相空间中非保守系统的Herglotz广义变分原理及其Noether定理

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。%Compared with the classical variational principle, the generalized variational principle of Herglotz based upon the action defined by differential equations gives a variational description of nonconservative dynamical system. The principle can describe all dynamical processes and nonconservative or dissipative systems. In the present study, the principle is extended to phase space, and the generalized variational principle of Herglotz type for non-conservative mechanical system in phase space is given and Noether’s theorem and its inverse of the system are studied. Firstly, the generalized variational principle of Herglotz type in phase space is presented, a variational description of non-conservative system in phase space is given, and the corresponding Hamilton canonical equations are deduced. Secondly, based upon the relation between non-isochronal variation and isochronal variation, two basic formulae for the variation of Hamilton-Herglotz action in phase space are obtained. Thirdly, the definition and the criterion of Noether symmetry are given, and Noether’s theorem and its inverse of nonconservative system for the variational problem of Herglotz type in phase space are proposed and proved, and the inner relation between the Noether symmetry and the conserved quantity for mechanical systems in phase space is revealed. The generalized variational principle of Herglotz type reduces to the classical variational principle under classical conditions, and Noether’s theorem for the variational problem of Herglotz type reduces to the classical Noether’s theorem of Hamilton system. In the end of the paper, we take the famous Emden equation and damping oscillator with second power as examples to illustrate the application of the results.