Jacobi Structures in $\mathbb{R}^3$

TL;DR

This paper classifies all Jacobi brackets in three-dimensional space, identifying two classes based on rank, and constructs the corresponding Hamiltonian vector fields, advancing the understanding of Jacobi structures.

Contribution

It provides a complete classification of Jacobi brackets in -dimensional space and explicitly constructs the associated Hamiltonian vector fields.

Findings

Two classes of Jacobi brackets identified based on rank

Explicit construction of Hamiltonian vector fields

Complete solution of Jacobi identity equations in D

Abstract

The most general Jacobi brackets in $\mathbb{R}^3$ are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed.