On the bi-Hamiltonian structure of the trigonometric spin Ruijsenaars--Sutherland hierarchy

TL;DR

This paper explores the bi-Hamiltonian structure of the trigonometric spin Ruijsenaars--Sutherland hierarchy, providing a direct proof of the second Poisson bracket's form derived from Poisson reduction of geodesic motion on U(n).

Contribution

It offers a new direct proof of the second Poisson bracket structure in the hierarchy, clarifying its form in suitable variables.

Findings

Explicit form of the second Poisson bracket

Validation of the hierarchy's bi-Hamiltonian structure

Connection to geodesic motion on U(n)

Abstract

We report on the the trigonometric spin Ruijsenaars--Sutherland hierarchy derived recently by Poisson reduction of a bi-Hamiltonian hierarchy associated with free geodesic motion on the Lie group U(n). In particular, we give a direct proof of a previously stated result about the form of the second Poisson bracket in terms of convenient variables.