Superintegrability of rational Ruijsenaars-Schneider systems and their action-angle duals. (English) Zbl 1273.81116

A classical Hamiltonian system on a 2\(n\)-dimensional symplectic manifold is Liouville integrable if there exist \(n\) independent constants of the motion \(h_1,\cdots,h_n\) in involution with respect to the Poisson bracket, such that the Hamiltonian can be expressed as a smooth function of these constants and the flows of the \(h_i\) are complete. The system is maximally superintegrable if there exist \(n-1\) additional constants of the motion \(f_1,\cdots,f_{n-1}\), globally defined, such that the full set of \(2n-1\) constants of the motion is functionally independent. It is easy to show that for any Hamiltonian system there always exist a maximal \(2n-1\) functionally independent constants of the motion, but these are usually only locally defined. Maximal superintegrability is very special. For natural Hamiltonian systems in mechanics it is often required for superintegrability that the constants of the motion be polynomial in the momenta, even more special, but that is not appropriate here where the rational Ruijsenaars-Schneider Hamiltonian is not polynomial in the momenta. The authors exploit, Ruijsenaars-duality, the fact that there is an action-angle construction for the R-S system such that 1) the action-angle variables are globally defined, 2) the momenta and particle positions of the R-S system convert to action-angle variables of the hyperbolic Sutherland system and 3) the momenta and particle positions of the hyperbolic Sutherland system convert to action-angle variables for the R-S system. They study the geometry behind this relationship and use it to show that both systems are superintegrable. The main novelty of the paper is that only Ruijsenaars-duality is used to show this and there is no resort to scattering theory or other mechanisms. They also discuss generalizations of this geometrical approach.


81R12 Groups and algebras in quantum theory and relations with integrable systems
70H05 Hamilton’s equations
81V70 Many-body theory; quantum Hall effect
70H20 Hamilton-Jacobi equations in mechanics
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