Stability of equilibrium of conservative systems with two degrees of freedom

This article intends to study the Liapounof's stability of an equilibrium of conservative Lagrangian systems with two degrees of freedom. We consider Omega subset of R-2 an open neighborhood of the origin and the Lagrangian L = T - pi, where pi : Omega --> R of class l(2) is the potential energy with a critical point at the origin and T:Omega x R-2 --> R is the kinetic energy, of class l(2). We assume that pi has a jet of order k at the origin, and this jet shows that the potential energy does not have a minimum in 0. With these hypotheses we prove that (0;0) is an unstable equilibrium according to Liapounof for the Lagrange equations of L. We achieve this by proving that there is an asymptotic trajectory to the origin. (C) 2003 Elsevier Inc. All rights reserved.