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The Herglotz variational problem is also known as Herglotz generalized variational principle whose action functional is defined by differential equation. Unlike the classical variational principle, the Herglotz variational principle gives a variational description of a holonomic non-conservative system. The Herglotz variational principle can describe not only all physical processes that can be described by the classical variational principlen, but also the problems that the classical variational principle is not applicable for. If the Lagrangian or Hamiltonian does not depend on the action functional, the Herglotz variational principle reduces to the classical integral variational principle. In this work, in order to describe the dynamical behavior of complex non-conservative system more accurately, we extend the Herglotz variational principle to the fractional order model, and study the adiabatic invariant for fractional order non-conservative Lagrangian system. Firstly, based on the Herglotz variational problem, the differential variational principle of Herglotz type and the differential equations of motion of the fractional non-conservative Lagrangian system are derived. Secondly, according to the relationship between the isochronal variation and the nonisochronal variation, the transformation of invariance condition of Herglotz differential variational principle is established and the exact invariants of the system are derived. Thirdly, the effects of small perturbations on fractional non-conservative Lagrangian systems are studied, the conditions for the existence of adiabatic invariants for the Lagrangian systems of Herglotz type based on Caputo derivatives are established, and the adiabatic invariants of Herglotz type are obtained. In addition, the exact invariant and adiabatic invariant of fractional non-conservative Hamiltonian system can be obtained by Legendre transformation. When
$ \alpha \to 1$ , the Herglotz differential variational principle for fractional non-conservative Lagrangian system degrades into classical Herglotz differential variational principle, and the corresponding exact invariants and adiabatic invariants also degenerate into the classical exact invariants and adiabatic invariants of Herglotz type. At the end of the paper, the fractional order damped oscillator of Herglotz type is discussed as an example to demonstrate the results.-
Keywords:
- non-conservative Lagrangian system /
- Herglotz generalized variational principle /
- invariants /
- fractional calculus
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[1] Herglotz G 1979 Gesammelte Schriften (Göttingen: Vandenhoeck & Ruprecht) p1
[2] Georgieva B, Guenter R 2002 Topol. Methods Nonlinear Anal. 20 261
Google Scholar
[3] Georgieva B, Guenter R, Bodurov T 2003 J. Math. Phys. 44 3911
Google Scholar
[4] Santos S P S, Martins N, Torres D F M 2014 Vietnam J. Math. 42 409
Google Scholar
[5] Santos S P S, Martins N and Torres D F M 2015 Discrete Contin. Dyn. Syst. 35 4593
Google Scholar
[6] Zhang Y, Tian X 2019 Phys. Lett. A 383 691
[7] Zhang Y 2017 Acta Mech. 228 1481
Google Scholar
[8] Zhang Y, Tian X 2018 Chin. Phys. B 27 090502
Google Scholar
[9] 张毅 2019 南京理工大学学报 43 759
Google Scholar
Zhang Y 2019 J. Nanjing Univ. Sci. Technol. 43 759
Google Scholar
[10] 张毅 2016 力学学报 48 1382
Google Scholar
Zhang Y 2016 Chin. J.Theor. Appl. Mech. 48 1382
Google Scholar
[11] Tian X, Zhang Y 2018 Commun. Theor. Phys. 70 280
Google Scholar
[12] Tian X, Zhang Y 2019 Chaos, Solitons Fractals 119 50
Google Scholar
[13] 赵跃宇, 梅凤翔 1996 力学学报 28 45
Google Scholar
Zhao Y Y, Mei F X 1996 Chin. J.Theor. Appl. Mech. 28 45
Google Scholar
[14] Chen X W, Zhang R C, Mei F X 2000 Acta Mech. Sin. 16 282
Google Scholar
[15] Chen X W, Li Y M, Zhao Y H 2005 Phys. Lett. A 337 274
Google Scholar
[16] Chen X W, Zhao Y H, Li Y M 2005 Commun. Theor. Phys. 44 773
Google Scholar
[17] 张毅, 范存新, 梅凤翔 2006 55 3237
Google Scholar
Zhang Y, Fan C X, Mei F X 2006 Acta Phys. Sin. 55 3237
Google Scholar
[18] 张毅 2006 55 3833
Google Scholar
Zhang Y 2006 Acta Phys. Sin. 55 3833
Google Scholar
[19] Jiang W, Li L, Li Z, Luo S K 2012 Nonlinear Dyn. 67 1075
Google Scholar
[20] Luo S K, Chen X W, Guo Y X 2007 Chin. Phys. B 16 3176
Google Scholar
[21] Ding N, Fang J H 2010 Commun. Theor. Phys. 54 785
Google Scholar
[22] Yang M J, Luo S K 2018 Int. J. Non Linear Mech. 101 16
Google Scholar
[23] Luo S K, Yang M J, Zhang X T, Dai Y 2018 Acta Mech. 229 1833
Google Scholar
[24] Song C J, Zhang Y 2017 Int. J. Non Linear Mech. 90 32
Google Scholar
[25] Kilbas A A, Srivastava H M, Trujillo J J 2006 Theory and Applications of Fractional Differential Equations (Amsterdam: Elsevier B V) p1
[26] 陈文, 孙洪广, 李西成 2010 力学与工程问题的分数阶导数建模 (北京: 北京科学出版社) 第113−188页
Chen W, Sun H G, Li X C 2010 Fractional Derivative Modeling of Mechanical and Engineering Problems (Beijing: Beijing Science Press) pp113−188 (in Chinese)
[27] Vujanovic B D, Jones S E 1989 Variational Methods in Nonconservative Phenomena (Boston: Academic Press) p80
[28] Scarlet W, Cantrijn F 1981 SIAM Rev. 23 467
Google Scholar
[29] Tian X, Zhang Y 2020 Theor. Math. Phys. 202 126
Google Scholar
[30] Balachandran K, Govindaraj V, Rivero M, Trujillo J J 2015 Appl. Math. Comput. 257 66
[31] Garra R, Taverna G S, Torres D F M 2017 Chaos, Solitons Fractals 102 94
Google Scholar
[32] Stanislavsky A A 2004 Phys. Rev. E 70 051103
Google Scholar
[33] Stanislavsky A A 2005 Physica A 354 101
Google Scholar
[34] Narahari B N, Hanneken J W, Enck T, Clarke T 2001 Physica A 297 361
Google Scholar
[35] Narahari B N, Hanneken J W, Clarke T 2002 Physica A 309 275
Google Scholar
[36] Kang Y G, Zhang X E 2010 Physica B 405 369
Google Scholar
[37] Tofighi A 2003 Physica A 329 29
Google Scholar
[38] Ryabov Y E, Puzenko A 2002 Phys. Rev. B 66 184201
Google Scholar
[39] Georgieva B 2010 Ann. Sofia Univ. Fac. Math. Inf. 100 113
[40] Zhang Y 2020 Trans. Nanjing Univ. Aero. Astro. 37 13
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