Fractional time-scales Noether theorem with Caputo<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"><mml:mstyle mathvariant="normal"><mml:mi>Δ</mml:mi></mml:mstyle></mml:math>derivatives for Hamiltonian systems

Tian, Xiaobin; Zhang, Yi

doi:10.1016/j.amc.2020.125753

Cited by 8 publications

(4 citation statements)

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“…Frederico and Torres [41] first presented fractional Noether's theorems. Since then, studies on fractional Noether symmetry and conservation laws have been extensively developed [42][43][44][45][46][47][48][49]. In addition, Torres and Frederico studied Noether's theorems of fractional action-like variation problems [50,51].…”

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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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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“…Recent developments in the fields of science, engineering, economics, bioengineering, and applied mathematics have demonstrated that many phenomena in nature are modelled more accurately using fractional derivatives [22,23]. Fractional Hamiltonian mechanics [24][25][26][27][28][29][30][31][32] and Noether theorems for them have also been established and investigated [33][34][35][36]. The fractional operators mainly referred to the left and right fractional Riemann-Liouville integrals, the left and right fractional Riemann-Liouville differential operators, the left and right fractional Caputo differential operators, the Riesz-Riemann-Liouville differential operator, and the Riesz-Caputo differential operator.…”

Section: Introductionmentioning

confidence: 99%

Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Song

Yao

2021

Advances in Mathematical Physics

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Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.

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“…Based on the second Euler-Lagrange equations, they proposed another method to find the Noether conserved quantity. Afterwards, according to these two methods, many scholars have obtained some results have been obtained in the study of variational principle, dynamical equations, and Noether symmetries for the different mechanical systems, such as references [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Time-Scale Version of Generalized Birkhoffian Mechanics and Its Symmetries and Conserved Quantities of Noether Type

Chen

Zhang

2021

Advances in Mathematical Physics

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The time-scale version of Noether symmetry and conservation laws for three Birkhoffian mechanics, namely, nonshifted Birkhoffian systems, nonshifted generalized Birkhoffian systems, and nonshitfed constrained Birkhoffian systems, are studied. Firstly, on the basis of the nonshifted Pfaff-Birkhoff principle on time scales, Birkhoff’s equations for nonshifted variables are deduced; then, Noether’s quasi-symmetry for the nonshifted Birkhoffian system is proved and time-scale conserved quantity is presented. Secondly, the nonshifted generalized Pfaff-Birkhoff principle on time scales is proposed, the generalized Birkhoff’s equations for nonshifted variables are derived, and Noether’s symmetry for the nonshifted generalized Birkhoffian system is established. Finally, for the nonshifted constrained Birkhoffian system, Noether’s symmetry and time-scale conserved quantity are proposed and proved. The validity of the result is proved by examples.

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Fractional time-scales Noether theorem with CaputoΔderivatives for Hamiltonian systems

Cited by 8 publications

References 45 publications

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

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

Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Time-Scale Version of Generalized Birkhoffian Mechanics and Its Symmetries and Conserved Quantities of Noether Type

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