## Abstract

We formulate the problem of finding equilibrium configurations of N-point vortices in the plane in terms of a gradient flow on the smallest singular value of a skewsymmetric matrix M whose nullspace structure determines the (real) strengths, rotational frequency and translational velocity of the configuration. A generic configuration gives rise to a matrix with empty nullspace, and hence is not a relative equilibrium for any choice of vortex strengths. We formulate the problem as a gradient flow in the space of square covariance matrices MTM. The evolution equation for det(MTM) drives the configuration to one with a real nullspace, establishing the existence of an equilibrium for vortex strengths that are elements of the nullspace of the matrix. We formulate both the unconstrained gradient flow problem where the point vortex strengths are determined a posteriori by the nullspace of M and the constrained problem where the point vortex strengths are chosen a priori and one seeks configurations for which those strengths are elements of the nullspace.

Original language | English (US) |
---|---|

Pages (from-to) | 1687-1702 |

Number of pages | 16 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 466 |

Issue number | 2118 |

DOIs | |

State | Published - Jun 8 2010 |

## Keywords

- Gradient flow
- Interacting particle systems
- N-body problems
- Point vortex equilibria
- Relative equilibria

## ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)