Multi-particle dynamical systems and polynomials

TL;DR

This paper introduces a novel method for integrating polynomial multi-particle dynamical systems by solving related partial differential equations, leading to new solutions and families of orthogonal polynomials, applicable to various systems including Darboux-Halphen.

Contribution

The paper presents a new approach to integrate polynomial multi-particle systems via PDE solutions, expanding the class of integrable systems and deriving new orthogonal polynomial families.

Findings

Integrated wide class of two and three-particle systems

Derived new families of orthogonal polynomials

Applied method to systems like Darboux-Halphen

Abstract

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is described. The method enables one to integrate a wide class of polynomial multi--particle dynamical systems. The general solutions of certain dynamical systems related to linear second--order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived. Our approach is also applicable to dynamical systems that are not multi--particle by their nature but that can be regarded as multi--particle (for example, the Darboux--Halphen system and its generalizations). A wide class of two and three--particle polynomial dynamical systems is integrated.