On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems
Exactly Solvable and Integrable Systems
2017-10-10 v2 Differential Geometry
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 Fass\`o 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
Cite
@article{arxiv.1506.08675,
title = {On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems},
author = {Manuele Santoprete},
journal= {arXiv preprint arXiv:1506.08675},
year = {2017}
}