The reduction of the linear stability of elliptic EulerMoulton solutions of the nbody problem to those of 3body problems
Abstract
In this paper, we consider the elliptic collinear solutions of the classical nbody 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 EulerMoulton collinear solution. Here we prove that the corresponding linearized Hamiltonian system at such an elliptic EulerMoulton collinear solution of nbodies splits into (n1) independent linear Hamiltonian systems, the first one is the linearized Hamiltonian system of the Kepler 2body problem at Kepler elliptic orbit, and each of the other (n2) systems is the essential part of the linearized Hamiltonian system at an elliptic Euler collinear solution of a 3body problem whose mass parameter is modified. Then the linear stability of such a solution in the nbody problem is reduced to those of the corresponding elliptic Euler collinear solutions of the 3body problems, which for example then can be further understood using numerical results of Martínez et al. on 3body Euler solutions in 20042006. As an example, we carry out the detailed derivation of the linear stability for an elliptic EulerMoulton solution of the 4body problem with two small masses in the middle.
 Publication:

Celestial Mechanics and Dynamical Astronomy
 Pub Date:
 April 2017
 DOI:
 10.1007/s105690169732x
 arXiv:
 arXiv:1511.00070
 Bibcode:
 2017CeMDA.127..397Z
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematical Physics;
 70F10;
 70H14;
 34C25
 EPrint:
 28 pages. arXiv admin note: text overlap with arXiv:1510.06822