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New compact forms of the trigonometric Ruijsenaars-Schneider system. (English) Zbl 1285.70005

Summary: The reduction of the quasi-Hamiltonian double of \(SU(n)\) that has been shown to underlie Ruijsenaars’ compactified trigonometric n-body system is studied in its natural generality. The constraints contain a parameter y, restricted in previous works to \(0<y<{\pi}/n\) because Ruijsenaars’ original compactification relies on an equivalent condition. It is found that allowing generic \(0<y<{\pi}/2\) results in the appearance of new self-dual compact forms of two qualitatively different types depending on the value of y. The type (i) cases are similar to the standard case in that the reduced phase space comes equipped with globally smooth action and position variables, and turns out to be symplectomorphic to \(\mathbb{C}\text{P}^{n-1}\) as a Hamiltonian toric manifold. In the type (ii) cases both the position variables and the action variables develop singularities on a nowhere dense subset. A full classification is derived for the parameter y according to the type (i) versus type (ii) dichotomy. The simplest new type (i) systems, for which \({\pi}/n<y<{\pi}/(n-1)\), are described in some detail as an illustration.

MSC:

70H05 Hamilton’s equations
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
70F10 \(n\)-body problems
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
81R12 Groups and algebras in quantum theory and relations with integrable systems

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SageMath; polymake
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