Invariant dynamical systems embedded in the NN-vortex problem on a sphere with pole vortices

We are concerned with the system of NN vortex points on a sphere with two fixed vortex points at the poles. This article gives a reduction method of the system to invariant dynamical systems when all the vortex points have the same strength. It is carried out by considering the invariant property of the system with respect to the shift and pole reversal transformations, for which the polygonal ring configuration of the NN vortex points at the line of latitude, called the “NN-ring”, remains unchanged. We prove that there exists a 2p2p-dimensional invariant dynamical system reduced by the pp-shift transformation for an arbitrary factor pp of NN. The pp-shift invariant system is equivalent to the pp-vortex-points system generated by the averaged Hamiltonian with the modified pole vortices. It is also shown that the system can be reduced by the pole reversal transformation when the pole vortices are identical. Since the reduced dynamical systems are defined in the linear space spanned by the eigenvectors given in the linear stability analysis for the NN-ring, we obtain the inclusion relation among the invariant reduced dynamical systems. This allows us to decompose the system of a large number of vortex points into a collection of invariant reduced subsystems.