Homogeneous bi-Hamiltonian structures and integrable contact systems
Leonardo Colombo, Manuel de León, María Emma Eyrea Irazú, Asier López-Gordón
TL;DR
The article analyzes the role of bi-Hamiltonian structures in integrable systems within the Jacobi/contact framework. It shows a no-go result: the recursion operator from two compatible Jacobi structures cannot produce a maximal set of functionally independent quantities in involution for completely integrable contact systems. However, by symplectising the contact system to a homogeneous symplectic system, one can recover a second Poisson structure whose eigenvalues provide a maximal involutive set in the symplectic setting, as demonstrated through a detailed example. The work clarifies limitations of direct Jacobi-based bi-Hamiltonian constructions while offering a practical symplectic route and outlines future directions involving Haantjes structures and non-Abelian Lie systems for broader applicability.
Abstract
Bi-Hamiltonian structures can be utilised to compute a maximal set of functions in involution for certain integrable systems, given by the eigenvalues of the recursion operator relating both Poisson structures. We show that the recursion operator relating two compatible Jacobi structures cannot produce a maximal set of functions in involution. However, as we illustrate with an example, bi-Hamiltonian structures can still be used to obtain a maximal set of functions in involution on a contact manifold, at the cost of symplectisation.