Stability of equilibrium of conservative systems with two degrees of freedom

This article intends to study the Liapounofʹs stability of an equilibrium of conservative Lagrangian systems with two degrees of freedom. We consider Ω R2 an open neighborhood of the origin and the Lagrangian , where π:Ω→R of class is the potential energy with a critical point at the origin and T:Ω×R2→R is the kinetic energy, of class . We assume that π 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 . We achieve this by proving that there is an asymptotic trajectory to the origin.