Three and four-body systems in one dimension: integrability, superintegrability and discrete symmetries

TL;DR

This paper reviews recent findings on the superintegrability of one-dimensional three- and four-body systems, highlighting how discrete symmetries like dihedral and Platonic symmetries underpin their integrability properties.

Contribution

It introduces a new perspective linking superintegrability in one-dimensional multi-body systems to their geometric symmetries in higher-dimensional Euclidean spaces.

Findings

Superintegrability of three-body systems is linked to dihedral symmetries.

Four-body systems' integrability relates to Platonic symmetries in four dimensions.

Discrete symmetries determine the polynomial degree of first integrals.

Abstract

Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of integrability and superintegrability of four-body systems in one dimension by interpreting them as one-body systems with the symmetries of the Platonic polyhedra in the four-dimensional Euclidean space. The paper is intended as a short review of recent results in the sector, emphasizing the relevance of discrete symmetries for the superintegrability of the systems considered.