Herglotz’s Variational Problem for Non-Conservative System with Delayed Arguments under Lagrangian Framework and Its Noether’s Theorem

Because Herglotz’s variational problem achieves the variational representation of non-conservative dynamic processes, its research has attracted wide attention. The aim of this paper is to explore Herglotz’s variational problem for a non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Firstly, we derive the non-isochronous variation formulas of Hamilton–Herglotz action containing delayed arguments. Secondly, for the Hamilton–Herglotz action case, we define the Noether symmetry and give the criterion of symmetry. Thirdly, we prove Herglotz type Noether’s theorem for non-conservative system with delayed arguments. As a generalization, Birkhoff’s version and Hamilton’s version for Herglotz type Noether’s theorems are presented. To illustrate the application of our Noether’s theorems, we give two examples of damped oscillators.


Introduction
Time delay is a common phenomenon in nature and engineering. Although time delays have often been ignored in the past and many problems have been solved, with the increasingly precise requirements for the dynamical behavior and control of complex systems, the effects of time delays on the system need to be considered. It has been shown that even millisecond delay can lead to complex dynamical behavior of the system. In addition, for many delayed systems, if the time delay is ignored, it will lead to a completely wrong conclusion. Therefore, the study of the dynamical characteristics of time-delayed systems is not only extremely important to the understanding of these systems themselves, but also to the research of biology, ecology, neural network, physics, electronics and information science, mechanical engineering, and other research fields [1][2][3][4]. For the variational problem in the case of delay, El'sgol'c first mentioned its extremum characteristic in [5]. Hughes derived the necessary conditions for a time-delayed variational problem in 1968 [6], which is similar to the classical one. Frederico and Torres [7] were the first to propose and prove the extension of Noether's theorem to time-delay variational problems and optimal control. In 2013, in reference [8], we extended the results of [7] in three aspects: from Lagrange system to general non-conservative system; from a group of point transformations corresponding to generalized coordinates and time to a group of transformations that depend on generalized velocities; from Noether symmetry to Noether quasi-symmetry. In recent years, Noether's theorems with time delay have been extended to high-order variational problems [9], fractional systems [10], Hamilton systems [11], nonholonomic systems [12], Birkhoff systems [13,14], and dynamics on time scales [15,16], etc. Although some important results have been obtained in the dynamics modeling of time-delay systems and its Noether's theorems, in general, the research in this field is still in the preliminary stage and is still an open topic.

HGVP for Non-Conservative Dynamics with Delayed Arguments
Considering a non-conservative mechanical system with delayed arguments, we assume that its configuration is described by q s (s = 1, 2, · · · , n). We now define Herglotz's variational problem of the non-conservative system with delayed arguments as: Suppose that functional z is determined by a first order differential equation Determine the trajectory q s (t) that satisfy the boundary conditions and initial condition so as to extremize the value z (t 1 ) → extr. Here, L = L (t, q s ,q s , q sτ ,q sτ , z) is the Lagrangian in the sense of Herglotz. f s (t) is a given function on [t 0 − τ, t 0 ] , which is piecewise smooth. τ is the delay quantity, and τ < t 1 − t 0 , which is a given positive real number. Here, q s1 and z 0 are constants.
We call a functional z Hamilton-Herglotz action with delayed arguments. The Herglotz's variational problem above can be called the HGVP for non-conservative system with delayed arguments.
For a non-conservative system with delayed arguments, it is easy from the above principle to obtain the Euler-Lagrange equations of Herglotz type, and we get where

Non-Isochronous Variation of Hamilton-Herglotz Action with Delayed Arguments
Consider the infinitesimal transformations that depend not only on generalized coordinates, and time, but also on generalized velocities, that is, or their expansiont where ξ σ 0 and ξ σ s are the generators, and ε σ (σ = 1, 2, · · · , r) are the infinitesimal parameters. The function z (t) is transformed by the infinitesimal transformation (5) intoz (t), and the relationship between them is as follows:z (t) = z (t) + ∆z (t) (7) where ∆z is the non-isochronous variation. By calculating the non-isochronous variation of Equation (1), Note that, for any differentiable function F, the following formulae hold [43]: Thus, we have From Equation (10), we get where ∆z (t 0 ) = 0 . By performing variable substitution operations t = θ + τ for the fourth and fifth items in Equation (11), and noting the boundary condition (2), we have Substituting Equation (12) into Equation (11), we get Equation (11) can also be written as By performing variable substitution operations t = θ + τ for the terms in Equation (14) with delay τ, and using condition (2), we get and t t 0 From Equations (15) and (16), we can rewrite Equation (14) as Since Substituting Equation (18) into Equations (13) and (17), we get and Equations (19) and (20) are the non-isochronous variation formulas of Hamilton-Herglotz action with delayed arguments.

Herglotz Type Noether's Theorem for Non-Conservative Systems with Delayed Arguments
If Hamilton-Herglotz action remains unchanged through the infinitesimal transformation of the group, namely ∆z (t 1 ) = 0, it is known as Noether symmetry for non-conservative mechanical system with delayed arguments.
According to Equation (19), we can obtain the criterion of Noether symmetry for the non-conservative system. That is, Criterion 1. If the generators ξ σ 0 and ξ σ s of infinitesimal transformation (5) make the following conditions true, when t ∈ [t 0 − τ, t 0 ), there is where s = 1, 2, · · · , n and σ = 1, 2, · · · , r, then the transformation corresponds to the Noether symmetry of non-conservative system with delayed arguments.
By Noether symmetry, we can find the conserved quantity, and we have the following results.
Theorem 1 is Herglotz type Noether's theorem for non-conservative system with delayed arguments, and the conserved quantity (24) given by the theorem can be called Herglotz type Noether conserved quantity.

Birkhoff Generalization of Herglotz Type Noether's Theorem
For the Birkhoff system with delayed arguments, the functional z can be defined by the differential Equation [35]: The corresponding Birkhoff's equations with delayed arguments of Herglotz type are We take the infinitesimal transformation of the group as follows: Then, the criterion of Noether symmetry for the Birkhoff system (31) can be expressed as Criterion 2. If the generators ξ σ 0 and ξ σ µ of infinitesimal transformation (32) make the following conditions true, when t ∈ [t 0 − τ, t 0 ), there is Then, the transformation corresponds to the Noether symmetry of Birkhoff system with delayed arguments.
Theorem 2. For Birkhoff system (31) with delayed arguments, if the infinitesimal transformation (32) corresponds to its Noether symmetry, then r linearly independent conserved quantities of Herglotz type exist, such as and and where σ = 1, 2, · · · , r and λ (t) = exp − In Reference [35], Herglotz type Noether's theorem for Birkhoff systems with delayed arguments was studied. However, the above Equations (33) and (36) were not obtained in [35] due to an error in calculating the non-isochronous variation in the interval t ∈ [t 0 − τ, t 0 ].

Hamilton Generalization of Herglotz Type Noether's Theorem
For the Hamilton system with delayed arguments, the functional z can be defined by the differential equation [30] dz The corresponding Hamilton's equations with delayed arguments of Herglotz type are Let the infinitesimal transformation bē , p s (t) = p s (t) + ε σ η σ s (t, q k , p k , z) , (s = 1, 2, · · · , n) .
Then, the criterion of Noether symmetry for the Hamilton system (40) can be expressed as Criterion 3. If the generators ξ σ 0 , ξ σ s and η σ s of infinitesimal transformation (41) make the following conditions true, when t ∈ [t 0 − τ, t 0 ), there is When t ∈ [t 0 , t 1 − τ], there is Then, the transformation corresponds to the Noether symmetry of Hamilton system with delayed arguments.
Theorem 3. For the Hamilton system (40) with delayed arguments, if the infinitesimal transformation (41) corresponds to its Noether symmetry, then r linearly independent conserved quantities of Herglotz type exist, such as and and where σ = 1, 2, · · · , r and λ (t) = exp In Reference [30], Herglotz type Noether's theorem for the Hamilton system with delayed arguments was studied. However, the above Equations (42) and (45) were not obtained in [30] due to an error in calculating the non-isochronous variation in the interval t ∈ [t 0 − τ, t 0 ] .

Examples
Example 1. Study the Noether symmetry and conserved quantity of a non-conservative system with delayed arguments. The Lagrangian of the system in the sense of Herglotz is Functional z satisfies the equation Equation (4) gives According to Criterion 1, when t ∈ [t 0 , t 1 − τ] , the criterion equation is There is a solution to Equation (51), which is when t ∈ (t 1 − τ, t 1 ] , the criterion equation is There is a solution to Equation (53), which is when t ∈ [t 0 − τ, t 0 ), from Equation (21), we have e t+τq τ (t + τ)q τ (t + τ) ξ 0 = 0 (55) Obviously, ξ 0 = 0 satisfies Equation (55). The generators (52) and (54) are associated with the Noether symmetry of the current system. According to Theorem 1, when t ∈ [t 0 , t 1 − τ] , we have when t ∈ (t 1 − τ, t 1 ] , we have Equations (56) and (57) are the conserved quantities of the system.

Example 2.
Consider a damped two-degree-of-freedom oscillator with time delay. The Lagrangian of Herglotz type is where m is the mass of the particle, k is the stiffness coefficient, and c the damping coefficient, and m, k, c are constants.