Noether Symmetry and Conserved Quantities of Fractional Birkhoffian System in Terms of Herglotz Variational Problem

Tian, Xiaobin; Zhang, Yi

doi:10.1088/0253-6102/70/3/280

Cited by 18 publications

(15 citation statements)

References 34 publications

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“…In this context, the generalization of Noether Theorem is fundamental to study conservative quantities in nonconservative systems described by Herglotz problems. In recent works, Noether's like theorems for several kinds of Herglotz variational problems are proposed [20,21,22,23,24,25,26,27]. In the present work, in order to generalize the Noether Theorem for our Action Principle given by the Fundamental Problem 1, we consider invariance transformations in the (x µ , φ)-space, depending on a real parameter ǫ.…”

Section: Gauge Invariance Of the Actionmentioning

confidence: 99%

“…A reason for this problem to be almost unknown is that a covariant generalization for several independent variables is not direct. Only recently the Herglotz variational problem gained more interest in the literature [19,20,21,22,23,24,25,26,27] and, in particular, in a recent work [16] we formulated a covariant generalization for the Herglotz problem to construct a non-conservative gravitational theory from the Lagrangian formalism. Furthermore, by following the ideas we introduced in [16], in [17] we formulated a general Action Principle for non-conservative systems for Lagrangian density functions depending itself on an action-density field.…”

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confidence: 99%

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Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems

2019

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In the present work, we formulate a generalization of the Noether Theorem for action-dependent Lagrangian functions. The Noether's theorem is one of the most important theorems for physics. It is well known that all conservation laws, e.g., conservation of energy and momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether theorem cannot be applied to study non-conservative systems because it is not possible to formulate physically meaningful Lagrangian functions for this kind of systems in the classical calculus of variation. On the other hand, recently it was shown that an Action Principle with action-dependent Lagrangian functions provides physically meaningful Lagrangian functions for a huge variety of non-conservative systems (classical and quantum). Consequently, the generalized Noether Theorem we present enable us to investigate conservation laws of nonconservative systems. In order to illustrate the potential of application, we consider three examples of dissipative systems and we analyze the conservation laws related to spacetime transformations and internal symmetries.

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Section: Gauge Invariance Of the Actionmentioning

confidence: 99%

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confidence: 99%

Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems

2019

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“…Zhang and Zhou [52] proposed the quasi-fractional Pfaff-Birkhoff principle and derived corresponding quasi-fractional Birkhoff's equations which is based on the quasi-fractional model given by [38]. Up to now, some results have been obtained on Noether symmetry of fractional or quasi-fractional Birkhoffian systems, such as [52,[59][60][61][62][63][64][65][66]. However, the results of these quasi-fractional Birkhoffian systems are limited to the Pfaff action containing only integral-order derivative terms.…”

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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“…Moreover, when these functions do not depend on the action functional, Herglotz variational principle can be reduced to the classical integral variational principle, which can deal with conservative problems. Since Herglotz type variational principle provides a new method for studying non-conservative systems, Herglotz type Noether theorems of mechanical systems have been investigated in recent decades, including non-conservative Lagrangian systems [19,20], non-conservative Hamiltonian systems [21], Birkhoffian systems [15,22], non-conservative non-holonomic systems [23] and other complex systems [24–31]. But so far, time-scales Herglotz variational principle is rarely studied, and the results are limited to Lagrangian formalism [32,33] and Hamiltonian formalism [34].…”

Section: Introductionmentioning

confidence: 99%

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

Tian

Zhang

2019

R. Soc. open sci.

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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.

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Noether Symmetry and Conserved Quantities of Fractional Birkhoffian System in Terms of Herglotz Variational Problem

Cited by 18 publications

References 34 publications

Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems

Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems

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

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

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