We consider linear Hamiltonian equations in R 2 n of the following type d γ d t ( t ) = J 2 n A ( t ) γ ( t ) , γ ( 0 ) ∈ Sp ( 2 n , R ) , where J = J 2 n = def [ 0 Id n − Id n 0 ] and A : t ↦ A ( t ) is a C 1 curve in the space of 2 n × 2 n real matrices which are symmetric. Then, t ↦ γ ( t ) is a C 2 curve in the space of 2 n × 2 n (real) symplectic matrices. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with algebraic multiplicity two and geometric multiplicity one. As a corollary, we obtain a simple formula about the derivative of the sum of the bifurcated eigenvalues at time t = 0 . In the end, we discuss possible potential applications for the linear stability of the elliptic Lagrangian solutions of the planar three-body problem.
