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Volume 56 Issue 11
Nov.  2024
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Dong Xinchang, Zhang Yi. Symmetry and Herglotz type conserved quantities for nonholonomic systems with time delay. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(11): 3302-3311. DOI: 10.6052/0459-1879-24-241
Citation: Dong Xinchang, Zhang Yi. Symmetry and Herglotz type conserved quantities for nonholonomic systems with time delay. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(11): 3302-3311. DOI: 10.6052/0459-1879-24-241
shu

SYMMETRY AND HERGLOTZ TYPE CONSERVED QUANTITIES FOR NONHOLONOMIC SYSTEMS WITH TIME DELAY

  • *.

    School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China

  • †.

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

  • Received Date: May 21, 2024
  • Accepted Date: July 21, 2024
  • Published Date: July 22, 2024
    Graphical Abstract
    • Abstract
  • Time delay is a common time delay phenomenon in nature and engineering practice, which has a profound impact on the dynamic behavior and basic properties of mechanical systems. The Herglotz type generalized variational principle extends the classical variational principle and can be used to study nonconservative systems. Therefore, using the Herglotz type generalized variational principle to study the symmetry and conserved quantity of nonholonomic systems with time delay is of great significance both in theory and application. In this paper, the Herglotz type Noether theorem is extended to nonholonomic systems with time delay. Firstly, the Herglotz type differential variational principle of the system with time delay is established. By means of the Lagrange multiplier method, the Routh-type differential equations of motion of the general nonholonomic system with time delay are derived. Secondly, based on the invariance of the Hamilton-Herglotz action with time delay under infinitesimal transformations, two basic formulas for the variation of the action are given, and then the Herglotz type Noether symmetry is defined and the Herglotz type Noether identity is given. Thirdly, the Noether theorem of Herglotz type for general nonholonomic systems with time delay is established. In addition, the Noether theorem in special cases is discussed. If all the constraints are holonomic, the theorem is reduced to the Herglotz type Noether theorem for holonomic systems with time delay. If the system is conservative, it is reduced to the Herglotz type Noether theorem for nonholonomic conservative systems with time delay. If the time delay is not considered, it is reduced to the Herglotz type Noether theorem for nonholonomic systems. Finally, the Noether inverse theorem of Herglotz type for nonholonomic systems with time delay is given. At the end of this paper, the theoretical analysis results are illustrated by solving an example with time delay, and the feasibility and correctness of the proposed method are verified by numerical simulation.
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  • [1]
    徐鉴, 裴利军. 时滞系统动力学近期研究进展与展望. 力学进展, 2006, 36(1): 17-30 (Xu Jian, Pei Lijun. Advances in dynamics for delayed systems. Advances in Mechanics, 2006, 36(1): 17-30 (in Chinese)

    Xu Jian, Pei Lijun. Advances in dynamics for delayed systems. Advances in Mechanics, 2006, 36(1): 17-30 (in Chinese)
    [2]
    胡海岩, 王在华. 论迟滞与时滞. 力学学报, 2010, 42(4): 740-746 (Hu Haiyan, Wang Zaihua. On hysteresis and retardation. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(4): 740-746 (in Chinese)

    Hu Haiyan, Wang Zaihua. On hysteresis and retardation. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(4): 740-746 (in Chinese)
    [3]
    El’sgol’c LE. Qualitative Methods in Mathematical Analysis. Providence: American Mathematical Society, 1964
    [4]
    Hughes DK. Variational and optimal control problems with delayed argument. Journal of Optimization Theory and Applications, 1968, 2(1): 1-14 doi: 10.1007/BF00927159
    [5]
    Frederico GSF, Torres DFM. Noether’s symmetry theorem for variational and optimal control problems with time delay. Numerical Algebra, Control and Optimization, 2012, 2(3): 619-630 doi: 10.3934/naco.2012.2.619
    [6]
    张毅, 金世欣. 含时滞的非保守系统动力学的Noether对称性. 物理学报, 2013, 62(23): 234502 (Zhang Yi, Jin Shixin. Noether symmetries of dynamics for non-conservative systems with time delay. Acta Physica Sinica, 2013, 62(23): 234502 (in Chinese) doi: 10.7498/aps.62.234502

    Zhang Yi, Jin Shixin. Noether symmetries of dynamics for non-conservative systems with time delay. Acta Physica Sinica, 2013, 62(23): 234502 (in Chinese) doi: 10.7498/aps.62.234502
    [7]
    Vladimir D, Roman K, Sergey M. Lagrangian formalism and Noether-type theorems for second-order delay ordinary differential equations. Journal of Physics A: Mathematical and Theoretical, 2023, 56(34): 345203 doi: 10.1088/1751-8121/ace5f6
    [8]
    Jin SX, Zhang Y. Noether symmetry and conserved quantity for a Hamilton system with time delay. Chinese Physics B, 2014, 23(5): 054501 doi: 10.1088/1674-1056/23/5/054501
    [9]
    Zhai XH, Zhang Y. Noether symmetries and conserved quantities for Birkhoffian systems with time delay. Nonlinear Dynamics, 2014, 77(1-2): 73-86 doi: 10.1007/s11071-014-1274-8
    [10]
    Noether AE. Invariante variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch Physikalische Klasse, 1918, KI(II): 235-257
    [11]
    Arnold VI, Kozlov VV, Neishtadt AI. Mathematical Aspects of Classical and Celestial Mechanics. Berlin: Springer, 2006
    [12]
    梅凤翔. 关于Noether定理——分析力学札记之三十. 力学与实践, 2020, 42(1): 66-74 (Mei Fengxiang. On the Noether’s theorem. Mechanics in Engineering, 2020, 42(1): 66-74 (in Chinese)

    Mei Fengxiang. On the Noether’s theorem. Mechanics in Engineering, 2020, 42(1): 66-74 (in Chinese)
    [13]
    傅景礼, 陆晓丹, 项春. 爬壁机器人系统的 Noether 对称性和守恒量. 力学学报, 2022, 54(6): 1680-1693 (Fu Jingli, Lu Xiaodan, Xiang Chun. Noether symmetries and conserved quantities of wall climbing robot system. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1680-1693 (in Chinese)

    Fu Jingli, Lu Xiaodan, Xiang Chun. Noether symmetries and conserved quantities of wall climbing robot system. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1680-1693 (in Chinese)
    [14]
    Zhang Y. Nonshifted dynamics of constrained systems on time scales under Lagrange framework and its Noether’s theorem. Communications in Nonlinear Science and Numerical Simulation, 2022, 108: 106214 doi: 10.1016/j.cnsns.2021.106214
    [15]
    Herglotz G. Gesammelte Schriften. Göttingen: Vandenhoeck & Ruprecht, 1979
    [16]
    Guenther RB, Guenther CM, Gottsch JA. The Herglotz lectures on contact transformations and Hamiltonian systems. Toruń: Juliusz Center for Nonlinear Studies, 1996
    [17]
    Bravetti A. Contact Hamilton dynamics: The concept and its use. Entropy, 2017, 19: 535 doi: 10.3390/e19100535
    [18]
    Zhan QY, Duan JQ, Li XF, et al. Numerical integration of stochastic contact Hamiltonian systems via stochastic Herglotz variational principle. Physica Scripta, 2023, 98(5): 055211 doi: 10.1088/1402-4896/acc984
    [19]
    de León M, Lainz M, Muñoz-Lecanda MC. Optimal control, contact dynamics and Herglotz variational problem. Journal of Nonlinear Science, 2023, 33(1): 9 doi: 10.1007/s00332-022-09861-2
    [20]
    Jordi G, Adrià M. Application of Herglotz’s variational principle to electromagnetic systems with dissipation. International Journal of Geometric Methods in Modern Physics, 2022, 19(10): 2250156 doi: 10.1142/S0219887822501560
    [21]
    Santos SPS, Martins N, Torres DFM. Variational problems of Herglotz type with time delay: Dubois-Reymond condition and Noether’s first theorem. Discrete and Continuous Dynamical Systems, 2015, 35(9): 4593-4610 doi: 10.3934/dcds.2015.35.4593
    [22]
    Santos SPS, Martins N, Torres DFM. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete and Continuous Dynamical Systems, Series S, 2018, 11(1): 91-102 doi: 10.3934/dcdss.2018006
    [23]
    Zhang Y. Herglotz’s variational problem for non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Symmetry, 2020, 12(5): 845 doi: 10.3390/sym12050845
    [24]
    Zhang Y. Noether’s theorem for a time-delayed Birkhoffian system of Herglotz type. International Journal of Non-Linear Mechanics, 2018, 101: 36-43 doi: 10.1016/j.ijnonlinmec.2018.02.010
    [25]
    Zhang Y. Noether symmetry and conserved quantity for a time-delayed Hamiltonian system of Herglotz type. Royal Society Open Science, 2018, 5(10): 180208 doi: 10.1098/rsos.180208
    [26]
    Huang LQ, Zhang Y. Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems with delayed arguments. Chaos, Solitons and Fractals, 2024, 182: 114854 doi: 10.1016/j.chaos.2024.114854
    [27]
    Zhang Y, Tian X. Conservation laws of nonconservative nonholonomic system based on Herglotz variational problem. Physics Letters A, 2019, 383(8): 691-696 doi: 10.1016/j.physleta.2018.11.034
    [28]
    Dong XC, Zhang Y. Herglotz-type principle and first integrals for nonholonomic systems in phase space. Acta Mechanica, 2023, 234(12): 6083-6095 doi: 10.1007/s00707-023-03707-y
    [29]
    梅凤翔, 吴惠彬, 李彦敏. 分析力学史略. 北京: 科学出版社, 2019 (Mei Fengxiang, Wu Huibin, Li Yanmin. A Brief History of Analytical Mechanics. Beijing: Science Press, 2019 (in Chinese)

    Mei Fengxiang, Wu Huibin, Li Yanmin. A Brief History of Analytical Mechanics. Beijing: Science Press, 2019 (in Chinese)
    [30]
    庄红超, 王柠, 董凯伦等. 非完整约束大负重比六足机器人多机动态协同编队避障控制策略. 机械工程学报, 2024, 60(1): 284-295 (Zhuang Hongchao, Wang Ning, Dong Kailun, et al. Obstacle avoidance control strategy of multi-robot dynamic cooperative formation of large-load-ratio six-legged robot under nonholonomic constraints. Journal of Mechanical Engineering, 2024, 60(1): 284-295 (in Chinese) doi: 10.3901/JME.2024.01.284

    Zhuang Hongchao, Wang Ning, Dong Kailun, et al. Obstacle avoidance control strategy of multi-robot dynamic cooperative formation of large-load-ratio six-legged robot under nonholonomic constraints. Journal of Mechanical Engineering, 2024, 60(1): 284-295 (in Chinese) doi: 10.3901/JME.2024.01.284
    [31]
    Xiong J, Wang N, Liu C. Bicycle dynamics and its circular solution on a revolution surface. Acta Mechanica Sinica, 2020, 36(1): 220-233 doi: 10.1007/s10409-019-00914-6
    [32]
    徐宏, 傅景礼. 基于伺服电机驱动的进给传动系统扭转振动的Lie群分析方法. 力学学报, 2023, 55(9): 2000-2009 (Xu Hong, Fu Jingli. Lie group analysis for torsional vibration of serve motor driven feeder driven system. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(9): 2000-2009 (in Chinese)

    Xu Hong, Fu Jingli. Lie group analysis for torsional vibration of serve motor driven feeder driven system. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(9): 2000-2009 (in Chinese)
    [33]
    Wang BF, Li S, Guo J, et al. Car-like mobile robot path planning in rough terrain using multi-objective particle swarm optimization algorithm. Neurocomputing, 2018, 282: 42-51 doi: 10.1016/j.neucom.2017.12.015
    [34]
    王囡囡, 熊佳铭, 刘才山. 自行车动力学建模及稳定性分析研究综述. 力学学报, 2020, 52(4): 917-927 (Wang Nannan, Xiong Jiaming, Liu Caishan. Review of dynamic modeling and stability analysis of a bicycle. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(4): 917-927 (in Chinese)

    Wang Nannan, Xiong Jiaming, Liu Caishan. Review of dynamic modeling and stability analysis of a bicycle. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(4): 917-927 (in Chinese)
    [35]
    Cara FE. Ordinary Differential Equations and Applications. Singapore: World Scientific Publishing Company, 2023
    [36]
    梅凤翔. 李群李代数对约束力学系统的应用. 北京: 科学出版社, 1999 (Mei Fengxiang. Applications of Lie Group Lie Algebras to Systems of Constrained Mechanics. Beijing: Science Press, 1999 (in Chinese)

    Mei Fengxiang. Applications of Lie Group Lie Algebras to Systems of Constrained Mechanics. Beijing: Science Press, 1999 (in Chinese)
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