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

Abstract

In this paper, we consider 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 the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3-body problem whose mass parameter is modified. Then the linear stability of such a solution in the n-body problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3-body problems, which for example then can be further understood using numerical results of Martinéz, Samà and Simó in [13] and [14] on 3-body Euler solutions in [2004][2005][2006]. As an example, we carry 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.

Linear stability of elliptic relative equilibria of four-body problem with two infinitesimal masses

“…To our knowledge, for the general n bodies, the elliptic Euler-Moulton solutions is the only case which has been well studied in [37]. It turns out that the stability of the elliptic Euler-Moulton solutions depends on (n − 1) parameters, namely the eccentricity e ∈ [0, 1) and the n − 2 mass parameters β 1 , β 2 , .…”

“…. , β n−2 which defined by (1.14) in [37]. For some special cases of n-body problem, the linear stability of ERE which raised from an n-gon or (1 + n)-gon central configurations with n equal masses was studied by Hu, Long and Ou in [5].…”

“…If the positions of m 1 , m 2 and the point L form a Euler central configuration, then the original central configuration of m 1 , m 2 , m 3 and m 4 must be collinear, i.e., an Euler-Moulton central configuration by [33] and [37]. The stability problem of such ERE was studied well in Section 3 of [37].…”

“…If the positions of m 1 , m 2 and the point L form a Euler central configuration, then the original central configuration of m 1 , m 2 , m 3 and m 4 must be collinear, i.e., an Euler-Moulton central configuration by [33] and [37]. The stability problem of such ERE was studied well in Section 3 of [37]. As a matter of fact, in the limiting case m 3 , m 4 → 0, by (3.69) bellow of [37], the stability problem is reduced to the linear stability problems of two restricted three-body problems, for which one has mass parameter β, and the other has mass parameter 3(β + 1) where β is given by (1.6).…”

Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem

“…Inspired by the analytic method, Q. Zhou and Y. Long in [14] studied the linear stability of elliptic triangle solutions of the charged three-body problem.Recently, in [15,16], Q. Zhou and Y. Long studied the linear stability of elliptic Euler-Moulton solutions of n-body problem for n = 3 and for general n ≥ 4, respectively.…”