N体问题共线解的简明数值方法

THE CONCISE NUMERICAL ANALYSIS FOR COLLINEAR SOLUTIONS OF N-BODY PROBLEM

研究N体问题共线解的数值方法.依照动力学和运动学原理,建立N体问题共线解所满足的条件方程,把解 微分方程组的问题转化为解非线性方程组的问题.当质量已知时,对条件方程组进行Taylor级数展开,使非线性方程组 转化为线性方程组,然后用牛顿迭代法解此方程组从而获得共线解.如果给定N体问题共线解中各质点之间的距离,那 么问题就变成求解满足这组给定轨道的质点的质量问题,此时的条件方程就是线性方程组,解此线性方程组就可以得到 答案.

The numerical methods of solving the collinear N-body problem are studied in detail. By the Newton's dynamic principle and the constraint equations that the collinear N-body systems satisfy, the problem of solving the set of differential equations is transformed into that of solving the nonlinear set of equations. Given mass of each object in N-body system, the nonlinear set of equations reduces to linear one by the Taylor series expansion of the constraint equation, and its collinear is obtained by Newtonian iteration. On the other hand, given the distance between any two masses, the problem we seek becomes that of finding the masses of N-body system. In this case, the constraint equations is naturally linear set of equations, which gives the masses of N-body system. A serious of numerical solutions to the collinear N-body problem are presented for both cases above.