时间尺度上事件空间中约束力学系统的Noether对称性与守恒量研究

发布时间：2018-05-25 01:34

本文选题：时间尺度 + 事件空间　；参考：《苏州科技大学》2017年硕士论文

【摘要】：力学系统的对称性与守恒量不仅具有着重要的数学意义,而且表现着深刻的物理规律。本文在时间尺度上事件空间中研究了约束力学系统的Noether对称性与守恒量。首先,本文分别介绍了位形空间中的Noether理论、事件空间中的Noether理论和时间尺度上的Noether理论的研究历史与进展,并概述本文研究的主要内容。然后,简要概述课本涉及到的时间尺度上的微积分知识。如向前跳跃算子、向后跳跃算子、步差函数和(35)-导数等。研究了时间尺度上事件空间中Lagrange系统的Noether对称性与守恒量。建立时间尺度上事件空间中的Lagrange系统的参数方程,给出时间尺度上事件空间中的Euler-Lagrange方程以及Euler-Lagrange变分方程。通过对Hamiltom作用量在无限小变换下的不变性,求得时间尺度上事件空间中的Noether对称关系式,再求得由对称性导致的守恒量。研究时间尺度上事件空间中Hamilton系统的Noether对称性与守恒量。给出时间尺度上事件空间中Lagrange函数,引入时间尺度上事件空间中广义动量和Hamilton函数,提出并建立时间尺度上事件空间中Hamilton系统的变分问题,求得时间尺度上事件空间中Hamilton正则方程。基于Hamiltom作用量在无限小变换下的不变性,给出了时间尺度上事件空间中Hamilton系统的Noether对称性的定义,利用时间重新参数化方法,求得时间尺度上事件空间中Hamilton系统的Noether对称性与守恒量。研究时间尺度上事件空间中Birkhoff系统的Noether对称性与守恒量。提出并建立时间尺度上事件空间中Birkhoff系统的变分问题;求得时间尺度上事件空间中Birkhoff系统的参数方程;基于Pfaff作用量在无限小变换下的不变性,给出了时间尺度上事件空间中Birkhoff系统的Noether对称性的定义,利用时间重新参数化方法,求得时间尺度上事件空间中Birkhoff系统的Noether对称性与守恒量。最后,我们对全文进行总结并展望未来。本文的创新点:(1)建立了时间尺度上事件空间中约束力学系统的参数方程;(2)得到了时间尺度上事件空间中约束力学系统的Noether对称性与守恒量;(3)推广了Noether定理,证明了位形空间的Noether定理、事件空间中的Noether定理、时间尺度上的Noether定理都是时间尺度上事件空间中的Noether定理的特例。
[Abstract]:The symmetries and conserved quantities of mechanical systems not only have important mathematical significance, but also exhibit profound physical laws. In this paper, Noether symmetries and conserved quantities of binding systems are studied in event space on a time scale. Firstly, this paper introduces the history and progress of Noether theory in configuration space, Noether theory in event space and Noether theory on time scale, and summarizes the main contents of this paper. Then, a brief overview of the textbooks involved in the time scale of calculus knowledge. For example, forward jump operator, backward jump operator, step difference function, and so on. The Noether symmetries and conserved quantities of Lagrange systems in event space on time scale are studied. The parameter equations of Lagrange system in event space on time scale are established. The Euler-Lagrange equation and Euler-Lagrange variational equation in event space on time scale are given. Through the invariance of Hamiltom action under infinitesimal transformation, the Noether symmetry relation in event space on time scale is obtained, and the conserved quantity caused by symmetry is obtained. The Noether symmetries and conserved quantities of Hamilton systems in event space on time scale are studied. The Lagrange function in event space on time scale is given. The generalized momentum and Hamilton functions in event space on time scale are introduced, and the variational problem of Hamilton system in event space on time scale is presented and established. The Hamilton canonical equation in event space on time scale is obtained. Based on the invariance of Hamiltom action under infinitesimal transformation, the definition of Noether symmetry of Hamilton system in event space on time scale is given, and the method of time reparameterization is used. The Noether symmetries and conserved quantities of Hamilton system in event space on time scale are obtained. The Noether symmetries and conserved quantities of Birkhoff systems in event space on time scale are studied. The variational problem of Birkhoff system in event space on time scale is proposed and established, the parameter equation of Birkhoff system in event space on time scale is obtained, based on the invariance of Pfaff action under infinitesimal transformation, The definition of Noether symmetry of Birkhoff system in event space on time scale is given, and the Noether symmetry and conserved quantity of Birkhoff system in event space on time scale are obtained by time reparameterization method. Finally, we summarize the full text and look forward to the future. In this paper, the author establishes the parameter equation of the binding system in event space on the time scale, and obtains the Noether symmetries and conserved quantities of the binding system in the event space on the time scale, which generalizes the Noether theorem. It is proved that the Noether theorem in the configuration space, the Noether theorem in the event space and the Noether theorem on the time scale are special cases of the Noether theorem in the event space on the time scale.
【学位授予单位】：苏州科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O316

【参考文献】