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

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 Title: Invariant dynamical systems embedded in the N-vortex problem on a sphere with pole vortices Authors: SAKAJO, Takashi Browse this author Keywords: Vortex points Flow on a sphere Reduction method Invariant dynamical systems Issue Date: 4-Nov-2005 Publisher: Department of Mathematics, Hokkaido University Journal Title: Hokkaido University Preprint Series in Mathematics Volume: 758 Start Page: 1 End Page: 20 Abstract: We are concerned with the system of the $N$ vortex points on a sphere with two fixed vortex points at the both poles. This article gives a reduction method of the system to invariant dynamical systems. It is accomplished by using the invariance of the system with respect to the shift and the pole reversal transformations, for which the polygonal ring configuration of the $N$ vortex points at the line of latitude, called $N$-ring'', remains unchanged. We prove that there exists the $2p$-dimensional invariant dynamical system reduced by the $p$-shift transformation for arbitrary factor $p$ of $N$, and the $p$-shift invariant system is equivalent to the $p$-vortex points system generated by the averaged Hamiltonian on the sphere 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 by the linear combination of the eigenvectors obtained in the linear stability analysis for the $N$-ring, we obtain the inclusion structure among the invariant reduced dynamical systems, which allows us to decompose the system of the large vortex points into a collection of small reduced systems. Type: bulletin (article) URI: http://hdl.handle.net/2115/69566 Appears in Collections: 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics