Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems

TL;DR

This paper introduces a new method for solving Jacobi PDEs in Poisson systems using time reparametrizations that preserve Poisson structures, enabling global canonical forms for arbitrary dimensions and ranks.

Contribution

It develops a novel, dimension- and rank-independent procedure for solving Jacobi PDEs via time reparametrizations, and applies it to obtain global Darboux forms for rank-two Poisson systems.

Findings

Constructed solution families for Jacobi PDEs using time reparametrizations.

Characterized two main classes of Poisson-preserving time reparametrizations.

Achieved global Darboux canonical forms for rank-two Poisson systems.

Abstract

The determination of solutions of the Jacobi partial differential equations (PDEs) for finite-dimensional Poisson systems is considered. In particular, a novel procedure for the construction of solution families is developed. Such a procedure is based on the use of time reparametrizations preserving the existence of a Poisson structure. As a result, a method which is valid for arbitrary values of the dimension and the rank of the Poisson structure under consideration is obtained. In this article two main families of time reprametrizations of this kind are characterized. In addition, these results lead to a novel application which is also developed, namely the global and constructive determination of the Darboux canonical form for Poisson systems of arbitrary dimension and rank two, thus improving the local result provided by Darboux theorem for such a case.