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PoissonGeometry

A Python module for (local) Poisson-Nijenhuis calculus on Poisson manifolds, with the following functions:

poisson_bracket	hamiltonian_vf	lichnerowicz_poisson_operator
modular_vf	curl_operator	flaschka_ratiu_bivector
sharp_morphism	bivector_to_matrix	jacobiator
one_forms_bracket	gauge_transformation	is_unimodular_homogeneous *
linear_normal_form_R3	isomorphic_lie_poisson_R3	is_in_kernel *
is_poisson_tensor *	is_casimir *	is_poisson_vf *
	is_poisson_pair *

Remark. We have indicated with an asterisk (*) the six methods whose implementations require testing whether a symbolic expression is zero. These are naturally limited by theoretical computational constraints.

This repository accompanies our paper 'On Computational Poisson Geometry I: Symbolic Foundations'.

Motivation

Some of the functions in this module have been used to obtain the results in these articles:

• L. C. Garcia-Naranjo, P. Suárez-Serrato & R. Vera,
  Poisson Structures on Smooth 4-Manifolds,
  Lett. Math. Phys. 105, 1533-1550 (2015)

• P. Suárez-Serrato & J. Torres-Orozco,
  Poisson Structures on Wrinkled Fibrations,
  Bol. Soc.Mat. Mex. 22, 263-280 (2016)

• P. Suárez-Serrato, J. Torres Orozco, & R. Vera,
  Poisson and Near-Symplectic Structures on Generalized Wrinkled Fibrations in Dimension 6,
  Ann. Glob. Anal. Geom. (2019) 55, 777-804 (2019)

• M. Evangelista-Alvarado, P. Suárez-Serrato, J. Torres-Orozco & R. Vera,
  On Bott-Morse Foliations and their Poisson Structures in Dimension 3,
  Journal of Singularities 19, 19-33 (2019)

🚀

• Run our tutorial on Colab English / Castellano

• Run on your local machine

  • Clone this repository on your local machine.
  • Open a terminal with the path where you clone this repository.
  • Create a virtual environment,(see this link)
  • Install the requirements:

    (venv_name) C:Users/dekstop/poisson$ pip install poissongeometry
    

  • Open python terminal to start:

    (venv_name) C:Users/dekstop/poisson$ python
    

  • Import PoissonGeometry module:

    >>> from poisson.poisson import PoissonGeometry
    

Bugs & Contributions

Our issue tracker is at https://github.com/appliedgeometry/poissongeometry/issues. Please report any bugs that you find. Or, even better, if you are interested in our project you can fork the repository on GitHub and create a pull request.

Licence 📄

MIT licence

Authors ✒️

This work is developed and maintained by:

• Miguel Evangelista Alvarado - @mevangelista-alvarado
• Jose C. Ruíz Pantaleón - @jcrpanta
• Pablo Suárez Serrato - @psuarezserrato

Thanks for citing our work if you use it! 🤓

@articleInfo{Evangelista-Alvarado Miguel Ángel 2021 Journal of Geometric Mechanics,
title = {On computational Poisson geometry I: Symbolic foundations},
journal = {Journal of Geometric Mechanics},
volume = {13},
number = {4},
pages = {607-628},
year = {2021},
issn = {1941-4889},
doi = {10.3934/jgm.2021018},
url = {/article/id/a7acc48a-54f1-4348-8b45-f5a31428bd29},
author = {Evangelista-Alvarado Miguel Ángel and Ruíz-Pantaleón José Crispín and Suárez-Serrato Pablo},
}

Acknowledgments

This work was partially supported by the grants CONACyT, “Programa para un Avance Global e Integrado de la Matemática Mexicana” CONACyT-FORDECYT 26566 and "Aprendizaje Geométrico Profundo" UNAM-DGAPA-PAPIIT-IN104819. JCRP wishes to also thank CONACyT for a postdoctoral fellowship held during the production of this work.

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