Variational methods on periodic and quasi-periodic solutions for the N-body problem

The purpose of this article is two fold. First, we show how quasi-periodic solutions for the N-body problem can be constructed by variational methods. We illustrate this by constructing uncountably many quasi-periodic solutions for the four- and six-body problems with equal masses. Second, we show by examples that a system of N masses can possess infinitely many simple or multiple choreographic solutions. In particular, it is shown that the four-body problem with equal masses has infinitely many double choreographic solutions and the six-body problem with equal masses has infinitely many simple and double choreographic solutions. Our approach is based on the technique of binary decomposition and some variational properties of Keplerian orbits.