98-554 Gareth E. Roberts

A Continuum of Relative Equilibria in the 5-Body Problem (18K, LATeX 2e) Aug 6, 98

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Abstract. It is generally believed that the set of relative equilibria equivalence classes in the Newtonian $n$-body problem, for a given set of positive masses, is finite. However, the result has only been proven for $n=3$ and remains a difficult, open question for $n \geq 4$ (Wintner~\cite{cc:wint}, Smale~\cite{cc:smale1}). We demonstrate that the condition for the masses being positive is a necessary one by finding a continuum of relative equilibria in the five-body problem which (unfortunately) includes one negative mass. This family persists in similar potential functions, including the logarithmic potential used to describe the motion of point vortices in a plane of fluid.

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