相空间中非保守系统的Herglotz广义变分原理及其Noether定理
发布时间：2018-01-03 11:38
本文关键词：相空间中非保守系统的Herglotz广义变分原理及其Noether定理 出处：《力学学报》2016年06期 论文类型：期刊论文
更多相关文章： Herglotz广义变分原理 Noether定理 非保守动力学 相空间
【摘要】：与经典变分原理相比,基于由微分方程定义的作用量的Herglotz广义变分原理给出了非保守动力学系统的一个变分描述,它不仅能够描述所有采用经典变分原理能够描述的动力学过程,而且能够应用于经典变分原理不能适用的非保守或耗散系统.将Herglotz广义变分原理拓展到相空间,研究相空间中非保守力学系统的Herglotz广义变分原理与Noether定理及其逆定理.首先,提出相空间中Herglotz广义变分原理,给出相空间中非保守系统的变分描述,导出相应的Hamilton正则方程;其次,基于非等时变分与等时变分之间的关系,导出相空间中Hamilton-Herglotz作用量变分的两个基本公式;再次,给出Noether对称变换的定义和判据,提出并证明相空间中非保守系统基于Herglotz变分问题的Noether定理及其逆定理,揭示了相空间中力学系统的Noether对称性与守恒量之间的内在联系.在经典条件下,Herglotz广义变分原理退化为经典变分原理,与之相应的相空间中的Noether定理退化为经典Hamilton系统的Noether定理.文末以著名的Emden方程和平方阻尼振子为例说明上述方法和结果的有效性.
[Abstract]:Compared with the classical variational principle, based on the Herglotz generalized variational principle defined by the differential equation, a variational description of a nonconservative dynamical system is given. It can not only describe all the dynamic processes that can be described by the classical variational principle. Moreover, it can be applied to non-conservative or dissipative systems where the classical variational principle is not applicable. The Herglotz generalized variational principle is extended to the phase space. The Herglotz generalized variational principle and Noether theorem and their inverse theorems for non-conservative mechanical systems in phase space are studied. Firstly, the Herglotz generalized variational principle in phase space is proposed. The variational description of non-conservative systems in phase space is given, and the corresponding Hamilton canonical equations are derived. Secondly, based on the relationship between non-equal-time-varying and equal-time-varying, two basic formulas of Hamilton-Herglotz action variation in phase space are derived. Thirdly, the definition and criterion of Noether symmetry transformation are given, and the Noether theorem and its inverse theorem for non-conservative systems in phase space based on Herglotz variational problem are presented and proved. The relationship between Noether symmetry and conserved quantities of mechanical systems in phase space is revealed. Under classical conditions, Herglotz generalized variational principle degenerates into classical variational principle. The corresponding Noether theorem in phase space degenerates to the Noether theorem of classical Hamilton system. At the end of this paper, the famous Emden equation and squared damped oscillator are taken as examples. The validity of the method and result is described.
【作者单位】： 苏州科技大学土木工程学院;
【基金】：国家自然科学基金资助项目(11272227,11572212)
【分类号】：O316
【正文快照】： 引言 Noether定理首次从变分学背景下揭示了对称性与守恒量之间的相互关系[1],即Hamilton作用量在关于广义坐标和时间的变换群的无限小变换下的不变性意味着沿着系统的动力学真实运动轨道存在一个守恒量.Noether定理阐释了牛顿力学的所有守恒量,如:时间的均R缘贾轮实阆档，
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