Stability of the Planar Equilibrium Solutions of a Restricted $1+N$ Body Problem

We started our studies with a planar Eulerian restricted four-body problem (ERFBP) where three masses move in circular orbits such that their configuration is always collinear. The fourth mass is small and does not influence the motion of the three primaries. In our model we assume that one of the primaries has mass 1 and is located at the origin and two masses of size $\mu$ rotate around it uniformly. The problem was studied in [3], where it was shown that there exist noncollinear equilibria, which are Lyapunov stable for small values of $\mu$. KAM theory is used to establish the stability of the equilibria. Our computations do not agree with those given in [3], although our conclusions are similar. The ERFBP is a special case of the $1+N$ restricted body problem with $N=2$. We are able to do the computations for any $N$ and find that the stability results are very similar to those for $N=2$. Since the $1+N$ body configuration can be stable when $N>6$, these results could be of more significance than for the case $N=2$.