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

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

References in corpus (4)

• Periodic solutions and regularization of a Kepler problem with time-dependent perturbation
• Scattering parabolic solutions for the spatial N-centre problem
• Periodic solutions to a forced Kepler problem in the plane
• Parabolic solutions for the planar $N$-centre problem: multiplicity and scattering