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).

中文翻译：

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多体问题的保守差分格式

提出了一种为多体问题构建任意阶差分格式的新方法，该方法保留了其所有代数积分。我们引入了附加变量，即物体之间的距离和倒数距离，并写下了关于坐标、速度和附加变量的微分方程组。在这种情况下，系统失去了它的哈密顿形式，但所考虑的多体问题的所有经典运动积分，以及描述物体坐标和附加变量之间关系的新积分都用线性或这些新变量中的二次多项式。因此，任何辛 Runge-Kutta 方案都精确地保留了这些积分。给出了所提议方法的证据。为了说明这个理论，