论文标题
在平面N体问题的周期性轨道上
On Periodic orbits of the Planar N-body Problem
论文作者
论文摘要
通过引入一个新的坐标系，我们证明了N体问题的相对平衡解决方案附近有丰富的新周期轨道。我们仅考虑三体问题的拉格朗日相对平衡和N体问题的Euler-Moulton相对均衡，尽管我们认为，对于大型相对平衡溶液的结果相似。所有这些周期性的轨道都在平面N体问题的2D中心歧管上。除了一个周期性轨道的一个参数家族外，众所周知的是Lyapunov的轨道或Weinstein的轨道外，我们还进一步证明了周期性的轨道出乎意料的轨道：一般而言的相对量度的相对衡量了一组周期性轨道的相对量度，这些周期性轨道近相对平衡液在2d二维中央歧管上的命名或contley in Bround or to to or conte and conte and conte to to to conte to to conte to conte to conte to contecors to concors to contecors to contecors to concors to conbits。基于Conley和Zender的扩展结果，该结果是对哈密顿量椭圆平衡点附近的周期性轨道的局部存在结果。特别是，结果提供了一些证据，以支持庞加莱关于N体问题的定期轨道的众所周知的主张。
By introducing a new coordinate system, we prove that there are abundant new periodic orbits near relative equilibrium solutions of the N-body problem. We consider only Lagrange relative equilibrium of the three-body problem and Euler-Moulton relative equilibrium of the N-body problem, although we believe that there are similar results for general relative equilibrium solutions. All of these periodic orbits lie on a 2d-dimensional central manifold of the planar N-body problem. Besides d one parameter family of periodic orbits which are well known as Lyapunov's orbits or Weinstein's orbits, we further prove that periodic orbits are unexpectedly abundant: generically the relative measure of the closure of the set of periodic orbits near relative equilibrium solutions on the 2d-dimensional central manifold is close to 1. These abundant periodic orbits are named Conley-Zender's orbits, since to find them is based on an extended result of Conley and Zender on the local existence result for periodic orbits near an elliptic equilibrium point of a Hamiltonian. In particular, the results provide some evidences to support the well known claim of Poincaré on the conjecture of periodic orbits of the N-body problem.
