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Эта публикация цитируется в 11 научных статьях (всего в 11 статьях)
On the stability of Thomson’s vortex configurations inside a circular domain
L. G. Kurakinab a Southern Mathematical Institute VNTs RAN, 362027 Vladikavkaz, Russia
b Faculty of mathematics, mechanics and computer science, Southern Federal University, Milchakova ul. 8a, 344090 Rostov-on-Don, Russia
Аннотация:
The paper is devoted to the analysis of stability of the stationary rotation of a system of $n$
identical point vortices located at the vertices of a regular $n$-gon of radius $R_0$ inside a circular domain of radius $R$.
Havelock stated (1931) that the corresponding linearized system has exponentially growing
solutions for $n\geqslant 7$ and in the case $2\leqslant n \leqslant6$ — only if the parameter $p={R_0^2}/{R^2}$ is greater than a certain
critical value: $p_{*n}<p<1$. In the present paper the problem of nonlinear stability is studied for all other cases $0<p\leqslant p_{*n},$ $n=2,\dots,6$.
Necessary and sufficient conditions for stability and instability for $n\neq 5 $ are formulated. A detailed proof for
a vortex triangle is presented. A part of the stability conditions is substantiated by the fact that the relative
Hamiltonian of the system attains a minimum on the trajectory of the stationary motion of the vortex triangle. The case
where the sign of the Hamiltonian is alternating requires a special approach. The analysis uses results of
KAM theory. All resonances up to and including the 4th order occurring here are enumerated and investigated. It has turned out
that one of them leads to instability.
Ключевые слова:
point vortices, stationary motion, stability, resonance.
Поступила в редакцию: 03.03.2009 Принята в печать: 17.05.2009
Образец цитирования:
L. G. Kurakin, “On the stability of Thomson’s vortex configurations inside a circular domain”, Regul. Chaotic Dyn., 15:1 (2010), 40–58
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd471 https://www.mathnet.ru/rus/rcd/v15/i1/p40
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