The reduction on 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 \cite{MSS1} and \cite{MSS2} on $3$-body Euler solutions in 2004-2006. 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.

28 pages. arXiv admin note: text overlap with arXiv:1510.06822

References in corpus (1)

• Collision index and stability of elliptic relative equilibria in planar n-body problem

Cited by in corpus (7)

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• Determination of the doubly-symmetric periodic orbits in the restricted three-body problem and Hill's lunar problem
• Linear Stability of Elliptic Relative Equilibria of Restricted Four-body Problem
• Linear Instability of Elliptic Rhombus Solutions to the Planar Four-body Problem
• The analytical aspect to the linear stability of elliptic equilibrium points of the Robe's restricted three-body problem
• Linear stability of elliptic relative equilibria of four-body problem with two infinitesimal masses
• Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases