Superintegrability of rational Ruijsenaars-Schneider systems and their action-angle duals

TL;DR

This paper demonstrates that the rational Ruijsenaars-Schneider systems and their duals are maximally superintegrable, providing new proofs and extending the results to BC(n) systems through symplectic reduction and duality insights.

Contribution

It offers a new direct proof of the symplectic structure in Ruijsenaars systems and extends superintegrability results to BC(n) generalizations using duality and reduction techniques.

Findings

Maximal superintegrability of rational Ruijsenaars-Schneider systems established.

A new direct proof of the Darboux form of the reduced symplectic structure provided.

Extension of results to BC(n) systems via duality and symplectic reduction.

Abstract

We explain that the action-angle duality between the rational Ruijsenaars-Schneider and hyperbolic Sutherland systems implies immediately the maximal superintegrability of these many-body systems. We also present a new direct proof of the Darboux form of the reduced symplectic structure that arises in the `Ruijsenaars gauge' of the symplectic reduction underlying this case of action-angle duality. The same arguments apply to the BC(n) generalization of the pertinent dual pair, which was recently studied by Pusztai developing a method utilized in our direct calculation of the reduced symplectic structure.