Noether symmetries and conserved quantities for fractional forced Birkhoffian systems

Jia, Qiuli; Wu, Huibin; Feng-Xiang, Mei

doi:10.1016/j.jmaa.2016.04.067

Cited by 26 publications

(10 citation statements)

References 49 publications

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“…Equations (26) and (27) are called generalized fractional Birkhoff equations with the Riesz-Riemann-Liouville fractional derivative and the Riesz-Caputo fractional derivative, respectively.…”

Section: Generalized Fractional Birkhoffian Systemmentioning

confidence: 99%

“…and 0 ] satisfy the Noether identity (50) for the generalized fractional Birkhoffian system (26), then this system has a conserved quantity (27), then this system has a conserved quantity…”

Section: Noether Symmetry and Conserved Quantitymentioning

confidence: 99%

“…Based on the Caputo fractional derivative, Muslih [24] extended the fractional Noether theorem of finite degree of freedom to the fractional field theory and presented the conserved quantity of the fractional Dirac field. In recent years, Zhang [25,26], Jia [27], and Zhang [28] studied the Noether symmetry and conserved quantity of the fractional Birkhoffian system. Fu [29] investigated the Lie symmetry of the fractional nonholonomic Hamiltonian system and the corresponding inverse problem.…”

Section: Introductionmentioning

confidence: 99%

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Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

Song

2018

Mathematical Problems in Engineering

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Perturbation to Noether symmetry and adiabatic invariants are investigated for the generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative, respectively. Firstly, differential equations of motion for the generalized fractional Birkhoffian system are established. Secondly, Noether symmetry and conserved quantity are studied. Thirdly, perturbation to Noether symmetry and adiabatic invariants are presented for the generalized fractional Birkhoffian mechanics. And finally, several applications are discussed to illustrate the results and methods.

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Section: Generalized Fractional Birkhoffian Systemmentioning

confidence: 99%

“…and 0 ] satisfy the Noether identity (50) for the generalized fractional Birkhoffian system (26), then this system has a conserved quantity (27), then this system has a conserved quantity…”

Section: Noether Symmetry and Conserved Quantitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

Song

2018

Mathematical Problems in Engineering

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“…generalidade a correspondência entre as simetrias de um problema variacional e as leis de conservação para as equações variacionais associadas [32,33].…”

Section: Neste Teorema Emmy Noether Formulou Com Completaunclassified

As Simetrias de Lie de um Pião

Basquerotto

Righetto

Silva

2017

Rev. Bras. Ensino Fís.

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A existência de simetrias em equações diferenciais pode gerar transformações em variáveis dependentes e independentes que facilitam a integração destas equações. Em especial, Sophus Lie desenvolveu no século XIX uma forma de extração de simetrias que podem ser usadas efetivamente para revelar as integrais primeiras, ou seja, as constantes de movimento, que muitas vezes podem estar escondidas. Estes invariantes podem em algumas situações ser identificados pelo teorema de Noether ou a partir de manipulações das próprias equações com transformações de Lie. Nos cursos iniciais de mecânica clássica, apesar de todo o formalismo em cima dos teoremas de conservação de energia e momento linear/angular, a relação disto com a existência de possíveis simetrias de Lie não é destacada de forma clara e objetiva. Neste sentido, o presente artigo busca apresentar uma introdução às simetrias de Lie usando uma linguagem acessível para um aluno de graduação de física, matemática ou engenharia com domínio básico em fundamentos de cálculo com várias variáveis. Para ilustrar a abordagem, considera-se um problema clássico de mecânica considerando um pião em regime de movimento com precessão estacionária. A partir das equações de movimento obtidas, as simetrias de Lie são identificadas e usadas na transformação para a redução da ordem. As integrais primeiras são obtidas a partir deste resultado com o Teorema de Noether, mostrando que neste exemplo e condição as simetrias de Lie também são simetrias de Noether. Por fim, a resolução das equações de movimento podem ser feitas usando funções elípticas de Jacobi para a obtenção dos ângulos de precessão, nutação e spin nas condições apresentadas.

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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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Noether symmetries and conserved quantities for fractional forced Birkhoffian systems

Cited by 26 publications

References 49 publications

Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

As Simetrias de Lie de um Pião

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

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