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On relative equilibria and integrable dynamics of point vortices in periodic domains

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Abstract

The motion of two point vortices defines an integrable Hamiltonian dynamical system in either singly or doubly periodic domains. The motion of three point vortices in these domains is also integrable when the net circulation is zero. The relative vortex motion in both domains can be reduced to advection of a passive particle by fixed vortices in an equivalent Hamiltonian system. A survey of the solutions for vortex motion in these systems is discussed. Some initial conditions lead to relative equilibria, or vortex configurations that move without change of shape or size. These configurations can be determined as stagnation points in the reduced problem or through explicit solution of the governing equations. These periodic point-vortex systems present a rich collection of interesting solutions despite the few degrees of freedom, and several questions on this subject remain open.

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  1. Department of Engineering Science and Mechanics, Virginia Tech, Mail Code 0219, Blacksburg, VA, 24061, USA

    Mark A. Stremler

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  1. Mark A. Stremler
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Communicated by H. Aref

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Stremler, M.A. On relative equilibria and integrable dynamics of point vortices in periodic domains. Theor. Comput. Fluid Dyn. 24, 25–37 (2010). https://doi.org/10.1007/s00162-009-0156-z

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