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On the stability of Thomson’s vortex configurations inside a circular domain. (English) Zbl 1229.37055

Summary: 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 \(n \geq 7\) and in the case \(2 \leq n \leq 6\) – only if the parameter \(p = R _{0}^{2}/R ^{2}\) is greater than a certain critical value: \(p _{\ast n } < p < 1\). In the present paper the problem of nonlinear stability is studied for all other cases \(0 < p \leq p _{\ast n }, n = 2, \dots , 6\). 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.

MSC:

37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76B47 Vortex flows for incompressible inviscid fluids
34D20 Stability of solutions to ordinary differential equations
70K30 Nonlinear resonances for nonlinear problems in mechanics
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