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On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additional assumptions, compatibility in the sense of Magri implies a notion of compatibility due to Fassò and Ratiu, that we dub bi-affine compatibility. We present two proofs of this fact. The first one uses the uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. The second proof uses Darboux–Nijenhuis coordinates and symplectic connections.
Keywords: bi-Hamiltonian systems; Lagrangian foliation; bott connection; symplectic connections. 
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     author = {Manuele Santoprete},
     title = {On the {Relationship} between {Two} {Notions} of {Compatibility} {for~Bi-Hamiltonian} {Systems}},
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Manuele Santoprete. On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a88/

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