Symbolic dynamics: from the $N$-centre to the $(N+1)$-body problem, a preliminary study

TL;DR

This paper proves the existence of infinitely many collision-free periodic solutions in a restricted $(N+1)$-body problem with homogeneous potentials, using a variational principle and symbolic dynamics to analyze the system.

Contribution

Introduces a Maupertuis' type variational principle and applies broken geodesics technique to establish new existence results for periodic solutions in a Finslerian setting.

Findings

Existence of infinitely many collision-free periodic solutions.

Application of symbolic dynamics to characterize the system.

Extension of variational methods to Finslerian functionals.

Abstract

We consider a restricted $(N+1)$-body problem, with $N \geq 3$ and homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collision-free periodic solutions with negative and small Jacobi constant and small values of the angular velocity, for any initial configuration of the centres. We will introduce a Maupertuis' type variational principle in order to apply the broken geodesics technique developed in the paper "N. Soave and S. Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete and Cont. Dynamical Systems A, 32 (2012)". Major difficulties arise from the fact that, contrary to the classical Jacobi length, the related functional does not come from a Riemaniann structure but from a Finslerian one. Our existence result allows us to characterize the associated dynamical system with a symbolic dynamics, where the symbols are given partitions of the centres in two non-empty sets.