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

TL;DR

This paper explores the relationship between two notions of compatibility in bi-Hamiltonian systems, showing that Magri compatibility implies Fassò-Ratiu bi-affine compatibility under certain conditions, with two different proofs provided.

Contribution

It establishes a link between Magri and Fassò-Ratiu compatibility notions in bi-Hamiltonian systems, enhancing understanding of their structural relationships.

Findings

Magri compatibility implies Fassò-Ratiu bi-affine compatibility under certain assumptions

Two proofs are provided: one using connection uniqueness, another using Darboux-Nijenhuis coordinates

Clarifies the structural relationship between different compatibility notions in integrable systems

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.