In this paper, we prove that the second order differential equation d^2x/dt^2＋x^2n_1f（x）＋p（t）=0with p（t ＋ 1） = p（t）, f（x ＋ T） = f（x） smooth and f（x） 〉 0, possesses Lagrangian stability despite of the fact that the monotone twist condition is violated.
In this paper, we study the planar Hamiltonian systemwhere where f is real analytic in x and θ,A（θ） is a 2×2real analytic symmetric matrix,J=（1^-1） and w is a Diophantine vector. Under the assumption that the unperturbed systemreducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system.
