Stability of Relative Equilibria in the Problem of N+1 Vortices

The stability of a regular polygon configuration of N vortices with a central vortex is investigated. When the strength of the central vortex has a value within a certain interval, it is shown that the configuration is locally Liapunov stable. When the stability of the configuration changes, new configurations bifurcate. Although the N+1 body problem of celestial mechanics looks similar, it has been shown there that the change of stability and the bifurcation of new configurations occur for different values of the central mass. Ever since the Adams Prize essay of Thomson, A Treatise on the Motion of Vortex Rings, it was known that the stability of the heptagon could not be decided by the linear terms. With methods from fluid mechanics G. J. Mertz had shown in 1978 that the heptagon is stable. By normalizing the Hamiltonian function we can show that except for rotational symmetry the heptagon is locally Liapunov stable.