Since the proof of the Figure-Eight solution by Chenciner and Montgomery, the direct methods of calculus of variations have had a great success in the classical Newtonian N-body problem. First in periodic solutions and more recently in parabolic, hyperbolic and hyperbolic-parabolic solutions. The foundation of many of these results is Marchal's lemma, which guarantee's fixed-end local minimizers are collision-free and hence classical solutions of the problem.

Meanwhile almost no result is available in the restricted problem through the direct methods of calculus of variations. Mainly due to the lack of a Marchal-type lemma.  In this talk,  we will first explain how to obtain  such lemmas for the restricted problem, then how they can be used to study periodic, hyperbolic and parabolic solutions. This is a joint work with Kuo-chang Chen.