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On computational Poisson geometry I: Symbolic foundations

  • * Corresponding author: José Crispín Ruíz-Pantaleón

    * Corresponding author: José Crispín Ruíz-Pantaleón 

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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  • 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.

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

    Citation:

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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]
     | Show Table
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