Physics – Mathematical Physics
Scientific paper
1999-10-07
Physics
Mathematical Physics
Minor typographical corrections and slightly revised introduction. 9 pages, 5 figures. To appear EQUIDIFF/99 proceedings
Scientific paper
The system of four point vortices in the plane has relative equilibria that behave as composite particles, in the case where three of the vortices have strength $-\Gamma/3$ and one of the vortices has strength $\Gamma$. These relative equilibria occur at nongeneric momenta. The reduction of this system, at those momenta, by continuous and then discrete symmetries, classifies the 4-vortex states which have been observed as products of collisions of two such composite particles. In this article I explicitly calculate these reductions, and show they are qualitatively identical one degree of freedom systems on a cylinder. The flows on these reduced systems all have one stable equilibrium and one unstable equilibrium, and all the orbits are periodic except for two homoclinic connections to the unstable equilibrium.
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