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.