Abstract
Abstract Periodic solutions of the N-body problem have been extensively studied for centuries. There is an extensive literature on the existence and nature of periodic solutions of the N-body problem. For the history and classical methods on this problem, we refer to the book K. Meyer [7]. In 2000 [4], Chenciner and Venturelli use calculus of variations to prove the existence of non-planar periodic solutions for the 4-body problem which they called the hip-hop solutions. In 2002, Chenciner show the existence of some generalized hip-hop solutions. For recent progress in this direction, we refer to Chenciner [5] and the references therein. In this paper we discuss the existence of hip-hop solutions for the nested N-body problem. In a paper by Barrabe’s, et al [1], they prove the existence of some hip-hop solutions for the 2N-body problem. In this article, they use the argument of analytic continuation to show that, by adding vertical oscillations to the circular motion of 2N bodies with equal mass sitting on vertices of a regular 2N-gon, the solution can be continued to 3-dimensional hip-hop solutions. These solutions ate periodic in a rotating frame. In this thesis, we also apply continuation method and conclude a more general case. In section 3, we discuss the case of two nested circles, each with 2N bodies sitting on vertices of a regular 2N- gon. We show that there exists a family of periodic solutions bifurcating from the planar solution. In the last section, we extend the number of circles and formulate the problem in a general form.