Homogeneous bi-Hamiltonian structures and integrable contact systems

TL;DR

This paper explores the limitations and potential of bi-Hamiltonian structures in integrable systems, demonstrating that while recursion operators may not always generate maximal involutive functions, contact manifolds offer a viable alternative.

Contribution

It reveals that recursion operators between compatible Jacobi structures cannot produce maximal involutive sets, but bi-Hamiltonian structures can still be effective on contact manifolds with symplectisation.

Findings

Recursion operators between Jacobi structures do not generate maximal involutive sets.

Bi-Hamiltonian structures can produce maximal involutive functions on contact manifolds.

Symplectisation is necessary to achieve maximal involution in contact systems.

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.