Quadratic choreographies

TL;DR

This paper studies quadratic particle systems using eigenvalue problems, providing solutions, convergence results, and numerical experiments for periodic and choreographic motions.

Contribution

It introduces a novel approach to solving quadratic Euler-Lagrange systems via quadratic eigenvalue problems and analyzes convergence for classical and discrete cases.

Findings

Solutions obtained through quadratic eigenvalue problems

Conditional convergence results established

Numerical experiments confirm convergence and periodic solutions

Abstract

This paper addresses the classical and discrete Euler-Lagrange equations for systems of $n$ particles interacting quadratically in $\mathbb{R}^d$. By highlighting the role played by the center of mass of the particles, we solve the previous systems via the classical quadratic eigenvalue problem (QEP) and its discrete transcendental generalization. The roots of classical and discrete QEP being given, we state some conditional convergence results. Next, we focus especially on periodic and choreographic solutions and we provide some numerical experiments which confirm the convergence.