The Falling Pencil and Superfluorescence: Macroscopic Indeterminacies After the Decay of Unstable Equilibria

Abstract

Think of a sharpened pencil placed vertically head-down on a horizontal plane. Will it fall? In practice, it always will. A student of classical mechanics should argue, though, that the upright position corresponds to an unstable equilibrium which is, once realized, infinitely long-lived. The said student could explain the fall by extraneous perturbations such as random drafts of air, vibrations of the ground,etc., or by the practical difficulties of realizing the precise upright position. A mathematician might point out that within the set of positions initially available to the pencil the strictly vertical one is a subset of measure zero. Since the tiniest initial deviation from verticality suffices to destroy the equilibrium it would thus seem hopeless that one would ever be able to realize unstable equilibrium.