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研究并证明时间尺度上非迁移Birkhoff系统的Mei对称性定理. 首先, 建立任意时间尺度上Pfaff-Birkhoff原理和广义Pfaff-Birkhoff原理, 由此导出时间尺度上非迁移Birkhoff系统(包括自由Birkhoff系统、广义Birkhoff系统和约束Birkhoff系统)的动力学方程. 其次, 基于非迁移Birkhoff方程中的动力学函数经历变换后仍满足原方程的不变性, 给出了时间尺度上Mei对称性的定义, 导出了相应的判据方程. 再次, 建立并证明了时间尺度上非迁移Birkhoff系统的Mei对称性定理, 得到了时间尺度上Birkhoff系统的Mei守恒量. 并通过3个算例说明了结果的应用.
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关键词:
- Birkhoff系统 /
- Mei对称性定理 /
- 时间尺度 /
- 非迁移变分学
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.-
Keywords:
- Birkhoffian system /
- Mei’s symmetry theorem /
- time scale /
- non-migrated variational calculus
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[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
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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)
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Google Scholar
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Google Scholar
[10] Zhang H B, Chen H B 2017 J. Math. Anal. Appl. 456 1442
Google Scholar
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Google Scholar
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Google Scholar
Xu X X, Zhang Y 2020 Acta Phys. Sin. 69 220401
Google Scholar
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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
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Google Scholar
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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
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Google Scholar
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Google Scholar
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[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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