On Computational Poisson Geometry II: Numerical Methods

TL;DR

This paper introduces twelve numerical methods for evaluating and computing various objects in Poisson geometry, providing detailed descriptions, code implementations, and experimental verifications in low dimensions.

Contribution

It presents a comprehensive suite of new numerical algorithms for Poisson geometry, including their implementation and complexity analysis.

Findings

Methods successfully evaluated on 2D and 3D examples.

Complexity of each method analyzed.

Experimental verification confirms effectiveness.

Abstract

We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1-forms, gauge transformations, and normal forms of Lie-Poisson structures on $\mathbf{R}^{3}$. The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.