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.

中文翻译：

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某些机械系统不稳定平衡的孤立稳定初始扰动

考虑了四个机械系统的不稳定平衡位置。选择以下系统：具有一个自由度的系统（一个倒立的数学摆），具有两个自由度的系统（一个倒立的球形摆），具有多个自由度的系统（一个弹性梁加载）大于欧拉临界载荷的压缩力）和连续系统（不可压缩的无粘性液体的重力不稳定两层系统）。对于这些情况中的每一个，我们发现一个孤立的扰动类，它们导致初始相空间中不稳定平衡的指数趋势或振荡状态的指数趋势。对于四个所考虑的系统，表征此类摄动的条件是相似的。