约束力学系统的梯度表示与非完整系统的数值分析

Study the problems of gradient representation for constrained mechanical systems and the numerical analysis of nonholonomic systems. The gradient system is a mathematical system whose differential equations are simple, and have very good properties which are specially fit for discussing stability. Constrained mechanical systems include holonomic constrained mechanical systems, nonholonomic constrained mechanical systems, Birkhoff systems, etc. Their differential equations are all very complex. Generally speaking, these differential equations mostly can not be expressed as equations of gradient system. In the first part of this project, we will discuss the condition under which the equations of a constrained mechanical system can become ones of a gradient system. When a constrained mechanical system becomes a gradient system, we can study the dynamical behaviours of the constrained mechanical system, particularly its stability, by using the property of gradient system. In the second part of this project, we will work on the approximate conserved quantity, the influence of constraint over bifurcation and so on for nonholonomic systems by using numerical calculation and qualitative analysis. Through the above research, we hope to establish the relation between the constrained mechanical system and the gradient system, find a new way for studying the motion stabilization, and introduce the numerical calculation into analytical mechanics.

研究约束力学系统的梯度表示与非完整系统的数值分析问题。梯度系统是一个数学系统，它的微分方程简单，但是有很好的性质，特别适合研究稳定性。约束力学系统，包括完整约束力学系统，非完整约束力学系统，Birkhoff系统等，这些系统的微分方程都很复杂。一般来说，这些微分方程大多不能表示为梯度系统的微分方程。本项目第一部分研究约束力学系统的方程成为梯度系统方程的条件。约束力学系统成为梯度系统之后，便可利用梯度系统的特性来研究约束力学系统的动力学行为，特别是力学系统的稳定性问题。本项目第二部分，利用数值计算与定性分析研究非完整系统的近似守恒量，约束对分叉的影响等。期望通过研究，建立约束力学系统与梯度系统的联系，找到研究运动稳定性的新途径，使数值计算进入分析力学。

首先，系统全面地研究了Lagrange系统、Hamilton系统、Birkhoff系统等十三类约束力学系统的梯度表示，包括梯度系统，约束力学系统与通常梯度系统，约束力学系统与斜梯度系统，约束力学系统与具有对称负定矩阵的梯度系统，约束力学系统与具有半负定矩阵的梯度系统，约束力学系统与组合梯度系统，约束力学系统与广义梯度系统，约束力学系统与广义斜梯度系统，及其逆问题等。借助梯度系统的性质，研究了约束力学系统的积分和解的稳定性等问题。. 其次，探讨了各种Birkhoff系统的离散变分算法，构造了相应的变分积分子，并用实例仿真验证了这种算法在稳定性、精度和保守恒量方面的优越性。. 最后，讨论了约束力学系统的对称性和守恒量等相关问题，取得了相应的结果。