open access

https://doi.org/10.14943/83908

https://hdl.handle.net/2115/69566

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

英語

Invariant dynamical systems

2005-11-04

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

758

1-20

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.

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