Fractional generalized Hamiltonian mechanics and Poisson conservation law in terms of combined Riesz derivatives

Shao-Kai, Luo; Li, Lin

doi:10.1007/s11071-013-0817-8

Cited by 39 publications

(14 citation statements)

References 45 publications

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“…Neglecting the terms of ε 2 and the higher order terms for (17), and using (1) and (19), the restriction equations of Mei symmetry for the nonholonomic constraint equations (1) under the infinitesimal transformations (10) are easily obtained as follows:…”

Section: Definitionmentioning

confidence: 99%

“…Research of symmetries and conserved quantities of constrained dynamical systems plays an important role in modern mechanical and mathematical sciences, and it is also a developing direction of modern mathematics, mechanics and physics [1][2][3]. Fruitful achievements have been gained in looking for conserved quantities by means of Noether symmetry, Lie symmetry and Mei symmetry [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Theories of the conformal invariance are classified into the gauge field theories in 1960s and 1970s, particularly are a hot topic of gravitational gauge field [22,23].…”

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confidence: 99%

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Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

et al. 2014

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For a nonholonomic system of Chetaev's type, the conformal invariance and the conserved quantity of Mei symmetry for Appell equations are investigated. First, under the infinitesimal one-parameter transformations of group and the infinitesimal generator vectors, Mei symmetry and conformal invariance of differential equations of motion for the system are defined, and the determining equation of Mei symmetry and conformal invariance for the system are given. Then, by means of the structure equation to which the gauge function is satisfied, the Mei-conserved quantity corresponding to the system is derived. Finally, an example is given to illustrate the application of the result.

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Section: Definitionmentioning

confidence: 99%

mentioning

confidence: 99%

Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

et al. 2014

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“…As is known to all, for a dynamical system with Lie algebraic structure, we can obtain the conserved quantities directly by using Poisson method. The fractional Lorentz-Dirac model (15) is a α-order fractional generalized Hamiltonian system, then the model possesses Lie algebraic structure and satisfies generalized Poisson conservation law [23]. Now, by using the generalized Poisson conservation law, we search for conserved quantities of the fractional Lorentz-Dirac model (15).…”

Section: Lie Algebraic Structure and Poisson Conserved Quantities Of mentioning

confidence: 99%

“…In the end of the 1970s, Mandelbrot [13] discovered a fact that a large number of fractional dimension examples existed in nature. Since then the study of the fractional dynamics has become a hot topic and won wide development, which include fractional Lagrangian mechanics, fractional Hamiltonian mechanics, fractional dynamics of nonholonomic system, fractional generalized Hamiltonian mechanics [14][15][16][17][18][19][20][21][22][23][24], and their application [25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Fractional Lorentz-Dirac Model and Its Dynamical Behaviors

Shao-Kai

2014

Int J Theor Phys

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In the paper, we construct a new kind of fractional dynamical model, i.e. the fractional Lorentz-Dirac model, and explore dynamical behaviors of the model. We find that the fractional Lorentz-Dirac model possesses Lie algebraic structure and satisfies generalized Poisson conservation law, and then a series of Poisson conserved quantities of the model are given. Further, the relation between conserved quantity and integral invariant of the model is studied, and it is proved that, using the Poisson conserved quantities, we can construct a series of integral invariants of the model. Finally, the stability for the manifolds of equilibrium state of the fractional Lorentz-Dirac model is studied.

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“…The symmetry under the Lie group transformation has its inherent applicability in classifying and reducing nonlinear differential equations as well as in finding out conservation laws [4,5,[41][42][43][44][45][46]. So applying the symmetry to the elastic rod and finding out its conserved quantities via the symmetry analysis will be helpful for its research.…”

Section: Introductionmentioning

confidence: 99%

Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod

Wang

Xue

2015

Nonlinear Dyn

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By Kirchhoff dynamic analogy, the thin elastic rod static equals to rotation of rigid body dynamic. The analytical mechanics methods reflect their advantages in the study of the modeling and equilibrium and stability of elastic rod static, especially for the constrained problems. The Lagrangian structure of the equation of motion for elastic rod is deduced from the integral variational principle. The definition of conformal invariance of Mei symmetry of elastic rod in Lagrangian form is given. The determining equation of conformal invariance of Mei symmetry is obtained based on the Lie point transformation group. The relation between conformal invariance of Mei symmetry and Mei symmetry is discussed. The structure equation and conserved quantity by using the Lagrangian structure along arc coordinate deduced from conformal invariance of Mei symmetry of elastic rod are constructed. Take rod with circular cross section as example to illustrate the application of the results get in this paper. These conserved quantities will be helpful in the study of exact solutions and stability, as well as the numerical simulation of the thin elastic rod nonlinear mechanics.

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Fractional generalized Hamiltonian mechanics and Poisson conservation law in terms of combined Riesz derivatives

Cited by 39 publications

References 45 publications

Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

Fractional Lorentz-Dirac Model and Its Dynamical Behaviors

Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod

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