分数阶约束力学系统的Noether对称性与守恒量研究

发布时间：2018-06-25 01:14

本文选题：Noether准对称性 + 分数阶Lagrange系统　；参考：《苏州科技大学》2017年硕士论文

【摘要】：动力学系统的对称性与守恒量在现代数学、力学、物理学等学科中占有重要的地位对其进行研究具有重要意义。本文利用时间重新参数法,分别在时间不变的特殊无限小变换群和时间变化的一般无限小变换群下研究了约束力学系统及分数阶约束力学系统的Noether准对称性定理。全文共分为五章。第一章绪论,简要论述了有关Noether对称性和分数阶微积分的发展概况及与课题相关的研究背景和意义,介绍了本文的主要研究内容和所做的工作。第二章预备知识,主要介绍了分数阶微积分和分数阶守恒量的定义、公式及性质。第三章分数阶Lagrange系统的Noether准对称性与守恒量。利用时间重新参数化方法证明了Lagrange系统的Noether准对称性与守恒量定理;在此基础上研究了分数阶Lagrange系统的Noether准对称性,给出分数阶Lagrange系统准对称性的定义,并由准对称性得到相应的守恒量。第四章分数阶Hamilton系统的Noether准对称性与守恒量。利用时间重新参数化方法证明了Hamilton系统的Noether准对称性与守恒量定理;在此基础上研究了分数阶Hamilton系统的Noether准对称性,给出分数阶Hamilton系统准对称性的定义,并由准对称性得到相应的守恒量。第五章分数阶广义Birkhoff系统的Noether准对称性与守恒量。利用时间重新参数化方法证明了广义Birkhoff系统的Noether准对称性与守恒量定理;在此基础上研究了分数阶广义Birkhoff系统的Noether准对称性,给出分数阶广义Birkhoff系统准对称性的定义,并由准对称性得到相应的守恒量。最后给出结论与展望。
[Abstract]:The symmetries and conserved quantities of dynamical systems play an important role in modern mathematics, mechanics, physics and so on. In this paper, the Noether's quasi-symmetry theorems for binding systems and fractional order systems are studied under the time-invariant special infinitesimal transformation group and the time-invariant general infinitesimal transformation group by means of time reparameterization method. The full text is divided into five chapters. In the first chapter, the development of Noether symmetry and fractional calculus is briefly discussed, and the research background and significance related to the subject are also discussed. The main research contents and work done in this paper are introduced. In the second chapter, we introduce the definitions, formulas and properties of fractional calculus and fractional conservation. Chapter 3 Noether quasi symmetry and conserved quantity of fractional Lagrange system. The Noether quasi-symmetry and conservation theorems of Lagrange system are proved by using the method of time re-parameterization, and the Noether quasi-symmetry of fractional Lagrange system is studied, and the definition of quasi-symmetry of fractional Lagrange system is given. The corresponding conserved quantities are obtained from quasi symmetry. Chapter 4 Noether quasi symmetry and conserved quantities of fractional order Hamiltonian systems. The Noether quasi-symmetry and conservation theorems of Hamilton system are proved by time reparameterization method, and the Noether quasi-symmetry of fractional Hamiltonian system is studied, and the definition of quasi-symmetry of fractional Hamiltonian system is given. The corresponding conserved quantities are obtained from quasi symmetry. Chapter 5 Noether quasi symmetry and conserved quantity of fractional order generalized Birkhoff system. The Noether quasi-symmetry and conservation theorems of generalized Birkhoff system are proved by time reparameterization method, and the Noether quasi-symmetry of fractional generalized Birkhoff system is studied, and the definition of fractional generalized Birkhoff system is given. The corresponding conserved quantities are obtained from quasi symmetry. Finally, the conclusion and prospect are given.
【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

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