ar X iv : n lin / 0 50 70 57 v 1 [ nl in . S I ] 2 6 Ju l 2 00 5 REDUCTION AND CHAOTIC BEHAVIOR OF POINT VORTICES ON A PLANE AND A SPHERE

We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincaré surface-of-section technique to perform the reduction a four-vortex system on a sphere. 1. Equations of motion and first integrals of a vortex system on a plane. Let us dwell briefly on the basic forms of the equations and first integrals of the point vortices' motion on a plane. A more complete account can be found in [1, 2], where the hydrodynamic assumptions are also given, under which these equations are valid.