负能量下各向异性$N$ -中心问题的符号动力学
Symbolic dynamics for the anisotropic $N$-centre problem at negative energies
摘要：
平面$N$ -中心问题描述了在$N$固定吸引中心的力场作用下在平面内运动的粒子的运动：\[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \]在本文中，我们证明了$N$ -中心问题在稍负能量下的符号动力学，其中势$V_j$是正的、各向异性的和均匀的$-α_j$：\[ V_j(x)=|x|^{-α_j}V_j\left(\frac{x}{|x|}\right). \]证明是基于一个破碎的测地线参数和轨迹的莫佩屠斯'功能的极值。与经典的$N$中心问题与开普勒潜力相比，一个主要的困难来自缺乏正则化的奇异性。我们将考虑碰撞动力学和非碰撞动力学。符号描述相关轨迹的几何和拓扑特征。
英文摘要：
The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \] In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-α_j$: \[ V_j(x)=|x|^{-α_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis' functional. Compared with the classical $N$-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.
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