On the influence of the kinetic energy on the stability of equilibria of natural Lagrangian systems
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Natural Lagrangian systems (T,Π) on R 2 described by the equation are considered, where is a positive definite quadratic form in and Π(q) has a critical point at 0. It is constructively proved that there exist a C ∞ potential energy Π and two C ∞ kinetic energies T and such that the equilibrium q(t)≡ 0 is stable for the system (T,Π) and unstable for the system . Equivalently, it is established that for C ∞ natural systems the kinetic energy can influence the stability. In the analytic category this is not true.