Stability of the Thomson vortex polygon with evenly many vortices outside a circular domain

Abstract

We consider the stability problem for the stationary rotation of a regular point vortex ngon lying outside a circular domain. After the article of Havelock (1931), the complete solution of the problem remains unclear in the case 2 ≤ n ≤ 6. We obtain the exhaustive results for evenly many vortices n = 2, 4, 6.

A survey of the stability criteria of Thomson’s vortex polygons outside a circular domain

“…The matrix A N is given for the case of N = 2, 4, 6 in [16] and for the case N = 3, 5 in [12] and [17] respectively. The quadratic terms (11) are represented as…”

“…The variable ζ N is cyclic variable [12,16,17]. The Hamiltonian E(r (ξ, ζ ), θ (ξ, ζ )) does not depend on this variable.…”

“…The similar approaches are used for normalization of quadratic terms of the Hamiltonian in cases N = 2, 4, 6 [16] and in case N = 5 [17].…”

“…, 6, when all eigenvalues of a linearization matrix lie on the imaginary axis. The results of nonlinear analysis were detailed in [12,16,17]. The approach developed in [14,15] was applied to the further solution of the Kelvin problem for vortices outside circular domain.…”

“…The approach developed in [14,15] was applied to the further solution of the Kelvin problem for vortices outside circular domain. The cases of N = 2, 4, 6 were analyzed in the framework of a unified theory [16]. Each of cases N = 3, 5 is divided into a series of the problems which require individual approaches, in particular, applying results of KAM theory and nonlinear analysis of all the resonances up to the fourth order.…”