Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 29, Number 3, June 2024 263 - 272 https://doi.org/10.1051/wujns/2024293263 03 July 2024

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

Noether proposed the Noether symmetry method in her paper "Invariante Variationsprobleme"[1] in 1918. The theorem for seeking conserved quantities through the Noether symmetry method is called Noether's theorem. It unveiled the correlation between the conserved quantities of mechanical systems and their inherent dynamical symmetries. Since then, the symmetry of mechanical systems has become one of the most effective ways to find the invariants of more complex mechanical systems, so the study of Noether's theory has long been a widely discussed subject[2-5].

Birkhoff mechanics is the subsequent evolution of classical mechanics following the advent of quantum mechanics, marking the fifth phase of the development of classical mechanics[6]. Therefore, the theory of Birkhoffian mechanics can be utilized in Hamilton mechanics, Lagrange mechanics, Newton mechanics[7], statistical mechanics, biophysics, and other domains[8]. However, the usual Birkhoff equation is challenging to construct. In contrast, the generalized Birkhoff's equation with only one additional term is easy to implement and has more degrees of freedom. Therefore, studying the dynamics of generalized Birkhoffian systems is vital. Mei et al generalized the Pfaff-Birkhoff principle and derived the generalized Birkhoff's equation[9,10] from it. In 2013, they concluded a study on the dynamics of generalized Birkhoffian systems, providing a comprehensive and systematic discussion of the system[11]. Subsequently, much progress in studying the symmetry and exact invariants of generalized Birkhoffian systems[12-14] followed.

Time scale is a mathematical theory proposed by German scholar Hilger in 1988[15]. It not only aids in uncovering the resemblances and disparities between continuous and discrete systems but also helps us understand the essential problems of complex dynamic systems more accurately and clearly, so it has become a popular tool in various scientific and engineering domains. In recent years, research on time-scale operation rules, variational problems, symmetries, and conserved quantities has attracted widespread attention[16-24]. The perturbation of symmetries and their adiabatic invariants in dynamic systems under small perturbation actions are closely related to the system's symmetry and specific dynamic characteristics. Hence, its research is also essential. Research on the perturbation of symmetries and adiabatic invariants on time scales is just beginning, and the research results on Noether symmetry are even rarer[25-28].

However, Anerot et al[29] highlighted the derivation of the second Euler-Lagrange equation on time scales in Ref. [30] is incorrect and provided their method to obtain the Noether theorem for Lagrangian system on time scales. Therefore, the correctness of the conclusions obtained by applying the second Euler-Lagrange equation in the references still needs to be explored. This paper restudy exact invariants of generalized Birkhoffian systems on time scales using the method given in Ref. [29], and the perturbations and adiabatic invariants are discussed on this basis.

## 1 Preliminaries

The definitions and properties listed below will be utilized throughout the entirety of this paper. A more detailed content and proof process of time-scale calculus, please see Ref. [16-18].

Time scale is an arbitrary closed subset of a real number set that is not empty, usually denoted as . Real number set , natural number set , integer set , and so on are all time scales.

Definition 1   On time scale , for , we declare the forward jump operator with and the backward jump operator with . In particular, we let and , with representing the empty set. If and , we say is left-dense. Also, if and , then we say is right-dense. Besides, when holds, is called dense.

Definition 2   Define the graininess function as the difference between the jump operator and , that is .

Definition 3   We define the set as

Definition 4   We call a function a rd-continuous function if is continuous at right-dense points and has a (finite) boundary at left-dense points in , with the set of being represented as , or . The set of functions , which are differentiable and have a derivative that is rd-continuous, can be represented as , , or .

Definition 5   Assume a function and is given. If there exists for and some such that

holds true for all , then we call the delta derivative of at . Additionally, if is present for all , we assert that has differentiability on .

Definition 6   A function is referred to as an antiderivative of when is satisfied for every . And antiderivatives are present in every rd-continuous function . Furthermore, if , then defined by for , acts as an antiderivative of .

Besides, assume , are differentiable at , the following formulas valid.

where , the map is an increasing function, and its image is a new time scale with symbolizes delta operator.

Lemma 1   (Dubois-Reymond) Let , and . Following that, for all with holds if and only if on for certain .

## 2 Exact Invariants

In this section, the equations and exact invariants of the generalized Birkhoffian system and the constrained Birkhoffian system on time scales will be presented.

### 2.1 Exact Invariant for Generalized Birkhoffian System

Generalized Pfaff-Birkhoff principle is

(1)

(2)

(3)

where Eq. (2) is the boundary condition and Eq. (3) is the commutative relationship, and the Birkhoffian , the Birkhoff's functions and the additional items all belong to , , , , .

From Eqs. (1)-(3), we have

Then from Lemma 1, we get

(4)

and then taking the derivative of Eq. (4), the equations of the generalized Birkhoffian system on time scales can be obtained as[24]

(5)

where , .

We introduce the following infinitesimal transformations regarding and

(6)

(7)

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

It is noteworthy that the generating function is usually a function of time and coordinates, and the transformation of coordinates and time that keep the motion equation unchanged forms Lie group. Sarlet and Cantrijn[31] discussed in detail the issue of function dependency in generating functions. Due to our research on the invariance of dynamic systems on time scales, we consider allowing for velocity-dependent transformations to expand the dimension of the generators.

Suppose that the map is an increasing function, and its image is a new time scale with and representing the delta operator and forward jump operator, respectively. And holds. Eqs. (6) and (7) are said to be Noether symmetric transformations for the generalized Birkhoffian system if and only if

(8)

holds. From Eq. (8), there is

which is to say,

(9)

By differentiating both sides of Eq. (9) with respect to the infinitesimal parameter , let , and applying [20], we obtain the Noether identity

(10)

Theorem 1   If and satisfy Eq. (10), then the generalized Birkhoffian system (5) has the exact invariant of the form

(11)

Proof   Making use of Eqs. (5) and (10), we get

Remark 1   For generalized Birkhoffian system Eq. (5), if there is , then Eq. (5) degenerates to Birkhoffian system, and the conclusion of Theorem 1 is still valid.

### 2.2 Exact Invariant for Constrained Birkhoffian System

If the variables in the Birkhoffian system have a correlation, and are constrained by the constraint equations

(12)

then do the isochronous variation for Eq. (12), we have

(13)

The constrained Birkhoff's equations with multiplier form on time scales can be obtained as[22]

(14)

where , , is called the constrained multiplier, . Then the constrained multipliers can be calculated by Eqs. (12) and (14). Substitute the constrained multipliers that has been calculated into Eq. (14), then Eq. (14) can be written as

(15)

where , . Eqs. (12) and (14) together are called constrained Birkhoff equations, and Eq. (15) is called the equation of the free Birkhoffian system corresponding to the constrained Birkhoffian system.

Eqs. (6) and (7) are said to be Noether symmetric transformations for the corresponding free Birkhoffian system Eq. (15) if and only if there holds

(16)

Therefore, for , Eq. (16) gives

(17)

Theorem 2   For the corresponding free Birkhoffian system (15), if and satisfy Eq. (17), then there exists the exact invariant

(18)

Proof   This proof process is similar to Theorem 1.

Remark 2   For the constrained Birkhoff Eqs. (12) and (14), if and satisfy the Noether identity (Eq. (17)) and constrained equation (Eq. (13)), then the system exists the same exact invariant (Eq. (18)).

This section studies the perturbation to symmetries and the corresponding adiabatic invariants of the disturbed generalized and constrained Birkhoffian system. First, the notion of adiabatic invariant is introduced.

Definition 7   If is a physical quantity on time scales of the mechanical system containing a small parameter with a maximum power of , and is proportional to , then is known as the z-th-order adiabatic invariant of the mechanical system on time scales.

In particular, the adiabatic invariant degenerates to the exact invariant when .

### 3.1 Adiabatic Invariant for Generalized Birkhoffian System

Assuming that generalized Birkhoffian system (5) is disturbed by the minor disturbance , thus

(19)

As a result of the effect of small perturbation , the primitive symmetry and exact invariant of the system will be altered correspondingly. Assuming that this change occurs on the basis of the system without disturbance, then

(20)

Therefore, we hold the theorem that follows.

Theorem 3   For disturbed generalized Birkhoffian system (19), if and satisfy

(21)

then this system has a z-th-order adiabatic invariant

(22)

where we let when .

Proof   Using Eqs. (19) and (21) can achieve

It can be seen that is proportional to . Therefore, according to Definition 7, it can be reached that Eq. (22) is a z-th-order adiabatic invariant of the disturbed generalized Birkhoffian system (19).

### 3.2 Adiabatic Invariant for Constrained Birkhoffian System

Assuming that the corresponding free Birkhoffian system (15) is disturbed by the minor disturbance , then we have

(23)

Theorem 4   For the corresponding free Birkhoffian system (23), which is disturbed by , if and satisfy the following equation

(24)

then this system has a z-th-order adiabatic invariant

(25)

where we let when .

Proof   A proof procedure similar to that of Theorem 3 can demonstrate the validity of this Theorem.

Remark 3   For the disturbed corresponding free Birkhoffian system (23), if and satisfy the Eq. (24) and constrained equation (Eq. (13)), so the symmetry perturbation of the corresponding free Birkhoffian system leads to the symmetry perturbation of the constrained Birkhoffian system, at which point the z-th-order adiabatic invariant of the disturbed constrained Birkhoffian system remains Eq. (25).

## 4 Examples

Example 1 We discuss the exact invariant and adiabatic invariant of the following system on the time scale . The generalized Birkhoffian , Birkhoff's functions and the additional items are

(26)

Firstly, it can be calculated that , . And from Eq. (5), the equations of the system can be given as

(27)

Eq. (10) gives

(28)

And the infinitesimal generators have been found

(29)

(30)

Therefore, from Eq. (11), it can be obtained that the exact invariants are

(31)

(32)

If , then , , we can get the classical results and of Eqs. (31) and (32)

(33)

(34)

Let , , , , we can plot the and and they are both constant according to Fig. 1.

Secondly, the adiabatic invariant of the system Eq. (27) is discussed. Suppose that the small disturbance forces are

(35)

Eq. (21) gives

(36)

Here the following solutions satisfy Eq. (36)

(37)

(38)

So the first-order adiabatic invariants are achievable according to Theorem 3

(39)

(40)

When , Eqs. (39) and (40) can also degenerate into classical case. Additionally, the adiabatic invariants of a higher order can also be acquired in a comparable manner.

Example 2 For Birkhoffian function

(41)

Birkhoff's functions

(42)

and the constraint equations

(43)

in which , are constant, we discuss the adiabatic invariant of the above system on time scales .

Eq. (14) gives

(44)

According to Eqs. (43) and (44), we get

(45)

Therefore, there is

(46)

and the equation for the corresponding free Birkhoffian system is

(47)

Eq. (17) gives

(48)

Eq. (48) has the following solutions

(49)

(50)

Also, Eq. (13) gives

(51)

Eq. (49) does not satisfy Eq. (51), so Eq. (49) corresponds to the Noether symmetric transformation of the free Birkhoffian system Eq. (47), and Eq. (50) satisfies Eq. (51), so, the corresponding transformation of Eq. (50) is the Noether symmetric transformation of the constrained Birkhoffian system.

From Eq. (18), it can be obtained that the exact invariants on this time scale are

(52)

(53)

If , then , , we can get the corresponding results and , and let , , , , we can also plot the exact invariants and , which are still constant as shown in Fig. 2.

Next, we calculate the first-order adiabatic invariants. Consider that the minor disturbance forces on the system are

(54)

and from Eq. (24) we can get

(55)

By calculation, it can be concluded that

(56)

(57)

are the solutions to Eq. (55). Also, Eq. (13) gives

(58)

Substituting Eq. (20) into Eq. (58) has

(59)

Eq. (57) satisfies Eq. (59), so Eq. (57) corresponds to the Noether symmetric transformation of the disturbed constrained Birkhoffian system, while Eq. (56) does not satisfy Eq. (59), so the corresponding transformation of Eq. (56) is the Noether symmetric transformation of the disturbed corresponding free Birkhoffian system. Then according to Eq. (25), the first-order adiabatic invariants are

(60)

(61)

Similarly, the system's higher order adiabatic invariants and those on other time scales are attainable.

## 5 Conclusion

Symmetry plays a pivotal and widespread role in mechanical systems, and Noether's theorem offers a pathway to discover additional conserved quantities. These conserved quantities hold significant importance in understanding mechanical systems' dynamics, stability, and computational aspects. But time scale and practical problems are very complex, so most of the current research on constrained mechanical systems on time scales is still in the stages of variational problems, symmetry problems, and conserved quantities and rarely involves the perturbation of symmetry and adiabatic invariant. In this paper, the exact invariants for the generalized Birkhoffian and constrained Birkhoffian systems and adiabatic invariants of the two systems under small perturbations are obtained. Each provided an example for verification and conducted numerical simulations on the exact invariants. Given the characteristics of unity and expansion exhibited by time scale, the approaches and results presented in this paper have the potential for broader application and extension to different systems.

## References

1. Noether A E. Invariante variationsprobleme [J]. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1918, KI(2): 235-257. [Google Scholar]
2. Mei F X. Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems [M]. Beijing: Science Press, 1999(Ch). [Google Scholar]
3. Zhao Y Y, Mei F X. Symmetries and Invariants of Mechanical Systems [M]. Beijing: Science Press, 1999(Ch). [Google Scholar]
4. Mei F X. Symmetries and Conserved Quantities of Constrained Mechanical Systems [M]. Beijing: Beijing Institute of Technology Press, 2004(Ch). [Google Scholar]
5. Mei F X. Analytical Mechanics II [M]. Beijing: Beijing Institute of Technology Press, 2013(Ch). [Google Scholar]
6. Mei F X. Advances in the symmetries and conserved quantities of classical constrained systems [J]. Advances in Mechanics, 2009, 39(1): 37-43(Ch). [Google Scholar]
7. Mei F X, Shi R C, Zhang Y F, et al. Birkhoff System Dynamics [M]. Beijing: Beijing Institute of Technology Press, 1996(Ch). [Google Scholar]
8. Santilli R M. Foundations of Theoretical Mechanics II[M]. Berlin: Springer-Verlag, 1983. [Google Scholar]
9. Mei F X, Zhang Y F, He G, et al. Fundamental framework of generalized Birkhoff system dynamics [J]. Transactions of Beijing Institute of Technology, 2007, 27(12): 1035-1038(Ch). [Google Scholar]
10. Mei F X. The Noether's theory of Birkhoffian systems [J]. Science in China, Serie A, 1993, 36(12): 1456-1467. [Google Scholar]
11. Mei F X. Generalized Birkhoff System Dynamics [M]. Beijing: Science Press, 2013(Ch). [Google Scholar]
12. Zhang Y. Birkhoff symmetries and conserved quantities of generalized Birkhoffian systems [J]. Acta Physica Sinica, 2009, 58(11): 7436-7439(Ch). [Google Scholar]
13. Song C J, Zhang Y. Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems [J]. International Journal of Non-Linear Mechanics, 2017, 90: 32-38. [NASA ADS] [CrossRef] [Google Scholar]
14. Zhou Y, Zhang Y. Noether symmetries for fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives [J]. Acta Mechanica, 2020, 231(7): 3017-3029. [Google Scholar]
15. Hilger S. Ein Maßkettenkalkül Mit Anwendung Auf Zentrumsmannigfaltigkeiten [D]. Würzburg: Universität Würzburg, 1988. [Google Scholar]
16. Bohner M, Peterson A C. Dynamic Equations on Time Scales: An Introduction with Applications [M]. Boston: Birkhäuser, 2001. [Google Scholar]
17. Bohner M, Georgiev S G. Multivariable Dynamic Calculus on Time Scales[M]. Cham: Springer International Publishing, 2016. [CrossRef] [Google Scholar]
18. Bohner M. Calculus of variations on time scales [J]. Dynamic Systems and Applications, 2004, 13(3-4): 339-349. [MathSciNet] [Google Scholar]
19. Zhai X H, Zhang Y. Noether theorem for non-conservative systems with time delay on time scales [J]. Communications in Nonlinear Science and Numerical Simulation, 2017, 52:32-43. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
20. Cai P P, Fu J L, Guo Y X. Noether symmetries of the nonconservative and nonholonomic systems on time scales [J]. Science China Physics, Mechanics and Astronomy, 2013, 56(5): 1017-1028. [CrossRef] [Google Scholar]
21. Song C J, Zhang Y. Noether theorem for Birkhoffian systems on time scales [J]. Journal of Mathematical Physics, 2015, 56(10): 102701. [CrossRef] [MathSciNet] [Google Scholar]
22. Zhang Y. Noether symmetries and conserved quantities of constrained Birkhoffian systems on time scales [J]. Journal of Dynamics and Control, 2019, 17(5): 482-486(Ch). [Google Scholar]
23. Song C J. Quasi-symmetry and conserved quantity for generalized Birkhoffian system on time scales [J]. Journal of Suzhou University of Science and Technology (Natural Science), 2020, 37(1): 12-17(Ch). [Google Scholar]
24. Song C J. The Study on Symmetries and Their Perturbation Theories for Constrained Mechanical Systems on Time Scales [D]. Nanjing: Nanjing University of Science and Technology, 2017(Ch). [Google Scholar]
25. Song C J, Zhang Y. Perturbation to symmetry and adiabatic invariant for Lagrangian system on time scale [J]. Journal of Nanjing University of Science and Technology, 2017, 41(2): 181-185(Ch). [Google Scholar]
26. Song C J. Perturbation to Noether quasi-symmetry and adiabatic invariants for nonholonomic systems on time scales [J]. Journal of Beijing Institute of Technology, 2019, 28(3): 469-476. [Google Scholar]
27. Jin S X, Li Y M. Perturbation and adiabatic invariants of Noether symmetry for generalized Chaplygin systems on time scales [J]. Chinese Quarterly of Mechanics, 2021, 42(3): 550-559(Ch). [Google Scholar]
28. Zhang Y. Perturbation to Noether symmetries and adiabatic invariants for nonconservative dynamic systems [J]. Acta Physica Sinica, 2013, 62(16): 164501. [Google Scholar]
29. Anerot B, Cresson J, Belgacem K H, et al. Noether's-type theorems on time scales [J]. Journal of Mathematical Physics, 2020, 61(11): 113502. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
30. Bartosiewicz Z, Martins N, Torres D F M. The second Euler-Lagrange equation of variational calculus on time scales [J]. European Journal of Control, 2011, 17(1): 9-18. [CrossRef] [MathSciNet] [Google Scholar]
31. Sarlet W, Cantrijn F. Generalizations of Noether's theorem in classical mechanics [J]. Siam Review, 1981, 23(4): 467-494. [Google Scholar]

## All Figures

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.