The Characterization of the Variational Minimizers for Spatial Restricted N+1-Body Problems

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restricted N+1-body problems with a zero mass on the vertical axis of the plane for N equal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a nonconstant periodic solution for any 2≤N≤472; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. This result contradicts with our intuition that the small mass should always be at the origin.