Perturbation to Lie symmetry and adiabatic invariants for Birkhoffian systems on time scales

Commun Nonlinear Sci Numer Simulat 75 (2019) 251–261

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Research paper

Perturbation to Lie symmetry and adiabatic invariants for Birkhoffian systems on time scales Yi Zhang a,∗, Xiang-Hua Zhai b a b

College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, P R China School of Science, Nanjing University of Science and Technology, Nanjing 210094, P R China

a r t i c l e

i n f o

Article history: Received 30 August 2018 Revised 19 March 2019 Accepted 3 April 2019 Available online 4 April 2019 Keywords: Birkhoffian system on time scales Lie symmetry Adiabatic invariant Perturbation

a b s t r a c t Lie symmetry of a dynamical system is the invariance of differential equations of motion under the infinitesimal transformations of a group and it can lead to invariants under certain conditions. Firstly, the Lie symmetry of Birkhoffian system on time scales is studied when there is no disturbance, the determining equations of Lie symmetry are established, and the exact invariants led by the Lie symmetry are given. Secondly, the perturbation to Lie symmetry and adiabatic invariants are studied when the system is subjected to small disturbance, and the determining equations of Lie symmetry of the disturbed system are established, and the condition of the Lie symmetry leading to adiabatic invariants and the form of adiabatic invariants are given. As an application of the results, we give the Lie symmetry theorems of Hamiltonian system on time scales. The results contain the exact invariants and adiabatic invariants of Lie symmetry for the classical continuous systems and discrete systems as their special cases. Two examples are given to illustrate the application of the results. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Lie symmetry is the invariance of differential equations under the action of one-parameter Lie group of point transformations [1,2]. In 1979, Lutzky [3] applied Lie symmetry theory to the differential equations of motion of dynamical systems. Prince et al. [4] studied the Lie symmetry of the classical Kepler problem. Zhao [5] studied the Lie symmetry of holonomic non-conservative systems. Mei established Lie symmetry theorem for nonholonomic non-conservative systems [6] and Birkhoffian systems [7]. In recent years, some progress has been made in the study of Lie symmetry of dynamical systems [8–13]. Dynamical symmetries are closely related to conserved quantities. A conserved quantity can be found through a dynamical symmetry. Conversely, the corresponding symmetry can be found through the conserved quantity. Symmetry is also called invariance, and conserved quantity can also be called invariant. In 1917, Burger [14] first proposed the concept of adiabatic invariant with respect to a special type of Hamiltonian system. After that, Kruskal [15], Djukic´ [16], Beulanov [17] and Notte et al. [18] obtained a series of important results. A classical adiabatic invariant is a certain physical quantity that changes more slowly than some parameter of the system that varies very slowly. In fact, the parameter varying very slowly is equivalent to the action of small disturbance. The change of symmetry under small disturbance and its corresponding

∗

Corresponding author. E-mail address: [email protected] (Y. Zhang).

https://doi.org/10.1016/j.cnsns.2019.04.005 1007-5704/© 2019 Elsevier B.V. All rights reserved.

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adiabatic invariants are closely related to the integrability of dynamical systems [19]. Therefore, the study of symmetry perturbation and adiabatic invariants of dynamical systems is always a hot topic, and some results [20–23] have been achieved recently. Birkhoffian mechanics is a natural development of Hamiltonian mechanics [24] and it represents a new stage in the development of analytical mechanics [25]. Birkhoffian mechanics is the most general mechanics constructed by Hamiltonian mechanics through the transformation theory, which can be applied to hadron physics, space mechanics, statistical mechanics, biophysics, and engineering, etc. Over the past two decades, some progress has been made in the research of Birkhoffian mechanics, especially in the Pfaff–Birkhoff principle, the integration theory and methods, the inverse problems of dynamics, the stability of motion, the geometric methods, the global analysis, and the symmetries and conserved quantities, and so on [26–35]. The calculus of time scales is a new mathematical theory put forward by Hilger [36,37] in order to unify discrete and continuous analysis. Using the theory of time scales, one can more clearly and more accurately characterize the physical essence of continuous and discrete systems and other complex dynamical systems, and the difference among them [38]. So, in recent years, the theory of time scales has been widely applied in many fields of science and engineering and has made some progress [39–45]. Recently, we studied the dynamics of Birkhoffian system and its Noether symmetry and conserved quantity on time scales [46]. In general, the study of the symmetries and invariants of dynamical systems on time scales is still an open research subject. In this paper, we will propose and study the perturbation to Lie symmetry and adiabatic invariants of the Birkhoffian system on time scales. The paper is organized as follows: In Section 2, the determining equations of Lie symmetry for the Birkhoffian system on time scales are given. The condition that Lie symmetry leads to conserved quantity and the form of conserved quantity are established. Since we are dealing in this section with the undisturbed case, the conserved quantity obtained is an exact invariant. In Section 3, the perturbation of Lie symmetry under small disturbance is studied, and the determining equations of Lie symmetry and the corresponding adiabatic invariant are given. Since Birkhoffian mechanics is a generalization of Hamiltonian mechanics, we apply the obtained results in Section 3 to Hamiltonian systems, and give the perturbation to Lie symmetry and adiabatic invariants of Hamiltonian systems on time scales in Section 4. In Section 5, two examples are given to illustrate the application of the main results. In the end, the conclusions are given. 2. Lie symmetry and exact invariants for Birkhoffian systems on time scales The definitions and basic properties of the calculus on time scales involved in this paper can be referred to Ref. [37]. Integral functional





S aμ (  ) =



t2

t1

 





σ Rν t, aσμ a ν − B t, aμ



t

(1)

can be called the Pfaff action on time scales, where aσμ (t ) = (aμ ◦ σ )(t ), a μ (t ) are the delta derivative of Birkhoff’s variables aμ (t ), B : R × R2n → R is the Birkhoffian and Rμ : R × R2n → R are Birkhoff’s functions. Suppose that these functions are C1rd and μ, ν = 1, 2, · · · , 2n. The isochronous variational principle

δS = 0

(2)

which satisfies the commutative conditions

  δ a μ = δ aμ , ( μ = 1, 2, · · · , 2n )

(3)

and the boundary conditions

δ aμ |t=t1 = δ aμ |t=t2 = 0, (μ = 1, 2, · · · , 2n)

(4)

can be called the Pfaff-Birkhoff principle on time scales. From the principle (2)–(4), the following equations can be derived

     σ  ∂ B t, aσρ ∂ Rv t, aσρ   av − Rμ t, aρ − = 0, (μ, ν, ρ = 1, 2, · · · , 2n). ∂ aσμ ∂ aσμ

(5)

Eq. (5) is Birkhoff’s equation on time scales [46]. We introduce the following one-parameter Lie group of point transformations











t¯ = t + ε ξ0 t, aρ , a¯ μ t¯ = aμ (t ) + ε ξμ t, aρ , (μ, ρ = 1, 2, · · · , 2n),

(6)

where ɛ is the infinitesimal parameter, ξ 0 and ξμ are the infinitesimal generators. The invariance of Birkhoff’s Eq. (5) on time scales under the infinitesimal transformations (6) comes down to the following conditions



X (1 )

      ∂ Rv    ∂B   ∂ Rv ( 1 ) R − X ( 1 ) a + − a ξ − X = 0, (μ = 1, 2, · · · , 2n), ξ ν v 0 μ ∂ aσμ v ∂ aσμ ∂ aσμ

(7)

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253

where [47]

∂ ∂ + ξμ , ∂t ∂ aμ

X ( 0 ) = ξ0

X (1 ) = X (0 ) +



 ξμ − a μ ξ0

(8)

 ∂ . ∂ a μ

(9)

Eq. (7) can be called the determining equations of the Lie symmetry for the Birkhoffian system (5) on time scales. Thus, we have the following definition. Definition 1. If the infinitesimal generators ξ 0 and ξμ of the transformations (6) satisfy the determining equations (7) of the Lie symmetry, the corresponding invariance is called a Lie symmetry for the Birkhoffian system (5) on time scales. A Lie symmetry on time scales can lead to a conserved quantity under certain conditions. We have the following theorem. Theorem 1. For the Birkhoffian system (5) on time scales, if the infinitesimal transformations (6) correspond to the Lie symmetry of the system, and there exists with a gauge function G(t, aσρ ) satisfying the following structure equation



Rμ a μ



−B



ξ0

+ X (1 )



Rμ a μ

  ∂ Rv  ∂ B   − B + μ(t ) a − a ξ = −G ∂ aσμ ν ∂ aσμ μ 0 

(10)

then the system has the conserved quantity of the form

  ∂ Rμ  ∂ B ξ + G = const.. I = Rμ ξμ − Bξ0 − μ(t ) a − ∂t μ ∂t 0

(11)

Proof. Let’s do the following calculation

 

∂ Rμ  ∂ B   I= Rμ ξμ − Bξ0 − μ(t ) aμ − ξ0 + G t t ∂t ∂t  

∂ Rμ  ∂ B σ + R ξ −  B + μ t ξ0σ − Bξ0 = R ξ a − ( ) μ μ μ μ t ∂t μ ∂t   ∂ Rμ  ∂ B  −μ(t ) aμ − ξ + G ∂t ∂t 0

(12)

In Ref. [41], Bartosiewicz et al. established the second Euler–Lagrange equation in the following form

   ∂L  ∂L ∂L −L + q + μ t =− ( ) t ∂t ∂t ∂ q

(13)

where L = L(t, qσ (t ), q (t )) is the Lagrangian on time scales. In fact, if the Lagrangian does not contain time t explicitly, then −L + ∂ L q = const. which is the energy integral of the system, so Eq. (13) can also be called the energy equation. ∂q

Similar to the proof of Eq. (13), for the time scale Birkhoffian system (5), it is easy to know

 

∂ Rμ  ∂ B ∂ Rμ  ∂ B  B + μ(t ) a − =− a + t ∂t μ ∂t ∂t μ ∂t

(14)

Eq. (14) can be called the energy equation of Birkhoffian system (5) on time scales. If the system is autonomous, it gives a law of conservation of quasi-energy, i.e. B = const. Substituting Eq. (14) into Eq. (12), and noting that





X ( 1 ) Rμ a μ −B =



   ∂ Rμ  ∂ B ∂ Rμ  ∂ B aμ − + ξν a − ∂t ∂t ∂ aσν μ ∂ aσν        ∂ Rμ  ∂ B    + ξν − a ξ R + μ t − a ξ a − ξ () ν ν ν 0 ν 0 ∂ aσν μ ∂ aσν

ξ0

(15)

We can easily obtain

 I= t



      ∂ Rv  ∂ B ( 1 ) R a − B − a + ξ σ + Rμ a μ μ μ − B ξ0 + X ∂ aσμ v ∂ aσμ μ   ∂ Rv  ∂ B   + μ(t ) a − a ξ + G ∂ aσμ ν ∂ aσμ μ 0 R μ

(16)

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By using structural Eq. (10) and Birkhoffian Eq. (5) on time scale, we get

 I=0 t

(17)

So the theorem is proved. The conserved quantity (11) is caused by the Lie symmetry when the Birkhoffian system is not disturbed. Therefore, it is an exact invariant of the Birkhoffian system (5) on time scales. 3. Perturbation to Lie symmetry and adiabatic invariants for Birkhoffian systems on time scales Assume that the Birkhoffian system (5) on time scales is disturbed by the small perturbation ε Qμ , where Qμ = Qμ (t, aσρ ) and ɛ is the small parameter, then the differential equations of motion of the system become

     σ  ∂ B t, aσρ   ∂ Rv t, aσρ   av − Rμ t, aρ − = ε Qμ t, aσρ , (μ = 1, 2, · · · , 2n). ∂ aσμ ∂ aσμ

(18)

Due to the action of the small perturbation, the original Lie symmetry and the invariants of the system will be changed correspondingly. If we use ξ00 (t, aρ ) and ξμ0 (t, aρ ) denote the generators of time and space when the system is not disturbed, and use ξ 0 (t, aρ ) and ξμ (t, aρ ) denote the generators of infinitesimal transformations after being disturbed, then

ξ0 = ξ00 + εξ01 + ε 2 ξ02 + · · · = ξ00 + ε l ξ0l , ξμ = ξμ0 + εξμ1 + ε 2 ξμ2 + · · · = ξμ0 + ε l ξμl , (l = 1, 2, · · ·),

(19)

that is, the disturbed generator is a small perturbation on the basis of the undisturbed generator. And the infinitesimal generator vector X(0) after being disturbed and its first order expansion X(1) are

X ( 0 ) = ξ0

∂ ∂ 0 0 + ξμ = X0( ) + ε l Xl( ) , ∂t ∂ aμ

X (1 ) = X (0 ) +



 ξμ − a μ ξ0

 ∂ 1 1 = X0( ) + ε l Xl( ) , ∂ a μ

(20)

(21)

where 0 Xl( ) = ξ0l

∂ ∂ + ξμl , ( l = 1 , 2 , · · · ), ∂t ∂ aμ

  1 0 l Xl( ) = Xl( ) + ξμl  − a μ ξ0

∂ , (l = 1, 2, · · ·). ∂ a μ

(22)

(23)

The invariance of the disturbed Birkhoff’s Eq. (18) on time scales under the infinitesimal transformations comes down to the determining equations of Lie symmetry as follows



X (1 )

        ∂ Rv    ∂B   ∂ Rv 1) 1) ( ( a + ξ ν − a v ξ0 −X Rμ − X = ε X ( 1 ) Qμ . ∂ aσμ v ∂ aσμ ∂ aσμ

(24)

Substituting Eqs. (19) and (21) into Eq. (24), and comparing the coefficients of ɛm on both sides of the equation, we obtain

        ∂ Rv   m ∂B 1) (1 )  (1 )  m ∂ Rv a + − a ξ − X R − X = Xm( −1 Qμ ξ m m v ν v 0 μ σ σ σ ∂ aμ ∂ aμ ∂ aμ (μ = 1, 2, · · · , 2n; m = 0, 1, 2, · · ·),

Xm( )



1

(25)

in which we let ξμ−1 = ξ0−1 = 0 when m = 0. Now, we introduce the concept of higher order adiabatic invariant. If Iz (t, aσρ , ε ) is a physical quantity of the Birkhoffian system on time scales containing the small parameter ɛ, in which the highest power is z, and Iz /t is directly in proportion to ε z+1 , then Iz is called a z-th order adiabatic invariant of the Birkhoffian system on time scales. It can be proved that, for the disturbed Birkhoff’s Eq. (18) on time scales, if the infinitesimal generators ξ0m and ξμm satisfy



    ∂ Rv  ∂ B  m − B + μ t a − a ξ ξ0m + Xm(1) Rμ a ( ) μ ∂ aσμ ν ∂ aσμ μ 0   m−1 σ −Qμ ξμm−1 − a = −G μ ξ0 m , ( m = 0 , 1 , 2 , · · · ), 

Rμ a μ −B

(26)

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where G = ε m Gm (t, aσρ ) is the gauge function, then the following formula

 

∂ Rμ  ∂ B m ξ0 + G m ε m Rμ ξμm − Bξ0m − μ(t ) aμ − ∂t ∂t m=0 z 

(27)

is the z-th order adiabatic invariant of the system. Here we use Iz to denote formula (27), then Iz /t is directly in proportion to ε z+1 . In fact, if we take the delta derivative of Iz with respect to time t, we get

 

z  ∂ Rμ  ∂ B   m  mσ m  mσ m I = ε R μ ξμ + R μ ξ μ − B ξ 0 − B ξ0 − a − μ(t ) ξ0mσ t z m=0 t ∂t μ ∂t  

∂ Rμ  ∂ B m ξ0 + G  −μ(t ) aμ − m . ∂t ∂t

(28)

Similar to Ref. [41], for the disturbed system (18), it’s easy to get that

 

∂ Rμ  ∂ B ∂ Rμ  ∂ B  σ B + μ(t ) aμ − =− a + − ε Qμ a μ . t ∂t ∂t ∂t μ ∂t

(29)

From Eq. (18), we have

R μ =

∂ Rv  ∂ B a − − ε Qμ . ∂ aσμ v ∂ aσμ

(30)

Substituting formulae (30) and (26) into formula (28) and utilizing formula (29), we obtain

 z        m−1 σ m σ σ mσ . Iz = ε m Qμ ξμm−1 − a − ε Qμ ξμm − a = −ε z+1 Qμ ξμmσ − a μ ξ0 μ ξ0 μ ξ0 t m=0

(31)

Therefore, Iz is the z-th order adiabatic invariant of the Birkhoffian system on time scales.  I = 0. Thus, the exact invariant of the Birkhoffian system on time It can be seen from Eq. (31) that, if Qμ = 0, then  t z scales is a special adiabatic invariant, but not vice versa. On the basis of the above discussion, we have the following theorem. Theorem 2. For the Birkhoffian system (18) on time scales, which is disturbed by the small perturbation ε Qμ , if the infinitesimal transformations (6) correspond to the Lie symmetry of the system, and there exists with a gauge function Gm (t, aσρ ) satisfying the following structure equation



    ∂ Rv  ∂ B  m − B + μ t a − a ξ ξ0m + Xm(1) Rμ a ( ) μ ∂ aσμ ν ∂ aσμ μ 0   m−1 σ −Qμ ξμm−1 − a = −G μ ξ0 m , ( m = 0, 1, 2, · · · ) 

Rμ a μ −B

in which we let ξμ−1 = ξ0−1 = 0 when m = 0, then

Iz =

z 



ε

m

m=0

 

∂ Rμ  ∂ B m Rμ ξμ − Bξ − μ(t ) a − ξ + Gm ∂t μ ∂t 0 m

m 0

is a z-th order adiabatic invariant of the system. If T = R, σ (t ) = t, μ(t ) = 0, Eq. (5) becomes



 ∂ Rμ ∂B ∂ Rν ∂ Rμ − a˙ − − = 0. ∂ aμ ∂ aν ν ∂ aμ ∂t

(32)

Eq. (32) is the classical Birkhoff’s equation. So in this case, from Theorem 2, we have the following corollary. Corollary 1. For the Birkhoffian system (32), which is disturbed by the small perturbation ε Qμ , if the infinitesimal transformations (6) correspond to the Lie symmetry of the system, and there exists with a gauge function Gm (t, aρ ) satisfying the following structure equation









1 Rμ a˙ μ − B ξ˙0m + Xm( ) Rμ a˙ μ − B − Qμ

in which we let ξμ = −1

Iz =

z 

ξ0−1



 ξμm−1 − a˙ μ ξ0m−1 = −G˙ m , (m = 0, 1, 2, · · ·),

(33)

= 0 when m = 0, then

  ε m Rμ ξμm − Bξ0m + Gm

m=0

is a z-th order adiabatic invariant of the system.

(34)

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If T = Z, σ (t ) = t + 1, μ(t ) = 1, Eq. (5) becomes

      ∂ B t, aρ (t + 1) ∂ Rv t, aρ (t + 1) aν (t ) − Rμ t, aρ (t + 1) − = 0. ∂ aμ (t + 1) ∂ aμ (t + 1)

(35)

Eq. (35) is Birkhoff’s equation of the discrete Birkhoffian system. So, Theorem 2 gives the following corollary. Corollary 2. For the discrete Birkhoffian system (35), which is disturbed by the small perturbation ε Qμ , if the infinitesimal transformations (6) correspond to the Lie symmetry of the system, and there exists with a gauge function Gm (t, aρ (t + 1 ) ) satisfying the following structure equation











Rμ t, aρ (t + 1 ) aμ (t ) − B t, aρ (t + 1 ) ξ0m (t )











+ Xm( ) Rμ t, aρ (t + 1 ) aμ (t ) − B t, aρ (t + 1 ) 1



    ∂ Rv t, aρ (t + 1) ∂ B t, aρ (t + 1) + aν (t ) − aμ (t )ξ0m (t ) ∂ aμ (t + 1) ∂ aμ (t + 1)     − Qμ ξμm−1 (t + 1 ) − aμ (t + 1 )ξ0m−1 (t + 1 ) = −Gm t, aρ (t + 1 ) ,

(36)

then

Iz =

     ε m Rμ t, aρ (t + 1) ξμm (t ) − B t, aρ (t + 1) ξ0m (t )

z  m=0

       ∂ Rμ t, aρ (t + 1) ∂ B t, aρ (t + 1) m + aμ (t ) − ξ0 (t ) + Gm t, aρ (t + 1) ∂t ∂t 

(37)

is a z-th order adiabatic invariant of the system. 4. Perturbation to Lie symmetry and adiabatic invariants for Hamiltonian systems on time scales Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics, the Hamilton principle is the special case of the Pfaff–Birkhoff principle and Hamilton’s canonical equations are the special cases of Birkhoff’s equations. Therefore, from Theorem 1, we can obtain the Lie symmetry theorem for undisturbed Hamiltonian systems on time scale, and from Theorem 2, we can obtain the Lie symmetry theorem for disturbed Hamiltonian systems on time scale. Let



aσμ =

qσμ , μ = 1, 2, · · · , n , pμ−n , μ = n + 1, n + 2, · · · , 2n

(38)

pμ , μ = 1, 2, · · · , n , 0, μ = n + 1, n + 2, · · · , 2n

(39)

 Rμ = B = H,

(40)

where qs ,ps (s = 1, 2, · · · , n) are the generalized coordinates and the generalized momenta respectively, and H = H (t, qσs , ps ) is the Hamiltonian. Then the Hamilton principle on time scales can be expressed as

δS = δ



t2

t1







σ ps q s − H t, qk , pk t

(41)

which satisfies the commutative conditions

δ q s =

 δ q , ( s = 1, 2, · · · , n ) t s

(42)

and the boundary conditions

δ qs |t=t1 = δ qs |t=t2 = 0, (s = 1, 2, · · · , n).

(43)

According to Eq. (5), we get

q s =

∂H  ∂H , p = − σ , (s = 1, 2, · · · , n), ∂ ps s ∂ qs

Eq. (44) is Hamilton’s equation on time scales [44]. In phase space, the infinitesimal transformations (6) become

(44)

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257

t¯ = t + ε ξ0 (t, qk , pk ),  q¯ s t¯ = qs (t ) + ε ξs (t, qk , pk ),



p¯ s t¯ = ps (t ) + ε ηs (t, qk , pk ), (s = 1, 2, · · · , n),

(45)

where ξ 0 , ξ s , ηs are generators of the infinitesimal transformations. According to the relationships (38)–(40), Eq. (7) gives

    ∂H  ∂H  ∂H ∂H 1)  1) ( ( ξs − ξ =X , ηs + ξ = −X , ( s = 1, 2, · · · , n ) ∂ ps 0 ∂ ps ∂ qσs 0 ∂ qσs 

(46)

where

X ( 0 ) = ξ0

∂ ∂ ∂ + ξs + ηs , ∂t ∂ qs ∂ ps

X (1 ) = X (0 ) +



 ξs − q s ξ0

(47)

 ∂   ∂  + ηs − p . s ξ0 ∂ q ∂ p s s

(48)

Eq. (46) can be called the determining equations of the Lie symmetry for the Hamiltonian system (44) on time scales. Thus, we have the following definition. Definition 2. If the generators ξ 0 , ξ s and ηs of the infinitesimal transformations (45) satisfy the determining Eq. (46) of the Lie symmetry, the corresponding invariance is called a Lie symmetry for the Hamiltonian system (44) on time scales. Thus, Theorem 1 is reduced to the Lie symmetry theorem for undisturbed Hamiltonian system (44) as follows. Theorem 3. For the Hamiltonian system (44) on time scales, if the infinitesimal transformations (45) correspond to the Lie symmetry of the system, and there exists with a gauge function G(t, qσk , pk ) satisfying the following structure equation

−H ξ0 − X (1 ) (H ) +

∂H ∂H η + ps ξs − μ(t ) σ q ξ  = −G ∂ ps s ∂ qs s 0

(49)

then the system has the conserved quantity of the form

I = ps ξs − H ξ0 + μ(t )

∂H ξ + G = const.. ∂t 0

(50)

The conserved quantity (50) is an exact invariant. Assume that the Hamiltonian system (44) on time scales is disturbed by the small perturbation ε Qs (t, qσk , pk ), then the Eq. (44) becomes

q s =

∂H  ∂H , p = − σ + ε Q s , ( s = 1 , 2 , · · · , n ). ∂ ps s ∂ qs

(51)

The determining equations of the Lie symmetry after being disturbed are

ξs −





∂H  (1 ) ∂ H , ∂ p s ξ0 = X ∂ ps

ηs +



∂H   (1 ) ∂ H ∂ qσs ξ0 − ε Qs ξ0 = −X ∂ qσs



+ ε X ( 1 ) ( Qs )

( s = 1 , 2 , · · · , n ).

(52)

Indeed, we have

ξsm −

    ∂ H m ∂ H m ∂H 1) (1 ) ∂ H ξ0 = Xm(1) , ηsm + ξ = −X + Qs ξ0m + Xm( −1 (Qs ), m ∂ ps ∂ ps ∂ qσs 0 ∂ qσs (s = 1, 2, · · · , n; m = 0, 1, 2, · · ·),

where

X (0 )

(0 )

= ε Xm = ε m

 m

 ∂ m ∂ m ∂ ξ + ξs + ηs , ∂t ∂ qs ∂ ps m 0



X (1 ) = ε m Xm( ) = ε m Xm( ) + 1

0



m ξsm − q s ξ0

 ∂  m  ∂  m + − p ξ . η s s 0 ∂ q ∂ p s s

(53)

(54)

(55)

So, Theorem 2 is reduced to the Lie symmetry theorem for disturbed Hamiltonian system (51) as follows. Theorem 4. For the Hamiltonian system (51) on time scales, which is disturbed by the small perturbation ɛQs , if the infinitesimal transformations (45) correspond to the Lie symmetry of the system, and there exists with a gauge function Gm (t, qσk , pk ) satisfying the following structure equation

−H ξ0m − Xm( ) (H ) + 1

  ∂H m ∂H m−1 σ η + ps ξsm − μ(t ) σ q ξ m − Qs ξsm−1 − q = −G s ξ0 m ∂ ps s ∂ qs s 0

(56)

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then

  ∂H ε m ps ξsm − H ξ0m + μ(t ) ξ0m + Gm ∂t m=0 z 

Iz =

(57)

is a z-th order adiabatic invariant of the system. If T = R, σ (t ) = t, μ(t ) = 0, Eq. (44) becomes

q˙ s =

∂H ∂H , p˙ = − , ( s = 1, 2, · · · , n ) ∂ ps s ∂ qs

(58)

Eq. (58) is the classical Hamilton canonical equation. So in this case, from Theorem 4, we have the following corollary. Corollary 3. For the Hamiltonian system (58), which is disturbed by the small perturbation ɛQs , if the infinitesimal transformations (45) correspond to the Lie symmetry of the system, and there exists with a gauge function Gm (t, qk , pk ) satisfying the following structure equation

−H ξ˙0m − Xm( ) (H ) + 1

  ∂H m ηs + ps ξ˙sm − Qs ξsm−1 − q˙ s ξ0m−1 = −G˙ m ∂ ps

(59)

then

Iz =

z 

ε m ( ps ξsm − H ξ0m + Gm )

(60)

m=0

is a z-th order adiabatic invariant of the system. If T = Z, σ (t ) = t + 1, μ(t ) = 1, Eq. (44) becomes

∂ H (t , qk (t + 1), pk (t )) ∂ H (t , qk (t + 1), pk (t )) ,  ps (t ) = − ∂ ps (t ) ∂ qs (t + 1)

qs (t ) =

(61)

Eq. (61) is Hamilton canonical equation of the discrete Hamiltonian system. So, Theorem 4 gives the following corollary. Corollary 4. For the discrete Hamiltonian system (61), which is disturbed by the small perturbation ɛQs , if the infinitesimal transformations (39) correspond to the Lie symmetry of the system, and there exists with a gauge function Gm (t, qk (t + 1 ), pk (t )) satisfying the following structure equation

−H (t , qk (t + 1 ), pk (t ))ξ0m (t ) − Xm( ) [H (t, qk (t + 1 ), pk (t ) )] 1

∂ H (t , qk (t + 1), pk (t )) m ∂ H (t , qk (t + 1), pk (t )) ηs (t ) + ps (t )ξsm (t ) − qs (t )ξ0m (t ) ∂ ps (t ) ∂ qs (t + 1)   − Qs ξsm−1 (t + 1 ) − qs (t + 1 )ξ0m−1 (t + 1 ) = −Gm (t ) +

(62)

then

Iz =

z 

ε m { ps (t )ξsm (t ) − H (t, qk (t + 1), pk (t ))ξ0m (t )

m=0

 ∂ H (t , qk (t + 1), pk (t )) m + ξ0 (t ) + Gm (t ) ∂t

(63)

is a z-th order adiabatic invariant of the system. 5. Examples Example 1 Let T = hZ = {hk : k ∈ Z}, h > 0 and consider the famous Hojman–Urrutia problem, the Pfaff action of which is

 S=

t2

t1



 1 σ 2 1 σ 2 σ σ σ σ (aσ2 + aσ3 )a (a3 ) + (a4 ) t . 1 + a4 a3 − a2 a3 − 2

2

(64)

Let’s study the Lie symmetry, the exact invariants and the adiabatic invariants of the system. The Eq. (5) gives 





σ  σ σ σ  σ −(aσ2 ) − (aσ3 ) = 0, a 1 − a3 = 0, a1 − ( a4 ) − a2 − a3 = 0, a3 + a4 = 0.

(65)

Here we take

ξ00 = 0, ξ10 = 1, ξ20 = ξ30 = ξ40 = 0.

(66)

The generators (66) satisfy the determining Eq. (7) obviously, so the generators (66) correspond to the Lie symmetry of the system.

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259

The structure Eq. (10) gives



 σ aσ − aσ aσ − 1 aσ 2 + 1 aσ 2 ξ 0 + a (aσ2 + aσ3 )a ( ) ( ) 1 4 3 2 3 0 2 3 2 4   1 1 1 σ σ σ σ + X0( ) (aσ2 + aσ3 )a (aσ3 )2 + (aσ4 )2 1 + a4 a3 − a2 a3 − 2 2        σ a + a − aσ − aσ a + a + aσ a = −G . + μ(t )ξ00 a − a 1 3 2 1 2 3 3 3 4 4 0

(67)

Substituting generators (66) into (67), we have G0 = 0. According to Theorem 1, we get the conserved quantity

I = aσ2 + aσ3 = const..

(68)

Formula (68) is an exact invariant of the system. Suppose that the system is disturbed by the small perturbations ɛQμ , that is

ε Q1 = 0, ε Q2 = −ε aσ3 , ε Q3 = 0, ε Q4 = ε aσ4 .

(69)

Eq. (18) gives 





σ σ  σ σ σ  σ σ −(aσ2 ) − (aσ3 ) = 0, a 1 − a3 = −ε a3 , a1 − (a4 ) − a2 − a3 = 0, a3 + a4 = ε a4 .

(70)

Take

ξ01 = 1, ξ11 = ξ21 = ξ31 = ξ41 = 0.

(71)

The generators (71) and (66) satisfy the determining Eq. (25) obviously, so the generators correspond to the Lie symmetry of the disturbed system (70). The structure Eq. (26) gives



 1 σ 2 1 σ 2 1 σ σ σ σ (aσ2 + aσ3 )a ( a 3 ) + ( a 4 ) ξ0 1 + a4 a3 − a2 a3 − 2 2   1 1 1 σ σ σ σ + X1( ) (aσ2 + aσ3 )a (aσ3 )2 + (aσ4 )2 1 + a4 a3 − a2 a3 − 2 2        σ a + a − aσ − aσ a + a + aσ a + μ(t )ξ01 a − a 1 3 2 1 2 3 3 3 4 4  0   0  σ  0 σ σ  0 σ  + a 3 ξ 2 − a 2 ξ0 − a 4 ξ4 − a 4 ξ0 = −G1 .

(72)

Substituting generators (71) and (66) into (72), we have G1 = 0. According to Theorem 2, we get



I1 = aσ2 + aσ3 − ε aσ2 aσ3 +



1 σ 2 1 σ 2 ( a ) − ( a4 ) . 2 3 2

(73)

Formula (73) is a first-order adiabatic invariant of the system. Example 2 Let T = {2m : m ∈ N0 } and the Hamiltonian of the system is

H=

 1 2 p + p22 + qσ2 . 2 1

(74)

Eq. (44) gives    q 1 = p1 , q2 = p2 , p1 = 0, p2 = −1.

(75)

From the determining Eq. (46) of Lie symmetry, we have

ξ1 − p1 ξ0 = η1 , ξ2 − p2 ξ0 = η2 , η1 = 0, η2 + ξ0 = 0.

(76)

And the structure Eq. (49) gives

−

 1 2 p1 + p22 ξ0 − qσ2 ξ0 − ξ2 − ξ2 μ + p1 ξ1 + p2 ξ2 = −G . 2

(77)

Eqs. (76) and (77) have a solution

ξ00 = 0, ξ10 = 1, ξ20 = η10 = η20 = 0, G0 = 0.

(78)

The generators (78) correspond to the Lie symmetry of the system. According to Theorem 3, the system exists with the conserved quantity of the form

I = p1 = const..

(79)

Suppose the system is disturbed by the small perturbations ɛQs , that is

ε Q1 = 0, ε Q2 = ε p2 ,

(80)

The differential equations of motion after being disturbed are    q 1 = p1 , q2 = p2 , p1 = 0, p2 = −1 + ε p2 .

(81)

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The determining Eq. (53) of the Lie symmetry gives

ξ11 − p1 ξ01 = η11 , ξ21 − p2 ξ01 = η21 , η11 = 0, η21 + ξ01 = p2 ξ00 + η20 .

(82)

Eq. (82) has a solution

ξ01 = 1, ξ11 = ξ21 = η11 = η21 = 0.

(83)

And the structure Eq. (56) gives

−

   1 2 0 σ p + p22 ξ01 − qσ2 ξ01 − ξ21 − ξ21 μ + p1 ξ11 + p2 ξ21 − p2 ξ20 − q = −G 2 ξ0 1 . 2 1

(84)

Substituting generators (83) and (78) into (84), we have G1 = 0. From Theorem 4, we have

I1 = p1 − ε

1 2





p21 + p22 + qσ2 .

(85)

Formula (85) is a first-order adiabatic invariant of the system. Similarly, the higher-order adiabatic invariants can be obtained. 6. Conclusions The theory of time scales has been widely used in many fields of science and engineering because of its unification and extension. In this paper, we proposed and studied the Lie symmetry and exact invariants of the undisturbed Birkhoffian system on time scales, and the perturbation to Lie symmetry and adiabatic invariants of the disturbed Birkhoffian system on time scales. Birkhoffian mechanics is a natural development of Hamiltonian mechanics. In this paper, we applied the obtained results of Birkhoffian systems on time scales to Hamiltonian systems, and established the theory of perturbation to Lie symmetry and adiabatic invariants of Hamiltonian systems on time scales. Compared with some previous studies on Birkhoffian systems, this paper not only allows the discrete result and the continuous result into a single model, but also achieves the more general model. Because of the feature of extension of time scales, the results are suitable in describing complex processing and also avoid some repetitive works between difference equations and differential equations. Recent work about the fractional calculus on time scales [48] potentiates research not only in the fractional calculus but also in solving fractional dynamical equations. The perturbation to Noether symmetries and adiabatic invariants for disturbed Hamiltonian systems with the fractional action-like variational approach [49] has been studied. This important approach is also worth to discuss in a time scales version for the perturbation to Lie symmetry and adiabatic invariants. The results and methods of this paper can be further extended to the generalized Birkhoffian systems on time scales, the generalized Hamiltonian systems on time scales and the nonholonomic systems on time scales, etc. Acknowledgments This work is supported by the National Natural Science Foundation of China (grant No. 11572212). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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