On the minimizing properties of the 8-shaped solution of the 3-body problem

The starting point of our study was the recent results of Alain Chenciner and Richard Montgomery concerning the discovery of the 8-shaped orbit of the planar 3-body problem with equal masses (in the sequel, we will call it just “the Eight,” 4]). Geometrically this orbit consists of 12 pieces such that each of them minimizes the Lagrangian action between Euler and isosceles configurations of the bodies. Our aim was to understand whether the larger pieces of the Eight are still solutions of some minimizing problem. The paper presents some preliminary analytical and numerical results on the minimizing properties of the Eight. Using the technique of the so-called Jacobi curves, we numerically show that the solution of Chenciner and Montgomery is no longer optimal after 0.52 of its period. Moreover, we find a better solution for the fixed endpoint problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.