Deformation of geometry and bifurcation of vortex rings

2012/10/20 by James Montaldi, Tadashi Tokieda, Montaldi, James +1
Engineering · Mathematics · #37J15 #37J20 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Material Science and Thermodynamics #math.DS #msc:37J15 #msc:37J20

paper · pdf · doi:10.48550/arxiv.1210.5662

26 pages

arxiv created 2012/10/20 · openalex publication_date 2012/10/20 · arxiv updated 2012/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.

Related

• Point vortices on the sphere: stability of relative equilibria
• Stability of Thomson's Configurations of Vortices on a Sphere
• Point vortices on a rotating sphere
• Point vortex on surfaces with continuous symmetry
• A Hamiltonian approach for point vortices on non-orientable surfaces