Nonplanar Periodic Solutions for Spatial Restricted 3-Body and 4-Body Problems

In this paper, we study the existence of non-planar periodic solutions for the following spatial restricted 3-body and 4-body problems: for $N=2or3$, given any masses $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N$, the mass points of $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N$ move on the $N$ circular obits centered at the center of masses, the sufficiently small mass moves on the perpendicular axis passing the center of masses. Using variational minimizing methods, we establish the existence of the minimizers of the Lagrangian action on anti-T/2 or odd symmetric loop spaces. Moreover, we prove these minimizers are non-planar periodic solutions by using the Jacobi’s Necessary Condition for local minimizers.