## Abstract

This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized as critical point of the Hamiltonian function restricted on the constant angular impulse hyper-surface in the phase space (topologically a pseudo-sphere whose coefficients are the circulation strengths of the vortices). Relative equilibria are generated by the circle action on the so-called shape pseudo-sphere (which generalize the standard shape sphere appearing in the study of the N-body problem). Inspired by the planar N-body problem, and after a geometrical and dynamical discussion of the problem, we investigate the relation intertwining the stability of relative equilibria and the inertia indices of the central configurations generating such equilibria. In the last section we applied our main results to some symmetric three and four vortices relative equilibria.

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

Article number | 76 |

Journal | Qualitative Theory of Dynamical Systems |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 2020 |

## Keywords

- Morse index
- N-vortex problem
- Relative equilibria
- Spectral stability

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics