This article is cited in 2 papers

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

Manuele Santoprete

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

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.

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

MSC: 70H06; 70G45; 37K10

Received: June 30, 2015; in final form November 3, 2015; Published online November 7, 2015

Language: English

DOI: 10.3842/SIGMA.2015.089

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ArXiv: 1506.08675