论文标题
关于多体问题的保守差异方案
On conservative difference schemes for the many-body problem
论文作者
论文摘要
提出了一种新的代数积分的多体问题的差异方案的构建新方法。我们介绍了其他变量，即身体之间的距离和相互距离，并写下了一个关于坐标，速度和其他变量的微分方程系统。在这种情况下，系统失去了哈密顿式形式，但是所考虑的多体问题的所有经典运动积分，以及描述身体坐标与其他变量之间关系的新积分。因此，任何符号runge-kutta方案都可以准确保留这些积分。给出了建议的方法的证据。为了说明该理论，呈现了平面上三体问题的数值实验的结果，并选择了与沿八个图的身体运动相对应的初始数据（编舞测试）。
A new approach to the construction of difference schemes of any order for the many-body problem that preserves all its algebraic integrals is proposed. We introduced additional variables, namely, distances and reciprocal distances between bodies, and wrote down a system of differential equations with respect to coordinates, velocities, and the additional variables. In this case, the system lost its Hamiltonian form, but all the classical integrals of motion of the many-body problem under consideration, as well as new integrals describing the relationship between the coordinates of the bodies and the additional variables are described by linear or quadratic polynomials in these new variables. Therefore, any symplectic Runge-Kutta scheme preserves these integrals exactly. The evidence for the proposed approach is given. To illustrate the theory, the results of numerical experiments for the three-body problem on a plane are presented with the choice of initial data corresponding to the motion of the bodies along a figure of eight (choreographic test).
