Isolated stable initial perturbations of unstable equilibria of some mechanical systems

The unstable equilibrium positions of four mechanical systems are considered. The following systems are chosen: the system with one degree of freedom (an inverted mathematical pendulum), the systems with two degrees of freedom (an inverted spherical pendulum), the system with a countable number of degrees of freedom (an elastic beam loaded by a compressive force greater than the Eulerian critical load), and a continual system (a gravitationally unstable two-layer system of incompressible inviscid liquids). For each of these cases, we find an isolated class of perturbations leading to an exponential tendency to the unstable equilibrium or to an exponential tendency to an oscillatory regime in the initial phase space. The conditions characterizing such a class of perturbations are similar for the four considered systems.