Abstract
Beginning with general remarks on vortex systems with zero net circulation, the three-vortex problem is considered and solved explicitly in the following special cases: 1. The distance between two vortices is small compared to the distance to the third, with two subcases: the neighboring vortices (i) have the same sign, (ii) have opposite signs. 2. The impulse of the system is small compared to the product of a typical vortex strength and the biggest distance found (at any time) between vortices. Formulas and rules for the construction of the orbits in case 2 are given; they include the circular paths that are found for zero impulse. These special cases are typical for the orbits in the different regions found between the separatrices of the Hamiltonian. Transitional paths are discussed in the case where the vortex circulations have the ratio 1∶1∶−2. In this case, it is shown (by inspection) that the Hamiltonian is a harmonic function. Actually, the Hamiltonian is a harmonic function whenever the sum of the three circulations is zero, as shown in an “Addendum” by Prof. Hassan Aref. This is an essential result for the full understanding of the problem under consideration.
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