Ruijsenaars duality for \(B, C, D\) Toda chains

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.