On conservative difference schemes for the many-body problem

TL;DR

This paper introduces a novel difference scheme for the many-body problem that preserves all algebraic integrals by using additional variables, enabling exact conservation with symplectic Runge-Kutta methods, demonstrated through numerical experiments.

Contribution

A new approach using extra variables to construct high-order difference schemes that exactly preserve all algebraic integrals of the many-body problem.

Findings

Exact preservation of all classical integrals demonstrated.

Numerical experiments confirm the scheme's effectiveness.

Applicable to complex motions like figure-eight trajectories.

Abstract

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).