Gareth E. Roberts
A Continuum of Relative Equilibria in the 5-Body Problem
(18K, LATeX 2e)
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.