The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by a class of examples with 3 degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we present a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds, and we give sufficient conditions such that the spectral invariants of the “quasi-Nijenhuis recursion operator” of a Poisson quasi-Nijenhuis manifold (obtained by deforming a Poisson-Nijenhuis structure) are in involution. Then we prove that the closed (or periodic) n-particle Toda lattice, along with its relation with the open (or non periodic) Toda system, can be framed in such a geometrical structure.
(2020). Poisson Quasi-Nijenhuis Manifolds and the Toda System [journal article - articolo]. In MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY. Retrieved from http://hdl.handle.net/10446/171144
Poisson Quasi-Nijenhuis Manifolds and the Toda System
Ortenzi, G.;Pedroni, M.
2020-01-01
Abstract
The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by a class of examples with 3 degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we present a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds, and we give sufficient conditions such that the spectral invariants of the “quasi-Nijenhuis recursion operator” of a Poisson quasi-Nijenhuis manifold (obtained by deforming a Poisson-Nijenhuis structure) are in involution. Then we prove that the closed (or periodic) n-particle Toda lattice, along with its relation with the open (or non periodic) Toda system, can be framed in such a geometrical structure.File | Dimensione del file | Formato | |
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