Computing hyperbolic choreographies

TL;DR

This paper introduces a numerical algorithm to compute choreographies in hyperbolic spaces with negative curvature, extending previous methods from Euclidean and spherical geometries, and employs optimization techniques in the Poincaré disk.

Contribution

It develops a novel algorithm for hyperbolic choreographies using stereographic projection, trigonometric polynomial approximation, and a two-phase optimization process, expanding the scope of choreographic solutions.

Findings

Discovered new hyperbolic choreographies analogous to previous solutions.

Successfully applied BFGS and Newton methods for high-precision computation.

Extended the computational framework to hyperbolic spaces with negative curvature.

Abstract

An algorithm is presented for numerical computation of choreographies in spaces of constant negative curvature in a hyperbolic cotangent potential, extending the ideas given in a companion paper for computing choreographies in the plane in a Newtonian potential and on a sphere in a cotangent potential. Following an idea of Diacu, P\'{e}rez-Chavela and Reyes Victoria, we use stereographic projection and study the problem in the Poincar\'{e} disk. Using approximation by trigonometric polynomials and optimization methods with exact gradient and exact Hessian matrix, we find new choreographies, hyperbolic analogues of the ones presented in the companion paper. The algorithm proceeds in two phases: first BFGS quasi-Newton iteration to get close to a solution, then Newton iteration for high accuracy.