Ruijsenaars duality for B, C, D Toda chains

TL;DR

This paper constructs dual integrable systems for generalized Toda chains of types B, C, D using Hamiltonian reduction, revealing their relation to rational Goldfish models and providing explicit formulas for their Hamiltonians.

Contribution

It introduces a method to derive Ruijsenaars dual systems for B, C, D Toda chains and connects them to rational Goldfish models via Hamiltonian reduction.

Findings

Dual systems are analogues of the rational Goldfish model for types B, C, D.

Explicit formulas for dual Hamiltonians are provided.

Higher Hamiltonians of Goldfish models are computed explicitly.

Abstract

We use the Hamiltonian reduction method to construct the Ruijsenaars dual systems to generalized Toda chains associated with the classical Lie algebras of types $B, C, D$. The dual systems turn out to be the $B, C$ and $D$ analogues of the rational Goldfish model, which is, as in the type $A$ case, the strong coupling limit of rational Ruijsenaars systems. We explain how both types of systems emerge in the reduction of the cotangent bundle of a Lie group and provide the formulae for dual Hamiltonians. We compute explicitly the higher Hamiltonians of Goldfish models using the Cauchy--Binet theorem.