几何动力学在非完整系统几何数值积分中的应用研究

项目名称： 几何动力学在非完整系统几何数值积分中的应用研究

项目编号： No.10872084

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 郭永新

作者单位： 辽宁大学

项目金额： 31万元

中文摘要： 运用几何动力学和几何数值积分方法研究了非完整系统的变分问题、基于非完整映射的Riemann-Cartan流形的几何构造及其应用问题以及非完整系统的Birkhoff表示、广义Birkhoff表示和几何数值积分问题。研究成果包括： (1)讨论了非自伴随动力学系统近Poisson结构的分解理论和基于非完整映射的Riemann-Cartan流形的几何构造及其在非完整力学系统中的应用问题； (2) 利用嵌入变分恒等式的非完整系统的变分原理，深入分析了非完整力学系统存在两种不等价动力学模型的根源和物理机制，为非完整力学系统的几何数值积分研究奠定理论基础； (3) 重点研究Chaplygin非完整系统的几何数值积分问题，尤其研究了特殊Chaplygin非完整系统的辛几何算法，并讨论了几何数值积分方法在一般Chaplygin非完整系统的两类动力学模型中的应用； (4) 讨论了非完整系统的Birkhoff表示与广义Birkhoff表示问题，在Birkhoff 动力学框架下构造非完整系统的一般辛结构和Birkhoff 方程,实现了Birkhoff积分子的构造并将其应用于研究非完整系统的几何数值积分问题。

中文关键词： 非完整系统;Riemann-Cartan流形;近Poisson分解;Birkhoff系统;几何数值积分

英文摘要： Geometric dynamics and geometric numerical integration methods are utilized to investigate the variations of nonholonomic systems, geometrical construction of the Riemann-Cartan manifolds based on the method of nonholonomic mapping and its applications, the Birkhoffian and generalized Birkhoffian representation of nonholonomic systems and its geometric numerical integration problems. The following four aspects are included in the research fruits: (1) The decomposition theory of almost Poisson structure for non-self-adjoint dynamical systems and the geometric construction problems of Riemann-Cartan manifolds based on the method of nonholonomic mapping are investigated in this project, which are utilized to the nonholonomic systems. (2) The variational principles embedded variation identity are utilized to study the dynamics of nonholonomic systems, which make us deeply understand the production of roots and physical mechanism by the existing two unequivalent dynamics models for nonholonomic systems.Therefore, a solid foundation for the geometrical numerical integration of nonholonomic systems is established by the above research results. (3) Especially the special Chaplygin's nonholonomic system,the geometric numerical integration problem of the Chaplygin's nonholonomic system is mainly investigated in this project, which includs the applications of geometric numerical integration in the vakonomic dynamics and nonholonomic dynamics of the general Chaplygin's nonholonomic system. (4) The problems on Birkhoffian and generalized Birkhoffian representation of nonholonomic systems are also discussed, from which the general symplectic struction and generalized symplectic structure are constructed. At the same time Birkhoff's equations and Birkhoffian integrator on nonholonomic systems in the framework of Birkhoffian dynamics are realized, which are used to investigate the geometric numerical integration problems of nonholonomic systmes.

英文关键词： nonholonomic systems;Riemann-Cartan manifolds;decomposition theory of almost-Poisson structure;Birkhoffian systems;geometric numerical integration