Variational Method For Solving The N-body Problem In Newtonian Periodic Solution

Based on the work of Zhang[30], we study planar cicular restricted 3-body and 4-body problems for the Newtonian gravitational potentials respectively, and under some restrictions on masses, we prove the existence of collision-free, non homographic symmetric solutions with some topological conditions. These solutions have some symmetry, which is introduced in Bessi-Coti Zelati([4]), and taking account of a topological constraint. The argument exploits the variational structure of the problem, and is based on the minimization of lagrangian action on a given class of paths.