Classical Geometric Phases: Foucault and Euler

Abstract

In the last chapter we saw how a quantum system can give rise to a Berry phase, by studying the adiabatic round trip of its quantum state on a certain parameter space. Rather than considering what happens to states in Hilbert space, we now turn to classical mechanics, where we are concerned instead with the evolution of the system in configuration space. As a first example, we are interested in the geometric phase of an oscillator that is constrained to a plane that is transported over some surface which moves along a certain path in three-dimensional space. Contrary to determining the Berry phase, there is no adiabatic approximation of the motion along the curve involved. The Foucault phase or the Euler phase are neither adiabatic nor approximate. Therefore they have nothing to do with Berry or with Hannay. In the case of the Foucault pendulum, the circular path around a fixed latitude is closed exactly and can be traversed independently of the speed of parametrization, just like the path in the case of the Eulerian circuit for the free top.