Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 3, June 2022



Page(s)  211  217  
DOI  https://doi.org/10.1051/wujns/2022273211  
Published online  24 August 2022 
Mathematics
CLC number: O 316
Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales
^{1}
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou
215009, Jiangsu, China
^{2}
College of Civil Engineering, Suzhou University of Science and Technology, Suzhou
215011, Jiangsu, China
^{†} To whom correspondence should be addressed. Email: zhy@mail.usts.edu.cn
Received:
6
January
2022
The Lie theorem for Birkhoffian systems with timescale nonshifted variational problems are studied, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system. First, the timescale nonshifted generalized PfaffBirkhoff principle is established, and the dynamical equations for three Birkhoffian systems under nonshifted variational problems are deduced. Afterwards, in the timescale nonshifted variational problems, by introducing the infinitesimal transformations, Lie symmetry for free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system are defined respectively. Then Lie symmetry theorems for three kinds of Birkhoffian systems are deduced and proved. In the end, three examples are given to explain the applications for the results.
Key words: time scales / Lie symmetry / conserved quantity / nonshifted Birkhoffian system
Biography: CHEN Jinyue, female, Master candidate, research direction: analytical mechanics. Email: 1291385301@qq.com
Foundation item: Supported by the National Natural Science Foundation of China (11972241, 11572212), the Natural Science Foundation of Jiangsu Province of China (BK20191454) and the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province of China (KYCX20_2744)
© Wuhan University 2022
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
Time scale provides effect mathematical tools for the research of complex dynamical systems. German scholar Hilger^{[1]} first unified and extended the theory of continuity and discreteness ― an analytical theory on time scales. Since then, many scholars^{[210] }have conducted detailed studies on time scales. Time scale is also widely used in practical problems. For example, in a simple series circuit composed of resistance, capacitance and selfinductive coil, when the capacitor is periodically closed at a fixed frequency, the current change rate in the circuit can be precisely described by the derivative on time scales^{[11]}, nonlinear partially defined systems on an arbitrary unbounded time scale are studied^{[12,13]}.
The research for symmetry and conservation law is an important direction in the development of modern analysis mechanics. Noether^{[14] }studied the invariance of action integrals in finite continuous groups and revealed the internal relationship between symmetries and conserved quantities in dynamic systems. Lutzky^{[15]} introduced the Lie symmetry method of mechanical system, that is, the equation remains unchanged after the introduction of infinitesimal group transformation. Mei^{[16,17]} studied the dynamics equations for holonomic systems, nonholonomic systems and Birkhoffian systems in detail, as well as the corresponding symmetries and conserved quantities for these systems. Afterwards, some progress has been made in the study of symmetry and conserved quantities for different mechanical systems^{[1822]}.
The symmetries and conserved quantities for different dynamical systems under timescale shifted variational problems are firstly studied. Take the Lagrange system as an example. In classical dynamical system, assuming that the configurations for the dynamics systems were determined by generalized coordinates , then Lagrangian functions were written as . On time scales, the Lagrangian functions were written as . On the shifted variational problem, Bartosiewicz and Torres^{[23]} derived Noether's theorem under the derivative by using the time reparameter method, and proved the variational problem of mechanical system. Bartosiewicz and his coworkers^{[24] }studied the timescale second EulerLagrange equation, and derived Noether's conserved quantities on time scales by using this equation. Subsequently, many scholars^{[2534] }began to carry out a series of studies on timescale the variational principle for constrained mechanical systems and Noether's theorem of different dynamic systems. However, on a time scale, there are few studies on Lie symmetry in mechanical systems. Cai et al ^{[35] }preliminarily studied Lie symmetry and timescale conserved quantity for Lagrangian systems. Zhang and his team^{[3639]} also discussed the timescale for different dynamic systems and its Lie theorem.
Recently, Bourdin^{[40]} studied the nonshifted problem on time scales. In the nonshifted variational principle, the classical Lagrangian function was written as on time scales. They also found the nonshifted variational problem in discrete conditions related to the structurepreserving algorithm. Anerot et al ^{[41] } rederived timescale Noether's theorem with shifted and nonshifted variational problems, and the correctness of the results was illustrated by numerical simulation. The calculation of the nonshifted variational principle is better than that of the shifted variational principle in the dynamic system. Song and Cheng^{[42] }studied Noether's symmetry for free Birkhoffian systems and Hamiltonian systems about timescale with nonshifted variational problems. Song^{[43]} also studied the adiabatic invariants for three kinds of dynamical systems under the nonshifted variational problem. Then Chen and Zhang^{[44]} researched Noether's quasisymmetry theorem of three kinds of Birkhoffian systems about timescale with nonshifted variational problems. Therefore, this paper will further research Lie symmetry theorem of three Birkhoffian systems about timescale with nonshifted variational problems, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system.
1 The Dynamical Equations
About timescale calculus and its properties, please refer to Ref. [2].
1.1 Free Birkhoffian System
The timescale nonshifted PfaffBirkhoff action is written as
where is the delta derivative of with respect to . The Birkhoffian , Birkhoff's functions . Let all functions are differential and their derivatives are rdcontinuous^{[2]}, .
The isochronous variation principle
and exchange relationship
with the endpoint conditions
are called the nonshifted PfaffBirkhoff principle.
We can obtain the timescale nonshifted Birkhoff's equations ^{[42]}:
1.2 Generalized Birkhoffian System
We extend the principle (2)(4) to the timescale nonshifted generalized PfaffBirkhoff principle
We have the timescale nonshifted generalized Birkhoff's equations ^{[44]}
1.3 Constrained Birkhoffian System
If the constraint equations are shown as
To calculate the variation in equation (8)
Form Eq. (9), we have
Integration by parts with equation (10), we get
By the principle and Eq. (11), we have the timescale nonshifted constrained Birkhoff's equations ^{[44]}
If the systems are nonsingular, by using Eq. (8) and (12), we can solve , the equation (12) can be written
where
Equation (13) should be called as the corresponding nonshifted free Birkhoffian system (5).
2 Structural Equations and Conserved Quantities
The infinitesimal transformations
where is an infinitesimal parameter, and are the infinitesimal generators.
Introduce the infinitesimal generated vectors ^{[17]}
And an extension of infinitesimal generator
2.1 Free Birkhoffian System
Definition 1 If the infinitesimal transformations (15) satisfy deterministic equation
and corresponding symmetry is said the Lie symmetry for nonshifted free Birkhoffian systems (5).
Theorem 1 If and satisfy the deterministic equation (18), and there exists satisfying structural equation
This system (5) has the conserved quantity, that is
Proof Take the  derivative for equation (20)
According to
Substituting equation (22) into equation (21), we can get
Therefore, the proof is completed.
2.2 Generalized Birkhoffian System
Definition 2 If and of the infinitesimal transformations (15) satisfy deterministic equation
then corresponding symmetry is said the Lie symmetry for the nonshifted generalized Birkhoffian system (7).
Theorem 2 If and satisfy the deterministic equation (24), and there exists satisfying the structural equation
This system (7) has the conserved quantity, that is
Proof Take the  derivative for equation (26)
And because
Substituting Eqs. (7) and (28) into equation (27), we can get
Therefore, the proof is completed.
2.3 Constrained Birkhoffian System
Definition 3 If and of the infinitesimal transformations (15) satisfy the Lie symmetry deterministic equation
then corresponding symmetry is said the Lie symmetry for nonshifted constrained Birkhoffian system (13).
Theorem 3 If and satisfy the deterministic equation (30), and exist satisfying the structural equation
then system (13) has the conserved quantity, that is
3 Example
Example 1 On time scales, the HojmanUrrutia problem can be written
According to the study of Santilli^{[45]}, the HojmanUrrutia problem admits a Birkhoffian representation. But since the equation itself is not selfadjoint, it has no Hamiltonian representation.
By the nonshifted Birkhoff's equation (5), we have
We take
The generator (35) satisfies determination equation (18), so the generator (35) corresponds to Lie symmetry in the system (5).
Form the structural equation (19), we can get
Substituting (35) into (36), we have .
By Theorem 1, we can get conserved quantity
Let , we have , then the conserved quantity
If the initial condition are , let the trajectory of motion and the conserved quantity are calculated, the results are shown in Fig. 1.
Fig. 1 Simulations of a_{1}, a_{2}, a_{3}, a_{4}, I on time scale 𝕋 = {2^{n}: n ∈ ℕ ∪ {0}} 
Let , we have , then the conserved quantity
Example 2 The nonshifted generalized Birkhoffian and Birkhoff's functions on time scales are
By the nonshifted generalized Birkhoff's equation (7), we have
We take
The generator (42) satisfies determination equation (24), so the generator (42) corresponds to Lie symmetry in this system (7).
According to the structural equation (25), we can get
Substituting (42) into (43), we have .
By Theorem 2, we can get conserved quantity
Let , we get , then the conserved quantity
Example 3 On time scales, let , the nonshifted constrained Birkhoffian and Birkhoff's functions are
The constraints equations are
By Eq. (13), we can get
From Eqs. (47) and (48), we have
Hence, we have
Let
The generator (51) satisfies determination equation (30), so the generator (51) corresponds to Lie symmetry in this system (13).
According to the structural equation (31), we can get
Substituting (51) into (52), we have .
By Theorem 3, we can get conserved quantity
4 Conclusion
Timescale theory has been generally used in many fields. When , the dynamic equation on time scale can be degraded into a continuous differential equation. When , the dynamic equation on time scale can be degraded into discrete difference equation. This paper mainly studies the timescale with nonshifted variational problem which Lie theorem for three kinds of Birkhoffian systems. In this paper, nonshifted generalized Birkhoff's equations and nonshifted constrained Birkhoff's equations are derived by using principle (6). By introducing infinitesimal transformation, the structural equation of Lie symmetry is derived, and then the Noether conserved quantities caused by Lie symmetries is deduced. Since the study about nonshifted variational problem on a time scale is just beginning, the research of paper has great theoretical and practical significance.
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All Figures
Fig. 1 Simulations of a_{1}, a_{2}, a_{3}, a_{4}, I on time scale 𝕋 = {2^{n}: n ∈ ℕ ∪ {0}} 

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