A geometric application of Nambu mechanics: the motion of three point vortices in the plane

Abstract

In this paper the dynamics of three point vortices in the plane is analysed in terms of Nambu mechanics and compared to the classical Hamiltonian dynamics. Two new aspects are introduced: (i) the motion of three point vortices can be classified as non-canonical Nambu mechanics and (ii) Nambu mechanics leads to a geometric representation of the trajectories in an adapted three-dimensional phase space without explicitly solving the differential equations. Thereby, the point vortex trajectory is given by the intersection of two conserved quantities as surfaces in the phase space. These constitutive quantities are the total energy and an angular momentum based Casimir function of the dynamical system. The topological structure of the last surface represents a one- or two- sheeted hyperboloid, a cone or an ellipsoid in the phase space. Examples of the periodic motion and a novel aspect of the collapse of three point vortices in the unbounded plane are discussed. Furthermore, an approach to generalize Nambu mechanics for an arbitrary number of point vortices is proposed.