Birkhoff 动力学的非完整几何积分子及对称性理论的研究

项目名称： Birkhoff 动力学的非完整几何积分子及对称性理论的研究

项目编号： No.11502071

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 夏丽莉

作者单位： 河南教育学院

项目金额： 20万元

中文摘要： 约束力学系统的数值计算方法在处理工程实际问题时非常关键。非完整动力学系统的约束子流形不是辛流形，这成为制约保辛数值计算方法在模拟非完整系统动力学行为方面的主要瓶颈。本项目旨在基于非完整力学系统在广义Birkhoff表示下的微分几何结构的基本概念，研究系统的非完整几何积分子和对称性理论。探讨非完整几何积分子得到的守恒律和离散对称性方法得到的守恒律的关系。重点研究基于Birkhoff形式下的非完整系统的辛结构形式，给出保系统约束和几何性质不变的非完整几何积分子。通过与离散动力学的Noether对称性理论得到的守恒量比较，探讨非完整几何积分子在保系统守恒律的有效性。应用到非完整动力学模型。验证非完整几何积分子在保系统结构、计算精度、稳定性等方面的合理性。该项目研究的开展将会发展和完善非完整动力学的数值算法和未知守恒律理论，为研究带有其他类型约束的动力学系统的数值算法提供理论参考。

中文关键词： Birkhoff系统；非完整几何积分子；离散化；数值计算；守恒律

英文摘要： Numerical method for constrained mechanical systems is one of the key topics of engineering dynamics. The constraint submanifold for the nonholonomic mechanical systems is not the symplectic manifold. This is the bottleneck on simulating the behaviors of the nonholonomic dynamical systems by symplectic algorithm. Based on the basic concepts of differential geometry for nonholonomic mechanical systems in generalized Birkhoff sense, the geometric nonholonomic integrators and the symmetries are studied.The relationships between the conservations derived from the symmetries and that from the geometric nonholonomic integrators are investigated. The preservation of the constraints and the geometry are highlighted via the symplectic forms of Birkhoff systems with nonholonomic constraints. The geometric nonholonomic integrators are effective by comparing the conservations with that from the discrete Noether symmetries.The integrators are applied to solve the dynamics with nonholonomic constraints. The reasonability of the structure-preserved, the accuracy and the stability for the new schemes are verified. The research in this project will be expected to contribute to the development and improvement of the theories about the unknown conservations and the algorithm of the nonholonomic mechanical systems. It provides specific theory references for the numerical calculations of other types of constraints in the engineering industry.

英文关键词： Birkhoff dynamical systems;Geometric nonholonomic integrators;Discretizations;Numerical calculation;Conservation laws