Symbolic dynamics for the anisotropic $N$-centre problem at negative energies

TL;DR

This paper establishes symbolic dynamics for the anisotropic N-centre problem at negative energies, using broken geodesics and Maupertuis' functional, addressing challenges from singularities without regularization.

Contribution

It introduces a novel approach to symbolic dynamics in anisotropic N-centre problems at negative energies, handling singularities without regularization.

Findings

Proves symbolic dynamics for anisotropic N-centre problem at negative energies.

Develops a broken geodesics argument for extremals of Maupertuis' functional.

Addresses both collisional and non-collisional trajectories.

Abstract

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 $-\alpha_j$: \[ V_j(x)=|x|^{-\alpha_j}V_j\left(\frac{x}{|x|}\right). \] 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.