Poisson systems as the natural framework for additional first integrals via Darboux invariant hypersurfaces

TL;DR

This paper extends the theory of Darboux polynomials and first integrals from Hamiltonian systems to general Poisson systems, broadening the applicability of these concepts to a wider class of dynamical systems.

Contribution

It generalizes the framework for Darboux polynomials and first integrals from Hamiltonian to Poisson systems, enabling analysis of more general vector fields.

Findings

Extension of Darboux polynomial theory to Poisson systems

Applicability to a broader class of vector fields

Examples illustrating the generalized framework

Abstract

In the literature, the existence of Darboux polynomials and additional polynomial first integrals has been considered in the case of Hamiltonian systems. In this article such problem is formulated in the more general framework of Poisson structures, which include Hamiltonian systems as a particular case. This generalization allows a natural extension of the previous results, which can now be applied to a larger class of vector fields and is valid for arbitrary diffeomorphisms (instead of canonical transformations). Examples are discussed.