Introduction

Abstract

Problems of stability appear for the first time in mechanics during the investigation of an equilibrium state of a system. A simple reflection may show that some equilibrium states of a system are stable with respect to small perturbations, whereas other balanced states, although available in principal, cannot be realized in practice. Thus, for instance, when a pendulum is in its lowest position any small perturbations will result only in its oscillation about this position. However, if after some effort we can set the pendulum at its highest position, then any push will cause its downfall. Certainly, the question of stability in this case is resolved in an elementary manner, but in general, the conditions under which the equilibrium state of a system will be stable are not always as clear. The criterion for stability of rigid bodies in equilibrium under gravitational forces was formulated by E. Torricelli in 1644. In 1788, G. Lagrange proved a theorem that defines sufficient conditions for stability of equilibrium of any conservative system (see Section 3.1).