S ep 2 01 6 Maslov-type indices and linear stability of elliptic Eulersolutions of the three-body problem
In this paper, we use the central configuration coordinate de composition to study the linearized Hamiltonian system near the 3-body elliptic Euler solution s. Then using the Maslov-type ω-index theory of symplectic paths and the theory of linear operators we compute theω-indices and obtain certain properties of linear stability of the Euler elliptic soluti ons of the classical three-body problem.
Ju l 2 01 6 The reduction on the linear stability of elliptic Euler-Moulton solutions of the n-body problem to those of 3-body problems
In this paper, we consider the elliptic collinear solutions f the classical n-body problem, where the n bodies always stay on a straight line, and each of them moves o n it wn elliptic orbit with the same eccentricity. Such a motion is called an elliptic Euler-Mou lt n 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 t he linearized Hamiltonian system of the Kepler 2-body problem at Kepler el liptic 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 li near stability of such a solution in the n-body problem is reduced to those of the corresponding ellip tic Euler collinear solutions of the 3-body problems, which for example then can be further understood u sing numerical results of Martinéz, Samà and Simó in [13] and [14] on 3-body Euler solutions in 2004-2 006. 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.
A Maslov-type index theory for symplectic paths
In this paper, we extend the Maslov-type index theory defined in [7], [15], [10], and [18] to all continuous degenerate symplectic paths, give a topological characterization of this index theory for all continuous symplectic paths, and study its basic properties. Suppose τ > 0. We consider an τ -periodic symmetric continuous 2n × 2n matrix function B(t), i.e. B ∈ C(Sτ ,Ls(R)) with Sτ = R/(τZ), L(R2n) being the set of all real 2n×2n matrices, and Ls(R) being the subset of all symmetric matrices. It is well-known that the fundamental solution γ of the linear first order Hamiltonian system