Illustrating Lyapunov’s analysis of asymptotic stability for a heavy particle affected by drag resistance

Abstract

The application of Lyapunov’s criterion for the asymptotic stability of equilibrium in mechanical systems is difficult in general because the natural candidate for the role of Lyapunov function, the mechanical energy, is suitable for determining simple stability but not asymptotic stability, unless the energy is modified in a way that is nontrivial, difficult to standardize, and whose physical interpretation is obscure. A simple class of mechanical systems is discussed for which the global asymptotic stability can be checked by means of simple physically reasonable Lyapunov functions. A comparison of alternative Lyapunov functions is also done when possible. The class of systems includes the motion of a heavy body moving through the atmosphere close to the Earth’s surface.