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

Received  March 2021 Revised  June 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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[39]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[39]

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

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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