On computational Poisson geometry I: Symbolic foundations

Evangelista-Alvarado Miguel Ángel; Ruíz-Pantaleón José Crispín; Suárez-Serrato Pablo

doi:10.3934/jgm.2021018

Article Contents

2021, Volume 13, Issue 4: 607-628. Doi: 10.3934/jgm.2021018

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On computational Poisson geometry I: Symbolic foundations

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Coyoacán, 04510, Mexico City, Mexico

^* Corresponding author: José Crispín Ruíz-Pantaleón
^* Corresponding author: José Crispín Ruíz-Pantaleón

Received: March 2021

Revised: June 2021

Early access: August 19, 2021

Published online: August 19, 2021

This research was partially supported by CONACyT, "Programa para un Avance Global e Integrado de la Matemática Mexicana" FORDECYT 265667 and UNAM-DGAPA-PAPIIT-IN104819. JCRP thanks CONACyT for a postdoctoral fellowship held during the production of this work.

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Abstract

We present a computational toolkit for (local) Poisson-Nijenhuis calculus on manifolds. Our Python module $\textsf{PoissonGeometry}$ implements our algorithms and accompanies this paper. Examples of how our methods can be used are explained, including gauge transformations of Poisson bivector in dimension 3, parametric Poisson bivector fields in dimension 4, and Hamiltonian vector fields of parametric families of Poisson bivectors in dimension 6.

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Mathematics Subject Classification: Primary: 68W30, 97N80, 53D17.

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Table 1. Functions, corresponding algorithms, and examples where each particular method can be or has been, used in the theory of Poisson geometry. Our methods perform the symbolic calculus that realize these ideas computationally

${\textbf{Function}}$	${\textbf{Algorithm}}$	${\textbf{Examples}}$
$\textsf{sharp_morphism}$	2.1	[13,27,7]
$\textsf{poisson_bracket}$	2.2	[27,7]
$\textsf{hamiltonian_vf}$	2.3	[7,36]
$\textsf{coboundary_operator}$	2.4	[32,2]
$\textsf{curl_operator}$	2.5	[12,2]
$\textsf{bivector_to_matrix}$	2.6	[13,27,7]
$\textsf{jacobiator}$	2.7	[13,27,7]
$\textsf{modular_vf}$	2.8	[1,21,2]
$\textsf{is_unimodular_homogeneous*}$	2.9	[12,27,2,7]
$\textsf{one_forms_bracket}$	2.10	[16,23]
$\textsf{gauge_transformation}$	2.11	[10,9]
$\textsf{linear_normal_form_R3}$	2.12	[32,7]
$\textsf{isomorphic_lie_poisson_R3}$	2.13	[32,7]
$\textsf{flaschka_ratiu_bivector}$	2.14	[12,18,35,15]
$\textsf{is_poisson_tensor*}$	2.15	[18,35,15]
$\textsf{is_in_kernel*}$	2.16	[18,35,15]
$\textsf{is_casimir*}$	2.17	[18,35,15]
$\textsf{is_poisson_vf*}$	2.18	[32,3]
$\textsf{is_poisson_pair*}$	2.19	[4,2]

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