On the influence of the kinetic energy on the stability of equilibria of natural Lagrangian systems

Natural Lagrangian systems (T, Pi) on R-2 described by the equation d/dt pa rtial derivative T/partial derivative q -partial derivative T/partial deriv ative q = - partial derivative Pi/partial derivative q are considered, wher e T(q, (q)over dot) is a positive definite quadratic form in (q)over dot an d Pi(q) has a critical point at 0. It is constructively proved that there e xist a C-infinity potential energy Pi and two C-infinity kinetic energies T and (T) over tilde such that the equilibrium q(t) drop 0 is stable for the system (T, Pi) and unstable for the system ((T) over tilde, Pi). Equivalen tly, it is established that for C-infinity natural systems the kinetic ener gy can influence the stability. In the analytic category this is not true.