Second order asymptotics for Krein indefinite multipliers with multiplicity two

We consider linear Hamiltonian equations in R2n of the following typedγdt(t)=J2nA(t)γ(t),γ(0)∈Sp(2n,R), where J=J2n=def[0Idn−Idn0] and A:t↦A(t) is a C1 curve space 2n×2n real matrices which are symmetric. Then, t↦γ(t) C2 (real) symplectic matrices. obtain second order asymptotics for eigenvalues bifurcated from non-real Krein indefinite with algebraic multiplicity two geometric one. As corollary, we simple formula about derivative sum at time t=0. In end, discuss possible potential applications stability elliptic Lagrangian solutions planar three-body problem.