非保守Lagrange系统的Herglotz型广义变分原理及其Noether理论  

Generalized variational principle of Herglotz type for non-conservative Lagrangian systems and its Noether’s theory

作　　者：田雪 张毅[2] Tian Xue;Zhang Yi(School of Sciences,Nanjing University of Science and Technology,Nanjing 210094,China;School of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,China)

机构地区：[1]南京理工大学理学院,江苏南京210094 [2]苏州科技大学土木工程学院,江苏苏州215011

出　　处：《南京理工大学学报》2019年第6期765-770,799,共7页Journal of Nanjing University of Science and Technology

基　　金：国家自然科学基金(11572212);江苏省普通高校研究生科研创新计划(KYZZ16_0479)

摘　　要：为了研究非保守动力学系统,该文利用Herglotz型广义变分原理研究非保守Lagrange系统的Noether定理及其逆定理。根据非保守Lagrange系统的Herglotz型广义变分原理及其动力学方程,给出Herglotz型Noether对称性的定义与判据,并导出Herglotz型Killing方程。建立了Herglotz型Noether定理及其逆定理,揭示了系统的Herglotz型Noether对称性与守恒量之间的内在联系。以Emden方程和二自由度系统为例,表明利用Herglotz型Noether对称性可以系统地研究保守和非保守问题的Noether理论。In order to study non-conservative dynamic systems,Noether’s theorem and inverse theorem for non-conservative Lagrangian systems are studied by using the generalized variational principle of Herglotz type.According to the generalized variational principle of Herglotz type for the non-conservative Lagrangian system and its dynamic equation,the definition and criteria of Noether symmetry of Herglotz type are given,and the Killing equations of Herglotz type are also derived.Noether’s theorem of Herglotz type and its inverse theorem are obtained,which reveal the inner relation between the Noether symmetry and conserved quantity of the system.Taking the Emden equation and a two-freedom system as examples,the results show the Noether symmetry of Herglotz type can be used to study the Noether theory of conservative and non-conservative problems systematically.

关 键 词：Herglotz型广义变分原理 非保守Lagrange系统 Noether定理 Noether逆定理 

分 类 号：O316[理学—一般力学与力学基础]