Action-angle map and duality for the open Toda lattice in the perspective of Hamiltonian reduction

An alternative derivation of the known action-angle map of the standard open Toda lattice is presented based on its identification as the natural map between two gauge slices in the relevant symplectic reduction of the cotangent bundle of GL(n,R). This then permits to interpret Ruijsenaarsʼ action-angle duality for the Toda system in the same group-theoretic framework which was established previously for Calogero type systems.