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时间尺度上Lagrange系统的Hojman守恒量

张毅 田雪 翟相华 宋传静

张毅, 田雪, 翟相华, 宋传静. 时间尺度上Lagrange系统的Hojman守恒量. 力学学报, 2021, 53(10): 2814-2822 doi: 10.6052/0459-1879-21-413
引用本文: 张毅, 田雪, 翟相华, 宋传静. 时间尺度上Lagrange系统的Hojman守恒量. 力学学报, 2021, 53(10): 2814-2822 doi: 10.6052/0459-1879-21-413
Zhang Yi, Tian Xue, Zhai Xiang-Hua, Song Chuan-Jing. Hojman conserved quantity for time scales Lagrange systems. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10): 2814-2822 doi: 10.6052/0459-1879-21-413
Citation: Zhang Yi, Tian Xue, Zhai Xiang-Hua, Song Chuan-Jing. Hojman conserved quantity for time scales Lagrange systems. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10): 2814-2822 doi: 10.6052/0459-1879-21-413

时间尺度上Lagrange系统的Hojman守恒量

doi: 10.6052/0459-1879-21-413
基金项目: 国家自然科学基金(11972241, 12002228, 11802193)和江苏省自然科学基金(BK20191454)资助项目
详细信息
    作者简介:

    张毅, 教授, 主要研究方向: 分析力学. E-mail: zhy@mail.usts.edu.cn

  • 中图分类号: O316

HOJMAN CONSERVED QUANTITY FOR TIME SCALES LAGRANGE SYSTEMS

  • 摘要: 利用对称性和守恒律, 可以简化动力学问题甚至求解力学系统的精确解, 更好地理解其动力学行为. 时间尺度分析将连续和离散动力学模型统一并拓展到时间尺度框架, 既避免了重复研究又可揭示两者之区别和联系. 因此, 通过对称性来探寻在时间尺度的框架下新的守恒定律很有必要. 本文首先建立了时间尺度上Lagrange方程, 利用时间尺度微积分性质导出了时间尺度上Lagrange系统的两个重要关系式; 其次, 依据微分方程在单参数Lie变换群下的不变性, 建立了时间尺度上Lie对称性的定义和确定方程; 最后, 建立了时间尺度上Lie对称性定理并利用上述关系式给出了证明, 得到了时间尺度上Lagrange系统的新守恒量. 当时间尺度取为实数集时, 该守恒量退化为著名的Hojman守恒量. 文末考察了一个两自由度时间尺度Lagrange系统, 在3种不同时间尺度情形下得到了该系统的Hojman守恒量, 数值计算结果验证了定理的正确性.

     

  • 图  1  时间尺度$ {{\mathbb{T}}} = {2^{{{\mathbb{N}}_0}}} $$ {q_1},{q_2},{I_1},{I_2} $的值

    Figure  1.  Simulations of $ {q_1},{q_2},{I_1},{I_2} $ on the time scale $ {{\mathbb{T}}} = {2^{{{\mathbb{N}}_0}}} $

    图  2  时间尺度${{\mathbb{T}}} = {{\mathbb{Z}}_ + }$$ {q_1},{q_2},{I_1},{I_2} $的值

    Figure  2.  Simulations of $ {q_1},{q_2},{I_1},{I_2} $ on the time scale ${{\mathbb{T}}} = {{\mathbb{Z}}_ + }$

    表  1  时间尺度$ {{\mathbb{T}}} = {2^{{{\mathbb{N}}_0}}} $$ {q_1},{q_2},{I_1},{I_2} $的值

    Table  1.   The values of $ {q_1},{q_2},{I_1},{I_2} $ on the time scale $ {{\mathbb{T}}} = {2^{{{\mathbb{N}}_0}}} $

    $ {\mathbb{T}}/{\rm{s}} $$ {q_1}/{\rm{m}} $$ {q_2}/{\rm{m}} $$ {I_1} $$ {I_2} $
    110−9.336
    221−9.336
    40−5−9.336
    8−20−81−9.336
    16−124−745−9.336
    32−588−6169−9.336
    64−2540−49785−9.336
    下载: 导出CSV

    表  2  时间尺度${{\mathbb{T}}} = {{\mathbb{Z}}_ + }$$ {q_1},{q_2},{I_1},{I_2} $的值.

    Table  2.   The values of $ {q_1},{q_2},{I_1},{I_2} $ on the time scale ${{\mathbb{T}}} = {{\mathbb{Z}}_ + }$

    $ {\mathbb{T}}/{\rm{s}} $$ {q_1} /{\rm{m}}$$ {q_2}/{\rm{m}} $$ {I_1} $$ {I_2} $
    110−44
    221−44
    320−44
    41−4−44
    5−1−12−44
    6−4−25−44
    7−8−44−44
    8−13−70−44
    9−19−104−44
    10−26−147−44
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-08-21
  • 录用日期:  2021-09-29
  • 网络出版日期:  2021-09-29
  • 刊出日期:  2021-10-26

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