# American Institute of Mathematical Sciences

May  2007, 19(2): 411-418. doi: 10.3934/dcds.2007.19.411

## N-vortex equilibrium theory

 1 Department of Aerospace & Mechanical Engineering and Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1191

Received  August 2006 Revised  February 2007 Published  July 2007

The problem of finding and classifying all relative equilibrium configurations of $N$-point vortices in the plane is first described as a classical variational principle and then formulated as a problem in linear algebra. Given a configuration of $N$ points in the plane, one must understand the structure of the $N(N-1)/2 \times N$ configuration matrix $A$ obtained by requiring that all interparticle distances remain fixed in time. If the determinant of the square, symmetric $N \times N$ covariance matrix $A^T A$ is zero, there is a non-trivial nullspace of $A$ and a basis set for this nullspace can be used to determine all vortex strengths $\vec{\Gamma} \in R^N$ for which the configuration remains rigid. Optimal basis sets are obtained by using the singular value decomposition of $A$ which allows one to categorize exact equilibria, approximate equilibria, and the distance between different equilibria in the appropriate vector space, as characterized by the Frobenius norm.
Citation: P.K. Newton. N-vortex equilibrium theory. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 411-418. doi: 10.3934/dcds.2007.19.411
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