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.
Linear Stability of the Elliptic Lagrangian Triangle Solutions in the Three-Body Problem
Abstract This paper concerns the linear stability of the well-known periodic orbits of Lagrange in the three-body problem. Given any three masses, there exists a family of periodic solutions for which each body is at the vertex of an equilateral triangle and travels along an elliptic Kepler orbit. Reductions are performed to derive equations which determine the linear stability of the periodic solutions. These equations depend on two parameters – the eccentricity e of the orbit and the mass parameter β=27(m1m2+m1m3+m2m3)/(m1+m2+m3)2. A combination of numerical and analytic methods is used to find the regions of stability in the βe-plane. In particular, using perturbation techniques it is rigorously proven that there are mass values where the truly elliptic orbits are linearly stable even though the circular orbits are not.
Brauer groups of genus zero extensions of number fields
We determine the isomorphism class of the Brauer groups of certain nonrational genus zero extensions of number fields. In particular, for all genus zero extensions E of the rational numbers Q that are split by Q(√2), Br(E) ≅ Br(Q(t)).