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Point vortices on a sphere: Stability of relative equilibria

Pekarsky, Sergey and Marsden, Jerrold E. (1998) Point vortices on a sphere: Stability of relative equilibria. Journal of Mathematical Physics, 39 (11). pp. 5894-5907. ISSN 0022-2488. doi:10.1063/1.532602.

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In this paper we analyze the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are worked out for the (integrable) case of three vortices. The system under consideration is SO(3) invariant; the associated momentum map generated by this SO(3) symmetry is equivariant and corresponds to the moment of vorticity. Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy-momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original (unreduced) phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle.

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Additional Information:©1998 American Institute of Physics. Received 27 January 1998; accepted 10 March 1998. We would like to thank Paul Newton for helpful discussions and for insightful remarks on vortex dynamics. We also thank Anthony Blaom, Serge Preston and Tudor Ratiu for their helpful comments and advice on this and related work.
Subject Keywords:vortices; flow instability; SO(3) groups; transforms; geometry
Issue or Number:11
Record Number:CaltechAUTHORS:PEKjmp98
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3610
Deposited By: Tony Diaz
Deposited On:21 Jun 2006
Last Modified:08 Nov 2021 20:10

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