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

Zhang, Yi

doi:10.3390/sym12050845

Cited by 8 publications

(4 citation statements)

References 38 publications

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“…Compared with some previous studies on time-delay mechanical systems [31][32][33][34][35][36][37][38][39][40], this paper not only takes into account the more realistic description of different time-delays between the generalized coordinates and the generalized velocities, but also achieves the more general Noether-type conserved quantities.…”

Section: Discussionmentioning

confidence: 99%

“…Frederico and Torres [27] preliminarily introduced the classical Noether's theory to the time-delay calculus of variations. Indeed, Noether's theory has been applied to various problems involving time-delay, such as non-smooth extremals of variational problems [28], isoperimetric variational problems [29], high-order variational problems [30], non-conservative systems [31], nonholonomic systems [32], Hamiltonian systems [33], Birkhoffian systems [34], generalized Herglotz variational problems [35][36][37], and dynamical systems in fractional [38,39] and time-scale frameworks [6,40].…”

Section: Introductionmentioning

confidence: 99%

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Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

Ji,

Yang,

Zhai

2024

Symmetry

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The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical system is investigated. Firstly, the multi-time-delay Hamilton principle is proposed. Then, multi-time-delay non-conservative dynamical equations are deduced. Secondly, depending on the infinitesimal group transformations, the invariance of the multi-time-delay Hamilton action is studied, and Noether symmetry, Noether quasi-symmetry, and generalized Noether quasi-symmetry are discussed. Finally, Noether-type conserved quantities for a multi-time-delay Lagrangian system and a multi-time-delay non-conservative mechanical system are obtained. Two examples in terms of a multi-time-delay non-conservative mechanical system and a multi-time-delay Lagrangian system are given.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

Ji,

Yang,

Zhai

2024

Symmetry

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“…The Herglotz variational problem [43] is a generalization of the usual variational problem, but instead of being given by an integral, the action functional is described by a differential equation. In this way, it can be more realistic to describe certain physical processes and can be applied to solve non-conservative problems for which the classical variational principle is not applicable [44,45]. It was formulated in the following way: Determine the curves x, z ∈ C 1 ([a, b], R) for which z(b) attains an extreme value, where x and z are related by the differential equation…”

Section: The Herglotz Problemmentioning

confidence: 99%

Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function

Almeida

2023

Fractal Fract

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In this paper, we investigate the necessary conditions to optimize a given functional, involving a generalization of the tempered fractional derivative. The exponential function is replaced by the Mittag–Leffler function, and the kernel depends on an arbitrary increasing function. The Lagrangian depends on time, the state function, its fractional derivative, and we add a terminal cost function to the formulation of the problem. Since this new fractional derivative is presented in a general form, some previous works are our own particular cases. In addition, for different choices of the kernel, new results can be deduced. Using variational techniques, the fractional Euler–Lagrange equation is proved, as are its associated transversality conditions. The variational problem with additional constraints is also considered. Then, the question of minimizing functionals with an infinite interval of integration is addressed. To end, we study the case of the Herglotz variational problem, which generalizes the previous one. With this work, several optimization conditions are proven that can be useful for different optimization problems dealing with various fractional derivatives.

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“…Recently, some new progress has been made in the study of these two symmetries (cf. [13][14][15][16][17][18][19][20][21][22][23][24] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Fractional Birkhoffian Mechanics Based on Quasi-Fractional Dynamics Models and Its Noether Symmetry

Jia

Zhang

2021

Mathematical Problems in Engineering

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This paper focuses on the exploration of fractional Birkhoffian mechanics and its fractional Noether theorems under quasi-fractional dynamics models. The quasi-fractional dynamics models under study are nonconservative dynamics models proposed by El-Nabulsi, including three cases: extended by Riemann–Liouville fractional integral (abbreviated as ERLFI), extended by exponential fractional integral (abbreviated as EEFI), and extended by periodic fractional integral (abbreviated as EPFI). First, the fractional Pfaff–Birkhoff principles based on quasi-fractional dynamics models are proposed, in which the Pfaff action contains the fractional-order derivative terms, and the corresponding fractional Birkhoff’s equations are obtained. Second, the Noether symmetries and conservation laws of the systems are studied. Finally, three concrete examples are given to demonstrate the validity of the results.

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

Cited by 8 publications

References 38 publications

Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function

Fractional Birkhoffian Mechanics Based on Quasi-Fractional Dynamics Models and Its Noether Symmetry

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