Symmetry of the Restricted 4+1 Body Problem with Equal Masses

We consider the problem of symmetry of the central configurations in the restricted 4+1 body problem when the four positive masses are equal and disposed in symmetric configurations, namely, on a line, at the vertices of a square, at the vertices of a equilateral triangle with a mass at the barycenter, and finally, at the vertices of a regular tetrahedron [1-3]. In these situations, we show that in order to form a non collinear central configuration of the restricted 4+1 body problem, the null mass must be on an axis of symmetry. In our approach, we will use as the main tool the quadratic forms introduced by A. Albouy and A. Chenciner [4]. Our arguments are general enough, so that we can consider the generalized Newtonian potential and even the logarithmic case. To get our results, we identify some properties of the Newtonian potential (in fact, of the function $\varphi(s) = -s^k$, with $k<0$) which are crucial in the proof of the symmetry.