The influence of the kinetic energy in equilibrium of Hamiltonian systems

We provide a simple and explicit example of the influence of the kinetic energy in the stability of the equilibrium of classical Hamiltonian systems of the type H(q,p)=+π(q). We construct a potential energy π of class Ck with a critical point at 0 and two different positive defined matrices B1andB2, both independent of q, and show that the equilibrium (0,0) is stable according to Lyapunov for the Hamiltonian H1=+π(q), while for H2=+π(q) the equilibrium is unstable. Moreover, we give another example showing that even in the analytical situation the kinetic energy has influence in the stability, in the sense that there is an analytic potential energy π and two kinetic energies, also analytic, T1 and T2 such that the attractive basin of (0,0) is a two-dimensional manifold in the system of Hamiltonian π+T1 and a one-dimensional manifold in the system of Hamiltonian π+T2.