非完整系统的约束子流形辛化及其在对称约化中的应用

Content: 1) The geometric structure on constrained submanifolds of nonholonomic systems is investigated and the closed (singular or regular) basic 2-form,i.e., the generalized symplectic form can be constructed in this project. In the framework of the generalized symplectic geometric structure of constrained submanifolds, a new dynamical representation of nonholonomic systems,i.e., the generalized Birkhoffian formulation of can be realized. 2) On the basis of analysing the distinguishment between generalized symplectic structure and presymplectic structure, the relation between generalized Birkhoffian systems and Dirac singular systems is investigated. 3) The conservation and evolvenment equations of momentum maps for generalized Birkhoffian systems with symmetry are discussed, which can be used to investigate the symmetry reduction of nonholonomic systems with symmetry...Meaning: The generalized symplectic geometric method is utilized to realize the closed geometric structure of nonholonomic systems, and the nonholonomicity of constrained equations can be represented by the singularity of generalized symplectic structure, which break through the study on nonholonomic systems by traditional geometric mechanics method which is only based on the almost-Poisson geometric structure. The generalized Birkhoffian systems are turned into almost-singular systems, which expand the research area of presymplectic geometry. At the same time, a new symmetry reduction approach, i.e. the generalized Birkhoffian symmetry reduction theory which can be used to investigate the symmetry reduction of nonholonomic systems will be realized in this project, which also can be as the foundation to study the geometric numerical integration and geometric control of nonholonomic systems.

内容：1）研究非完整系统约束子流形的几何结构，构造其闭合（或奇异或正规）的基本2-形式，即广义辛形式，并在此框架下构造具有广义辛几何结构的约束流形上非完整系统的新表示- - 广义伯克霍夫表示。2）深入分析广义辛几何结构与预辛几何结构的关系，并以此研究广义伯克霍夫动力学系统与狄拉克约束奇异动力学系统之间的关系。3）研究具有对称性的广义伯克霍夫系统动量映射的守恒性及其演化方程，并将其应用于研究具有对称性的非完整系统的对称约化问题。意义：运用广义辛几何方法，可以实现非完整系统的封闭几何结构，并把约束方程的非完整性表现为广义辛结构的奇异性，突破了在传统几何力学中仅采用近泊松几何结构研究非完整系统的局限，扩展了预辛几何的研究范围，使得广义伯克霍夫系统成为一种"准"奇异系统，同时可以实现非完整系统对称约化研究的新途径- - 广义伯克霍夫约化，为可以为非完整系统的几何数值积分和几何控制研究奠定新的理论基础。

项目摘要：运用几何动力学方法，研究非完整系统广义辛几何结构的构造，非完整系统的对称约化，非完整和非保守约束系统的辛算法。研究内容分为三个方面：.1．非完整系统广义结构的构造方面，建立了非完整约束系统的广义辛几何理论，以及非完整系统的Birkhoff和广义Birkhoff动力学函数组的构造方法。初步实现了Birkhoff和广义Birkhoff动力学的几何理论。.2．非完整系统的对称约化方面，通过构造非完整约束系统的广义辛几何结构，建立了广义辛约化理论，并在Birkhoff动力学和广义Birkhoff动力学框架内实现了非完整约束系统的广义辛约化问题研究。.3．非完整系统的辛算法方面，在Birkhoff几何动力学框架内构造了Birkhoff系统的辛算法，实现了非完整约束系统与非保守系统的保结构数值计算问题研究，并得到了比传统的Runge-Kutta法更精确的数值结果。