Poisson brackets with prescribed family of functions in involution

TL;DR

This paper develops an algorithm to construct Poisson brackets with a specified family of functions in involution, aiding the explicit integration of completely integrable systems and handling bi-Hamiltonian structures.

Contribution

It introduces a novel algorithm for constructing Poisson brackets with prescribed involutive functions, extending techniques for bi-Hamiltonian and integrable systems.

Findings

Algorithm successfully constructs Poisson brackets with desired involutive functions.

Enables explicit integration of equations of motion in integrable systems.

Facilitates handling of bi-Hamiltonian structures constructively.

Abstract

It is well known that functions in involution with respect to Poisson brackets have a privileged role in the theory of completely integrable systems. Finding functionally independent functions in involution with a given function $h$ on a Poisson manifold is a fundamental problem of this theory and is very useful for the explicit integration of the equations of motion defined by $h$. In this paper, we present our results on the study of the inverse, so to speak, problem. By developing a technique analogous to that presented in P. Damianou and F. Petalidou, Poisson brackets with prescribed Casimirs, Canad. J. Math., 2012, vol. 64, 991-1018, for the establishment of Poisson brackets with prescribed Casimir invariants, we construct an algorithm which yields Poisson brackets having a given family of functions in involution. Our approach allows us to deal with bi-Hamiltonian structures constructively and therefore allows us to also deal with the completely integrable systems that arise in such a framework.