Marchal-type lemmas for the restricted $(N+1)$-body problem with their applications

The direct methods of calculus of variations have had a great success in the classical Newtonian $N$-body problem in this century. First in finding periodic solutions and more recently in the existence of 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 $N$-body 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 paper such a lemma and its variation will be obtained for the restricted problem. We then apply them to the restricted problem to study unbounded (hyperbolic, parabolic, bi-hyperbolic and bi-parabolic) solutions, and bounded (periodic and quasi-periodic) solutions. We emphasize that the extension to the restricted case does not follow from the standard arguement for the unrestricted case.