VORTEX MOTION AND THE GEOMETRIC PHASE - PART I - BASIC CONFIGURATIONSAND ASYMPTOTICS

The geometric, or Hannay-Berry, phase is calculated for three canonica l point vortex configurations in the plane. The simplest configuration is the three-vortex problem with arbitrary (like signed) circulations , where two of the vortices are near each other compared to the distan ce between them and a third vortex. We show that the third (distant) v ortex induces a geometric phase in the relative angle variable between the two nearby vortices. The second configuration is a particle and v ortex in a circular domain. In this problem, the geometric phase is in duced on the particle by the presence of the boundary. The third confi guration is an infinite row of point vortices undergoing subharmonic p airing. In this case, a geometric phase is induced on a particle orbit ing one of the vortices as the vortex pairs orbit each other. In each case we derive the formula for the geometric phase using an asymptotic procedure, then we give it a geometric interpretation. For the asympt otic derivation, we show how the geometric phase can be interpreted as the product of two terms, one of which goes to zero, the other to inf inity. Because they go at rates that balance each other, there is a re sidual O(1) term in the limit epsilon --> 0. In this way, we can see t hat the epsilon --> 0 problem is fundamentally different from the epsi lon = 0 problem. For the geometric interpretation, we introduce a 1-fo rm gamma defined on the plane and show that the phase theta(g) in the appropriate angle variable can be constructed by taking the contour in tegral of this 1-form over the closed vortex path. In each case this g ives the simple formula theta(g) = closed integral gamma.