Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems

The time-scales theory provides a powerful theoretical tool for studying differential and difference equations simultaneously. With regard to Herglotz type variational principle, this generalized variational principle can deal with non-conservative or dissipative problems. Combining the two tools, this paper aims to study time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems. We introduce the time-scales Herglotz type variational problem of Birkhoffian systems firstly and give the form of time-scales Pfaff–Herglotz action for delta derivatives. Then, time-scales Herglotz type Birkhoff’s equations for delta derivatives are derived by calculating the variation of the action. Furthermore, time-scales Herglotz type Noether symmetry for delta derivatives of Birkhoffian systems are defined. According to this definition, time-scales Herglotz type Noether identity and Noether theorem for delta derivatives of Birkhoffian systems are proposed and proved, which can become the ones for delta derivatives of Hamiltonian systems or Lagrangian systems in some special cases. Therefore, it is shown that the results of Birkhoffian formalism are more universal than Hamiltonian or Lagrangian formalism. Finally, the time-scales damped oscillator and a non-Hamiltonian Birkhoffian system are given to exemplify the superiority of the results.


YZ, 0000-0002-7703-1185
The time-scales theory provides a powerful theoretical tool for studying differential and difference equations simultaneously. With regard to Herglotz type variational principle, this generalized variational principle can deal with non-conservative or dissipative problems. Combining the two tools, this paper aims to study time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems. We introduce the time-scales Herglotz type variational problem of Birkhoffian systems firstly and give the form of time-scales Pfaff-Herglotz action for delta derivatives. Then, time-scales Herglotz type Birkhoff's equations for delta derivatives are derived by calculating the variation of the action. Furthermore, time-scales Herglotz type Noether symmetry for delta derivatives of Birkhoffian systems are defined. According to this definition, time-scales Herglotz type Noether identity and Noether theorem for delta derivatives of Birkhoffian systems are proposed and proved, which can become the ones for delta derivatives of Hamiltonian systems or Lagrangian systems in some special cases. Therefore, it is shown that the results of Birkhoffian formalism are more universal than Hamiltonian or Lagrangian formalism. Finally, the time-scales damped oscillator and a non-Hamiltonian Birkhoffian system are given to exemplify the superiority of the results.

Introduction
In 1988, Hilger proposed the definition of a time scale T, which is an arbitrary non-empty closed subset of the real numbers R, in order to analyse continuous and discrete systems uniformly [1]. For instance, if we choose a continuous time scale, i.e. T ¼ R, this time-scale calculus is the same as the calculus of the Birkhoff's equations for delta derivatives are deduced; then, the time-scales Herglotz type Noether identity and theorem for delta derivatives of Birkhoffian systems are formulated. In §4, the results of Hamiltonian systems and Lagrangian systems are listed to account for the relationship of Hamiltonian, Lagrangian and Birkhoffian systems. Section 5 gives the time-scales damped oscillator of Birkhoffian system and a non-Hamiltonian system as examples. Finally, we offer some conclusions in §6.

Time-scales preliminaries
A time scale T is an arbitrary non-empty closed subset of the set R of real numbers. Let T be a time scale, for t [ T, the forward jump operator s : If σ(t) > 0, σ(t) = 0, ρ(t) > 0, or ρ(t) = 0, then t is called right-scattered, rightdense, left-scattered and left-dense, respectively. The graininess function m : Definition 2.2. A function f : T ! R is called rd-continuous provided it is continuous at the rightdense points in T and its left-sided limits exist (finite) at all left-dense points in T. The set of rd-continuous functions f : T ! R will be denoted by C rd ¼ C rd (T) ¼ C rd (T, R). The set of functions f : T ! R that are differentiable and whose derivative is rd-continuous is denoted by where c is an arbitrary constant. And the definite integral of f is defined by holds for all t [ T k .
the exponential function is defined by Here, the set of rd-continuous and regressive functions f : and t 0 [ T is fixed, we list the following properties of exponential functions: and the unique solution of the initial value problem The above definitions, lemmas and the specific proof processes of lemmas can be referred to in the literature [1].

Main results
First, we indicate that the time-scales Herglotz variational problem for delta derivatives of Birkhoffian systems is a functional extremum problem of determining the function a n (t) that extremizes z(t 2 ), where the action z(t) is a solution of with the boundary conditions a n (t)j t¼t1 ¼ a n1 , a n (t)j t¼t2 ¼ a n2 , (n ¼ 1, 2, . . . , 2n) ( 3 :2) and the initial condition Considering the exchange relationships [10] According to the condition (3.3), equation (3.5) satisfies the initial value condition Let g(t) ¼ @Rn @z a D n À @B @z , by lemma 2.8 and the properties (2.6), (2.7), (2.8), the solution of equations (3.5) and From the boundary conditions (3.2), we have δz(t 2 ) = 0. And consider that the action z(t) yields its extremum at t = t 2 , so that ð t2 ((@R n =@z)_ a n À (@B=@z))du). Thus, equations (3.14) become the Herglotz type Birkhoff's equations in the continuous case [22] exp À ð t t1 @R n @z _ a n À @B @z , then γ(t) = 0, e s g (t 1 , t) ¼ 1. Thus, equations (3.14) change to the time-scales Birkhoff's equations based on the traditional variational problem [11] Then, we study the time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems. Let U be a set of C 1 rd functions a n : [t 1 , t 2 ] ! R n . We introduce the infinitesimal transformations of the one-parameter group with respect to time t on U t ¼ t þ 1t (t, a v , z) and a n ( t) ¼ a n (t) þ 1j n (t, a v , z), (3:17) where τ, j n are infinitesimal generators, and ɛ is an infinitesimal parameter. Then, under the transformations (3.17), we can write the time-scales Pfaff-Herglotz action z(t) as z( t) ¼ z(t) þDz(t), whereD denotes total variation. According to the literature [10], we knowDq ¼ dq þ q DD t.
holds for all t ∈ [t 1 , t 2 ]. Formula (3.18) is called the time-scales Herglotz type Noether identity for delta derivatives of Birkhoffian systems.
Proof. On the basis of definition 3.4, we knowDz(t b ) ¼ 0. Then, from equation (3.1), we obtaiñ  , a v , z). Note the initial conditionDz(t a ) ¼ 0 that the solution of equation (3.21) isD Then, when t = t b , we have According to the arbitrariness of integral interval, we obtain e s g (t a , t) ▪ Since e s g (t a , t) . 0, theorem 3.5 is proved.
Theorem 3.6. If the transformations (3.17) correspond to the time-scales Herglotz type Noether symmetry for delta derivatives of Birkhoffian systems, then there exists a conserved quantity in the form of Proof.
royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191248 D Dt I N ¼ 0: Integrating the above formula, therefore the theorem is proved. ▪ Remark 3.7. If T ¼ R, i.e. σ(t) = t, μ(t) = 0, then formula (3.18) becomes the Herglotz type Noether identity of Birkhoffian systems in the continuous case [22] @R n @t _ a n À @B @t t þ @R n @a v _ a n À @B @a v j v þ R n j n À B_ t ¼ 0: (3:26) And the conserved quantity (3.25) changes to the Herglotz type Noether conserved quantity of classical Birkhoffian systems [22] I N ¼ exp À ð t ta @R n @z _ a n À @B @z du (R n j n À Bt) ¼ const: (3.18) can be written as And the conserved quantity (3.25) changes to Formulae (3.28) and (3.29) are the Herglotz type Noether identity and Noether conserved quantity of Birkhoffian systems in the discrete case.
royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191248 Theorem 4.5. If the transformations (4.14) correspond to the time-scales Herglotz type Noether symmetry for delta derivatives of Lagrangian systems, then there exists a conserved quantity in the form of Remark 4.6. If T ¼ R, i.e. σ(t) = t, μ(t) = 0, e s g (t a , t) ¼ exp (À Ð t ta (@L=@z) dt), then formula (4.15) becomes the Herglotz type Noether identity of Lagrangian systems in the continuous case And the conserved quantity (4.16) changes to the Herglotz type Noether conserved quantity of Lagrangian systems @L @z du @L @ _ q s (j s À _ q s t) þ Lt ¼ const: (4:18)