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- Title
The reduction of the linear stability of elliptic Euler-Moulton solutions of the n-body problem to those of 3-body problems.
- Authors
Zhou, Qinglong; Long, Yiming
- 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 Martínez et al. 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.
- Subjects
MANY-body problem; STABILITY of linear systems; ELLIPTICAL orbits; HAMILTONIAN systems; KEPLER problem
- Publication
Celestial Mechanics & Dynamical Astronomy, 2017, Vol 127, Issue 4, p397
- ISSN
0923-2958
- Publication type
Article
- DOI
10.1007/s10569-016-9732-x