基于Herglotz型微分变分原理研究相空间中非保守系统的守恒律  

On the Conservation Law for Nonconservative System in Phase Space Based on Herglotz Differential Variational Principle

作　　者：张毅[1] ZHANG Yi(College of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,Jiangsu,China)

机构地区：[1]苏州科技大学土木工程学院,江苏苏州215011

出　　处：《力学季刊》2018年第4期681-688,共8页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(11572212;11272227)

摘　　要：为了进一步探究非保守系统的Herglotz变分问题与其守恒律之间的关系,该文提出并研究基于Herglotz型微分变分原理构建相空间中非保守力学系统的守恒律.首先,基于相空间中非保守系统的Herglotz变分问题,建立该系统的Herglotz型微分变分原理;其次,利用广义变分与经典等时变分之间的关系,给出微分变分原理不变性条件的变换,并建立非保守系统的守恒定理,得到了该系统基于Herglotz变分问题的守恒量及其存在条件;再次,导出守恒定理的逆定理,由相空间中非保守系统的已知守恒量可找到无限小变换的空间和时间的生成元.文末举例说明结果的应用.In order to further study the relationship between Herglotz variation of the non-conservative system and its conservation laws, we propose and study how to construct the conservation laws of the non-conservative system in phase space based on the differential variational principle of Herglotz type. Firstly, based on the variational problem of Herglotz type for the non-conservative system in phase space, the Herglotz differential variational principle of the system is established. Secondly, by using the relation between the generalized variation and the classical isochronal variation, the transformation for the invariance of the differential variational principle is given, and the conservation theorem for the non-conservative system is established. In addition, the conserved quantity based on Herglotz variational problem and the conditions for its existence are obtained. Thirdly, the inversed theorem of the conservation theorem is derived. From a given conserved quantity of the non-conservative system in phase space, one can find the spatial and time generators of the infinitesimal transformation. An example is given to illustrate the application of the results.

关 键 词：微分变分原理 Herglotz变分问题 守恒律 非保守系统 相空间 

分 类 号：O316[理学—一般力学与力学基础]