Geometric control methods in the study of Mañe perturbations of the linearized Poincare maps

First, we introduce the concept of Mañe perturbation of a time-independent smooth Tonelli Hamiltonian defined on cotangent bundle of a manifold. Such a perturbation is to add a smooth small potential — with respect to $C^\infty$ topology — to a given Hamiltonian; Where by a potential we mean a function that depends only on the base manifold. Then with applying geometric control theory methods, we prove that under Mañe perturbations, the linearized Poincare map of a given closed orbit of a Hamiltonian vector field, reaches an open dense subset in the set of symplectic linear maps. This fact is crucially needful to obtain the bumpy metric like theorem in the sense of Mañe in a given regular energy level. We also will see that in what extend the mentioned assertion is true for non-convex Hamiltonians.