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.

Falqui, G., Mencattini, I., Ortenzi, G., Pedroni, M. (2020). Poisson Quasi-Nijenhuis Manifolds and the Toda System. MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY, 23(3) [10.1007/s11040-020-09352-4].

Poisson Quasi-Nijenhuis Manifolds and the Toda System

Falqui, G;Ortenzi, G;Pedroni, M
2020

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.
Articolo in rivista - Articolo scientifico
Bi-Hamiltonian manifolds; Integrable systems; Poisson quasi-Nijenhuis manifolds; Toda lattices
English
2020
17
Falqui, G., Mencattini, I., Ortenzi, G., Pedroni, M. (2020). Poisson Quasi-Nijenhuis Manifolds and the Toda System. MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY, 23(3) [10.1007/s11040-020-09352-4].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/286051
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