Hyperbolicity of elliptic Lagrangian orbits in the planar three body problem

The linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three body problem depends on the mass parameter β = 27(m ₁ m ₂ +m ₂ m ₃ +m ₃ m ₁)/(m ₁ +m ₂ +m ₃)² ∈ 0, 9] and the eccentricity e ∈ 0, 1). In this paper we use Maslov-type index to study the stability of these solutions and prove that the elliptic Lagrangian solutions is hyperbolic for β > 8 with any eccentricity.