Variational methods and periodic solutions of Newtonian N-body problems

The objective of this thesis is to study periodic solutions of the Newtonian N-body problem by variational methods. Variational methods had been successfully applied to the N-body problem in recent years. The most remarkable success is the "figure-8" orbit found by C. Moore, A. Chenciner and R. Montgomery. On this orbit all masses chase each other along a eight-shaped circuit without collision, and the solution curve passes through every Euler's configuration periodically. A. Chenciner and R. Montgomery proved that each piece of the curve connecting Euler's configuration to the nearby isosceles triangular configurations is the minimizer of the action functional on some path spaces. In this thesis we will provide a simpler proof for their result. Another application of variational methods is on the parallelogram four-body problem. We will prove the existence of a periodic solution whose configuration changes from square to collinear periodically and remains a parallelogram for all time. This orbit consists of two identical star-shaped simple closed curves, these two closed curves intersect at vertices of a square with their major axis perpendicular to each other. Two masses travel on one closed curve while the other two masses travel on the other curve along opposite orientation. The proof is also based on a minimization problem for the action functional over a suitable path space. This approach makes it possible to show analytically the existence of many new classes of periodic solutions found by computer simulations.