Construction of the classical $R$-matrices for the Toda and Calogero models

TL;DR

This paper constructs classical R-matrices for Toda and Calogero-Moser models, revealing constant R-matrices for Toda and dynamical R-matrices for Calogero-Moser, based on Hamiltonian reduction techniques.

Contribution

It introduces a unified method to derive R-matrices for both models using Hamiltonian reduction, highlighting the difference between constant and dynamical R-matrices.

Findings

Toda models have constant R-matrices.

Calogero-Moser models possess dynamical R-matrices.

The construction links geometric reduction to integrable structures.

Abstract

We use the definition of the Calogero-Moser models as Hamiltonian reductions of geodesic motions on a group manifold to construct their $R$-matrices. In the Toda case, the analogous construction yields constant $R$-matrices. By contrast, for Calogero-Moser models they are dynamical objects.