Computing planar and spherical choreographies

TL;DR

This paper introduces an algorithm for numerically computing choreographies in planar and spherical settings, utilizing stereographic projection, trigonometric polynomial approximation, and advanced optimization techniques, resulting in new spherical choreographies.

Contribution

The paper presents a novel algorithm for computing choreographies on the sphere and plane, incorporating stereographic projection and precise optimization methods.

Findings

New choreographies on the sphere are discovered.

The algorithm effectively computes choreographies in both planar and spherical geometries.

Utilizes exact gradient and Hessian for optimization, improving accuracy.

Abstract

An algorithm is presented for numerical computation of choreographies in the plane in a Newtonian potential and on the sphere in a cotangent potential. It is based on stereographic projection, approximation by trigonometric polynomials, and quasi-Newton and Newton optimization methods with exact gradient and exact Hessian matrix. New choreographies on the sphere are presented.