本文選題：時(shí)間尺度 + 事件空間��； 參考：《蘇州科技大學(xué)》2017年碩士論文

【摘要】：力學(xué)系統(tǒng)的對(duì)稱性與守恒量不僅具有著重要的數(shù)學(xué)意義,而且表現(xiàn)著深刻的物理規(guī)律。本文在時(shí)間尺度上事件空間中研究了約束力學(xué)系統(tǒng)的Noether對(duì)稱性與守恒量。首先,本文分別介紹了位形空間中的Noether理論、事件空間中的Noether理論和時(shí)間尺度上的Noether理論的研究歷史與進(jìn)展,并概述本文研究的主要內(nèi)容。然后,簡(jiǎn)要概述課本涉及到的時(shí)間尺度上的微積分知識(shí)。如向前跳躍算子、向后跳躍算子、步差函數(shù)和(35)-導(dǎo)數(shù)等。研究了時(shí)間尺度上事件空間中Lagrange系統(tǒng)的Noether對(duì)稱性與守恒量。建立時(shí)間尺度上事件空間中的Lagrange系統(tǒng)的參數(shù)方程,給出時(shí)間尺度上事件空間中的Euler-Lagrange方程以及Euler-Lagrange變分方程。通過對(duì)Hamiltom作用量在無限小變換下的不變性,求得時(shí)間尺度上事件空間中的Noether對(duì)稱關(guān)系式,再求得由對(duì)稱性導(dǎo)致的守恒量。研究時(shí)間尺度上事件空間中Hamilton系統(tǒng)的Noether對(duì)稱性與守恒量。給出時(shí)間尺度上事件空間中Lagrange函數(shù),引入時(shí)間尺度上事件空間中廣義動(dòng)量和Hamilton函數(shù),提出并建立時(shí)間尺度上事件空間中Hamilton系統(tǒng)的變分問題,求得時(shí)間尺度上事件空間中Hamilton正則方程。基于Hamiltom作用量在無限小變換下的不變性,給出了時(shí)間尺度上事件空間中Hamilton系統(tǒng)的Noether對(duì)稱性的定義,利用時(shí)間重新參數(shù)化方法,求得時(shí)間尺度上事件空間中Hamilton系統(tǒng)的Noether對(duì)稱性與守恒量。研究時(shí)間尺度上事件空間中Birkhoff系統(tǒng)的Noether對(duì)稱性與守恒量。提出并建立時(shí)間尺度上事件空間中Birkhoff系統(tǒng)的變分問題;求得時(shí)間尺度上事件空間中Birkhoff系統(tǒng)的參數(shù)方程;基于Pfaff作用量在無限小變換下的不變性,給出了時(shí)間尺度上事件空間中Birkhoff系統(tǒng)的Noether對(duì)稱性的定義,利用時(shí)間重新參數(shù)化方法,求得時(shí)間尺度上事件空間中Birkhoff系統(tǒng)的Noether對(duì)稱性與守恒量。最后,我們對(duì)全文進(jìn)行總結(jié)并展望未來。本文的創(chuàng)新點(diǎn):(1)建立了時(shí)間尺度上事件空間中約束力學(xué)系統(tǒng)的參數(shù)方程;(2)得到了時(shí)間尺度上事件空間中約束力學(xué)系統(tǒng)的Noether對(duì)稱性與守恒量;(3)推廣了Noether定理,證明了位形空間的Noether定理、事件空間中的Noether定理、時(shí)間尺度上的Noether定理都是時(shí)間尺度上事件空間中的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.
【學(xué)位授予單位】：蘇州科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O316

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