### Abstract

We give a detailed analytical description of the global dynamics of N points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group Dl of order 2l. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as McGehee transformation. After performing this change of coordinates that regularizes the total collision, we study the rest-points of the ow, the invariant manifolds and, with the help of a computer algebra system, we derive interesting information about the global dynamics for l = 2. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.

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

Pages (from-to) | 3011-3042 |

Number of pages | 32 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 33 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2013 |

### Fingerprint

### Keywords

- Dihedral N-vortex filaments
- Global dynamics
- Logarithmic potential
- McGehee coordinates
- N-body problems

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*33*(7), 3011-3042. https://doi.org/10.3934/dcds.2013.33.3011

**Geometry of stationary solutions for a system of vortex filaments : A dynamical approach.** / Paparella, Francesco; Portaluri, Alessandro.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems- Series A*, vol. 33, no. 7, pp. 3011-3042. https://doi.org/10.3934/dcds.2013.33.3011

}

TY - JOUR

T1 - Geometry of stationary solutions for a system of vortex filaments

T2 - A dynamical approach

AU - Paparella, Francesco

AU - Portaluri, Alessandro

PY - 2013/7/1

Y1 - 2013/7/1

N2 - We give a detailed analytical description of the global dynamics of N points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group Dl of order 2l. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as McGehee transformation. After performing this change of coordinates that regularizes the total collision, we study the rest-points of the ow, the invariant manifolds and, with the help of a computer algebra system, we derive interesting information about the global dynamics for l = 2. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.

AB - We give a detailed analytical description of the global dynamics of N points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group Dl of order 2l. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as McGehee transformation. After performing this change of coordinates that regularizes the total collision, we study the rest-points of the ow, the invariant manifolds and, with the help of a computer algebra system, we derive interesting information about the global dynamics for l = 2. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.

KW - Dihedral N-vortex filaments

KW - Global dynamics

KW - Logarithmic potential

KW - McGehee coordinates

KW - N-body problems

UR - http://www.scopus.com/inward/record.url?scp=84872101355&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872101355&partnerID=8YFLogxK

U2 - 10.3934/dcds.2013.33.3011

DO - 10.3934/dcds.2013.33.3011

M3 - Article

AN - SCOPUS:84872101355

VL - 33

SP - 3011

EP - 3042

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 7

ER -