It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter $bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2in [0,9]$ and the eccent
ricity $ein [0,1)$. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle $[0,9]times [0,1)$, besides perturbation methods for $e>0$ small enough, blow-up techniques for $e$ sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full $(bb,e)$ range $[0,9]times [0,1)$ via the $om$-index theory of symplectic paths for $om$ belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the $om$-index decreasing property of the solutions in $bb$ for fixed $ein [0,1)$, we prove the existence of three curves located from left to right in the rectangle $[0,9]times [0,1)$, among which two are -1 degeneracy curves and the third one is the right envelop curve of the $om$-degeneracy curves for $om ot=1$, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter $(bb,e)$ passes through each of these three curves. Interesting symmetries of these curves are also observed. The singular case when the eccentricity $e$ approaches to 1 is also analyzed in details concerning the linear stability.
Let $M$ be an exact symplectic manifold with contact type boundary such that $c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya category of $M$ has the structure of an involutive Lie bialgebra. Inspired by a work of Cieliebak-
Latschev we show that there is a Lie bialgebra homomorphism from the linearized contact homology of $M$ to the cyclic cohomology of the Fukaya category. Our study is also motivated by string topology and 2-dimensional topological conformal field theory.
Relative orbifold Gromov-Witten theory is set-up and the degeneration formula is given.
We study the relationship between the masses and the geometric properties of central configurations. We prove that in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of
the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.
