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In this paper, we study the existence of non-planar periodic solutions forthe following spatial restricted 3-body and 4-body problems: for $N=2 or 3$,given any masses $m_{1},...,m_{N}$, the mass points of $m_{1},...,m_{N}$ moveon the $N$ circular obits centered at the center of masses, the sufficientlysmall mass moves on the perpendicular axis passing the center of masses. Usingvariational minimizing methods, we establish the existence of the minimizers ofthe Lagrangian action on anti-T/2 or odd symmetric loop spaces. Moreover, weprove these minimizers are non-planar periodic solutions by using the Jacobi'sNecessary Condition for local minimizers.

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