamic behavior.

In the past decades, time-scale variational problems and their symmetry problems have been explored and perfected, such as optimal control problems [16,17], Hamilton Jacobi method [18],generalized canonical transformation [19], time-delay dynamics[20,21], fractional variational problems [22–24], Herglotz variational problem [25,26], Lie symmetry [27–29], and Noether theorem [30–33]. The above time-scale variational problems and symmetries are almost based on shifted systems. However, a shifted variational problem does not cover the very important area of discrete calculus. In the case of discrete non-shifted systems, the time-scale variational method becomes the classical discrete variational method, and the time-scale numerical calculation scheme of non-shifted variational method has the characteristic of preserving structure [34]. Moreover, due to the combination ofΔand? derivatives in the time-scale non-shifted system, a higher order of convergence can be obtained in numerical analysis [34].In recent years, Song and Cheng [35] obtained Noether theorem for time-scale non-shifted dynamical systems preliminarily.Then, Chen and Zhang [36] studies Noether symmetry theorem for non-shifted generalized Birkhoffian mechanics on time scales.And Zhang researched Mei symmetries and conserved laws for three kinds of time-scale non-shifted mechanical systems, i.e., Lagrangian systems [37], Hamiltonian systems [38], and Birkhoffian systems [39]. Zhang [40] also studies Noether symmetry theory of time-scale non-shifted Lagrangian systems, non-shifted general holonomic systems and non-shifted nonholonomic systems. However, it is very difficult or even impossible to find general solutions for general time-scale equations. Therefore, there are some results from the time-scale equations to analyze some properties of their solutions, for instance, existence, boundedness, vibration, etc. [41–43]. But up to now, there is no research on the algorithm of solving the time-scale non-shifted dynamic equations.

It is well known that the Hamilton principle induces Hamilton canonical equations with symplectic structure in the continuous case. Furthermore, the time-scale Hamilton equations induced by the time-scale Hamilton principle also have the symplectic property. If the time-scale Hamilton principle is discretized, the time-scale discrete Hamilton equations can be derived from the time-scale discrete Hamilton principle. As the product of the discrete variational principle, the time-scale discrete Hamilton equations inherit the symplectic geometric properties of the time-scale non-shifted system when used as a numerical difference scheme.In addition, the momentum of the discrete system is conserved,which satisfies the Noether theorem of the time-scale non-shifted Hamiltonian system. In this letter, we will study the structurepreserving algorithm for time-scale non-shifted Hamiltonian systems.

For the convenience of the reader, some time-scale concepts are listed below. Please refer to Ref. [13] for specific time-scale definitions and properties.

Table 1 Examples on time scales.

Proof. Since the infinitesimal transformations (24) are the Noether symmetric transformations of the system, from Definition 1, we knowΔ*SD=0. According to Criterion 2, we obtain

then the infinitesimal transformations (23) are the Noether quasisymmetric transformations of the time-scale non-shifted Hamiltonian system.

Fig. 1. The q1 displacements of the Kepler system on T=0.1N.

Formula (54) is the conservation of energy. Let the time scale be T=hN, i.e.,μ(t)=h. Assume the initial conditionsq1,0=0.003 m, q2,0=0.9 m, q3,0=0.01 m, p1,1=0.1 m, p2,1=0.1 m, p3,1=0.8 m, and the constantK2=0.5625 kg·m3/s2. Ifa=0,h=0.1 andN=1000, according to Eq. (50), compared with the traditional 4th order Runge-Kutta method and 4th-5th order Runge-Kutta method, theq1displacements of the system are given in Fig. 1. The step size of the 4th order Runge-Kutta method is consistent with that of the time-scale discrete variational method,which ish=0.1. Since 4th-5th order Runge-Kutta method can achieve variable step size calculation automatically, this procedure is calculated by MATLAB with its own ode45 function. From Fig. 1, the time-scale discrete variational method and the 4th order Runge-Kutta method can describe the motion trajectory of the system stably. In addition, the solution curve of the system of time-scale discrete variational method, 4th order Runge-Kutta method and 4th-5th order Runge-Kutta method are given, respectively, as shown in Fig. 2. As can be seen from Fig. 2, the results obtained by the time-scale discrete variational method and the 4th order Runge-Kutta method are more stable than those obtained by the 4th-5th order Runge-Kutta method. Furthermore, these three methods calculate the conserved quantity (54), and the results are shown in Fig. 3. In Fig. 3, the Noether conserved quantity (54) obtained by the time-scale discrete variational method is a constant,while the Noether conserved quantities obtained by the 4th order Runge-Kutta method and the 4th-5th order Runge-Kutta method have larger fluctuations. To further compare the time-scale discrete variational method with the 4th order Runge-Kutta method,we set the time scale as T=0.5N, i.e., the step size ish=0.5,and the initial conditions remain unchanged. Then, the results of the displacements, motion trajectories and Noether conserved quantities of the Kepler system obtained by the time-scale discrete variational method and the 4th order Runge-Kutta method are compared.

As can be seen from Figs. 4–6, whent=332 s,μ=0.5, the 4th order Runge-Kutta method begins to diverge, while the time-scale discrete variational method tends to be stable due to the preserving structure. Moreover, when we take a time scale with variable step size, such as the time scale T=λN, the time-scale discrete variational method is also structure-preserving, and the time-scale calculus can handle the volume subsystem. By settingλ=1.0001,N ∈[0,50000], we can obtain the displacement, motion trajectory and Noether conserved quantity of the Kepler system, as shown in Fig. 7.

Not only can the time-scale discrete variational method solve time-scale non-shifted dynamic equations, but also it can solve continuous equations when the graininess function is small enough. Next, we give the damped oscillator as an example to illustrate that it provides an effective method for solving equations.

Fig. 2. The motion trajectories of Kepler system on T=0.1N.

Fig. 3. The Noether conserved quantities of the Kepler system on T=0.1N.

Fig. 4. The q1 displacements of the Kepler system on T=0.5N.

Fig. 6. The Noether conserved quantities of the Kepler system on T=0.5N.

From Criterion 2, its Noether identity is Let its time scale be T=hZ. Ifh→0 i.e., T=R, then the problems of continuous systems can be studied by the time-scale discrete variational method. At this time, we can compare the exact solution of the system with the result obtained by the proposed method to prove the effectiveness. Let the step size beh=0.001, and the initial conditions beq0=1,p1=1. Then The exact solution of the coordinateqand the solutions obtained by the time-scale discrete variational method and 4th order Runge-Kutta method are given in Fig. 8. According to Fig. 8, the solution ofqobtained by the time-scale discrete variational method is almost identical to the one by the 4th order Runge-Kutta method.Both results are highly consistent with the exact solution. Furthermore, we draw the phase diagrams by the time-scale discrete variational method and the 4th order Runge-Kutta method in Fig. 9, and give the Noether conserved quantity (64) in Fig. 10.Figures 9 and 10 show that the results by the time-scale discrete variational method are stable. There is a tiny deviation between the conserved quantity and the exact one for the continuous equation since this method still solves the discrete equation even ifh=0.001. The result is consistent with the real value if and only ifhtends to zero.

Fig. 5. The motion trajectories of Kepler system on T=0.5N.

Fig. 7. T=λN time-scale discrete variational method.

Fig. 8. The q displacements of the damped oscillator with h=0.001.

Fig. 9. The phase diagrams of the damped oscillator with h=0.001.

Fig. 10. The Noether conserved quantities of the damped oscillator with h=0.001.

The time-scale discrete Hamilton equations (18) are obtained by discretization of time-scale Hamilton principle and variation of time-scale discrete Hamilton principle (10), (12). Thus, the symplectic numerical algorithm scheme (19) is determined. The definition and criterion of the discrete Noether symmetric transformations are given for the time-scale non-shifted Hamiltonian system, and the time-scale discrete Noether theorem is obtained,namely Theorem 1. In addition, the definition and criterion of discrete Noether quasi-symmetric transformations are given, and Theorem 2 is obtained, which shows that more conserved quantities may be obtained by taking different infinitesimal generators and appropriate gauge functions. The Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems are taken as examples to illustrate that the time-scale discrete variational method is a structure-preserving algorithm.

Due to the advantage of time-scale theory, the new algorithm proposed in this letter has a host of strengths.

1) The new algorithm provides a numerical method for solving time-scale non-shifted dynamic equations and continuous equations.

2) Unlike the traditional discrete variational method with fixed step sizes, the proposed method can also be calculated with variable step sizes and even segmented time step sizes under certain conditions to improve the computational speed.

3) Since the forward jump operatorσ(or backward jump operatorρ) and forward graininess functionμ(or backward graininess functionν) represent the forward jump (or backward jump) of time, the equations obtained by the time-scale discrete variational method are uniform in format and elegant in form. Thus,at a discrete timetk, the time of all variables corresponds totk,which is not easy to cause time confusion.

However, the structure-preserving algorithms for time-scale non-shifted systems are still at the preliminary stage. The timescale discrete variational method can extend to quasi-Hamiltonian systems and Birkhoffian systems. It will be our future research work to apply the time-scale method to other constrained mechanical systems to study their structure-preserving algorithms.Besides, it remains to investigate whether the combination of timescale theory and other numerical algorithms can improve computational accuracy, efficiency or stability.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 11972241, 11572212); the Natural Science Foundation of Jiangsu Province (No. BK20191454); and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX20_0251).