Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type

doi:10.1088/1674-1056/ab6d51

中国物理B ›› 2020, Vol. 29 ›› Issue (4): 44501-044501.doi: 10.1088/1674-1056/ab6d51

• ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS • 上一篇下一篇

Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type

Juan-Juan Ding(丁娟娟), Yi Zhang(张毅)

1 School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China;
2 College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China

收稿日期:2019-12-19 修回日期:2020-01-16 出版日期:2020-04-05 发布日期:2020-04-05
通讯作者: Yi Zhang E-mail:zhy@mail.usts.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11972241, 11572212, and 11272227), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20191454), and the Innovation Program for Postgraduade in Higher Education Institutions of Jiangsu Province, China (Grant No. KYCX19_2013).

Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type

Juan-Juan Ding(丁娟娟)¹, Yi Zhang(张毅)²

1 School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China;
2 College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China

Received:2019-12-19 Revised:2020-01-16 Online:2020-04-05 Published:2020-04-05
Contact: Yi Zhang E-mail:zhy@mail.usts.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11972241, 11572212, and 11272227), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20191454), and the Innovation Program for Postgraduade in Higher Education Institutions of Jiangsu Province, China (Grant No. KYCX19_2013).

摘要/Abstract

摘要： In order to further study the dynamical behavior of nonconservative systems, we study the conserved quantities and the adiabatic invariants of fractional Brikhoffian systems with four kinds of different fractional derivatives based on Herglotz differential variational principle. Firstly, the conserved quantities of Herglotz type for the fractional Brikhoffian systems based on Riemann-Liouville derivatives and their existence conditions are established by using the fractional Pfaff-Birkhoff-d'Alembert principle of Herglotz type. Secondly, the effects of small perturbations on fractional Birkhoffian systems are studied, the conditions for the existence of adiabatic invariants for the Birkhoffian systems of Herglotz type based on Riemann-Liouville derivatives are established, and the adiabatic invariants of Herglotz type are obtained. Thirdly, the conserved quantities and adiabatic invariants for the fractional Birkhoffian systems of Herglotz type under other three kinds of fractional derivatives are established, namely Caputo derivative, Riesz-Riemann-Liouville derivative and Riesz-Caputo derivative. Finally, an example is given to illustrate the application of the results.

关键词: fractional Birkhoffian system, Pfaff-Birkhoff-d'Alembert principle, adiabatic invariant, Herglotz generalized variational principle

Abstract: In order to further study the dynamical behavior of nonconservative systems, we study the conserved quantities and the adiabatic invariants of fractional Brikhoffian systems with four kinds of different fractional derivatives based on Herglotz differential variational principle. Firstly, the conserved quantities of Herglotz type for the fractional Brikhoffian systems based on Riemann-Liouville derivatives and their existence conditions are established by using the fractional Pfaff-Birkhoff-d'Alembert principle of Herglotz type. Secondly, the effects of small perturbations on fractional Birkhoffian systems are studied, the conditions for the existence of adiabatic invariants for the Birkhoffian systems of Herglotz type based on Riemann-Liouville derivatives are established, and the adiabatic invariants of Herglotz type are obtained. Thirdly, the conserved quantities and adiabatic invariants for the fractional Birkhoffian systems of Herglotz type under other three kinds of fractional derivatives are established, namely Caputo derivative, Riesz-Riemann-Liouville derivative and Riesz-Caputo derivative. Finally, an example is given to illustrate the application of the results.

Key words: fractional Birkhoffian system, Pfaff-Birkhoff-d'Alembert principle, adiabatic invariant, Herglotz generalized variational principle

中图分类号: (Perturbation and fractional calculus methods)

45.10.Hj

11.25.Db (Properties of perturbation theory)

丁娟娟, 张毅. Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type[J]. 中国物理B, 2020, 29(4): 44501-044501.

Juan-Juan Ding(丁娟娟), Yi Zhang(张毅). Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type[J]. Chin. Phys. B, 2020, 29(4): 44501-044501.

参考文献 44

[1]	Machado J T, Kiryakova V and Mainardi F 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 1140 doi: 10.1016/j.cnsns.2010.05.027
[2]	Oldham K B and Spanier J 1974 The Fractional Calculus (San Diego: Academic Press)
[3]	Podlubny I 1999 Fractional Differential Equations (New York: Academic Press)
[4]	Kilbas A A, Srivastava H M and Trujillo J J 2006 Theory and Applications of Fractional Differential Equations (Amsterdam: Elsevier B V)
[5]	Malinowska A B and Torres D F M 2012 Introduction to the Fractional Calculus of Variations (London: Imperial College Press)
[6]	Hilfer R 2000 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
[7]	El-Nabulsi R A 2013 Indian J. Phys. 87 835 doi: 10.1007/s12648-013-0295-3
[8]	Riewe F 1996 Phys. Rev. E 53 1890 doi: 10.1103/PhysRevE.53.1890
[9]	Riewe F 1997 Phys. Rev. E 55 3581 doi: 10.1103/PhysRevE.55.3581
[10]	Frederico G S F and Torres D F M 2007 J. Math. Anal. Appl. 334 834 doi: 10.1016/j.jmaa.2007.01.013
[11]	Cresson J 2007 J. Math. Phys. 48 033504 doi: 10.1063/1.2483292
[12]	Atanacković T M, S Konjik, Pilipović S and Simić S 2009 Nonlinear Anal. Theor. Methods Appl. 71 1504 doi: 10.1016/j.na.2008.12.043
[13]	Luo S K, Yang M J, Zhang X T and Dai Y 2018 Acta Mech. 229 1833 doi: 10.1007/s00707-017-2040-z
[14]	Zhou Y and Zhang Y 2014 Chin. Phys. B 23 124502 doi: 10.1088/1674-1056/23/12/124502
[15]	Luo S K and Xu Y L 2015 Acta Mech. 226 829 doi: 10.1007/s00707-014-1230-1
[16]	Zhang Y and Zhai X H 2015 Nonlinear Dyn. 81 465
[17]	Yan B and Zhang Y 2016 Acta Mech. 227 2439 doi: 10.1007/s00707-016-1622-5
[18]	Zhang Y 2017 Journal of Suzhou University of Science and Technology 34 1
[19]	Tian X and Zhang Y 2018 Commun. Theor. Phys. 70 280 doi: 10.1088/0253-6102/70/3/280
[20]	Herglotz G 1930 Berührungs transformationen (Lectures at the University of Göttingen)
[21]	Georgieva B 2010 Ann. Sofia Univ. Fac. Math. Inf. 100 113
[22]	Santos S P S, Martins, Natália and Torres D F M 2014 Vietnam J. Math. 42 409 doi: 10.1007/s10013-013-0048-9
[23]	Georgieva B and Guenther R 2002 Topol. Methods Nonlinear Anal. 20 261 doi: 10.12775/TMNA.2002.036
[24]	Zhang Y 2017 Acta Mech. 228 1481 doi: 10.1007/s00707-016-1758-3
[25]	Tian X and Zhang Y 2018 Int. J. Theor. Phys. 57 887 doi: 10.1007/s10773-017-3621-2
[26]	Almeida R and Malinowska A B 2014 Discrete Contin. Dyn. Syst. Ser. B 19 2367
[27]	Almeida R 2017 J. Optimiz. Theory Appl. 174 276 doi: 10.1007/s10957-016-0883-4
[28]	Satntos S P S, Martins N and Torres D F M 2015 Discrete Contin. Dyn. Syst. Ser. A 35 4593 doi: 10.3934/dcds.2015.35.4593
[29]	Zhang Y 2018 Chin. Quart. Mech. 39 681
[30]	Kruskal M 1961 Adiabatic Invariants (Princeton: Princeton University Press)
[31]	Bulanov S V and Shasharina S G 1992 Nucl. Fus. 32 1531 doi: 10.1088/0029-5515/32/9/I03
[32]	Notte J, Fajans J, Chu R and Wurtele J S 1993 Phys. Lett. 70 3900 doi: 10.1103/PhysRevLett.70.3900
[33]	Cveticanin L 1994 Int. J. Nonlinear Mech. 29 799 doi: 10.1016/0020-7462(94)90072-8
[34]	Cveticanin L 1995 J. Sound Vib. 183 881 doi: 10.1006/jsvi.1995.0292
[35]	Zhao Y Y and Mei F X 1996 Acta Mech. Sin. 28 207
[36]	Jiang W A, Liu K, Zhao G L and Chen M 2018 Acta Mech. 229 4771 doi: 10.1007/s00707-018-2257-5
[37]	Chen X W, Li Y M and Zhao Y H 2007 Phys. Lett. A 337 274
[38]	Jiang W and Luo S 2012 Nonlinear Dyn. 67 475 doi: 10.1007/s11071-011-9996-3
[39]	Zhang K J, Fang J H and Li Y 2011 Chin. Phys. B 20 054501 doi: 10.1088/1674-1056/20/5/054501
[40]	Xu X X and Zhang Y 2019 Chin. Phys. B 28 120402 doi: 10.1088/1674-1056/ab5210
[41]	Wang P 2012 Nonlinear Dyn. 68 53 doi: 10.1007/s11071-011-0203-3
[42]	Song C J and Zhang Y 2017 Int. J. Nonlinear Mech. 90 32 doi: 10.1016/j.ijnonlinmec.2017.01.003
[43]	Song C J and Zhang Y 2019 Indian J. Phys. 93 1057 doi: 10.1007/s12648-018-01362-x
[44]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)

Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type

Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type

RichHTML

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 44

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	夏丽莉, 戈新生, 陈立群. Discrete symmetrical perturbation and variational algorithm of disturbed Lagrangian systems[J]. 中国物理B, 2019, 28(3): 30201-030201.
[2]	徐鑫鑫, 张毅. A new type of adiabatic invariants for disturbednon-conservative nonholonomic system[J]. 中国物理B, 2019, 28(12): 120402-120402.
[3]	周燕, 张毅. Noether's theorems of a fractional Birkhoffian system within Riemann–Liouville derivatives[J]. 中国物理B, 2014, 23(12): 124502-124502.
[4]	刘成周 . Area and entropy spectra of black holes via an adiabatic invariant[J]. 中国物理B, 2012, 21(7): 70401-070401.
[5]	李慧玲. Area spectrum of the three-dimensional Gödel black hole[J]. Chin. Phys. B, 2012, 21(12): 120401-120401.
[6]	张克军, 方建会, 李燕. Perturbation to Mei symmetry and Mei adiabatic invariants for discrete generalized Birkhoffian system[J]. 中国物理B, 2011, 20(5): 54501-054501.
[7]	丁宁, 方建会. Perturbation of symmetries for super-long elastic slender rods[J]. 中国物理B, 2011, 20(12): 120201-120201.
[8]	张明江, 方建会, 张小妮, 路凯. Perturbation to Mei symmetry and Mei adiabatic invariants for mechanical systems in phase space[J]. 中国物理B, 2008, 17(6): 1957-1961.
[9]	丁宁, 方建会, 陈相霞. Perturbation to Lie symmetry and another type of Hojman adiabatic invariants for Birkhoffian systems[J]. 中国物理B, 2008, 17(6): 1967-1971.
[10]	丁宁, 方建会. Perturbation to Mei symmetry and adiabatic invariants for Hamilton systems[J]. 中国物理B, 2008, 17(5): 1550-1553.
[11]	罗绍凯, 蔡建乐, 贾利群. Adiabatic invariants of generalized Lutzky type for disturbed holonomic nonconservative systems[J]. 中国物理B, 2008, 17(10): 3542-3548.
[12]	夏丽莉, 李元成. Perturbation to symmetries and Hojman adiabatic invariant for nonholonomic controllable mechanical systems with non-Chetaev type constraints[J]. 中国物理B, 2007, 16(6): 1516-1520.
[13]	罗绍凯, 陈向炜, 郭永新. Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems[J]. 中国物理B, 2007, 16(11): 3176-3181.
[14]	张毅. A new type of adiabatic invariants for nonconservative systems of generalized classical mechanics[J]. 中国物理B, 2006, 15(9): 1935-1940.
[15]	陈向炜, 刘翠梅, 李彦敏. Lie symmetries, perturbation to symmetries and adiabatic invariants of Poincaré equations[J]. 中国物理B, 2006, 15(3): 470-474.