申请辽宁大学硕士学位论文
辛向量空间上的Hamilton系统Hamiltonian Systems on Symplectic Vector Spaces 星光大道白婧作者：李蕊上海物理教育网
指导教师：郭永新教授
专业：粒子物理与原子核物理
答辩日期：2018年5月26日
二○一八年五月·中国辽宁
摘要
摘要
奚正平
辛向量空间是被赋予了辛结构的向量空间，辛结构是一种斜称耦对的结构，最早在数学领域被Abel研究过，并与复结构混名，后被Weyl于1938年正式更为此名，中文翻译大约在1944年由华罗庚先生音译而得。其实，分析力学中的相空间就是辛空间，只是经典分析力学大师们多关注于力学系统运动的解析表达，并给出了今天研究辛结构局部特征仍很实的解析方法，而较少关注其全局几何结构而已。而且分析力学相空间上的动力学理论——Hamilton力学正是由于具有辛结构才使得Hamilton力学的应用领域从物理学的经典理论拓展到现代理论，从物理学的宏观领域拓展到微观和宇观领域，从物理学的确定性问题拓展到随机性问题，从物理学的连续性问题拓展到离散性问题，从物理学的完整约束问题拓展到非完整约束问题。
本文详细研究了向量空间的辛几何结构和泊松几何结构，以及辛向量空间和泊松向量空间子空间的几何结构。并基于辛矢量空间和泊松矢量空间的几何结构研究了线性Hamilton系统几何结构和动力学问题。主要包括如下研究内容：首先，研究了辛矢量空间的几何结构以及辛向量空间的子空间理论，并研究了辛向量空间的约化问题。
其次，主要研究了泊松向量空间的几何结构以及泊松向量空间的子空间理论，并研究了泊松向量空间的约化问题，
最后，研究了辛向量空间和泊松向量空间上的线性Hamilton力学，详细讨论了辛向量空间上的Hamilton向量场和光滑函数的李代数结构，并讨论了Hamilton系统的正则变换理论。
关键词：辛向量空间，泊松向量空间，辛向量空间的约化，Hamilton系统，正则变换鼠咬热
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Abstract
ABSTRACT
The symplectic vector space is a kind of vector space with the symplectic structure.The symplectic structure is skew symmetric coupling structure,which is the first studied by Abel.Because the skew symmetric structure is confused with the name of complex structure,Weyl named the skew symmetric coupling structure as symplectic structure in1938.The Chinese name of symplectic structure is translated according to the pronunciation of symplectic by Luogeng Hua in1944.In fact,the phase space of analytical mechanics is exact the symplectic space.But most of the great masters of classical mechanics pay more attention to the analytic formulations not the global formulation of dynamical systems.And lots of very useful analytic methods are be given which can be used to study the local characteristic of symplectic structure.Hamiltonian mechanics is a kind of classical theory on phase space of dynamical systems.The application fields of Hamiltonian mechanics are expanded because the geometric structure of Hamiltonian mechanics is symplectic structure. Such as Hamiltonian mechanics is used not only in classical physics but also in modern physics,not only in macroscopic fields but also in microscopic fields and cosmoscopic fields,not only in deterministic problems but also in stochastic problems, not only in continuous problems but also in
discrete problems,not only in holonomically constrianed problems but also in nonholonomically constrained problems.
In this paper,the symplectic geometric structure and Poisson geometric structure in vector space are studied,as well as the geometric structure of symplectic vector space and Poisson vector space and its subspace.Based on the geometric structure of symplectic vector space and Poisson vector space,the geometric structure and dynamics of linear Hamilton system are studied.The research contents are following: Firstly,the geometric structure of symplectic vector space,its subspace theory, and the reduction problem are studied.
Secondly,geometric structure of Poisson vector space,its subspace theory,and the reduction problem are studied.中华医学会
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