### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 089, 11 pages      arXiv:1506.08675      https://doi.org/10.3842/SIGMA.2015.089
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

### On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems

Manuele Santoprete
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

Received June 30, 2015, in final form November 03, 2015; Published online November 07, 2015

Abstract
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.

Key words: bi-Hamiltonian systems; Lagrangian foliation; bott connection; symplectic connections.

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