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 引用本文: 董欣畅, 张毅. 含时滞非完整系统的对称性与Herglotz型守恒量. 力学学报, 待出版.
Dong Xinchang, Zhang Yi. Symmetry and herglotz type conserved quantities for nonholonomic systems with time delay. Chinese Journal of Theoretical and Applied Mechanics, in press.
 Citation: Dong Xinchang, Zhang Yi. Symmetry and herglotz type conserved quantities for nonholonomic systems with time delay. Chinese Journal of Theoretical and Applied Mechanics, in press.

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

• 摘要: 时滞是自然界和工程实践中常见的一种时间滞后现象, 对力学系统的动力学行为及基本性质都具有深刻的影响. Herglotz广义变分原理推广了经典变分原理, 可用于非保守系统的研究. 故利用Herglotz广义变分原理研究含时滞的非完整系统的对称性与守恒量在理论和应用上均具有重要意义. 文章将Herglotz型Noether定理拓展到含时滞的非完整系统. 首先, 建立含时滞系统的Herglotz型微分变分原理, 利用拉格朗日乘子法, 推导出含时滞的一般非完整系统的Herglotz型Routh方程. 其次, 基于含时滞的Hamilton-Herglotz作用量在无限小变换下的不变性, 给出作用量变分的两个基本公式, 进而定义含时滞非完整系统的Herglotz型Noether对称性, 给出Herglotz型Noether等式. 再次, 建立含时滞一般非完整系统的Herglotz型Noether定理. 讨论特殊情形下的Noether定理: 如果约束都是完整的, 定理退化为含时滞完整系统的Herglotz型Noether定理; 如果系统是保守的, 退化为含时滞非完整保守系统的Herglotz型Noether定理; 如果不考虑时滞, 则退化为非完整系统的Herglotz型Noether定理. 最后, 给出含时滞非完整系统的Herglotz型Noether逆定理. 文末通过对含时滞的算例求解, 说明了理论分析结果, 并通过数值模拟验证方法的可行性及正确性.

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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