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

TL;DR

This paper proves the existence of infinitely many collision-free periodic solutions with negative energy in the planar N-centre problem, using topological, variational, and geometric methods, and characterizes the dynamics via symbolic dynamics.

Contribution

It introduces a new approach to establish infinitely many solutions for the N-centre problem at negative energies, with a novel symbolic dynamics characterization.

Findings

Existence of infinitely many collision-free periodic solutions.

Solutions exist for any distribution of centres within a compact set.

Characterization of the dynamical system through symbolic dynamics.

Abstract

We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets.