open access

https://hdl.handle.net/2115/8557

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

英語

Invariant dynamical systems

2006-05

Elsevier

Physica D: Nonlinear Phenomena

217(2)

142-152

PACS: 47.32.Cc 47.20.Ky,05.45.-a

We are concerned with the system of N 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 N vortex points at the line of latitude, called "N-ring", remains unchanged. We prove that there exists a 2p-dimensional invariant dynamical system reduced by the p-shift transformation for arbitrary factor p of N. The p-shift invariant system is equivalent to the p-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 N-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.

学術雑誌論文

Accepted Manuscript

0167-2789

AA00362211