一类强非线性二阶微分方程的多模态近似解析解研究

利用自治力学系统的哈密顿函数为守恒量的性质,提出一种求非线性二阶微分方程多模态近似解析解的方法,称为哈密顿函数法.首先,介绍哈密顿函数法求多模态近似解的基本理论.其次,以质点在旋转的抛物线上运动为模型建立强非线性二阶微分方程.最后,用哈密顿函数法求得在给定初始条件和参数下强非线性二阶微分方程的三模态近似解析解表达式,作出三模态近似解析解的解曲线,并与直接用Mathematica软件作出的解曲线进行比较,讨论三模态近似解析解的精确性.结果表明：用哈密顿函数法求得的三模态近似解析解的解曲线与直接用Mathematica软件作出的解曲线十分吻合.

Study on multimode approximate analytical solution of a class of strongly nonlinear second order differential equations

Hamiltonian is a conservation quantity of autonomous mechanical system, and the multimode approximate analytical solutions of a nonlinear second order differential equation can be obtained by using characteristics of the conservation quantity, which is called Hamiltonian method. First, the basic theory of Hamiltonian method was introduced. Second, a strong nonlinear second order differential equation for the motion of a particle on a rotating parabola was established. Finally, the three mode approximate analytical solutions of the strong nonlinear second order differential equation were obtained under given initial conditions and parameters. The approximate solution using Hamiltonian method was verified by the numerical solution using Mathematics software, which showed that the approximate solution is in good agreement with the numerical one.