Mei Symmetry and New Conserved Quantities of Time-Scale Birkhoff’s Equations

Zhai, Xiang-Hua; Zhang, Yi

doi:https://doi.org/10.1155/2020/1691760

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Research Article | Open Access

Volume 2020 | Article ID 1691760 | https://doi.org/10.1155/2020/1691760

Mei Symmetry and New Conserved Quantities of Time-Scale Birkhoff’s Equations

Xiang-Hua Zhai^1,2and Yi Zhang¹

Academic Editor: Yan-Ling Wei

Received03 Oct 2019

Revised10 Dec 2019

Accepted17 Dec 2019

Published25 Jan 2020

Abstract

The time-scale dynamic equations play an important role in modeling complex dynamical processes. In this paper, the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations are studied. The definition and criterion of the Mei symmetry of the Birkhoffian system on time scales are given. The conditions and forms of new conserved quantities which are found from the Mei symmetry of the system are derived. As a special case, the Mei symmetry of time-scale Hamilton canonical equations is discussed and new conserved quantities for the Hamiltonian system on time scales are derived. Two examples are given to illustrate the application of results.

1. Introduction

In 1988, Hilger [1] proposed the calculus on time scales to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. This theory is really important and plays a useful role in modeling complex dynamic processes and has a tremendous potential for applications [2–4].

Symmetry is an important aspect in studying dynamic equations. From a symmetry, we can find a conserved quantity or the first integral of dynamic equations. The famous Noether symmetry theory which reveals the relationships between symmetries and conversation laws has been applied to a time-scale analogue of analytical mechanics, modern theoretical physics, and engineering [5–10]. Except for the Noether symmetry method, the other two popular symmetry methods, the Lie symmetry method [11] and the Mei symmetry method [12], are widely applied in studying dynamic systems. The Mei symmetry means that when the dynamic functions are replaced by the transformed functions under the infinitesimal transformations of group, the forms of the differential equations of motion keep invariant. Here, the dynamic functions are the Lagrangian, the Hamiltonian, the Birkhoffian, Birkhoff’s functions, the generalized forces, the generalized constraint reactions, etc. New kinds of conserved quantity can be led by Mei symmetry. This method has been successfully applied in equations of motion for Lagrangian systems, Hamiltonian systems, Birkhoffian systems, the motion of charged particles in an electromagnetic field, the equation of nonmaterial volumes, the equation of thin elastic rod, etc. [13–21].

The Birkhoffian mechanics [22] represents a new stage in the development of analytical mechanics. It is a natural development of the Hamiltonian mechanics and has valuable applications in hadronic physics, spatial mechanics, statistical mechanics, biophysics, and engineering [23–26]. Over the past two decades, important achievements, including the symmetries and conserved quantities of Birkhoffian systems, the Birkhoffian dynamic inverse problems, the stability of motion of Birkhoff’s equations, the Birkhoffian systems with time delay, and the fractional Birkhoffian systems, have been made [27–38].

The Pfaff–Birkhoff principle and Birkhoff’s equations were studied in a time-scale analogue, and their corresponding Noether symmetry and Lie symmetry as well as the conserved quantities have been studied [39,40] recently. However, the Mei symmetry of Birkhoffian systems on time scales has not been done. It is worth going a step further to find more new conserved quantities of Birkhoff’s equations by another symmetry method. In this paper, we will study the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations.

This paper is organized as follows: in Section 2, we first give the definition and criterion of Mei symmetry for the Birkhoffian system on time scales. Three kinds of conserved quantities led by Mei symmetry of the system on time scales are given in Section 3. As a special case, Section 4 deals with the Mei symmetry of time-scale Hamilton canonical equations and the corresponding new conserved quantities for the Hamiltonian system on time scales. Two examples are given in Section 5 to show the application of results. In the end, conclusions and future works are given.

2. Mei Symmetry of Time-Scale Birkhoff’s Equations

For basic knowledge about the calculus on time scales, the readers can refer to References [2, 3].

The Pfaff–Birkhoff action on time scales isAccording to the Pfaff–Birkhoff principle on time scales,with boundary conditionsand exchange relationsthe time-scale Birkhoff’s equations can be derived, i.e.,where is the Birkhoffian function, are Birkhoff’s functions, are the delta derivatives of the Birkhoffian variables with respect to , and . This result can be found in Ref. [39].

Introduce the one-parameter infinitesimal transformations of time and variables aswhere and are infinitesimals and the infinitesimal parameter . The corresponding infinitesimal generator isand the first extended infinitesimal generator is

The Birkhoffian and Birkhoff’s functions become the new Birkhoffian and the new Birkhoff’s functions under transformations (6), that is,By performing Taylor series expansion at the point , we have

Definition 1. If the form of equation (5) keeps invariant when the Birkhoffian and Birkhoff’s functions are replaced by and , respectively, i.e.,then this invariance is called the Mei symmetry of the Birkhoffian system on time scales.
From Definition 1 and formulae (10) and (11), we have the following.

Criterion 1. For the Birkhoffian system (5) on time scales, if the infinitesimals and satisfythen the corresponding invariance is Mei symmetry.

3. New Conserved Quantities of Birkhoffian Systems on Time Scales

For the Birkhoffian system (5) on time scales, new conserved quantities can be found from Mei symmetry.

Theorem 1. If the infinitesimals and of the Mei symmetry of the Birkhoffian system (5) and the gauge function satisfy the following structural equation:then the Mei symmetry of the system can lead to the new conserved quantity

Proof. To prove that formula (15) is a conserved quantity, we need to proveTherefore, we haveHere, for system (5), the following energy equation [40]:holds. After the infinitesimal transformations, the new energy equationholds. Noting formulae (10) and (11), we getThus, combining equation (14) with (20) yieldsThe proof is completed.
Theorem 1 gives the Mei conserved quantity (15) on time scales led directly by Mei symmetry of system (5) with considering the structure equation (14). When , the conserved quantity (15) becomes the classical Mei conserved quantity.

Remark 1. If , then and , and the conserved quantity (15) becomes the classical one [14]

Remark 2. If , then and , and the conserved quantity (15) becomes the quantum onewhere .

Theorem 2. If the infinitesimals and of the Mei symmetry of the Birkhoffian system (5) and the gauge function satisfy the following condition:then the Mei symmetry of the system can lead to the new conserved quantity

With taking note of equation (13) and condition (24), we can derive that formula (25) is a conserved quantity of system (5).

Theorem 3. If the infinitesimals and of the Mei symmetry of the Birkhoffian system (5) and the gauge function satisfy the following condition:then the Mei symmetry of the system can lead to the new conserved quantity

With taking note of formulae (20) and (26), we can derive that formula (27) is a conserved quantity of system (5).

Theorems 2 and 3 give the other two kinds of conserved quantities (25) and (27) on time scales also led by Mei symmetry with considering conditions (24) and (26).

4. Mei Symmetry and Conserved Quantities of Time-Scale Hamilton Canonical Equations

The Birkhoffian mechanics is a natural development of the Hamiltonian mechanics; if we takethen we can obtain the time-scale Hamiltonian canonical equations [8]

Now, we introduce the infinitesimal transformationswhere , , and are infinitesimals and the infinitesimal parameter . Undergoing transformation (30), the Hamiltonian becomes . Then,where

Definition 2. If the form of equations (29) keep invariant when the Hamiltonian is replaced by , i.e.,then this invariance is called the Mei symmetry of the Hamiltonian system on time scales.

Criterion 2. For the Hamiltonian system (29) on time scales, if the infinitesimals , , and satisfythen the corresponding invariance is Mei symmetry.
Thus, we have the following three kinds of conserved quantities led by Mei symmetry of the Hamiltonian system.

Theorem 4. If the infinitesimals , , and of the Mei symmetry of the Hamiltonian system (29) and the gauge function satisfy the following structural equation:wherethen the Mei symmetry of the system can lead to the new conserved quantity

Remark 3. If , then and , and the conserved quantity (37) becomes the classical one [16]

Remark 4. If , then and , and the conserved quantity (37) becomes the quantum one

Theorem 5. If the infinitesimals , , and of the Mei symmetry of the Hamiltonian system (29) and the gauge function satisfy the following condition:then the Mei symmetry of the system can lead to the new conserved quantity

Theorem 6. If the infinitesimals , , and of the Mei symmetry of the Hamiltonian system (29) and the gauge function satisfy the following condition:then the Mei symmetry of the system can lead to the new conserved quantity

Theorems 5 and 6 give the other two kinds of conserved quantities (41) and (43) on time scales also led by Mei symmetry with considering conditions (40) and (42).

5. Examples

Example 1. Birkhoff’s functions and Birkhoffian of a fourth-order Birkhoffian system on time scales arewhich leads us to study the Mei symmetry and the conserved quantity of the system.
The equations of the system areIf we choose the infinitesimalswe can verify that infinitesimals (47) and (48) satisfy the criterion equation (13); therefore, the corresponding invariance is Mei symmetry.
Substituting infinitesimal (47) into the structure equation (14), we have the gauge function . According to Theorem 1, we haveSubstituting infinitesimal (48) into the structure equation (14), we have the gauge function . From Theorem 1, we haveThe conserved quantities (49) and (50) show that different infinitesimals correspond to different forms of conserved quantities.
From formulae (47) and (24), we obtainAccording to Theorem 2, we haveThis conserved quantity is a trivial one.
From formulae (48) and (26), we obtainAccording to Theorem 3, we haveConserved quantity (54) is led by the Mei symmetry of the system.

Example 2. The Hamiltonian of a system on time scales iswhich leads us to study the Mei symmetry and the conserved quantity of the system.
The canonical equations of the system areTaking the calculation, we haveIf we choose the infinitesimals asthenSubstituting (58) into (35), we obtain . According to Theorem 4, we haveSubstituting (59) into (35), we obtain . From Theorem 4, we haveThe conserved quantities (61) and (62) show that different infinitesimals correspond to different forms of conserved quantities.
Substituting (58) and (59) into (40), respectively, we obtain . From Theorem 5, we get a trivial conserved quantity .
Taking note of formula (60), we obtain . From Theorem 6, we also get a trivial conserved quantity .

6. Conclusions

In recent years, the theory of time-scale calculus becomes widely available in describing plenty systems that contain both continuous-time and discrete-time domains. Symmetry theory is an important method to find a conserved quantity or the first integral of the equations of dynamical systems. This paper presented and studied the Mei symmetry of Birkhoffian systems on time scales. Three new forms of conserved quantities on time scales were derived from the Mei symmetry. In addition, as a special case, the Mei symmetry and conserved quantities for the Hamiltonian system are studied. The examples given illustrated the effectiveness of the results. The results of this paper show a method to find the conserved quantity or the first integral of dynamical equations on time scales. The method is universal. The continuous results and discrete results are special cases. By choosing different time scales, general results can be obtained directly without repeated proof.

Further works about the Mei symmetry for general holonomic systems, nonholonomic systems, Hamiltonian systems on time scales, and Birkhoffian systems on time scales, as well as its relation to Lie symmetry on time scales, are still worth doing.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding this research work.

Authors’ Contributions

All authors contributed equally to this manuscript and have read and approved the final version.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant numbers: 11972241 and 11572212) and the Natural Science Foundation of Jiangsu Province (grant number BK20191454).

References

S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Universtät Würzburg, Würzburg, Germany, 1988, Ph.D. thesis.
M. Bohner and A. Peterson, Dynamic Equations on Time Scale: An Introduction with Applications, Birkhäuser, Boston, MA, USA, 2001.
M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer International Publishing, Cham, Switzerland, 2016.
R. P. Agarwal, M. Bohner, D. O’Regan, and A. Peterson, “Dynamic equations on time scales: a survey,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.
View at: Publisher Site | Google Scholar
Z. Bartosiewicz and D. F. M. Torres, “Noether’s theorem on time scales,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1220–1226, 2008.
View at: Publisher Site | Google Scholar
N. Martins and D. F. M. Torres, “Noether’s symmetry theorem for nabla problems of the calculus of variations,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1432–1438, 2010.
View at: Publisher Site | Google Scholar
A. B. Malinowska and M. R. S. Ammi, “Noether’s theorem for control problems on time scales,” International Journal of Difference Equations, vol. 9, no. 1, pp. 87–100, 2014.
View at: Google Scholar
Y. Zhang, “Noether theory for Hamiltonian system on time scales,” Chinese Quarterly of Mechanics, vol. 37, no. 2, pp. 214–224, 2016.
View at: Google Scholar
X.-H. Zhai and Y. Zhang, “Noether theorem for non-conservative systems with time delay on time scales,” Communications in Nonlinear Science and Numerical Simulation, vol. 52, pp. 32–43, 2017.
View at: Publisher Site | Google Scholar
Y. Zhang and X.-H. Zhai, “Perturbation to Lie symmetry and adiabatic invariants for Birkhoffian systems on time scales,” Communications in Nonlinear Science and Numerical Simulation, vol. 75, pp. 251–261, 2019.
View at: Publisher Site | Google Scholar
M. Lutzky, “Dynamical symmetries and conserved quantities,” Journal of Physics A: Mathematical and General, vol. 12, no. 7, pp. 973–981, 1979.
View at: Publisher Site | Google Scholar
F.-X. Mei, “Form invariance of Lagrange system,” Journal of Beijing Institute of Technology, vol. 9, no. 2, pp. 120–124, 2000.
View at: Google Scholar
F.-X. Mei, Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems, Science Press, Beijing, China, 1999.
F.-X. Mei, Analytical Mechanics II, Beijing Institute Technology Press, Beijing, China, 2013.
J.-H. Fang, P. Wang, and N. Ding, “Noether-Mei symmetry of mechanical system in phase space,” Communications in Theoretical Physics, vol. 45, no. 5, pp. 882–884, 2006.
View at: Google Scholar
S.-K. Luo, “Mei symmetry, Noether symmetry and Lie symmetry of Hamiltonian system,” Acta Physica Sinica, vol. 52, no. 12, pp. 2941–2944, 2003.
View at: Google Scholar
Y. Zhang and W.-K. Ge, “A new conservation law from Mei symmetry for the relativistic mechanical system,” Acta Physica Sinica, vol. 54, no. 4, pp. 1464–1467, 2005.
View at: Google Scholar
Z.-M. Lou, “Mei symmetry for two-dimensional charged particle in motion,” Acta Physica Sinica, vol. 54, no. 3, pp. 1015–1017, 2005.
View at: Google Scholar
W.-A. Jiang, K. Liu, Z.-W. Xia, and M. Chen, “Mei symmetry and new conserved quantities for non-material volumes,” Acta Mechanica, vol. 229, no. 9, pp. 3781–3786, 2018.
View at: Publisher Site | Google Scholar
S.-K. Luo, M.-J. Yang, X.-T. Zhang, and Y. Dai, “Basic theory of fractional Mei symmetrical perturbation and its applications,” Acta Mechanica, vol. 229, no. 4, pp. 1833–1848, 2018.
View at: Publisher Site | Google Scholar
P. Wang and Y. Xue, “Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod,” Nonlinear Dynamics, vol. 83, no. 4, pp. 1815–1822, 2016.
View at: Publisher Site | Google Scholar
R. M. Santilli, Foundations of Theoretical Mechanics II, Springer, New York, NY, USA, 1983.
F.-X. Mei, H.-B. Wu, Y.-M. Li, and X.-W. Chen, “Advances in research on Birkhoffian mechanics,” Chinese Journal of Theoretical and Applied Mechanics, vol. 48, no. 2, pp. 263–268, 2016.
View at: Google Scholar
F.-X. Mei, R.-C. Shi, Y.-F. Zhang, and H.-B. Wu, Dynamics of Birkhoffian Systems, Beijing Institute of Technology Press, Beijing, China, 1996.
F.-X. Mei, Dynamics of Generalized Birkhoffian Systems, Science Press, Beijing, China, 2013.
A. S. Galiullin, G. G. Gafarov, R. P. Malaishka, and A. M. Khwan, Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems, UFN, Moscow, Russia, 1997.
F.-X. Mei, “On the Birkhoffian mechanics,” International Journal of Non-Linear Mechanics, vol. 36, no. 5, pp. 817–834, 2001.
View at: Publisher Site | Google Scholar
Y.-X. Guo, S.-K. Luo, M. Shang, and F.-X. Mei, “Birkhoffian formulations of nonholonomic constrained systems,” Reports on Mathematical Physics, vol. 47, no. 3, pp. 313–322, 2001.
View at: Publisher Site | Google Scholar
J.-L. Fu and L.-Q. Chen, “Perturbation of symmetries of rotational relativistic Birkhoffian systems and its inverse problem,” Physics Letters A, vol. 324, no. 2-3, pp. 95–103, 2004.
View at: Publisher Site | Google Scholar
Y. Zhang, “Poisson theory and integration method of Birkhoffian systems in the event space,” Chinese Physics B, vol. 19, no. 8, Article ID 080301, 2010.
View at: Publisher Site | Google Scholar
H.-B. Wu and F.-X. Mei, “Type of integral and reduction for a generalized Birkhoffian system,” Chinese Physics B, vol. 20, no. 10, Article ID 104501, 2011.
View at: Publisher Site | Google Scholar
Y. Zhang, “Stability of motion for generalized Birkhoffian systems,” Defence Technology, vol. 6, no. 3, pp. 161–165, 2010.
View at: Google Scholar
Y. Zhang, “The method of variation on parameters for integration of a generalized Birkhoffian system,” Acta Mechanica Sinica, vol. 27, no. 6, pp. 1059–1064, 2011.
View at: Publisher Site | Google Scholar
Y. Zhang and Y. Zhou, “Symmetries and conserved quantities for fractional action-like Pfaffian variational problems,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 783–793, 2013.
View at: Publisher Site | Google Scholar
Y. Zhang and X.-H. Zhai, “Noether symmetries and conserved quantities for fractional Birkhoffian systems,” Nonlinear Dynamics, vol. 81, no. 1-2, pp. 469–480, 2015.
View at: Publisher Site | Google Scholar
X.-H. Zhai and Y. Zhang, “Noether symmetries and conserved quantities for Birkhoffian systems with time delay,” Nonlinear Dynamics, vol. 77, no. 1-2, pp. 73–86, 2014.
View at: Publisher Site | Google Scholar
B. Yan and Y. Zhang, “Noether’s theorem for fractional Birkhoffian systems of variable order,” Acta Mechanica, vol. 227, no. 9, pp. 2439–2449, 2016.
View at: Publisher Site | Google Scholar
S.-K. Luo and Y.-L. Xu, “Fractional Birkhoffian mechanics,” Acta Mechanica, vol. 226, no. 3, pp. 829–844, 2015.
View at: Publisher Site | Google Scholar
C.-J. Song and Y. Zhang, “Noether theorem for Birkhoffian systems on time scales,” Journal of Mathematical Physics, vol. 56, no. 10, Article ID 102701, 2015.
View at: Publisher Site | Google Scholar
X.-H. Zhai and Y. Zhang, “Lie symmetry analysis on time scales and its application on mechanical systems,” Journal of Vibration and Control, vol. 25, no. 3, pp. 581–592, 2019.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Xiang-Hua Zhai and Yi Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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