无阻尼振动方程解的稳定性

The Stability of the Solutions to the Un-damped Oscillation Equation

在无干扰力的环境中,定性分析无阻尼振动方程解的稳定性,可归结为下述几个问题:牛顿第二运动定律应用于振动建模描述力与运动的关系,线性化方程是求解微分方程的有效方法;单摆振动的等时性与非等时性特征表明,微分方程的解不仅决定于方程本身,而且也决定于解的初值;微分方程定性理论,特别是李雅普诺夫第二方法,是研究非线性微分方程解的稳定性的有效手段;如何构造李雅普诺夫函数,至今仍是一个吸引人的研究课题.

Under conditions without disturbance forces, the stability of the solutions to the an-damped oscillation equation by qualitative analysis can be conduced as follows： Newton＇s second law of motion is applied to modeling pendulum, describing the relations of force and motion; Linearing equation is a effective method to solve differential equation; The isochronicle and anisochronicle character of single pendulum states, the solutions of differential equation are determined by not only the equation itself, but also its initial value; the stable theory of differential equation ,especial Liapounov＇s second method, is a effective approach to research the stability of the solutions to nonlinear differential equation; How to construct Liapounov function is still an attractive problem at present.