Nonlinear stability analysis of a regular vortex pentagon outside a circle

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

“…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].…”

“…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…”

“…Regarding the results of the common stability theory of the equilibrium positions of Hamiltonian systems mentioned here and below we refer to the book [19] and review [10]. The vortex triangle and pentagon stability are studied in detail correspondingly in [12] and [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.…”