【作者】 张宏彬；

【导师】 陈立群；

【作者基本信息】 上海大学 ， 一般力学与力学基础， 2005， 博士

【摘要】 本文对动力学系统Lie对称性和守恒量的有关问题进行了研究,其中包括动力学系统的Lie对称性与Hojman守恒量、动力学系统Lie对称性与守恒量的逆问题和离散动力学系统的变分原理和离散Noether定理等。 第一章,绪论:简要介绍了近年来动力学系统Lie对称性和守恒量有关研究的进展,包括非Noether守恒量理论的研究、Lie对称性与守恒量逆问题的研究和离散力学系统的对称性和守恒量的研究等。 第二章,Hojman定理和Lutzky定理的统一形式:首先,引入一般意义下的Lie变换群(即位型变量q_s和时间变量t同时变换),给出系统的Lie对称性确定方程,提出一个新的守恒律,Hojman定理与Lutzky定理则分别是这个新守恒律在两个特殊情况下的推论,导出一个可排除平凡Hojman守恒量的定理,并分别讨论了Birkhoff系统和非完整系统的Lie对称性和Hojman守恒量,最后,讨论了Hamilton系统的梅对称性与Lie对称性的关系,给出了由梅对称性求Hojman守恒量的方法。 第三章,动力学系统Lie对称性与守恒量逆问题:将Katzin和Levine在研究二阶微分方程含速度无限小对称变换的特征函数结构时使用的方法进行了推广,并分别研究一阶非完整约束系统和Birkhoff系统的无限小对称变换的特征函数结构,讨论了非完整系统非等时变分方程的特解与其第一积分的联系,给出非完整系统Lie对称逆问题的一个解法。 第四章,位型空间中离散力学系统的对称性与第一积分:首先,将位型空间离散变分原理进行了推广,并分别应用于非保守系统和一阶线性非完整系统,得到了它们的离散运动方程和离散的Noether定理;接着,将位型空间中的离散变分原理推广至相空间,给出了Hamilton形式的离散变分原理、得到了Hamilton形式的离散运动方程、讨论了Hamilton形式的离散对称性和第一积分。 第五章,事件空间中离散力学的对称性与离散第一积分:首先,将位型空间中的离散变分原理推广到事件空间中,并分别应用于完整保守系统和Birkhoff系统,得到了它们的离散运动方程,并讨论了它们的离散对称性和第一积分,不仅给出了系统的离散“动量”积分,而且还得到了系统的离散“能量”积分。 第六章,总结与展望,说明本文所得到的主要结果以及未来研究的一些想法。更多还原

【Abstract】 The present dissertation treats Lie symmetries and conserved quantities for dynamical systems, including the Lie symmetries and Hojman’s conserved quantities of dynamical systems, the inverse problem of Lie symmetries and conserved quantities, and the variational principle of discrete dynamical systems and the discrete Noethers theorem. The dissertation consists of six chapters.The first chapter surveyed briefly the resent progresses in the theory of non-Noethers conserved quantities, the inverse problem of Lie symmetries and conserved quantities, and the symmetries and conserved quantities of discrete mechanical systems.Chapter two proposes the unified form of Hojman’s conservation law and Lutzky’s conservation law. Firstly, the author introduces the general Lie group of transformations that the variations of both the time and the generalized coordinates are considered, derives the determining equation of Lie symmetry for the system, presents a new conservation law, which contains the Hojman’s and the Lutzky’s conservation law as two special cases, and obtains a condition to exclude trivial Hojman’s conserved quantities. Next, the author investigates the Lie symmetries and Hojman’s conserved quantities for the Birkhoff systems and the nonholonomic systems respectively. Finally, the author discusses the relation between Mei’s symmetry and Lie symmetry for Hamilton systems, and develop a method to find Hojman’s conserved quantities by using Mei’s symmetries.Chapter three analyzes the inverse problem of Lie symmetries and conserved quantities for dynamical systems. Firstly, the author generalizes the method used to find the characteristic functional structure of velocity-dependent infinitesimal symmetry transformations for second order differential equations by Katzin and Levine, and studies the characteristic functional structure of infinitesimal symmetry transformations for the first order nonholonomic constrained systems and the Birkhoff systems respectively. Next, the author studies the connection of first integrals with particular solutions of the nonsimultaneous variational equations for nonholonomic systems, and presents a new approach to find the inverse problem of Lie symmetry for nonholonomic systems.Chapter four deals with the symmetries and first integrals of discrete mechanics in更多还原