Maslov-Type Indices and Linear Stability of Elliptic Euler Solutions of the Three-Body Problem

Abstract

In this paper, we use the central configuration coordinate decomposition to study the linearized Hamiltonian system near the 3-body elliptic Euler solutions. Then using the Maslov-type ω-index theory of symplectic paths and the theory of linear operators we compute the ω-indices and obtain certain properties of linear stability of the Euler elliptic solutions of the classical three-body problem.

Linear stability of elliptic relative equilibria of restricted four-body problem

“…We paraphrase their results in our notations. For details, readers may refer to [38] for details. When α, β > 0, the operator A(α, β, e) can be written as…”

“…Similarly, we have ±1 ∈ σ(M 1 ). Then by Lemma 5.6 in Appendix 5.2 of [38], we have M 1 = D(−2) or M 1 = R(θ) for some θ ∈ (0, π) ∪ (π, 2π). If M 1 = D(−2), by the properties of splitting Theorem 7.11.…”

“…In [6] and [38], the authors used Maslov-type index and operator theory to study the linear stability, and obtained a full description of the bifurcation diagram. For the near-collision Euler solutions of 3-body problem, the linear stability was studied by Hu and Ou in [8].…”

Trace estimation of a family of periodic Sturm-Liouville operators with application to Robe’s restricted three-body problem

“…We will consider a family of periodic Sturm-Liouville operators A(β, e) = − d 2 dt 2 − 1 + Then it is self-adjoint. Such an operator was studied in [3], [11] and [13]. We conclude their main results here: (1) If β > 1, A(β, e) is positive definite on D(ω, 2π); (2) A(1, e) is positive definite on D(ω, 2π) when ω = 1;…”

“…We prove first that i ω (γ β,e ) = 0 when β is near 1. By Lemma 4.1(ii) in [13], A(1, e) is positive definite on D(ω, 2π) when ω = 1. Therefore, there exists an ǫ > 0 small enough, which may depends on e and ω, such that A(β, e) is also positive definite on D(ω, 2π) when 1 − ǫ < β ≤ 1.…”