Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 3, June 2024



Page(s)  263  272  
DOI  https://doi.org/10.1051/wujns/2024293263  
Published online  03 July 2024 
Mathematics
CLC number: O316
Perturbation to Symmetries and Adiabatic Invariants for Generalized Birkhoffian Systems on Time Scales
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
^{†} Corresponding author. Email: songchuanjingsun@mail.usts.edu.cn
Received:
9
October
2023
Time scale is a new and powerful tool for dealing with complex dynamics problems. The main result of this study is the exact invariants and adiabatic invariants of the generalized Birkhoffian system and the constrained Birkhoffian system on time scales. Firstly, we establish the differential equations of motion for the above two systems and give the corresponding Noether symmetries and exact invariants. Then, the perturbation to the Noether symmetries and the adiabatic invariants for the systems mentioned above under the action of slight disturbance are investigated, respectively. Finally, two examples are provided to show the practicality of the findings.
Key words: time scale / generalized Birkhoffian system / Noether symmetry / perturbation and adiabatic invariants
Cite this article: HOU Shuang, SONG Chuanjing. Perturbation to Symmetries and Adiabatic Invariants for Generalized Birkhoffian Systems on Time Scales[J]. Wuhan Univ J of Nat Sci, 2024, 29(3): 263272.
Biography: HOU Shuang, female, Master candidate, research direction: analytical mechanics. Email: hous00309280@163.com
Fundation item: Supported by the National Natural Science Foundation of China (12172241, 12272248, 11972241, 12002228) and Qing Lan Project of Colleges and Universities in Jiangsu Province
© Wuhan University 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^{[25]}.
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 PfaffBirkhoff 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^{[1214]} 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 timescale operation rules, variational problems, symmetries, and conserved quantities has attracted widespread attention^{[1624]}. 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^{[2528]}.
However, Anerot et al^{[29]} highlighted the derivation of the second EulerLagrange 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 EulerLagrange 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 timescale calculus, please see Ref. [1618].
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 leftdense. Also, if and , then we say is rightdense. 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 rdcontinuous function if is continuous at rightdense points and has a (finite) boundary at leftdense points in , with the set of being represented as , or . The set of functions , which are differentiable and have a derivative that is rdcontinuous, 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 rdcontinuous 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 (DuboisReymond) 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 PfaffBirkhoff principle is
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 , , , , .
Then from Lemma 1, we get
and then taking the derivative of Eq. (4), the equations of the generalized Birkhoffian system on time scales can be obtained as^{[24]}
where , .
We introduce the following infinitesimal transformations regarding and
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 velocitydependent 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
holds. From Eq. (8), there is
which is to say,
By differentiating both sides of Eq. (9) with respect to the infinitesimal parameter , let , and applying ^{[20]}, we obtain the Noether identity
Theorem 1 If and satisfy Eq. (10), then the generalized Birkhoffian system (5) has the exact invariant of the form
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
then do the isochronous variation for Eq. (12), we have
The constrained Birkhoff's equations with multiplier form on time scales can be obtained as^{[22]}
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
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
Therefore, for , Eq. (16) gives
Theorem 2 For the corresponding free Birkhoffian system (15), if and satisfy Eq. (17), then there exists the exact invariant
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)).
3 Adiabatic Invariants
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 zthorder 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
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
Therefore, we hold the theorem that follows.
Theorem 3 For disturbed generalized Birkhoffian system (19), if and satisfy
then this system has a zthorder adiabatic invariant
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 zthorder 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
Theorem 4 For the corresponding free Birkhoffian system (23), which is disturbed by , if and satisfy the following equation
then this system has a zthorder adiabatic invariant
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 zthorder 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
Firstly, it can be calculated that , . And from Eq. (5), the equations of the system can be given as
Eq. (10) gives
And the infinitesimal generators have been found
Therefore, from Eq. (11), it can be obtained that the exact invariants are
If , then , , we can get the classical results and of Eqs. (31) and (32)
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
Eq. (21) gives
Here the following solutions satisfy Eq. (36)
So the firstorder adiabatic invariants are achievable according to Theorem 3
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
Birkhoff's functions
and the constraint equations
in which , are constant, we discuss the adiabatic invariant of the above system on time scales .
Eq. (14) gives
According to Eqs. (43) and (44), we get
Therefore, there is
and the equation for the corresponding free Birkhoffian system is
Eq. (17) gives
Eq. (48) has the following solutions
Also, Eq. (13) gives
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
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 firstorder adiabatic invariants. Consider that the minor disturbance forces on the system are
and from Eq. (24) we can get
By calculation, it can be concluded that
are the solutions to Eq. (55). Also, Eq. (13) gives
Substituting Eq. (20) into Eq. (58) has
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 firstorder adiabatic invariants are
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.
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