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 beta = 27(m (1) m (2) + m (2) m (3) + m (3) m (1))/(m (1) + m (2) + m (3))(2) a [0, 9] and the eccentricity e a [0, 1). Based on new techniques for evaluating the hyperbolicity and the recently developed trace formula for Hamiltonian systems [9], we identify regions for (beta, 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.