Asymptotic motions and the inversion of the lagrange-dirichlet theorem

Abstract

The motions of natural mechanical systems which tend to an equilibrium position as time increases without limit are studied. The degenerate case when several frequencies of small oscillations vanish is explained. An existence theorem is proved for asymptotic trajectories on the assumption that the Maclaurin series for the potential energy has the form V 2 + V m + V m + 1 + …( V 8 is a homogeneous form of degree s) and the function V 2 + V m does not have a local minimum at the equilibrium position. We proved earlier a claim /1, 2/ about the asymptotic motions for the special case when V 2  0. This theorem is used to solve the question of the existence of asymptotic trajectories in the case of simple and unimodal singularities of the potential energy, for which “canonical” normal forms are known. Similar assertions also hold for the equilibrium positions of gradient dynamic systems. The existence of a trajectory, asymptotic to the equilibrium position, naturally implies that this position is unstable in Lyapunov's sense.