New solution family of the Jacobi equations: Characterization, invariants, and global Darboux analysis

TL;DR

This paper introduces a new, versatile family of solutions to the Jacobi equations for finite-dimensional Poisson systems, providing explicit invariants and a global Darboux reduction method applicable across various dimensions and nonlinearities.

Contribution

It characterizes a broad family of skew-symmetric solutions to Jacobi equations, including explicit invariants and a global Darboux analysis, extending previous limited local approaches.

Findings

Defined for arbitrary dimension and rank

Explicitly determined Casimir invariants

Global Darboux reduction method

Abstract

A new family of skew-symmetric solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is characterized and analyzed. Such family has some remarkable properties. In first place, it is defined for arbitrary values of the dimension and the rank. Secondly, it is described in terms of arbitrary differentiable functions, namely it is not limited to a given degree of nonlinearity. Additionally, it is possible to determine explicitly the fundamental properties of those solutions, such as their Casimir invariants and the algorithm for the reduction to the Darboux canonical form, which have been reported only for a very limited sample of finite-dimensional Poisson structures. Moreover, such analysis is carried out globally in phase space, thus improving the usual local scope of Darboux theorem.