Open Access
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

© Wuhan University 2022

Licence Creative CommonsThis 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[2-10] 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 self-inductive 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[18-22].

The symmetries and conserved quantities for different dynamical systems under time-scale 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 co-workers[24] studied the time-scale second Euler-Lagrange equation, and derived Noether's conserved quantities on time scales by using this equation. Subsequently, many scholars[25-34] began to carry out a series of studies on time-scale 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 time-scale conserved quantity for Lagrangian systems. Zhang and his team[36-39] also discussed the time-scale 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 structure-preserving algorithm. Anerot et al [41] rederived time-scale 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 time-scale 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 quasi-symmetry theorem of three kinds of Birkhoffian systems about time-scale with nonshifted variational problems. Therefore, this paper will further research Lie symmetry theorem of three Birkhoffian systems about time-scale with nonshifted variational problems, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system.

1 The Dynamical Equations

About time-scale calculus and its properties, please refer to Ref. [2].

1.1 Free Birkhoffian System

The time-scale nonshifted Pfaff-Birkhoff action is written as

(1)

where is the delta derivative of with respect to . The Birkhoffian , Birkhoff's functions . Let all functions are differential and their derivatives are rd-continuous[2], .

The isochronous variation principle

(2)

and exchange relationship

(3)

with the endpoint conditions

(4)

are called the nonshifted Pfaff-Birkhoff principle.

We can obtain the time-scale nonshifted Birkhoff's equations [42]:

(5)

1.2 Generalized Birkhoffian System

We extend the principle (2)-(4) to the time-scale nonshifted generalized Pfaff-Birkhoff principle

(6)

We have the time-scale nonshifted generalized Birkhoff's equations [44]

(7)

1.3 Constrained Birkhoffian System

If the constraint equations are shown as

(8)

To calculate the variation in equation (8)

(9)

Form Eq. (9), we have

(10)

Integration by parts with equation (10), we get

(11)

By the principle and Eq. (11), we have the time-scale nonshifted constrained Birkhoff's equations [44]

(12)

If the systems are nonsingular, by using Eq. (8) and (12), we can solve , the equation (12) can be written

(13)

where

(14)

Equation (13) should be called as the corresponding nonshifted free Birkhoffian system (5).

2 Structural Equations and Conserved Quantities

The infinitesimal transformations

(15)

where is an infinitesimal parameter, and are the infinitesimal generators.

Introduce the infinitesimal generated vectors [17]

(16)

And an extension of infinitesimal generator

(17)

2.1 Free Birkhoffian System

Definition 1   If the infinitesimal transformations (15) satisfy deterministic equation

(18)

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

(19)

This system (5) has the conserved quantity, that is

(20)

Proof   Take the - derivative for equation (20)

(21)

According to

(22)

Substituting equation (22) into equation (21), we can get

(23)

Therefore, the proof is completed.

2.2 Generalized Birkhoffian System

Definition 2   If and of the infinitesimal transformations (15) satisfy deterministic equation

(24)

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

(25)

This system (7) has the conserved quantity, that is

(26)

Proof   Take the - derivative for equation (26)

(27)

And because

(28)

Substituting Eqs. (7) and (28) into equation (27), we can get

(29)

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

(30)

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

(31)

then system (13) has the conserved quantity, that is

(32)

3 Example

Example 1 On time scales, the Hojman-Urrutia problem can be written

(33)

According to the study of Santilli[45], the Hojman-Urrutia problem admits a Birkhoffian representation. But since the equation itself is not self-adjoint, it has no Hamiltonian representation.

By the nonshifted Birkhoff's equation (5), we have

(34)

We take

(35)

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

(36)

Substituting (35) into (36), we have .

By Theorem 1, we can get conserved quantity

(37)

Let , we have , then the conserved quantity

(38)

If the initial condition are , let the trajectory of motion and the conserved quantity are calculated, the results are shown in Fig. 1.

thumbnail Fig. 1

Simulations of a1, a2, a3, a4, I on time scale 𝕋 = {2n: n ∈ ℕ ∪ {0}}

Let , we have , then the conserved quantity

(39)

Example 2 The nonshifted generalized Birkhoffian and Birkhoff's functions on time scales are

(40)

By the nonshifted generalized Birkhoff's equation (7), we have

(41)

We take

(42)

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

(43)

Substituting (42) into (43), we have .

By Theorem 2, we can get conserved quantity

(44)

Let , we get , then the conserved quantity

(45)

Example 3 On time scales, let , the nonshifted constrained Birkhoffian and Birkhoff's functions are

(46)

The constraints equations are

(47)

By Eq. (13), we can get

(48)

From Eqs. (47) and (48), we have

(49)

Hence, we have

(50)

Let

(51)

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

(52)

Substituting (51) into (52), we have .

By Theorem 3, we can get conserved quantity

(53)

4 Conclusion

Time-scale 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 time-scale 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

thumbnail Fig. 1

Simulations of a1, a2, a3, a4, I on time scale 𝕋 = {2n: n ∈ ℕ ∪ {0}}

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