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.
Women's health in prison: urgent need for improvement in gender equity and social justice.
The needs of women held in detention have received little attention and con-tinue to be neglected by health systems and prison authorities. One reason for this, too readily accepted until now, is that women prisoners are a clear minor-ity group within prisons all over the world. It is only recently that attention has been drawn to the fact that a minor-ity status does not justify the widespread ignorance of women’s basic rights and the considerable gender insensitivity still dominating criminal justice systems.In Europe alone, there are about 100 000 women and girls in prison.