JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2021018 ©American Institute of Mathematical Sciences ON COMPUTATIONAL POISSON GEOMETRY I: SYMBOLIC FOUNDATIONS Miguel Angel´ Evangelista-Alvarado, Jose´ Crisp´ın Ru´ız-Pantaleon´ ∗ and Pablo Suarez-Serrato´ Instituto de Matem´aticas Universidad Nacional Aut´onomade M´exico Circuito Exterior, Ciudad Universitaria, Coyoac´an,04510, Mexico City, Mexico (Communicated by Juan C. Marrero) Abstract. We present a computational toolkit for (local) Poisson{Nijenhuis calculus on manifolds. Our Python module 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 di- mension 3, parametric Poisson bivector fields in dimension 4, and Hamiltonian vector fields of parametric families of Poisson bivectors in dimension 6. 1. Introduction. The origin of the concepts in this paper is the analysis of me- chanical systems of Sim´eonDenis Poisson in 1809 [33]. A Poisson manifold is a pair (M; Π), where M is a smooth manifold and Π a contravariant 2{tensor field (bivector field) on M satisfying the equation [[Π; Π]] = 0; (1) with respect to the Schouten{Nijenhuis bracket [[ ; ]] for multivector fields [31, 13]. Suppose m = dim M, and fix a local coordinate system x = (U; x1; : : : ; xm) on M. Then, Π has the following coordinate representation [28, 37]: @ @ X @ @ Π = 1 Πij ^ = Πij ^ (2) 2 @xi @xj @xi @xj 1≤i<j≤m ij 1 Here, the functions Π 2 CU are called the coefficients of Π, they are skew{ symmetric, Πji = −Πij, and f@=@xig is the canonical basis for vector fields on U ⊆ M. The Poisson bivector field, and its associated bracket, are essential elements in the comprehension of Hamiltonian dynamics [27, 13]. We recommend interested readers consult the available surveys of this field [39, 23]. 2020 Mathematics Subject Classification. Primary: 68W30, 97N80, 53D17. Key words and phrases. Poisson structures, Poisson{Nijenhuis calculus, symbolic computation, Python. This research was partially supported by CONACyT, \Programa para un Avance Global e Inte- grado de la Matem´aticaMexicana" FORDECYT 265667 and UNAM-DGAPA-PAPIIT-IN104819. JCRP thanks CONACyT for a postdoctoral fellowship held during the production of this work. ∗ Corresponding author: Jos´eCrisp´ınRu´ız-Pantale´on. 1 2 EVANGELISTA{ALVARADO, RU´IZ{PANTALEON´ AND SUAREZ{SERRATO´ Table1 below compiles the functions in our Python module PoissonGeometry1, their corresponding algorithm, and examples where such objects are used in the references. We describe all of our algorithms in section 2. Finally, in section 3 we present two applications that illustrate the usefulness of our computational methods. These are, a new result about gauge transformations of Poisson bivector fields in dimension 3 (Proposition 3.1), and a description of parametric families of Poisson bivectors in dimension 4 (Lemma 3.2). 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. Function Algorithm Examples sharp morphism 2.1 [13, 27,7] poisson bracket 2.2 [27,7] hamiltonian vf 2.3 [7, 36] coboundary operator 2.4 [32,2] curl operator 2.5 [12,2] bivector to matrix 2.6 [13, 27,7] jacobiator 2.7 [13, 27,7] modular vf 2.8 [1, 21,2] is unimodular homogeneous * 2.9 [12, 27,2,7] one forms bracket 2.10 [16, 23] gauge transformation 2.11 [10,9] linear normal form R3 2.12 [32,7] isomorphic lie poisson R3 2.13 [32,7] flaschka ratiu bivector 2.14 [12, 18, 35, 15] is poisson tensor * 2.15 [18, 35, 15] is in kernel * 2.16 [13, 27,2,7] is casimir * 2.17 [12, 18, 35, 15] is poisson vf * 2.18 [32,3] is poisson pair * 2.19 [4,2] Remark 1. We have indicated with an asterisk (*) the six methods whose im- plementations require testing whether a symbolic expression is zero. Theoretical computational constraints naturally limit these. The following diagram illustrates functional dependencies in PoissonGeometry. 1Our code repository is found at: https://github.com/appliedgeometry/poissongeometry. ON COMPUTATIONAL POISSON GEOMETRY I 3 one forms bracket sharp morphism is in kernel hamiltonian vf is casimir poisson bracket jacobiator coboundary operator is poisson tensor is poisson vf flaschka ratiu bivector is poisson pair curl operator modular vf bivector to matrix is unimodular homogeneous gauge transformation linear normal form R3 isomorphic lie poisson R3 2. Syntax and implementation of PoissonGeometry methods. PoissonGeom- etry is our Python module for local calculus on Poisson manifolds. First, we define symbolic variables that emulate local coordinates on a finite smooth manifold M. By default, these symbolic variables are just the juxtaposition of the symbol x and an index of the set f1; : : : ; m = dim Mg: (x1, . , xm). Scalar Functions. A local representation of a scalar function is written in Pois- sonGeometry using string literal expressions. For example, the function f = a(x1)2 + b(x2)2 + c(x3)2 should be written exactly as follows: `a * x1**2 + b * x2**2 + c * x3**2'. All characters that are not local coordinates are treated as (symbolic) pa- rameters: a, b and c for the previous example. Our standing assumption is that the functions involved in all the algorithms below are computable. Multivector Fields and Differential forms. Both multivector fields and differ- ential forms are written using dictionaries with tuples of integers as keys and string type values. If the coordinate expression of a (non{trivial) a{multivector field A on M, with a 2 N, is given by X @ @ @ A = Ai1i2···ia ^ ^ · · · ^ ;Ai1···ia = Ai1···ia (x); @xi1 @xi2 @xia 1≤i1<i2<···<ia≤m then A should be written using a dictionary, as follows: 1···a i1···ia m−a+1···m (1; :::; a): A ; :::; (i1; :::; ia): A ; :::; (m − a + 1; :::; m): A : Here, each key (i1; : : : ; ia) is a tuple containing ordered indices 1 ≤ i1 < ··· < ia ≤ m, and the corresponding value Ai1···ia is the string expression of the scalar function (coefficient) Ai1···ia of A. 4 EVANGELISTA{ALVARADO, RU´IZ{PANTALEON´ AND SUAREZ{SERRATO´ The syntax for differential forms is the same. It is important to remark that we only need to write the keys and values of non{zero coefficients. See the documen- tation for more details. Now we will describe the implementation of all functions of the module Pois- sonGeometry. 2.1. Key functions. This subsection contains functions that serve as a basis for the implementation of almost all functions of PoissonGeometry. 2.1.1. Sharp morphism. The function sharp morphism computes the image of a dif- ferential 1{form under the vector bundle morphism Π] : T∗M ! TM, induced by a bivector field Π on M, defined by β; Π](α) := Π(α; β); (3) for any α; β 2 T∗M [13, 27]. Here, h ; i is the natural pairing for differential 1{forms and vector fields. ] Equivalently, Π (α) = iαΠ, where i• is the interior product of multivector fields and differential forms defined by the rule iβ^β0 := iβ ◦ iβ0 [24]. Analogously for j vector fields. In local coordinates, if α = αj dx , j = 1; : : : ; m, then m X @ X @ @ Π](α) = α Πij = α Πij − α Πij : (4) i @xj i @xj j @xi i;j=1 1≤i<j≤m Algorithm 2.1 sharp morphism(bivector, one form) Input: a (Poisson) bivector field Π and a differential 1{form α Output: a vector field, which is the image of α under the vector bundle morphism in (3) induced by Π 1: procedure 2: m dimension of the manifold 3: if m = 1 then . bivector fields are trivial in this case 4: return a variable encoding the trivial vector field 5: end if 6: bivector a variable encoding the (Poisson) bivector field 7: one form a variable encoding the differential 1{form 8: sharp array a variable encoding the computation of the vector field Π](α) in (4), using bivector and one form 9: return sharp array 10: end procedure Observe that the morphism in (3) is defined, in particular, for Poisson bivector fields. So the function sharp morphism can be applied to this class of bivector fields. 2.1.2. Poisson brackets. A Poisson bracket on M is a Lie bracket structure f ; g on 1 the space of smooth functions CM which is compatible with the pointwise product by the Leibniz rule [13, 27]. Explicitly, the Poisson bracket induced by a Poisson bivector field Π on M is given by the formula i @g ff; gg = dg; Π](df) = Π](df) ; f; g 2 C 1; (5) Π @xi M for i = 1; : : : ; m. ON COMPUTATIONAL POISSON GEOMETRY I 5 The function poisson bracket computes the Poisson bracket, induced by a Poisson bivector field, of two scalar functions. Algorithm 2.2 poisson bracket(bivector, function 1, function 2 ) Input: a Poisson bivector field and two scalar functions Output: the Poisson bracket, induced by the Poisson bivector field, of the two scalar functions 1: procedure 2: m dimension of the manifold 3: if m = 1 then 4: return 0 . bivector fields are trivial in this case 5: end if 6: bivector a variable encoding the Poisson bivector field 7: function 1, function 2 variables encoding the scalar functions 8: if function 1= function 2 then 9: return 0 . if f = g in (5), then ff; ggΠ = 0 10: end if 11: diff function 1 a variable encoding the differential of function 1 12: sharp function 1 sharp morphism(bivector, diff function 1) . see, Algorithm 2.1 13: bracket a variable encoding the pairing of the differential of function 2 with sharp function 1 .