Generalized variational problems and Birkhoff equations

Zhang, Hongbin; Chen, Haibo

doi:10.1007/s11071-015-2331-7

Cited by 11 publications

(5 citation statements)

References 32 publications

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“…For Equation (24), the integer generalized momentum, the fractional generalized momentum, and the Hamiltonian can be defined as…”

Section: Fractional Primary Constraintmentioning

confidence: 99%

“…Since Riewe [11,12] found that fractional derivatives can be used to express dissipative forces, fractional calculus of variations with different fractional derivatives, such as the Riemann-Liouville fractional derivative [13][14][15], the Caputo fractional derivative [16,17], the symmetric fractional derivative [18], the Riesz fractional derivative [19][20][21], Agrawal's new operators [22][23][24], the combined fractional derivative [25][26][27], the mixed integer and fractional derivatives [28,29], and so on [30][31][32][33][34][35][36], have been investigated. It is noted that the combined fractional derivative is more general than most other fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

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Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Song

2022

Fractal Fract

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Singular systems, which can be applied to gauge field theory, condensed matter theory, quantum field theory of anyons, and so on, are important dynamic systems to study. The fractional order model can describe the mechanical and physical behavior of a complex system more accurately than the integer order model. Fractional singular systems within mixed integer and combined fractional derivatives are established in this paper. The fractional Lagrange equations, fractional primary constraints, fractional constrained Hamilton equations, and consistency conditions are analyzed. Then Noether and Lie symmetry methods are studied for finding the integrals of the fractional constrained Hamiltonian systems. Finally, an example is given to illustrate the methods and results.

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“…For Equation (24), the integer generalized momentum, the fractional generalized momentum, and the Hamiltonian can be defined as…”

Section: Fractional Primary Constraintmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Song

2022

Fractal Fract

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“…Fractional Noether theorem has been extended to the fractional Lagrange system, Hamilton system, generalized Hamilton system, Birkhoff system, and so on. [23][24][25][26][27][28][29][30][31][32][33] However, there is no research on fractional Noether symmetry in mechano-electrophysiological coupling equations of neuron dynamics. Considering the visco-elasticity of neuron membranes, we will adopt a fractional derivative of variable orders.…”

Section: Introductionmentioning

confidence: 99%

Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

Wang

2023

Chinese Phys. B

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Noether theorem is applied into a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics. The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics, is given. The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given. The forms of variable orders fractional Noether conserved quantities correspond to Noether symmetry generators solutions of the model under different conditions are detailed discussed, and it is found that the expressions of variable orders fractional Noether conserved quantities are closely depending on the external nonconservative forces and material parameters of neuron.

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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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Generalized variational problems and Birkhoff equations

Cited by 11 publications

References 32 publications

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

A New Fractional Gradient Representation of Birkhoff Systems

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