QUASI-SYMMETRY AND NOETHER'S THEOREM FOR FRACTIONAL BIRKHOFFIAN SYSTEMS IN TERMS OF CAPUTO DERIVATIVES

doi:10.6052/0459-1879-16-350

Abstract

Abstract:

The dynamical behavior and physical process of a complex system can be described and studied more accurately by using a fractional model, at the same time the Birkhoffian mechanics is a generalization of Hamiltonian mechanics, and therefore, the study of dynamics of fractional Birkhoffian systems is of great significance. Fractional Noether's theorem reveals the intrinsic relation between the Noether symmetric transformation and the fractional conserved quantity, but when the transformation is replaced by the Noether quasi-symmetric transformation, the corresponding extension of Noether's theorem is very difficult. In this paper, the Noether quasi-symmetry and the conserved quantity for fractional Birkhoffian systems in terms of Caputo derivatives are presented and studied by using a technique of timereparametrization. Firstly, the technique is applied to the study of the Noether quasi-symmetry and the conserved quantity for classical Birkhoffian systems and Noether's theorem in its general form is established. Secondly, the definitions and criteria of Noether quasi-symmetric transformations for fractional Birkhoffian systems are given which are based on the invariance of fractional Pfaff action under one-parameter infinitesimal group of transformations without transforming the time and with transforming the time, respectively. Based on the definition of fractional conserved quantity proposed by Frederico and Torres, Noether's theorem for fractional Birkhoffian systems is established by using the method of timereparametrization. The theorem reveals the inner relationship between Noether quasi-symmetry and fractional conserved quantity and contains Noether's theorem for the symmetry of fractional Birkhoffian system and Noether's theorem for classical Birkhoffian system as its specials. Finally, we take the Hojman-Urrutia problem as an example to illustrate the application of the results.

Key words:

fractional Birkhoffian system|Noether quasi-symmetry|Frederico-Torres fractional conserved quantity|caputo derivative

CLC Number:

O316

Zhang Yi. QUASI-SYMMETRY AND NOETHER'S THEOREM FOR FRACTIONAL BIRKHOFFIAN SYSTEMS IN TERMS OF CAPUTO DERIVATIVES[J]. Chinese Journal of Theoretical and Applied Mechani, 2017, 49(3): 693-702.

References

1 Oldham KB, Spanier J. The Fractional Calculus. San Diego:Academic Press, 1974
2 Podlubny I. Fractional Differential Equations. San Diego:Academic Press, 1999
3 Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier B V, 2006
4 Riewe F. Nonconservative Lagrangian and Hamiltonian mechanics. Physical Review E, 1996, 53(2):1890-1899
5 Riewe F. Mechanics with fractional derivatives. Physical Review E, 1997, 55(3):3581-3592
6 Agrawal OP. Formulation of Euler-Lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications, 2002, 272(1):368-379
7 Agrawal OP. Fractional variational calculus in terms of Riesz fractional derivatives. Journal of Physics A:Mathematical and Theoretical, 2007, 40(24):6287-6303
8 Baleanu D, Avkar T. Lagrangians with linear velocities within Riemann-Liouville fractional derivatives. Nuovo Cimento B, 2003, 119(1):73-79
9 Baleanu D, Trujillo JJ. A new method of finding the fractional EulerLagrange and Hamilton equations within Caputo fractional derivatives. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(5):1111-1115
10 Atanackovi? TM, Konjik S, Pilipovi? S. Variational problems with fractional derivatives:Euler-Lagrange equations. Journal of Physics A:Mathematical and Theoretical, 2008, 41(9):095201
11 Atanackovi? TM, Konjik S, Oparnica Lj, et al. Generalized Hamilton's principle with fractional derivatives. Journal of Physics A:Mathematical and Theoretical, 2010, 43(25):255203
12 Almeida R, Torres DFM. Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(3):1490-1500
13 Malinowska AB, Torres DFM. Introduction to the Fractional Calculus of Variations. London:Imperial College Press, 2012
14 Frederico GSF, Torres DFM. A formulation of Noether's theorem for fractional problems of the calculus of variations. Journal of Mathematical Analysis and Applications, 2007, 334(2):834-846
15 Frederico GSF, Torres DFM. Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense. Reports on Mathematical Physics, 2013, 71(3):291-304
16 Frederico GSF. Fractional optimal control in the sense of Caputo and the fractional Noether's theorem. International Mathematical Forum, 2008, 3(10):479-493
17 Frederico GSF, Torres DFM. Fractional Noether's theorem in the Riesz-Caputo sense. Applied Mathematics and Computation, 2010, 217(3):1023-1033
18 Frederico GSF, Lazo MJ. Fractional Noether's theorem with classical and Caputo derivatives:constants of motion for non-conservative systems. Nonlinear Dynamics, 2016, 85(2):839-851
19 Atanackovi? TM, Konjik S, Pilipovi? S, et al. Variational problems with fractional derivatives:invariance conditions and Noether's theorem. Nonlinear Analysis:Theory, Methods & Applications, 2009, 71(5-6):1504-1517
20 Odzijewicz T, Malinowska AB, Torres DFM. Noether's theorem for fractional variational problems of variable order. Central European Journal of Physics, 2013, 11(6):691-701
21 Zhang SH, Chen BY, Fu JL. Hamilton formalism and Noether symmetry for mechanico-electrical systems with fractional derivatives. Chinese Physics B, 2012, 21(10):100202
22 Zhou S, Fu H and FuJL. Symmetry theories of Hamiltonian systems with fractional derivatives. Science China Physics, Mechanics & Astronomy, 2011, 54(10):1847-1853
23 Luo SK, Li L. Fractional generalized Hamiltonian equations and its integral invariants. Nonlinear Dynamics, 2013, 73(1):339-346
24 Luo SK, Li L. Fractional generalized Hamiltonian mechanics and Poisson conservation law in terms of combined Riesz derivatives. Nonlinear Dynamics, 2013, 73(1):639-647
25 Jia QL, Wu HB, Mei FX. Noether symmetries and conserved quantities for fractional forced Birkhoffian systems. Journal of Mathematical Analysis and Applications, 2016, 442(2):782-795
26 Zhang Y, Zhou Y. Symmetries and conserved quantities for fractional action-like Pfaffian variational problems. Nonlinear Dynamics, 2013, 73(1-2):783-793
27 Zhang Y, Zhai XH. Noether symmetries and conserved quantities for fractional Birkhoffian systems. Nonlinear Dynamics, 2015, 81(1-2):469-480
28 Long ZX, Zhang Y. Noether's theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space. Acta Mechanica, 2014, 225(1):77-90
29 El-Nabulsi RA. Fractional variational symmetries of Lagrangians, the fractional Galilean transformation and the modified Schrödinger equation. Nonlinear Dynamics, 2015, 81(1):939-948
30 Zhai XH, Zhang Y. Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay. Communications in Nonlinear Science and Numerical Simulation, 2016, 36:81-97
31 Yan B, Zhang Y. Noether's theorem for fractional Birkhoffian systems of variable order. Acta Mechanica, 2016, 227(9):2439-2449
32 梅凤翔. 李群和李代数对约束力学系统的应用. 北京:科学出版社, 1999 (Mei Fengxiang. Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems. Beijing:Science Press,1999 (in Chinese))
33 梅凤翔, 吴惠彬, 李彦敏等. Birkhoff 力学的研究进展. 力学学报, 2016, 48(2):263-268 (Mei Fengxiang, Wu Huibin, Li Yanmin, et al. Advances in research on Birkhoffian mechanics. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(2):263-268 (in Chinese))
34 张毅. 相空间中非保守系统的 Herglotz 广义变分原理及其 Noether 定理. 力学学报, 2016, 48(6):1382-1389 (Zhang Yi. Generalized variational principle of Herglotz type for nonconservative system in phase space and Noether's theorem. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6):1382-1389 (in Chinese))
35 Luo SK, Xu YL. Fractional Birkhoffian mechanics. Acta Mechanica, 2015, 226(3):829-844

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