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

Abstract

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.

The computations presented in the paper were performed in Sage computer algebra system (www.sagemath.org). The contribution of E. A. Ayryan (Investigation), M. D. Malykh (Investigation, proofs of theorems), and L. A. Sevastianov (Conceptualization, writing) is supported by the Russian Science Foundation (grant no. 20-11-20257).

Appendix A   Investigation of the Set V in Sage

We have tried unsuccessfully to investigate the algebraic set V introduced above in Sect. 4 of Sage, using the tools developed by W. Stein for working with ideals in multidimensional polynomial rings. Using the notation of (19), we have rewritten system (S) as 1) three equations

$$ (x_i-x_i)^2 + (y_i-y_j)^2=a_{ij}^2, $$

2) two systems (20) and (21), and 3) three equations

$$ (\textit{\textbf{r}}_i -\textit{\textbf{r}}_j )\cdot (\textit{\textbf{v}}_i -\textit{\textbf{v}}_j) = 0 ,\quad i, j = 1 ,\dots , n;\, i\not = j, $$

which in our case are equivalent to six equations that approximate Eqs. (10) and (11). Systems (20) and (21) are linear in variables marked with hats. We have solved these systems in Sage by means of standard function solve:

Then we have constructed an ideal of the algebraic set V (in our code called Lagrange ideal):

For simplicity, we have added to the equations that determine the set V a condition of coincidence of the gravity center and the origin of coordinates. Thus we obtained an ideal J in the ring

$$ \mathbb {Q} [a, b, c, dt, x_1, x_2, x_3, y_1, y_2, y_3, u_1, u_2, u_3, v_1, v_2, v_3]. $$

Sage is unable to answer even such trivial questions about this ideal as belonging of element $y_1 + y_2 + y_3$ to ideal J: