基于非标准Lagrange函数的变分问题的对称性摄动理论

发布时间：2018-01-09 12:11

本文关键词：基于非标准Lagrange函数的变分问题的对称性摄动理论　出处：《苏州科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：自然界中最普遍问题大多是关于非线性非保守动力学系统的问题,非标准Lagrange函数具有一些标准Lagrange函数不具有的一些性质,它能描述非线性非保守问题,因此对非标准Lagrange函数的研究有很重要意义和价值。本文主要是对指数Lagrange函数和幂律Lagrange函数下动力学系统的Noether对称性、Lie对称性和Mei对称性这三种对称性摄动与绝热不变量问题的研究。本文第一部分,首先,列出指数Lagrange函数和幂律Lagrange函数以及El-Nabulsi模型下指数Lagrange函数和幂律Lagrange函数下的Noether型精确不变量。其次,再从高阶绝热不变量的定义出发,继续探究小扰动作用下指数Lagrange函数和幂律Lagrange函数以及El-Nabulsi模型下指数Lagrange函数和幂律Lagrange函数下的Noether对称性摄动与其导致的Noether型绝热不变量之间的关系。最后,再由高阶绝热不变量存在的条件和形式,建立了相应的摄动定理。本文第二部分,首先,列出指数Lagrange函数和幂律Lagrange函数下的Lie对称性间接导致的Noether守恒量和直接导致的Hojman守恒量。其次,再从高阶绝热不变量的定义出发,继续探究小扰动作用下指数Lagrange函数和幂律Lagrange函数下的Lie对称性摄动与其间接导致的Noether型和直接导致的Hojman型绝热不变量之间的关系。最后,再由高阶绝热不变量存在的条件和形式,建立了相应的摄动定理。本文第三部分,首先,列出指数Lagrange函数和幂律Lagrange函数下的Mei精确不变量。其次,再从高阶绝热不变量的定义出发,继续探究小扰动作用下指数Lagrange函数和幂律Lagrange函数下的Mei对称性摄动与其直接导致的Mei绝热不变量之间的关系。最后,再由高阶绝热不变量存在的条件和形式,建立了相应的摄动定理。
[Abstract]:The most common problem in nature is the problem of nonlinear non-conservative dynamical system. The nonstandard Lagrange function has some properties that the standard Lagrange function does not have. It can describe nonlinear nonconservative problems. Therefore, the study of nonstandard Lagrange functions is of great significance and value. This paper focuses on the Noe of dynamical systems under exponential Lagrange functions and power law Lagrange functions. Ther symmetry. Lie symmetry and Mei symmetry are three kinds of symmetry perturbation and adiabatic invariants. The first part of this paper, first. The exponential Lagrange function, the power law Lagrange function and the exponential Lagrange function and the power law Lagrange function under the El-Nabulsi model are listed. Oether exact invariant. Second. Then the definition of high order adiabatic invariants is given. Continue to explore the exponential Lagrange function and power law Lagrange function under small perturbation, and the exponential Lagrange function and power law Lagra under El-Nabulsi model. The relation between the Noether symmetry perturbation under nge function and the Noether type adiabatic invariants. Finally. Then according to the condition and form of the existence of high order adiabatic invariants, the corresponding perturbation theorems are established. The conserved quantity of Noether and the conserved quantity of Hojman caused by Lie symmetry under exponential Lagrange function and power law Lagrange function are listed. Then the definition of high order adiabatic invariants is given. Further study on the Lie Symmetry perturbation of exponential Lagrange function and Power Law Lagrange function under small perturbation and its indirect Noether Type and Hojm. The relationship between an type adiabatic invariants. Then from the condition and form of the existence of high order adiabatic invariants, the corresponding perturbation theorems are established. The exact invariants of Mei under exponential Lagrange function and power law Lagrange function are listed. Secondly, the definition of higher order adiabatic invariants is given. The relationship between the Mei symmetry perturbation of exponential Lagrange function and power law Lagrange function and the adiabatic invariant of Mei directly caused by small perturbation is further explored. Finally. Based on the conditions and forms of the existence of higher order adiabatic invariants, the corresponding perturbation theorem is established.
【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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