时间尺度上非迁移Birkhoff系统的Mei对称性定理

张毅

doi:10.7498/aps.70.20210372

摘要

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.

关键词:

Abstract

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:

作者及机构信息

张毅

苏州科技大学土木工程学院, 苏州　215011

通信作者: 张毅, zhy@mail.usts.edu.cn

基金项目: 国家自然科学基金(批准号: 11972241, 11572212)和江苏省自然科学基金(批准号: BK20191454)资助的课题.

Authors and contacts

Zhang Yi

College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China

Corresponding author: Zhang Yi, zhy@mail.usts.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11972241, 11572212) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20191454).

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参考文献

[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
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[8]	Wang P, Xue Y, Liu Y L 2012 Chin. Phys. B 21 070203 Google Scholar
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[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
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[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
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[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

施引文献

[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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