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The Mei symmetry and its corresponding conserved quantities for non-migrated Birkhoffian systems on a time scale are proposed and studied. Firstly, the dynamic equations of non-migrated Birkhoffian systems (including free Birkhoffian systems, generalized Birkhoffian systems and constrained Birkhoffian systems) on a time scale are derived based on the time-scale Pfaff-Birkhoff principle and time-scale generalized Birkhoff principle. Secondly, based on the fact that the dynamical functions in the non-migrated Birkhoff’s equations still satisfy the original equations after they have been transformed, the definitions of Mei symmetry on an arbitrary time scale are given, and the corresponding criterion equations are derived. Thirdly, Mei’s symmetry theorems for non-migrated Birkhoffian systems on a time scales are established and proved, and Mei conserved quantities of Birkhoffian systems on a time scale are obtained. The results are illustrated by three examples.
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Keywords:
- Birkhoffian system /
- Mei’s symmetry theorem /
- time scale /
- non-migrated variational calculus
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Google Scholar
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Google Scholar
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[1] Birkhoff G D 1927 Dynamical Systems (Providence: AMS College Publ. ) pp59–96
[2] Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag) pp1–280
[3] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) pp1–228
[4] Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems (Moscow: UFN) pp118–226
[5] Mei F X 2013 Dynamics of Generalized Birkhoffian Systems (Beijing: Science Press) pp1–206
[6] 梅凤翔, 吴惠彬, 李彦敏, 陈向炜 2016 力学学报 48 263
Google Scholar
Mei F X, Wu H B, Li Y M, Chen X W 2016 J. Theor. Appl. Mech. 48 263
Google Scholar
[7] 梅凤翔 2004 约束力学系统的对称性与守恒量 (北京: 北京理工大学出版社) 第1—482页
Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) pp1–482 (in Chinese)
[8] Wang P, Xue Y, Liu Y L 2012 Chin. Phys. B 21 070203
Google Scholar
[9] Zhang Y, Zhai X H 2015 Nonlinear Dyn. 81 469
Google Scholar
[10] Zhang H B, Chen H B 2017 J. Math. Anal. Appl. 456 1442
Google Scholar
[11] Zhang Y 2018 Int. J. Non-Linear Mech. 101 36
Google Scholar
[12] 徐鑫鑫, 张毅 2020 物理学报 69 220401
Google Scholar
Xu X X, Zhang Y 2020 Acta Phys. Sin. 69 220401
Google Scholar
[13] Zhang L J, Zhang Y 2020 Commun. Nonlinear Sci. Numer. Simul. 91 105435
Google Scholar
[14] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[15] Liu S X, Liu C, Guo Y X 2011 Chin. Phys. B 20 034501
Google Scholar
[16] Chen X W, Li Y M 2013 Acta Mech. 224 1593
Google Scholar
[17] Luo S K, He J M, Xu Y L 2016 Int. J. Non-Linear Mech. 78 105
Google Scholar
[18] 刘世兴, 刘畅, 郭永新 2011 物理学报 60 064501
Google Scholar
Liu S X, Liu C, Guo Y X 2011 Acta Phys. Sin. 60 064501
Google Scholar
[19] Liu S X, Liu C, Hua W, Guo Y X 2016 Chin. Phys. B 25 114501
Google Scholar
[20] Kong X L, Wu H B, Mei F X 2013 Appl. Math. Comput. 225 326
[21] 孔新雷, 吴惠彬 2017 物理学报 66 084501
Google Scholar
Kong X L, Wu H B 2017 Acta Phys. Sin. 66 084501
Google Scholar
[22] He L, Wu H B, Mei F X 2017 Nonlinear Dyn. 87 2325
Google Scholar
[23] Hilger S 1990 Results Math. 18 18
Google Scholar
[24] Bohner M, Peterson A 2001 Dynamic Equations on Time Scales (Boston: Birkhäuser) pp1–353
[25] Bohner M, Georgiev S G 2016 Multivariable Dynamic Calculus on Time Scales (Switzerland: Springer International Publishing AG) pp1–600
[26] Georgiev S G 2018 Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales (Switzerland: Springer International Publishing AG) pp1–357
[27] Atici F M, Biles D C, Lebedinsky A 2006 Math. Comput. Modell. 43 718
Google Scholar
[28] Bohner M 2004 Dyn. Syst. Appl. 13 339
[29] Bartosiewicz Z, Torres D F M 2008 J. Math. Anal. Appl. 342 1220
Google Scholar
[30] Benkhettou N, Brito da Cruz A M C, Torres D F M 2015 Signal Process. 107 230
Google Scholar
[31] Dryl M, Torres D F M 2017 Springer Proceedings in Mathematics & Statistics 195 223
[32] 韩振来, 孙书荣 2014 时间尺度上动态方程振动理论 (济南: 山东大学出版社) 第1—232页
Han Z L, Sun S R 2014 Oscillation Theory of Dynamic Equations on Time Scales (Jinan: Shandong University Press) pp1–232 (in Chinese)
[33] Bourdin L 2014 J. Math. Anal. Appl. 411 543
Google Scholar
[34] Anerot B, Cresson J, Belgacem K H, Pierret F 2020 J. Math. Phys. 61 113502
Google Scholar
[35] Song C J, Zhang Y 2015 J. Math. Phys. 56 102701
Google Scholar
[36] Song C J, Zhang Y 2017 J. Nonlinear Sci. Appl. 10 2268
Google Scholar
[37] Song C J, Cheng Y 2020 Appl. Math. Comput. 374 125086
[38] Zhang Y 2019 Chaos, Solitons Fractals 128 306
Google Scholar
[39] Zhang Y, Zhai X H 2019 Commun. Nonlinear Sci. Numer. Simul. 75 251
Google Scholar
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