Noether's symmetry and conserved quantity for a time-delayed Hamiltonian system of Herglotz type

The variational problem of Herglotz type and Noether's theorem for a time-delayed Hamiltonian system are studied. Firstly, the variational problem of Herglotz type with time delay in phase space is proposed, and the Hamilton canonical equations with time delay based on the Herglotz variational problem are derived. Secondly, by using the relationship between the non-isochronal variation and the isochronal variation, two basic formulae of variation of the Hamilton–Herglotz action with time delay in phase space are derived. Thirdly, the definition and criterion of the Noether symmetry for the time-delayed Hamiltonian system are established and the corresponding Noether's theorem is presented and proved. The theorem we obtained contains Noether's theorem of a time-delayed Hamiltonian system based on the classical variational problem and Noether's theorem of a Hamiltonian system based on the variational problem of Herglotz type as its special cases. At the end of the paper, an example is given to illustrate the application of the results.


Introduction
It is well known that variational principles play an important role in the fields of mechanics, physics and engineering, etc. However, with the classical variational principle it is difficult to describe non-conservative or dissipative physical processes. Herglotz proposed a class of generalized variational principle, which generalizes the classical variational principle by defining its functional whose extreme is sought by a differential equation [1]. Compared with the classical variational principle, the generalized variational principle of Herglotz type can give a variational description of a non-conservative dynamic system. It can describe not only all dynamic processes that the classical variational principle can, but also many others for which the classical variational principle is not applicable [2]. The principle is a promotion of classical variational principle. In 2002, Georgieva and Guenther proposed and proved the first Noether-type theorem for the generalized variational principle of Herglotz [2], and extended the results to one with several independent variables [3]. Donchev applied the generalized variational principle of Herglotz type and its Noether's theorem to Bö cher equation and nonlinear Schrö dinger equation, etc. that these equations do not have a variational description with the classical variational principle [4]. Santos et al. studied the variational problem of Herglotz type with time delay and obtained the corresponding Noether's theorem [5]. The research on the variation problems of Herglotz type and the corresponding symmetries and the conserved quantities has attracted strong attention of scholars and obtained a series of important results [6 -14]. In this paper, we will extend the variational problem of Herglotz type to a time-delayed Hamiltonian system and study the Noether symmetry and the conserved quantity for the system based on the generalized variational principle of Herglotz type.

Generalized variational principle for a time-delayed Hamiltonian system of Herglotz type
According to the generalized variational principle of Herglotz [1,2], the variational problem of Herglotz type with time delay can be defined as follows: Determine the trajectories q s (t), p s (t) to extremize the value z(t 1 ) where the functional z is defined by the differential equation ð2:1Þ and satisfying a given initial condition and boundary conditions where H(t,q s (t),p s (t),q s (t À t),p s (t À t),z(t)) W H(t,q s ,p s ,q st ,p st ,z) can be called the Hamiltonian for the variational problem of Herglotz type with time delay; t is a given positive real number such that t , t 1 À t 0 ; f s (t), g s (t) are given piecewise smooth functions on the interval [t 0 À t,t 0 ]; q s (t 1 ),p s (t 1 ),z 0 are fixed real numbers. Here and later, we use the Einstein summation convention on repeated indices. The functional z determined by formula (2.1) can be called the Hamilton -Herglotz action with time delay. The above variational problem can be referred to as the generalized variational principle for a time-delayed Hamiltonian system of Herglotz type.

Hamilton canonical equations with time delay based on the variational problem of Herglotz type
The isochronal variation of the differential equation (2.1) gives d_ z ¼ _ q s dp s þ p s d_ q s þ _ q st dp st þ p st d_ q st À @H @q s dq s À @H @p s dp s À @H @q st dq st À @H @p st dp st À @H @z dz: ð3:1Þ Using the commutative relation d_ z ¼ ðd=dtÞdz, we can write equation (3.1) as where A ¼ _ q s dp s þ p s d_ q s þ _ q st dp st þ p st d_ q st À @H @q s dq s À @H @p s dp s À @H @q st dq st À @H @p st dp st : ð3:3Þ The solution of equation (3.2) is From the initial condition (2.2), and noting that z(t) reaches its extreme value at t ¼ t 1 , we have As the equation (3.4) holds for all t [ ½t 0 ,t 1 , particularly, take t ¼ t 1 , we obtain ð t1 t0 exp ð t t0 @H @z du _ q s dp s þ p s d_ q s þ _ q st dp st þ p st d_ q st À @H @q s dq s À @H @p s dp s À @H @q st dq st À @H @p st dp st dt:

ð3:6Þ
By performing a linear change of variable for the terms involving time delay in equation (3.6), and considering the boundary conditions (2.3), we have Making integration by parts for the terms corresponding to dq s and dp s in equation (3.12), and using the conditions (2.3) and (2.4), we have Substituting formulae (3.9) and (3.10) into equation (3.8), we obtain According to the basic lemma [15] of the calculus of variation, from equation (3.11), we have ð3:12Þ Taking the derivative of equation (3.12) with respect to t, we have

Variation of Hamilton -Herglotz action with time delay
Introduce an r-parameter Lie group of transformations with respect to time t, generalized coordinates q s and generalized momenta p s , i.e. ð4:3Þ For any function F, there is a relationship between the non-isochronal variation DF and the isochronal variation dF as follows [16]: It is obvious that Dz(t 0 ) ¼ 0. By performing a linear change of variable for the terms involving time delay in the above equation, and using the boundary conditions (2.3), we have Substituting equation (4.10) into equation (4.9), we obtain DtÀ @H @t (t)Dt À @H @q s (t)Dq s À @H @p s (t)Dp s dt: Equation (4.9) can also be expressed as þ l(t) À _ p s À @H @q s À p s @H @z (Dq s À _ q s Dt)þ _ q s À @H @p s (Dp s À _ p s Dt) By performing a linear change of variable for the terms involving time delay in equation (4.12), and using the boundary conditions (2.3), we have Substituting formulae (4.13) and (4.14) into equation (4.12), we have  and

Definition and criterion of Noether's symmetry of a time-delayed
Hamiltonian system of Herglotz type The classical Noether symmetry refers to the invariance of Hamilton action under the infinitesimal transformation with respect to the generalized coordinates and time. In this section, we establish the definition and the criterion of the Noether symmetry of a time-delayed Hamiltonian system of Herglotz type. for t 0 t t 1 À t, and l(t) _ q s (t)h s s þ p s (t) _ j s s À H(t)_ t s À @H @t (t)t s À @H @q s (t)j s s À @H @p s (t)h s s ¼ 0, (s ¼ 1,2, Á Á Á ,n), ð5:3Þ for t 1 À t t t 1 , then the transformations (4.2) are the Noether symmetry transformations of the time-delayed Hamiltonian system (3.13) of Herglotz type.