The linear stability of elliptic Euler-Moulton solutions of the n-body problem via those of 3-body problems

Qinglong Zhou, Yiming Long

Published 2015-10-31Version 1

In this paper, we study the linear stability of the elliptic collinear solutions of the classical $n$-body problem, where the $n$ bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler-Moulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic Euler-Moulton collinear solution of $n$-bodies splits into $(n-1)$ independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler $2$-body problem at Kepler elliptic orbit, and each of the other $(n-2)$ systems is the essential part of some linearized Hamiltonian system at an elliptic Euler collinear solution of the $3$-body problem whose mass parameter is modified. Then using analytical results proved by Zhou and Long in [21] and by Hu and Ou in [5] on $3$-body Euler solutions, the linear stability of such a solution in the $n$-body problem is reduced to those of the corresponding elliptic Euler collinear solutions in the $3$-body problems. As an example, we carried out the detailed derivation of the linear stability for an elliptic Euler-Moulton solution of the $4$-body problem with two small masses in the middle.