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
[1]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855

[2]

James Montaldi. Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry. Journal of Geometric Mechanics, 2014, 6 (2) : 237-260. doi: 10.3934/jgm.2014.6.237

[3]

Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439

[4]

David Rojas, Pedro J. Torres. Bifurcation of relative equilibria generated by a circular vortex path in a circular domain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 749-760. doi: 10.3934/dcdsb.2019265

[5]

Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373

[6]

Miguel Rodríguez-Olmos. Continuous singularities in hamiltonian relative equilibria with abelian momentum isotropy. Journal of Geometric Mechanics, 2020, 12 (3) : 525-540. doi: 10.3934/jgm.2020019

[7]

Florian Rupp, Jürgen Scheurle. Classification of a class of relative equilibria in three body coulomb systems. Conference Publications, 2011, 2011 (Special) : 1254-1262. doi: 10.3934/proc.2011.2011.1254

[8]

Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073

[9]

D. J. W. Simpson. On the stability of boundary equilibria in Filippov systems. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3093-3111. doi: 10.3934/cpaa.2021097

[10]

Alain Albouy, Holger R. Dullin. Relative equilibria of the 3-body problem in $ \mathbb{R}^4 $. Journal of Geometric Mechanics, 2020, 12 (3) : 323-341. doi: 10.3934/jgm.2020012

[11]

Marshall Hampton, Anders Nedergaard Jensen. Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations. Journal of Geometric Mechanics, 2015, 7 (1) : 35-42. doi: 10.3934/jgm.2015.7.35

[12]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363

[13]

Carlos Hervés-Beloso, Emma Moreno-García. Market games and walrasian equilibria. Journal of Dynamics & Games, 2020, 7 (1) : 65-77. doi: 10.3934/jdg.2020004

[14]

Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537

[15]

Alessia Marigo. Equilibria for data networks. Networks & Heterogeneous Media, 2007, 2 (3) : 497-528. doi: 10.3934/nhm.2007.2.497

[16]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067

[17]

PaweŁ Hitczenko, Georgi S. Medvedev. Stability of equilibria of randomly perturbed maps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 369-381. doi: 10.3934/dcdsb.2017017

[18]

Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, 2021, 8 (1) : 69-99. doi: 10.3934/jdg.2021002

[19]

Jean-Bernard Baillon, Guillaume Carlier. From discrete to continuous Wardrop equilibria. Networks & Heterogeneous Media, 2012, 7 (2) : 219-241. doi: 10.3934/nhm.2012.7.219

[20]

Sylvain Sorin, Cheng Wan. Finite composite games: Equilibria and dynamics. Journal of Dynamics & Games, 2016, 3 (1) : 101-120. doi: 10.3934/jdg.2016005

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]