Topological manifestations in classical mechanics: discrete allowed and forbidden states of motion

Abstract

Consequences of the topology of the configuration space of a Hamiltonian dynamical system are considered for a coherent system of trajectories. It is shown that when the space is multiply-connected and therefore the action integral is multivalued, the allowed states of motion (labeled by the initial data) are constrained to a discrete set by the requirement that the action be single-valued. One thus obtains a quantum-like discretization of allowed states of motion even in classical mechanics. Such discrete "allowed" and "forbidden" states have indeed been observed in the classical mechanical system of charged particles in a magnetic field. The relationship of this formalism with a Schrödinger-like formalism for the latter problem given earlier is discussed.