Geometric Phases

Abstract

In this chapter we introduce the Hannay angle, adiabatic Hannay angle, or geometric phase ¹for classical systems and describe how it arises in vortex dynamics problems. The chapter opens with three simple examples exhibiting a classical Hannay angle. We first outline the use of multiscale perturbation methods to compute phases in slowly varying Hamiltonian systems. These kinds of calculations go back at least to Volosov (1963), who developed averaging techniques in this and other contexts. In Section 5.3 we give a formal definition of the Hannay angle and adiabatic Hannay angle following the work of Golin (1988), Golin and Marmi (1990), and Golin, Knauf, and Marmi (1990). Our main example in this section is the restricted 3-vortex problem described in Newton (1994), where multiscale perturbation methods are used to derive a geometric phase formula. Section 5.4 details the Hannay angle calculation for the general 3-vortex problem following the work of Shashikanth and Newton (1998). We also give an interpretation based on the limiting adiabatic geometry of the problem, which leads to a formula for the geometric phase reminiscent of the bead-on-hoop formula. The final Section 5.5 describes two applications of these ideas. We first describe work developed in Shashikanth and Newton (1999) in which the growth rate of a passive fluid interface in certain vortex configurations can be related to the geometric phase.