2016/08/24 by Thomas Bartsch, Bartsch, Thomas, Matteo Sacchet +1
Mathematics · #34C25 #37E40 #37J45 #37N10 #76B47 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:34C25 #msc:37E40 #msc:37J45 #msc:37N10 #msc:76B47
paper · pdf · doi:10.48550/arxiv.1608.06775
17 pages
arxiv created 2017/07/14 · arxiv updated 2017/07/17
We consider Hamiltonian systems with two degrees of freedom of point vortex type κj zj = J ∇zj HΩ(z1,z2), j=1,2, for z1,z2 in a domain Ω⊂ℝ2. In the classical point vortex context the Hamiltonian HΩ is of the form HΩ(z1,z2) = -\fracκ1 κ2π log |z1-z2| - 2κ1 κ2g(z1,z2) - κ12 h(z1) - κ22 h(z2), where g:Ω×Ω→ℝ is the regular part of a hydrodynamic Green function in Ω, h:Ω→ℝ is the Robin function: h(z)=g(z,z), and κ1, κ2 are the vortex strengths. We prove the existence of infinitely many periodic solutions with prescribed minimal period that are superpositions of a slow motion of the center of vorticity close to a star-shaped level line of h and of a fast rotation of the two vortices around their center of vorticity. The proofs are based on a recent higher dimensional version of the Poincaré-Birkhoff theorem due to Fonda and Ureña.