Congruence method for global Darboux reduction of finite-dimensional Poisson systems

TL;DR

This paper introduces a matrix congruence-based method for globally constructing Casimir invariants and Darboux forms in finite-dimensional Poisson systems, eliminating the need for prior PDE integration.

Contribution

It presents a novel algebraic approach that simultaneously finds Darboux coordinates and Casimir invariants without solving PDEs, improving existing reduction techniques.

Findings

The method successfully constructs Darboux forms and Casimir invariants in examples.

It simplifies the reduction process by avoiding PDE integration.

The approach is applicable to various finite-dimensional Poisson systems.

Abstract

A new procedure for the global construction of the Casimir invariants and Darboux canonical form for finite-dimensional Poisson systems is developed. This approach is based on the concept of matrix congruence and can be applied without the previous determination of the Casimir invariants (recall that their prior knowledge is unavoidable for the standard reduction methods, thus requiring either the integration of a system of PDEs or solving some equivalent problem). Well the opposite, in the new congruence method, both the Darboux coordinates and the Casimir invariants arise simultaneously as the outcome of the reduction algorithm. In fact, the congruence algorithm proceeds only in terms of matrix-algebraic transformations and direct quadratures, thus avoiding the need of previously integrating a system of PDEs and therefore improving previously known approaches. Physical examples illustrating different aspects of the theory are provided.