奇异力学系统的矩阵分析与几何研究

Matrix Analysis and Geometric Study of Singular Mechanical Systems

本文研究了奇异Lagrange系统(后文统称“奇异力学系统”)的Hesse矩阵求逆、泛函极值、Jacobi场共轭点等问题,利用矩阵分析、变分学及微分几何等一系列数学方法,创造性地推导了奇异力学系统的矩阵解法,获得了该类系统测地线的某些邻域行为以及测地线的极值特性等结论。相关结果推广了矩阵分析、微分几何等有关理论在动力学系统中的应用,使矩阵分析与几何研究相融合,为进一步研究奇异力学系统提供了参考的基础。

In this paper, we study the inverse of Hesse matrix, functional extremum and Jacobi conjugate point of singular Lagrange system, by using a series of mathematical methods such as matrix analysis, variational calculus and differential geometry, the matrix solution of singular mechanical system is deduced creatively, and some conclusions about the neighborhood behavior of geodesic and the extreme value characteristics of geodesic are obtained. The related results extend the application of the theories of matrix analysis and differential geometry in the field of dynamical system mechanics, and make the matrix analysis and geometry research merge, which provides a reference for the further study of singular mechanical systems.