December  2021, 13(4): 607-628. doi: 10.3934/jgm.2021018

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

Received  March 2021 Revised  June 2021 Published  December 2021 Early access  August 2021

Fund Project: 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

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.

Citation: Miguel Ángel Evangelista-Alvarado, José Crispín Ruíz-Pantaleón, Pablo Suárez-Serrato. On computational Poisson geometry I: Symbolic foundations. Journal of Geometric Mechanics, 2021, 13 (4) : 607-628. doi: 10.3934/jgm.2021018
References:
[1]

A. Abouqateb and M. Boucetta, The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation, C. R. Math., 337 (2003), 61-66.  doi: 10.1016/S1631-073X(03)00254-1.

[2]

M. AmmarG. KassM. Masmoudi and N. Poncin, Strongly r-matrix induced tensors, Koszul cohomology, and arbitrary-dimensional quadratic Poisson cohomology, Pacific J. Math., 245 (2010), 1-23.  doi: 10.2140/pjm.2010.245.1.

[3]

M. Avendaño-Camacho, J. A. Vallejo and Y. Vorobiev, A perturbation theory approach to the stability of the Pais-Uhlenbeck oscillator, J. Math. Phys., 58 (2017), 093501 doi: 10.1063/1.5000382.

[4]

M. Avendaño-Camacho and Y. Vorobiev, Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems, Int. J. Geom. Methods Mod. Phys, 14 (2017), 1750086. doi: 10.1142/S0219887817500864.

[5]

F. BayenM. FlatoC. FronsdalA. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Phys., 111 (1978), 61-110.  doi: 10.1016/0003-4916(78)90224-5.

[6]

F. BayenM. FlatoC. FronsdalA. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. II. Physical applications, Ann. Phys., 111 (1978), 111-151.  doi: 10.1016/0003-4916(78)90225-7.

[7]

E. Bayro-Corrochano, Geometric Algebra Applications Vol. I. Computer Vision, Graphics And Neurocomputing, 1$^{st}$ edition, Springer International Publishing AG, part of Springer Nature, 2019. doi: 10.1007/978-3-319-74830-6.

[8]

M. A. BurrM. Schmoll and C. Wolf, On the computability of rotation sets and their entropies, Ergodic Theory Dynam. Systems, 40 (2020), 367-401.  doi: 10.1017/etds.2018.45.

[9]

H. Bursztyn, On gauge transformations of Poisson structures, In Quantum Field Theory and Noncommutative Geometry, Lect. Notes Phys., Springer, Berlin, 662 (2005), 89-112. doi: 10.1007/11342786_5.

[10]

H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier, 53 (2003), 309-337.  doi: 10.5802/aif.1945.

[11]

R. Cardona, E. Miranda, D. Peralta-Salas and F. Presas, Constructing Turing complete Euler flows in dimension 3, Proc. Nat. Acad. Sci., 118 (2021). doi: 10.1073/pnas.2026818118.

[12]

P. A. Damianou and F. Petalidou, Poisson brackets with prescribed Casimirs, Canad. J. Math., 64 (2012), 991-1018.  doi: 10.4153/CJM-2011-082-2.

[13]

J. P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, 242. Birkhäuser Verlag, Basel, 2005. doi: 10.1007/b137493.

[14]

M. Evangelista-AlvaradoJ. C. Ruíz-Pantaleón and P. Suárez-Serrato, On computational Poisson geometry II: Numerical methods, J. Comput. Dyn., 8 (2021), 273-307. 

[15]

M. Evangelista-AlvaradoP. Suárez-SerratoJ. Torres-Orozco and R. Vera, On Bott-Morse foliations and their Poisson structures in dimension 3, J. Singul., 19 (2019), 19-33.  doi: 10.5427/jsing.2019.19b.

[16]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179.  doi: 10.1006/aima.2001.2070.

[17]

M. FlatoA. Lichnerowicz and D. Sternheimer, Déformations 1-différentiables des algèbres de Lie attachées à une variété symplectique ou de contact, Compos. Math., 31 (1975), 47-82. 

[18]

L. C. García-NaranjoP. Suárez-Serrato and R. Vera, Poisson structures on smooth 4-manifolds, Lett. Math. Phys., 105 (2015), 1533-1550.  doi: 10.1007/s11005-015-0792-8.

[19]

V. L. Ginzburg and A. Weinstein, Lie-Poisson structure on some Poisson Lie groups, J. Am. Math. Soc., 5 (1992), 445-453.  doi: 10.1090/S0894-0347-1992-1126117-8.

[20]

L. Grabowski, Vanishing of $l^{2}$-cohomology as a computational problem, Bull. Lond. Math. Soc., 47 (2015), 233-247.  doi: 10.1112/blms/bdu114.

[21]

V. GuilleminE. Miranda and A. R. Pires, Codimension one symplectic foliations and regular Poisson stuctures, Bull. Braz. Math. Soc., (N.S.), 42 (2011), 607-623.  doi: 10.1007/s00574-011-0031-6.

[22]

B. Kostant, Orbits, symplectic structures, and representation theory, In Collected Papers, (eds. A. Joseph, S. Kumar and M. Vergne), Springer, New York, NY, (2009), 482-482. doi: 10.1007/b94535_20.

[23]

Y. Kosmann-Schwarzbach, Poisson manifolds, Lie algebroids, modular classes: A survey, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), 5-34.  doi: 10.3842/SIGMA.2008.005.

[24]

J. L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, The Mathematical Heritage of Élie Cartan (Lyon, 1984). Astérisque, Numéro Hors Série, (1985), 257-271.

[25]

V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 31. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-78393-7.

[26]

M. KrögerM. Hütter and H. C. Öttinger, Symbolic test of the Jacobi identity for given generalized 'Poisson' bracket, Comput. Phys. Commun., 137 (2001), 325-340.  doi: 10.1016/S0010-4655(01)00161-8.

[27]

C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 347. Springer, Heidelberg, 2013.

[28]

A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom., 12 (1977), 253-300. 

[29]

Z. J. Liu and P. Xu, On quadratic Poisson structures, Lett. Math. Phys., 26 (1992), 33-42.  doi: 10.1007/BF00420516.

[30]

A. Meurer et al., SymPy: Symbolic computing in Python, PeerJ Comput. Sci., 3 (2017). doi: 10.7717/peerj-cs.103.

[31]

P. W. Michor, Topics in Differential Geometry, vol. 93, American Mathematical Society, 2008. doi: 10.1090/gsm/093.

[32]

N. Nakanishi, On the structure of infinitesimal automorphisms of linear Poisson manifolds I, J. Math. Kyoto Univ., 31 (1991), 71-82.  doi: 10.1215/kjm/1250519890.

[33]

S. D. Poisson, Sur la variation des constantes arbitraires dans les questions de mécanique, J. Ecole Polytechnique, 8 (1809), 266-344. 

[34]

P. Severa and A. Weinstein, Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl., 144 (2001), 145-154. 

[35]

P. Suárez-Serrato and J. Torres-Orozco, Poisson structures on wrinkled fibrations, Bol. Soc. Mat. Mex., 22 (2016), 263-280.  doi: 10.1007/s40590-015-0072-8.

[36]

S. Takato and J. A. Vallejo, Hamiltonian dynamical systems: Symbolical, numerical and graphical study, Math. Comput. Sci., 13 (2019), 281-295.  doi: 10.1007/s11786-019-00396-6.

[37]

A. Weinstein, The local structure of Poisson manifolds, J. Differ. Geom., 18 (1983), 523-557. 

[38]

A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379-394.  doi: 10.1016/S0393-0440(97)80011-3.

[39]

A. Weinstein, Poisson geometry, Diff. Geom. Appl., 9 (1998), 213-238.  doi: 10.1016/S0926-2245(98)00022-9.

show all references

References:
[1]

A. Abouqateb and M. Boucetta, The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation, C. R. Math., 337 (2003), 61-66.  doi: 10.1016/S1631-073X(03)00254-1.

[2]

M. AmmarG. KassM. Masmoudi and N. Poncin, Strongly r-matrix induced tensors, Koszul cohomology, and arbitrary-dimensional quadratic Poisson cohomology, Pacific J. Math., 245 (2010), 1-23.  doi: 10.2140/pjm.2010.245.1.

[3]

M. Avendaño-Camacho, J. A. Vallejo and Y. Vorobiev, A perturbation theory approach to the stability of the Pais-Uhlenbeck oscillator, J. Math. Phys., 58 (2017), 093501 doi: 10.1063/1.5000382.

[4]

M. Avendaño-Camacho and Y. Vorobiev, Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems, Int. J. Geom. Methods Mod. Phys, 14 (2017), 1750086. doi: 10.1142/S0219887817500864.

[5]

F. BayenM. FlatoC. FronsdalA. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Phys., 111 (1978), 61-110.  doi: 10.1016/0003-4916(78)90224-5.

[6]

F. BayenM. FlatoC. FronsdalA. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. II. Physical applications, Ann. Phys., 111 (1978), 111-151.  doi: 10.1016/0003-4916(78)90225-7.

[7]

E. Bayro-Corrochano, Geometric Algebra Applications Vol. I. Computer Vision, Graphics And Neurocomputing, 1$^{st}$ edition, Springer International Publishing AG, part of Springer Nature, 2019. doi: 10.1007/978-3-319-74830-6.

[8]

M. A. BurrM. Schmoll and C. Wolf, On the computability of rotation sets and their entropies, Ergodic Theory Dynam. Systems, 40 (2020), 367-401.  doi: 10.1017/etds.2018.45.

[9]

H. Bursztyn, On gauge transformations of Poisson structures, In Quantum Field Theory and Noncommutative Geometry, Lect. Notes Phys., Springer, Berlin, 662 (2005), 89-112. doi: 10.1007/11342786_5.

[10]

H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier, 53 (2003), 309-337.  doi: 10.5802/aif.1945.

[11]

R. Cardona, E. Miranda, D. Peralta-Salas and F. Presas, Constructing Turing complete Euler flows in dimension 3, Proc. Nat. Acad. Sci., 118 (2021). doi: 10.1073/pnas.2026818118.

[12]

P. A. Damianou and F. Petalidou, Poisson brackets with prescribed Casimirs, Canad. J. Math., 64 (2012), 991-1018.  doi: 10.4153/CJM-2011-082-2.

[13]

J. P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, 242. Birkhäuser Verlag, Basel, 2005. doi: 10.1007/b137493.

[14]

M. Evangelista-AlvaradoJ. C. Ruíz-Pantaleón and P. Suárez-Serrato, On computational Poisson geometry II: Numerical methods, J. Comput. Dyn., 8 (2021), 273-307. 

[15]

M. Evangelista-AlvaradoP. Suárez-SerratoJ. Torres-Orozco and R. Vera, On Bott-Morse foliations and their Poisson structures in dimension 3, J. Singul., 19 (2019), 19-33.  doi: 10.5427/jsing.2019.19b.

[16]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179.  doi: 10.1006/aima.2001.2070.

[17]

M. FlatoA. Lichnerowicz and D. Sternheimer, Déformations 1-différentiables des algèbres de Lie attachées à une variété symplectique ou de contact, Compos. Math., 31 (1975), 47-82. 

[18]

L. C. García-NaranjoP. Suárez-Serrato and R. Vera, Poisson structures on smooth 4-manifolds, Lett. Math. Phys., 105 (2015), 1533-1550.  doi: 10.1007/s11005-015-0792-8.

[19]

V. L. Ginzburg and A. Weinstein, Lie-Poisson structure on some Poisson Lie groups, J. Am. Math. Soc., 5 (1992), 445-453.  doi: 10.1090/S0894-0347-1992-1126117-8.

[20]

L. Grabowski, Vanishing of $l^{2}$-cohomology as a computational problem, Bull. Lond. Math. Soc., 47 (2015), 233-247.  doi: 10.1112/blms/bdu114.

[21]

V. GuilleminE. Miranda and A. R. Pires, Codimension one symplectic foliations and regular Poisson stuctures, Bull. Braz. Math. Soc., (N.S.), 42 (2011), 607-623.  doi: 10.1007/s00574-011-0031-6.

[22]

B. Kostant, Orbits, symplectic structures, and representation theory, In Collected Papers, (eds. A. Joseph, S. Kumar and M. Vergne), Springer, New York, NY, (2009), 482-482. doi: 10.1007/b94535_20.

[23]

Y. Kosmann-Schwarzbach, Poisson manifolds, Lie algebroids, modular classes: A survey, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), 5-34.  doi: 10.3842/SIGMA.2008.005.

[24]

J. L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, The Mathematical Heritage of Élie Cartan (Lyon, 1984). Astérisque, Numéro Hors Série, (1985), 257-271.

[25]

V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 31. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-78393-7.

[26]

M. KrögerM. Hütter and H. C. Öttinger, Symbolic test of the Jacobi identity for given generalized 'Poisson' bracket, Comput. Phys. Commun., 137 (2001), 325-340.  doi: 10.1016/S0010-4655(01)00161-8.

[27]

C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 347. Springer, Heidelberg, 2013.

[28]

A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom., 12 (1977), 253-300. 

[29]

Z. J. Liu and P. Xu, On quadratic Poisson structures, Lett. Math. Phys., 26 (1992), 33-42.  doi: 10.1007/BF00420516.

[30]

A. Meurer et al., SymPy: Symbolic computing in Python, PeerJ Comput. Sci., 3 (2017). doi: 10.7717/peerj-cs.103.

[31]

P. W. Michor, Topics in Differential Geometry, vol. 93, American Mathematical Society, 2008. doi: 10.1090/gsm/093.

[32]

N. Nakanishi, On the structure of infinitesimal automorphisms of linear Poisson manifolds I, J. Math. Kyoto Univ., 31 (1991), 71-82.  doi: 10.1215/kjm/1250519890.

[33]

S. D. Poisson, Sur la variation des constantes arbitraires dans les questions de mécanique, J. Ecole Polytechnique, 8 (1809), 266-344. 

[34]

P. Severa and A. Weinstein, Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl., 144 (2001), 145-154. 

[35]

P. Suárez-Serrato and J. Torres-Orozco, Poisson structures on wrinkled fibrations, Bol. Soc. Mat. Mex., 22 (2016), 263-280.  doi: 10.1007/s40590-015-0072-8.

[36]

S. Takato and J. A. Vallejo, Hamiltonian dynamical systems: Symbolical, numerical and graphical study, Math. Comput. Sci., 13 (2019), 281-295.  doi: 10.1007/s11786-019-00396-6.

[37]

A. Weinstein, The local structure of Poisson manifolds, J. Differ. Geom., 18 (1983), 523-557. 

[38]

A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379-394.  doi: 10.1016/S0393-0440(97)80011-3.

[39]

A. Weinstein, Poisson geometry, Diff. Geom. Appl., 9 (1998), 213-238.  doi: 10.1016/S0926-2245(98)00022-9.

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]
${\textbf{Function}}$ ${\textbf{Algorithm}}$ ${\textbf{Examples}}$
$\textsf{sharp_morphism}$ 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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