On Periodic Approximate Solutions of the Three-Body Problem Found by Conservative Difference Schemes

The possibility of using implicit difference schemes to study the qualitative properties of solutions of dynamical systems, primarily the periodicity of the solution, is discussed. An implicit difference scheme for the many-body problem that preserves all algebraic integrals of motion is presented based on the midpoint scheme. In this concern, we consider the finite-difference analogue of the Lagrange problem: using the midpoint scheme to find all approximate solutions of the three-body problem on a plane in which the distances between the bodies do not change. It is shown that this problem can be solved by purely algebraic methods. Two theorems are proved that reduce this problem to the study of the midpoint scheme properties for a system of coupled oscillators. It is shown that in the case when the bodies form a regular triangle, the approximate solution inherits the periodicity property of the exact Lagrange solution. © 2020, Springer Nature Switzerland AG.