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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2015
• Main Author: Santoprete, M.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2015
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/147160
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems / M. Santoprete // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 15 назв. — англ.

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.