Nonholonomic constraints with fractional derivatives

Tarasov, Vasily E.; Zaslavsky, G. M.

doi:10.1088/0305-4470/39/31/010

Cited by 56 publications

(37 citation statements)

References 77 publications

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“…As a further step, Riewe [1,2] formulated a version of the Euler-Lagrange equation for problems of calculus of variation with fractional derivatives. Recently, further studies concerning the fractional Euler-Lagrange equations can be found in the works of Agrawal and coworkers [5][6][7][8], Baleanu and coworkers [9][10][11][12][13][14][15], Tarasov and Zaslavsky [16,17] and others [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

“…In last decades, much interest was devoted to apply fractional calculus to almost every field of science, engineering and mathematics [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The awareness of the importance of this type of equation has grown continuously include for viscoelasticity and rheology, image processing, mechanics, mechatronics, physics, and control theory, see for instance [23][24][25].…”

Section: Introductionmentioning

confidence: 99%

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Time-fractional KdV equation: formulation and solution using variational methods

et al. 2010

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In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Time-fractional KdV equation: formulation and solution using variational methods

et al. 2010

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“…In the end of the 1970s, Mandelbrot [26] discovered that a large number of fractional dimension examples exists in nature. Since then, the study of the fractional dynamics has become a hot topic, and won wide development in theories and applications, including fractional Lagrangian mechanics, fractional Hamiltonian mechanics, fractional dynamics of nonholonomic system and fractional generalized Hamiltonian mechanics [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], and their application [44][45][46][47][48][49][50]. In order to better solve the fractional dimension problems in science and engineering, it is necessary to propose a fractional dynamical theory of Birkhoffian systems.…”

Section: Introductionmentioning

confidence: 99%

Fractional Birkhoffian mechanics

Shao-Kai

2014

Acta Mech

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In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff-Birkhoff principle, and give four kinds of fractional Pfaff-Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff-Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman-Urrutia model, the fractional Hénon-Heiles model and the fractional Emden model.

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“…This is possible through a powerful tool known as the Euler-Lagrange equation [33]. Recently the theory of the calculus of variations has been considered in the fractional context [7,8,10,11,12,13,14,23,25,27,32]. The fractional calculus allows to generalize the ordinary differentiation and integration to an arbitrary (non-integer) order, and provides a powerful tool for modeling and solving various problems in science and engineering [28,29,31].…”

Section: Introductionmentioning

confidence: 99%

Isoperimetric problems of the calculus of variations with fractional derivatives

Almeida

Ferreira

Torres

2012

Acta Mathematica Scientia

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Nonholonomic constraints with fractional derivatives

Abstract: We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle.

Cited by 56 publications

References 77 publications

Time-fractional KdV equation: formulation and solution using variational methods

Time-fractional KdV equation: formulation and solution using variational methods

Fractional Birkhoffian mechanics

Isoperimetric problems of the calculus of variations with fractional derivatives

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