The Classical $r$-Matrix for the Relativistic Ruijsenaars-Schneider System

TL;DR

This paper derives the classical r-matrix for the relativistic Ruijsenaars-Schneider system across all relativistic parameters, linking it to known non-relativistic and soliton limits, thus advancing integrable systems theory.

Contribution

It provides the explicit classical r-matrix for the relativistic Ruijsenaars-Schneider model at all speeds, connecting it to non-relativistic and soliton cases.

Findings

Derived the r-matrix for all relativistic parameters

Connected relativistic r-matrix to non-relativistic Calogero-Moser case

Linked the model to sine-Gordon soliton limit

Abstract

We compute the classical $r$-matrix for the relativistic generalization of the Calogero-Moser model, or Ruijsenaars-Schneider model, at all values of the speed-of-light parameter $\lambda$. We connect it with the non-relativistic Calogero-Moser $r$-matrix $(\lambda \rightarrow -1)$ and the $\lambda = 1$ sine-Gordon soliton limit.