Perturbation to Symmetries and Adiabatic Invariants for Generalized Birkhoffian Systems on Time Scales

: 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 invari‐ ants. 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.


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 Noethers 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 Noethers theory has long been a widely discussed subject [2][3][4][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 Birkhoffs 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 Birkhoffs equation [9,10] from it.In 2013, they concluded a study on the dynamics of general-ized 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][13][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][17][18][19][20][21][22][23][24] .The perturbation of symmetries and their adiabatic invariants in dynamic systems under small perturbation actions are closely related to the systems 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][26][27][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.

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 T. Real number set R, natural number set N, integer set Z, and so on are all time scales.Definition 1 On time scale T, for s Î T, we declare the forward jump operator σ: T ® T with σ (s) = inf {r Î T: r > s} and the backward jump operator ρ: T ® T with ρ (s) = sup{r Î T: r < s}.In particular, we let inf AE = sup T and sup AE = inf T, with AE representing the empty set.If s > inf T and ρ (s) = s, we say s is left-dense.Also, if s < sup T and σ (s) = s, then we say s is right-dense.Besides, when ρ (s) = s = σ (s) holds, s is called dense.Definition 2 Define the graininess function μ: T ® ) [0¥ as the difference between the jump operator σ (s) and s, that is μ (s) = σ (s) -s.

Definition 3
We define the set T κ as

Definition 4
We call a function f: T ® R a rd-continuous function if f is continuous at right-dense points and has a (finite) boundary at left-dense points in T, with the set of f: T ® R being represented as C rd ( TR ) , C rd (T) or C rd .
The set of functions f: T ® R, which are differentiable and have a derivative that is rd-continuous, can be represented as Definition 5 Assume a function f: T ® R and s Î T κ is given.If there exists U = ( s -δs + δ ) Ç T for "ε > 0 and some δ > 0 such that holds true for all t Î U, then we call f D (s) the delta derivative of f at s. Additionally, if f D (s) is present for all s Î T κ , we assert that f has differentiability on T κ .
Definition 6 A function F: T ® R is referred to as an antiderivative of f: for every s Î T κ .And antiderivatives are present in every rd-continuous function f: T ® R. Furthermore, if s 0 Î T, then F defined by F (s) = ∫ s 0 s f ( ) τ Dτ for s Î T, acts as an antiderivative of f.
Besides, assume H, F: T ® R are differentiable at t Î T κ , the following formulas valid.
where H σ = H  σ, the map α: [ ab] Ç T ® R is an increasing C 1 rd function, and its image is a new time scale with D ˉ symbolizes delta operator.

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.

Exact Invariant for Generalized Birkhoffian System
Generalized Pfaff-Birkhoff principle is where Eq. ( 2) is the boundary condition and Eq. ( 3) is the commutative relationship, and the Birkhoffian B ( From Eqs. ( 1)-( 3), we have 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] ¶R ν ( ) where We introduce the following infinitesimal transformations regarding t and a ν where υ is an infinitesimal parameter, ξ 0 B0 and ξ 0 Bν 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
rd function, and its image is a new time scale with D ˉ and σ ˉ representing the delta operator and forward jump operator, respectively.And σ ˉ α = α  σ 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 By differentiating both sides of Eq. ( 9) with respect to the infinitesimal parameter υ, let υ = 0, and applying δa l = Da la D l Dt [20] , we obtain the Noether identity Theorem 1 If ξ 0 B0 ( ta ϖ ) and ξ 0 Bν ( ta ϖ ) 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 Λ l = 0, then Eq. ( 5) degenerates to Birkhoffian system, and the conclusion of Theorem 1 is still valid.

Exact Invariant for Constrained Birkhoffian System
If the variables a σ ϖ 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 Birkhoffs equations with multiplier form on time scales can be obtained as [22] ¶R l ( ) where t Î [t 1 t 2 ] κ , νlϖ = 122n, λ β ( ta σ ϖ ) is called the constrained multiplier, β = 122k.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 ¶R l ( ) where κ .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 t Î [t 1 t 2 ] κ , Eq. ( 16) gives Theorem 2 For the corresponding free Birkhoffian system (15), if ξ 0 B0 ( ta ϖ ) and ξ 0 Bν ( ta ϖ ) 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 ξ 0 B0 ( ta ϖ ) and ξ 0 Bν ( ta ϖ ) satisfy the Noether identity (Eq.( 17)) and constrained equation (Eq.( 13)), then the system exists the same exact invariant (Eq.( 18)).

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.
) is a physical quantity on time scales of the mechanical system containing a small parameter ε with a maximum power of z, and DI z Dt is proportional to ε z + 1 , then I z 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 z = 0.

Adiabatic Invariant for Generalized Birkhoffian System
Assuming that generalized Birkhoffian system ( 5) is disturbed by the minor disturbance εW Bl ( ta σ ϖ ) , thus ¶R ν ( ) As a result of the effect of small perturbation εW Bl , 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 ξ m B0 ( ta ϖ ) and ξ m Bν ( ta ϖ ) satisfy (21)   then this system has a z-th-order adiabatic invariant where we let ξ m -1 Proof Using Eqs. ( 19) and ( 21) can achieve D Dt It can be seen that DI GBz Dt is proportional to ε z + 1 .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).

Adiabatic Invariant for Constrained Birkhoffian System
Assuming that the corresponding free Birkhoffian system (15) is disturbed by the minor disturbance ε B W Cl ( ta σ ϖ ) , then we have ¶R ν ( ) Theorem 4 For the corresponding free Birkhoffian system (23), which is disturbed by Bν ( ta ϖ ) satisfy the following equation (24)   then this system has a z-th-order adiabatic invariant where we let ξ m -1 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 ξ m B0 ( ta ϖ ) and ξ m Bν ( ta ϖ ) 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).

Examples
Example 1 We discuss the exact invariant and adiabatic invariant of the following system on the time scale T = {2 n + 1 :n Î N}.The generalized Birkhoffian B, Birkhoffs functions R μ and the additional items Λ l are Firstly, it can be calculated that σ (t) = 2t, μ (t) = t.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 T = R, then σ (t) = t, μ (t) = 0, we can get the classical results I C0 and I C1 of Eqs. ( 31) and ( 32) (34) Let a 1 (1) = 1, a 2 (1) = 0, a 3 (1) = 1 2 , a 4 (1) = 0, we can plot the I C0 and I C1 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 first-order adiabatic invariants are achievable according to Theorem 3 (40) When T = R, 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 Birkhoffs functions , we can also plot the exact invariants I CC0 and I CC1 , 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 Dτ , (60) Similarly, the systems higher order adiabatic invariants and those on other time scales are attainable.

Conclusion
Symmetry plays a pivotal and widespread role in mechanical systems, and Noethers 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.

Fig. 1
Fig. 1 Trace of I C0 and I C1 Fig. 2 Trace of I CC0 and I CC1