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 中文摘要數百年以來有關N 體問題的週期解已經被廣大的研討。現今而言，存在了許多文獻討論某些解的存在性。就歷史方面相關的知識和古典方法我們可以參考K. Meyer 的書[7]。在2000年, Chenciner 和 Venturelli 用變分法證明了四體問題非平面週期解的存在性，在那時候他們稱這種解叫 hip-hop solution [4]。到了2002年，Chenciner 又證明了一些廣義 hip-hop solution 解存在性 [2]。而有關多體問體這方面領域最近的發展我們可參考 Chenciner 所寫的文獻 [5]。在這篇論文裡，我們主要討論的是巢狀多體問題的擾動解。最近Barrabe’s, … [1] 等人用龐加萊延拓法共同證明了如果對於在正2N邊形頂點上的2N個質點給一個適當鉛直方向的擾動，則這2N個質點由原本平面的圓周運動變為三度空間週期的運動。在這論文裡我們主要也是運用龐加萊延拓法來分析並且試著得到更一般的結果。前兩節就這篇論文會用到的定理作一些簡介。在第三節裡我們就4N個質點分別落在同心圓上的正2N邊形上討論，並說明在對這4N個質點擾動後會有週期解產生。而在第四，第五和第六節我們把圓的個數增加並且系統地闡述問題的一般性。
 AbstractPeriodic 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.
 Contents1. The N-body problem 1.1 Newton's Equations and Integrals of Motions 1.2 Choreographic Solution and the Hip-hop solution2. Some result of ordinary differential equations 2.1 Linearization 2.2 Continuation of the periodic solution and cross section3. The hip-hop solutions of the 4N-body problem(1) 3.1 Existence 3.2 Hip-hop solution of the 4N-body problem 3.2.1 The equations of motion 3.2.2 Linearization 3.2.3 The existence of periodic solutions after small perturbations4. The hip-hop solutions of the 4N-body problem(2)5. The hip-hop solutions of the 2N-body problem6. The nested 2N-body problem7 Reference
 Reference[1] Esther , Josep Maria Cors, Conxita Pinyol and JaumeSoler. Hip-Hop solutions of the 2N-body problem, toappear in Celestial Mechanics and Dynamical Astronomy[2] Chenciner A., Action minimizing solutions of theNewtonian n-body problem: from homology to symmetry.Proceedings of the International Congress ofMathematicians (Beijing, 2002). Vol.III, 279-294.[3] Chenciner, A.; Montgomery, R “A remarkable periodicsolution of the three body problem in the case of equalmasses”, Annals of Math. 152(2000), 881-901.[4] Chenciner, A.; Venturelli, A., Minima de d’actiondu newtonien de 4 corps de masses dans : orbites‘‘hip-hop’’, Celestial Mech. Dynam. Astronomy. 77(2)(2001), 139-152.[5] Chenciner, A., Symmetries and ‘‘simple’’ solutionsof the classical N-body problem. Proceedings of theICMP, Lisbonne, 20003, World Scientific.[6] W.A. Coppel: Stability and Asymptotic Behavior ofDifferential Equations.[7] Kenneth R. Meyer, Periodic Solutions of the N-bodyProblem, Lecture notes in Mathematics, Springer-Verlag.1993.[8] Kenneth R. Meyer and Hall, G. R, Introduction to Hamiltonian dynamical systems and the N-body problem, Vol. 90 of Applied Mathematical Sciences, Springer-Verrlag, New York. 1992.[9] Carles . New families of solutions in N-body problems, European Congress of Mathematics, Vol. I (Barcelona, 2000). Progr. Math., 201 , Basel. PP.101-115, 2001.
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