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

Jia, Yun-Die; Zhang, Yi

doi:10.1155/2021/6694709

Cited by 6 publications

(3 citation statements)

References 54 publications

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“…Symmetry and conserved quantity have always been a hot spot in analytical mechanics [6][7][8] . Recently, Zhang and his collaborators have studied Noether theorems for systems with non-standard Lagrangians [9,10] , non-standard Hamiltonians [11] , non-standard Birkhoffians [12] , and Lie symmetry [13,14] , Mei symmetry [15] , first integral and method of reduction [16,17] . There have been some results about nonlinear dynamical equations and their symmetries with non-standard Lagrangians [18][19][20][21][22][23][24][25][26] , but canonical transformation and Poisson theory for non-standard Lagrangian dynamics have not been involved.…”

Section: Introductionmentioning

confidence: 99%

Canonical Transformations and Poisson Theory for Dynamics with Non-Standard Lagrangians

ZHU,

ZHANG

2024

Wuhan Univ. J. Nat. Sci.

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The canonical transformation and Poisson theory of dynamical systems with exponential, power-law, and logarithmic non-standard Lagrangians are studied, respectively. The criterion equations of canonical transformation are established, and four basic forms of canonical transformations are given. The dynamic equations with non-standard Lagrangians admit Lie algebraic structure. From this, we establish the Poisson theory, which makes it possible to find new conservation laws through known conserved quantities. Some examples are put forward to demonstrate the use of the theory and verify its effectiveness.

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Section: Introductionmentioning

confidence: 99%

Canonical Transformations and Poisson Theory for Dynamics with Non-Standard Lagrangians

ZHU,

ZHANG

2024

Wuhan Univ. J. Nat. Sci.

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“…(B4) Generalizations of the Noether theorem for fractional Lagrangian and Hamiltonian systems is considered in [51,86,[100][101][102][103][104][105][106][107][108][109][110][111][112][113][114][115][116][117][118]. (B5) The Noether theorem for fractional Birkhoffian systems is considered in [119][120][121][122][123].…”

Section: Introductionmentioning

confidence: 99%

General Fractional Noether Theorem and Non-Holonomic Action Principle

Tarasov

2023

Mathematics

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Using general fractional calculus (GFC) of the Luchko form and non-holonomic variational equations of Sedov type, generalizations of the standard action principle and first Noether theorem are proposed and proved for non-local (general fractional) non-Lagrangian field theory. The use of the GFC allows us to take into account a wide class of nonlocalities in space and time compared to the usual fractional calculus. The use of non-holonomic variation equations allows us to consider field equations and equations of motion for a wide class of irreversible processes, dissipative and open systems, non-Lagrangian and non-Hamiltonian field theories and systems. In addition, the proposed GF action principle and the GF Noether theorem are generalized to equations containing general fractional integrals (GFI) in addition to general fractional derivatives (GFD). Examples of field equations with GFDs and GFIs are suggested. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for general fractional non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory.

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“…Since then, fractional calculus has been applied in many elds such as physics, mechanics, etc. [2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

A New Fractional Gradient Representation of Birkhoff Systems

Wang

2022

Mathematical Problems in Engineering

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A fractional generalization of the gradient system to the Birkhoff mechanics, that is, a new fractional gradient representation of the Birkhoff system is investigated, in this paper. The definitions of the fractional gradient system are generalized to Birkhoff mechanics. Based on the definition, a general condition that a Birkhoff system can be a fractional gradient system is derived, the former studies for fractional gradient representation of the Birkhoff system are special cases of this paper. As applications of the results, the Birkhoff equations and fractional gradient expression of several classical nonlinear models are derived, such as the Hénon–Heiles equation, the Duffing oscillator model, and the Hojman–Urrutia equations. The results indicate, different from the former studies, that only gave the second-order gradient (integer order) expression for the Birkhoff system, an arbitrary fractional order gradient representation, exists for the Birkhoff system. The fractional potential function obtained from the general condition can determine the stationary states of these models in Birkhoffian expression.

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

Cited by 6 publications

References 54 publications

Canonical Transformations and Poisson Theory for Dynamics with Non-Standard Lagrangians

Canonical Transformations and Poisson Theory for Dynamics with Non-Standard Lagrangians

General Fractional Noether Theorem and Non-Holonomic Action Principle

A New Fractional Gradient Representation of Birkhoff Systems

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