An estimation for the hyperbolic region of elliptic Lagrangian solutions in the planar three-body problem

It is well known that the linear stability of elliptic Lagrangian solutions 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). Based on new techniques for evaluating the hyperbolicity and the recently developed trace formula for Hamiltonian systems 9], we identify regions for (β, e) such that elliptic Lagrangian solutions are hyperbolic. Consequently, we have proven that the elliptic relative equilibrium of square central configurations is hyperbolic with any eccentricity.