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July  2021, 8(3): 273-307. doi: 10.3934/jcd.2021012

On computational Poisson geometry II: Numerical methods

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Coyoacán, 04510, Mexico City, Mexico

* Corresponding author: J. C. Ruíz–Pantaleón

Received  October 2020 Revised  March 2021 Published  July 2021 Early access  May 2021

Fund Project: This work is partially supported by UNAM–DGAPA–PAPIIT grant IN104819

We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1–forms, gauge transformations, and normal forms of Lie–Poisson structures on $ {\mathbf{R}^{{3}}} $. The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.

Citation: Miguel Ángel Evangelista-Alvarado, José Crispín Ruíz-Pantaleón, Pablo Suárez-Serrato. On computational Poisson geometry II: Numerical methods. Journal of Computational Dynamics, 2021, 8 (3) : 273-307. doi: 10.3934/jcd.2021012
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 Yu. Vorobiev, A perturbation theory approach to the stability of the Pais–Uhlenbeck oscillator, J. Math. Phys., 58 (2017), 093501, 13 pp. doi: 10.1063/1.5000382.  Google Scholar

[4]

P. Balseiro and L. C. García–Naranjo, Gauge transformations, twisted poisson brackets and hamiltonization of nonholonomic systems, Arch. Ration. Mech. Anal., 205 (2012), 267-310.  doi: 10.1007/s00205-012-0512-9.  Google Scholar

[5]

P. Battaglia, R. Pascanu, M. Lai, D. J. Rezende and K. Kavukcuoglu, Interaction networks for learning about objects, relations and physics, in Proc. of the 30th International Conference on Neural Information Proc. Systems (eds. D. D. Lee, U. Luxburg, R. Garnett, M. Sugiyama and I. Guyon), Curran Associates Inc., (2016), 4509–4517. Google Scholar

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P. G. BreenC. N. FoleyT. Boekholt and S. P. Zwart, Newton versus the machine: Solving the chaotic three–body problem using deep neural networks, MNRAS, 494 (2020), 2465-2470.  doi: 10.1093/mnras/staa713.  Google Scholar

[7]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, 1$^{st}$ edition, Springer–Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[8]

H. Bursztyn, On gauge transformations of Poisson structures, in Quantum Field Theory and Noncommutative Geometry, Lecture Notes in Phys. (eds. U. Carow–Watamura, Y. Maeda and S. Watamura), Springer, Berlin Heidelberg, (2005), 89–112. doi: 10.1007/11342786_5.  Google Scholar

[9]

C. Caracciolo and U. Locatelli, Computer–assisted estimates for Birkhoff normal forms, J. Comput. Dyn., 7 (2020), 425-460.  doi: 10.3934/jcd.2020017.  Google Scholar

[10]

Z. Chen, J. Zhang, M. Arjovsky and L. Bottou, Symplectic recurrent neural networks, in International Conference on Learning Representations, (2020). Google Scholar

[11]

M. Cranmer, A. Sanchez–Gonzalez, P. Battaglia, R. Xu, K. Cranmer, D. Spergel and S. Ho, Discovering symbolic models from deep learning with inductive biases, in Advances in Neural Information Processing Systems (eds. H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan and H. Lin), Curran Associates Inc., (2020), 17429–17442. Google Scholar

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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

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P. A. M. Dirac, Quelques problèmes de mécanique quantique, Ann. Inst. Henri Poincaré, 1 (1930), 357-400.   Google Scholar

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M. de la CruzN. GasparL. Jiménez–Lara and R. Linares, Classification of the classical ${SL(2,\mathbf{R}^{{}})}$ gauge transformations in the rigid body, Ann. Physics, 379 (2017), 112-130.  doi: 10.1016/j.aop.2017.02.016.  Google Scholar

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J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, 1$^{st}$ edition, Birkhäuser Basel, 2005.  Google Scholar

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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

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M. A. Evangelista–Alvarado, J. C. Ruíz–Pantaleón and P. Suárez–Serrato, On computational Poisson geometry I: Symbolic foundations, to appear in J. Geom. Mech. Google Scholar

[19]

L. Falorsi, P. De Haan, T. R. David son and P. Forré, Reparameterizing distributions on Lie groups, in Proceedings of the Twenty–Second International Conference on Artificial Intelligence and Statistics (eds. K. Chaudhuri and M. Sugiyama), PMLR, (2019), 3244–3253. Google Scholar

[20]

P. Frejlich and I. Marcut, The homology class of a Poisson transversal, Int. Math. Res. Not., 2020 (2020), 2952-2976.  doi: 10.1093/imrn/rny105.  Google Scholar

[21]

B. Fuchssteiner, The Lie algebra structure of degenerate Hamiltonian and bi–Hamiltonian systems, Prog. Theor. Phys., 68 (1982), 1082-1104.  doi: 10.1143/PTP.68.1082.  Google Scholar

[22]

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

[23]

V. L. Ginzburg and A. Weinstein, Lie–Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc., 5 (1992), 445-453.  doi: 10.2307/2152773.  Google Scholar

[24]

J. GrabowskiG. Marmo and A. M. Perelomov, Poisson structures: Towards a classification, Modern Phys. Lett. A, 08 (1993), 1719-1733.  doi: 10.1142/S0217732393001458.  Google Scholar

[25]

J. Grabowski, Brackets, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360001, 45 pp. doi: 10.1142/S0219887813600013.  Google Scholar

[26]

S. Greydanus, M. Dzamba and J. Yosinski, Hamiltonian neural networks, in Advances in Neural Information Processing Systems (eds. H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché–Buc, E. Fox and R. Garnett), NeurIPS, (2019), 15379–15389. Google Scholar

[27]

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

[28]

D. Jimenez, G. Papamakarios, S. Racaniere, M. S. Albergo, G. Kanwar, P. E. Shanahan and K. Cranmer, Normalizing flows on tori and spheres, in Proceedings of the Thirty–Seven International Conference on Machine Learning (eds. N. Lawrence and M. Reid), PMLR, (2020), 8083–8092. Google Scholar

[29]

D. Jimenez, S. Racaniere, I. Higgins and P. Toth, Equivariant Hamiltonian flows, in CoRR, JCD, (2019). Google Scholar

[30]

M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), 157-216.  doi: 10.1023/B:MATH.0000027508.00421.bf.  Google Scholar

[31]

Y. Kosmann–Schwarzbach, Poisson manifolds, Lie algebroids, modular classes: A survey, SIGMA, 4 (2008), 30 pp. doi: 10.3842/SIGMA.2008.005.  Google Scholar

[32]

J.–L. Koszul, Crochet de Schouten–Nijenhuis et cohomologie, in Astérisque, Société mathématique de France, (1985), 257–271.  Google Scholar

[33]

V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, 1$^{st}$ edition, Springer–Verlag, Berlin Heidelberg, 1996. doi: 10.1007/978-3-642-78393-7.  Google Scholar

[34]

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

[35]

M. Lainz, C. Sardón and A. Weinstein, Plasma in a monopole background does not have a twisted Poisson structure, Phys. Rev. D, 100 (2019), 105016, 5 pp. doi: 10.1103/PhysRevD.100.105016.  Google Scholar

[36]

C. Laurent–Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures, 1$^{st}$ edition, Springer–Verlag, Berlin Heidelberg, 2013. doi: 10.1007/978-3-642-31090-4.  Google Scholar

[37]

S.-H. Li, C.-X. Dong, L. Zhang and L. Wang, Neural canonical transformation with symplectic flows, Phys. Rev. X, 10 (2020), 021020, 1–13. doi: 10.1103/PhysRevX.10.021020.  Google Scholar

[38]

A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom., 12 (1977), 253-300.  doi: 10.4310/jdg/1214433987.  Google Scholar

[39]

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

[40]

Y.–A. MaY. ChenC. JinN. Flammarion and M. I. Jordan, Sampling can be faster than optimization, Proc. Natl. Acad. Sci. USA, 116 (2019), 20881-20885.  doi: 10.1073/pnas.1820003116.  Google Scholar

[41]

I. Marcut and F. Zeiser, The Poisson cohomology of ${\mathfrak{sl}^{\ast}_2(\mathbf{R}^{{}})}$, preprint, arXiv: 1911.11732 [math.SG]. Google Scholar

[42]

P. W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, AMS, 2008. doi: 10.1090/gsm/093.  Google Scholar

[43]

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

[44]

A. PedrozaE. Velasco–Barreras and Yu. Vorobiev, Unimodularity criteria for Poisson structures on foliated manifolds, Lett. Math. Phys., 108 (2018), 861-882.  doi: 10.1007/s11005-017-1014-3.  Google Scholar

[45]

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

[46]

A. Sanchez–Gonzalez, V. Bapst, K. Cranmer and P. Battaglia, Hamiltonian graph networks with ODE integrators, in CoRR, JCD, (2019). Google Scholar

[47]

P. Ševera and A. Weinstein, Poisson geometry with a 3–form background, Prog. Theor. Phys., Suppl., 144 (2001), 145-154.  doi: 10.1143/PTPS.144.145.  Google Scholar

[48]

Y. Sheng, Linear Poisson structures on ${\mathbf{R}^{{4}}}$, J. Geom. Phys., 57 (2007), 2398-2410.  doi: 10.1016/j.geomphys.2007.08.008.  Google Scholar

[49]

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

[50]

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

[51]

D. Tamayo, M. Cranmer, S. Hadden, H. Rein, P. Battaglia, A. Obertas, P. Armitage, S Ho, D. N. Spergel, C. Gilbertson, N. Hussain, A. Silburt, D. Jontof–Hutter and K. Menou, Predicting the long-term stability of compact multiplanet systems, in Proceedings of the National Academy of Sciences (ed. M. R. Berenbaum), PNAS, (2020), 18194–18205. Google Scholar

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P. Toth, D.J. Rezende, A. Jaegle, S. Racanière, A. Botev and I. Higgins, Hamiltonian generative networks, in International Conference on Learning Representations, (2020). Google Scholar

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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 Yu. Vorobiev, A perturbation theory approach to the stability of the Pais–Uhlenbeck oscillator, J. Math. Phys., 58 (2017), 093501, 13 pp. doi: 10.1063/1.5000382.  Google Scholar

[4]

P. Balseiro and L. C. García–Naranjo, Gauge transformations, twisted poisson brackets and hamiltonization of nonholonomic systems, Arch. Ration. Mech. Anal., 205 (2012), 267-310.  doi: 10.1007/s00205-012-0512-9.  Google Scholar

[5]

P. Battaglia, R. Pascanu, M. Lai, D. J. Rezende and K. Kavukcuoglu, Interaction networks for learning about objects, relations and physics, in Proc. of the 30th International Conference on Neural Information Proc. Systems (eds. D. D. Lee, U. Luxburg, R. Garnett, M. Sugiyama and I. Guyon), Curran Associates Inc., (2016), 4509–4517. Google Scholar

[6]

P. G. BreenC. N. FoleyT. Boekholt and S. P. Zwart, Newton versus the machine: Solving the chaotic three–body problem using deep neural networks, MNRAS, 494 (2020), 2465-2470.  doi: 10.1093/mnras/staa713.  Google Scholar

[7]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, 1$^{st}$ edition, Springer–Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[8]

H. Bursztyn, On gauge transformations of Poisson structures, in Quantum Field Theory and Noncommutative Geometry, Lecture Notes in Phys. (eds. U. Carow–Watamura, Y. Maeda and S. Watamura), Springer, Berlin Heidelberg, (2005), 89–112. doi: 10.1007/11342786_5.  Google Scholar

[9]

C. Caracciolo and U. Locatelli, Computer–assisted estimates for Birkhoff normal forms, J. Comput. Dyn., 7 (2020), 425-460.  doi: 10.3934/jcd.2020017.  Google Scholar

[10]

Z. Chen, J. Zhang, M. Arjovsky and L. Bottou, Symplectic recurrent neural networks, in International Conference on Learning Representations, (2020). Google Scholar

[11]

M. Cranmer, A. Sanchez–Gonzalez, P. Battaglia, R. Xu, K. Cranmer, D. Spergel and S. Ho, Discovering symbolic models from deep learning with inductive biases, in Advances in Neural Information Processing Systems (eds. H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan and H. Lin), Curran Associates Inc., (2020), 17429–17442. 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]

P. A. M. Dirac, Quelques problèmes de mécanique quantique, Ann. Inst. Henri Poincaré, 1 (1930), 357-400.   Google Scholar

[14]

M. de la CruzN. GasparL. Jiménez–Lara and R. Linares, Classification of the classical ${SL(2,\mathbf{R}^{{}})}$ gauge transformations in the rigid body, Ann. Physics, 379 (2017), 112-130.  doi: 10.1016/j.aop.2017.02.016.  Google Scholar

[15]

S. DuaneA. D. KennedyB. J. Pendleton and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B, 195 (1987), 216-222.  doi: 10.1016/0370-2693(87)91197-x.  Google Scholar

[16]

J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, 1$^{st}$ edition, Birkhäuser Basel, 2005.  Google Scholar

[17]

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

[18]

M. A. Evangelista–Alvarado, J. C. Ruíz–Pantaleón and P. Suárez–Serrato, On computational Poisson geometry I: Symbolic foundations, to appear in J. Geom. Mech. Google Scholar

[19]

L. Falorsi, P. De Haan, T. R. David son and P. Forré, Reparameterizing distributions on Lie groups, in Proceedings of the Twenty–Second International Conference on Artificial Intelligence and Statistics (eds. K. Chaudhuri and M. Sugiyama), PMLR, (2019), 3244–3253. Google Scholar

[20]

P. Frejlich and I. Marcut, The homology class of a Poisson transversal, Int. Math. Res. Not., 2020 (2020), 2952-2976.  doi: 10.1093/imrn/rny105.  Google Scholar

[21]

B. Fuchssteiner, The Lie algebra structure of degenerate Hamiltonian and bi–Hamiltonian systems, Prog. Theor. Phys., 68 (1982), 1082-1104.  doi: 10.1143/PTP.68.1082.  Google Scholar

[22]

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

[23]

V. L. Ginzburg and A. Weinstein, Lie–Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc., 5 (1992), 445-453.  doi: 10.2307/2152773.  Google Scholar

[24]

J. GrabowskiG. Marmo and A. M. Perelomov, Poisson structures: Towards a classification, Modern Phys. Lett. A, 08 (1993), 1719-1733.  doi: 10.1142/S0217732393001458.  Google Scholar

[25]

J. Grabowski, Brackets, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360001, 45 pp. doi: 10.1142/S0219887813600013.  Google Scholar

[26]

S. Greydanus, M. Dzamba and J. Yosinski, Hamiltonian neural networks, in Advances in Neural Information Processing Systems (eds. H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché–Buc, E. Fox and R. Garnett), NeurIPS, (2019), 15379–15389. Google Scholar

[27]

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

[28]

D. Jimenez, G. Papamakarios, S. Racaniere, M. S. Albergo, G. Kanwar, P. E. Shanahan and K. Cranmer, Normalizing flows on tori and spheres, in Proceedings of the Thirty–Seven International Conference on Machine Learning (eds. N. Lawrence and M. Reid), PMLR, (2020), 8083–8092. Google Scholar

[29]

D. Jimenez, S. Racaniere, I. Higgins and P. Toth, Equivariant Hamiltonian flows, in CoRR, JCD, (2019). Google Scholar

[30]

M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), 157-216.  doi: 10.1023/B:MATH.0000027508.00421.bf.  Google Scholar

[31]

Y. Kosmann–Schwarzbach, Poisson manifolds, Lie algebroids, modular classes: A survey, SIGMA, 4 (2008), 30 pp. doi: 10.3842/SIGMA.2008.005.  Google Scholar

[32]

J.–L. Koszul, Crochet de Schouten–Nijenhuis et cohomologie, in Astérisque, Société mathématique de France, (1985), 257–271.  Google Scholar

[33]

V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, 1$^{st}$ edition, Springer–Verlag, Berlin Heidelberg, 1996. doi: 10.1007/978-3-642-78393-7.  Google Scholar

[34]

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

[35]

M. Lainz, C. Sardón and A. Weinstein, Plasma in a monopole background does not have a twisted Poisson structure, Phys. Rev. D, 100 (2019), 105016, 5 pp. doi: 10.1103/PhysRevD.100.105016.  Google Scholar

[36]

C. Laurent–Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures, 1$^{st}$ edition, Springer–Verlag, Berlin Heidelberg, 2013. doi: 10.1007/978-3-642-31090-4.  Google Scholar

[37]

S.-H. Li, C.-X. Dong, L. Zhang and L. Wang, Neural canonical transformation with symplectic flows, Phys. Rev. X, 10 (2020), 021020, 1–13. doi: 10.1103/PhysRevX.10.021020.  Google Scholar

[38]

A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom., 12 (1977), 253-300.  doi: 10.4310/jdg/1214433987.  Google Scholar

[39]

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

[40]

Y.–A. MaY. ChenC. JinN. Flammarion and M. I. Jordan, Sampling can be faster than optimization, Proc. Natl. Acad. Sci. USA, 116 (2019), 20881-20885.  doi: 10.1073/pnas.1820003116.  Google Scholar

[41]

I. Marcut and F. Zeiser, The Poisson cohomology of ${\mathfrak{sl}^{\ast}_2(\mathbf{R}^{{}})}$, preprint, arXiv: 1911.11732 [math.SG]. Google Scholar

[42]

P. W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, AMS, 2008. doi: 10.1090/gsm/093.  Google Scholar

[43]

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

[44]

A. PedrozaE. Velasco–Barreras and Yu. Vorobiev, Unimodularity criteria for Poisson structures on foliated manifolds, Lett. Math. Phys., 108 (2018), 861-882.  doi: 10.1007/s11005-017-1014-3.  Google Scholar

[45]

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

[46]

A. Sanchez–Gonzalez, V. Bapst, K. Cranmer and P. Battaglia, Hamiltonian graph networks with ODE integrators, in CoRR, JCD, (2019). Google Scholar

[47]

P. Ševera and A. Weinstein, Poisson geometry with a 3–form background, Prog. Theor. Phys., Suppl., 144 (2001), 145-154.  doi: 10.1143/PTPS.144.145.  Google Scholar

[48]

Y. Sheng, Linear Poisson structures on ${\mathbf{R}^{{4}}}$, J. Geom. Phys., 57 (2007), 2398-2410.  doi: 10.1016/j.geomphys.2007.08.008.  Google Scholar

[49]

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

[50]

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

[51]

D. Tamayo, M. Cranmer, S. Hadden, H. Rein, P. Battaglia, A. Obertas, P. Armitage, S Ho, D. N. Spergel, C. Gilbertson, N. Hussain, A. Silburt, D. Jontof–Hutter and K. Menou, Predicting the long-term stability of compact multiplanet systems, in Proceedings of the National Academy of Sciences (ed. M. R. Berenbaum), PNAS, (2020), 18194–18205. Google Scholar

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Figure 2.1.  Left: Symplectic foliation of $ {\Pi_{\mathfrak{so}(3)}} $ in (5). Right: Modular vector field of $ \Pi $ in (15) relative to the Euclidean volume form on $ {\mathbf{R}^{{3}}} $. The color scale indicates the magnitude of the vectors
Figure 2.2.  Left: Symplectic foliation of $ {\Pi_{\mathfrak{sl}(2)}} $ in (7). Right: Vector field W in (14), tangent to the symplectic foliation of $ {\Pi_{\mathfrak{sl}(2)}} $ on $ {\mathbf{R}^{{3}}_{x} \setminus \{x_{3}{\rm{–axis}}\}} $. The color scale indicates the magnitude of the vectors
Figure 2.3.  Symplectic (open book) foliation of $ \Pi $ in (20)
Table 3.3, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom-graph in each plot due to the accumulation of runtime values">Figure 3.1.  Log–log graphs of the execution time in seconds versus the number of points in $ {10^{\kappa}} $ –point (irregular) meshes of the $\mathtt{NumPoissonGeometry}$ functions 1–10 in Table 3.3, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom-graph in each plot due to the accumulation of runtime values
Table 3.5, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom–graph in each plot due to the accumulation of runtime values">Figure 3.2.  Log–log graphs of the execution time in seconds versus the number of points in $ {10^{\kappa}} $ –point (irregular) meshes of the $\mathtt{NumPoissonGeometry}$ functions 1–5 and 8–11 in Table 3.5, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom–graph in each plot due to the accumulation of runtime values
Table 3.5, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom–graph in each plot due to the accumulation of runtime values">Figure 3.3.  Log–log graphs of the execution time in seconds versus the number of points in $ {10^{\kappa}} $ –point (irregular) meshes of the $\mathtt{NumPoissonGeometry}$ functions 6 and 7 in Table 3.5, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom–graph in each plot due to the accumulation of runtime values
Figure 3.4.  Log-log graph of the execution time in seconds versus the number of points in a $ {10^{\kappa}} $-point (irregular) mesh of the $\mathtt{NumPoissonGeometry}$ function $\mathtt{num\_modular\_vf}$, for $ {\kappa = 4, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime, with the corresponding determination coefficient (R-squared) indicated in the legend
Figure 3.5.  Log–log graph of the execution time in seconds versus the number of points in a $ {10^{\kappa}} $ –point (irregular) mesh of the $\mathtt{num\_flaschka\_ratiu\_bivector}$ method, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime, with the corresponding determination coefficient (R–squared) indicated in the legend. We include a zoom–graph due to the accumulation of runtime values
Table 1.1.  Our numerical methods, with their corresponding algorithms, and examples where they are used. The right column is an informal summary of the algorithmic complexities, computed and presented in detail in Section 3
Method Algorithm Examples Complexity
$\mathtt{num\_bivector\_field}$ 2.1 [16,7,31,36] O($ m^2 $)
$\mathtt{num\_bivector\_to\_matrix}$ 2.2 [16,7,31,36] O($ m^2 $)
$\mathtt{num\_hamiltonian\_vf}$ 2.3 [33,7,50,6,26] O($ m^2 $)
$\mathtt{num\_poisson\_bracket}$ 2.4 [34,16,31,36] O($ m^2 $)
$\mathtt{num\_sharp\_morphism}$ 2.5 [16,31,36] O($ m^2 $)
$\mathtt{num\_coboundary\_operator}$* 2.6 [43,16,31,2,41] O($ 2^m $)
$\mathtt{num\_modular\_vf}$* 2.7 [1,16,31,27,36,3,44] O($ 2^m $)
$\mathtt{num\_curl\_operator}$* 2.8 [24,12,16,36] O($ 2^m $)
$\mathtt{num\_one\_forms\_bracket}$* 2.9 [16,31,36,25] O($ m^2 $)
$\mathtt{num\_gauge\_transformation}$ 2.10 [8,4,14] O($ m^7 $)
$\mathtt{num\_linear\_normal\_form\_R3}$* 2.11 [39,43,23,16,48,7,36,14,41,20] O($ m $)
$\mathtt{num\_flaschka\_ratiu\_bivector}$* 2.12 [24,12,22,49,17] O($ m^6 $)
Method Algorithm Examples Complexity
$\mathtt{num\_bivector\_field}$ 2.1 [16,7,31,36] O($ m^2 $)
$\mathtt{num\_bivector\_to\_matrix}$ 2.2 [16,7,31,36] O($ m^2 $)
$\mathtt{num\_hamiltonian\_vf}$ 2.3 [33,7,50,6,26] O($ m^2 $)
$\mathtt{num\_poisson\_bracket}$ 2.4 [34,16,31,36] O($ m^2 $)
$\mathtt{num\_sharp\_morphism}$ 2.5 [16,31,36] O($ m^2 $)
$\mathtt{num\_coboundary\_operator}$* 2.6 [43,16,31,2,41] O($ 2^m $)
$\mathtt{num\_modular\_vf}$* 2.7 [1,16,31,27,36,3,44] O($ 2^m $)
$\mathtt{num\_curl\_operator}$* 2.8 [24,12,16,36] O($ 2^m $)
$\mathtt{num\_one\_forms\_bracket}$* 2.9 [16,31,36,25] O($ m^2 $)
$\mathtt{num\_gauge\_transformation}$ 2.10 [8,4,14] O($ m^7 $)
$\mathtt{num\_linear\_normal\_form\_R3}$* 2.11 [39,43,23,16,48,7,36,14,41,20] O($ m $)
$\mathtt{num\_flaschka\_ratiu\_bivector}$* 2.12 [24,12,22,49,17] O($ m^6 $)
Table 3.1.  Worst–case time complexity of $\mathtt{NumPoissonGeometry}$ methods. In the second column: $ m $ denotes the dimension of $ {\mathbf{R}^{{m}}} $, $ k $ is the number of points in a mesh on $ {\mathbf{R}^{{m}}} $, we denote by $ {[\cdot]} $ the integer part function and by $ \mathrm{comb} $ a combination
Method Time Complexity
1. $\mathtt{num\_bivector\_field}$ $ { \mathscr{O}(m^2k|bivector|)} $
2. $\mathtt{num\_bivector\_to\_matrix}$ $ { \mathscr{O}(m^{2}k|bivector|)} $
3. $\mathtt{num\_hamiltonian\_vf}$ $ { \mathscr{O}(mk(m|bivector| + \mathrm{len}(ham\_function)))} $
4. $\mathtt{num\_poisson\_bracket}$ $ { \mathscr{O}(mk(m|bivector| + \mathrm{len}(function\_1) + \mathrm{len}(function\_2)))} $
5. $\mathtt{num\_sharp\_morphism}$ $ { \mathscr{O}(m(mk|bivector| + |one\_form|))} $
6. $\mathtt{num\_coboundary\_operator}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|bivector|\mathrm{len}(function)(m^{5} + k))} $
7. $\mathtt{num\_modular\_vf}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|bivector|\mathrm{len}(function)(m+k))} $
8. $\mathtt{num\_curl\_operator}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|multivector|\mathrm{len}(function)(m+k))} $
9. $\mathtt{num\_one\_forms\_bracket}$ $ { \mathscr{O}(m^2k|bivector||one\_form\_1||one\_form\_2|)} $
10. $\mathtt{num\_gauge\_transformation}$ $ { \mathscr{O}(m^2k(m^5 + |bivector| + |two\_form|))} $
11. $\mathtt{num\_linear\_normal\_form\_R3}$ $ { \mathscr{O}(k|bivector|)} $
12. $\mathtt{num\_flaschka\_ratiu\_bivector}$ $ { \mathscr{O}(m^{6}k|bivector|)} $
Method Time Complexity
1. $\mathtt{num\_bivector\_field}$ $ { \mathscr{O}(m^2k|bivector|)} $
2. $\mathtt{num\_bivector\_to\_matrix}$ $ { \mathscr{O}(m^{2}k|bivector|)} $
3. $\mathtt{num\_hamiltonian\_vf}$ $ { \mathscr{O}(mk(m|bivector| + \mathrm{len}(ham\_function)))} $
4. $\mathtt{num\_poisson\_bracket}$ $ { \mathscr{O}(mk(m|bivector| + \mathrm{len}(function\_1) + \mathrm{len}(function\_2)))} $
5. $\mathtt{num\_sharp\_morphism}$ $ { \mathscr{O}(m(mk|bivector| + |one\_form|))} $
6. $\mathtt{num\_coboundary\_operator}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|bivector|\mathrm{len}(function)(m^{5} + k))} $
7. $\mathtt{num\_modular\_vf}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|bivector|\mathrm{len}(function)(m+k))} $
8. $\mathtt{num\_curl\_operator}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|multivector|\mathrm{len}(function)(m+k))} $
9. $\mathtt{num\_one\_forms\_bracket}$ $ { \mathscr{O}(m^2k|bivector||one\_form\_1||one\_form\_2|)} $
10. $\mathtt{num\_gauge\_transformation}$ $ { \mathscr{O}(m^2k(m^5 + |bivector| + |two\_form|))} $
11. $\mathtt{num\_linear\_normal\_form\_R3}$ $ { \mathscr{O}(k|bivector|)} $
12. $\mathtt{num\_flaschka\_ratiu\_bivector}$ $ { \mathscr{O}(m^{6}k|bivector|)} $
Table 3.2.  Input data used for the time performance tests of functions 1-10 in Table 3.3
Function Input Function Input
1 $ {\Pi_{0}} $ 6 $ {\Pi_{0}} $, $ {W = x_{2} \frac{\partial}{\partial x_{1}} - x_{1} \frac{\partial}{\partial x_{2}}} $
2 $ {\Pi_{0}} $ 7 $ {\Pi_{0}} $, $ {f=1} $
3 $ {\Pi_{0}} $, $ {h = x_{1}^{2} + x_{2}^{2}} $ 8 $ {\Pi_{0}} $, $ {f=1} $
4 $ {\Pi_{0}} $, $ {f = x_{1}^{2} + x_{2}^{2}} $, $ {g=x_{1} + x_{2}} $ 9 $ {\Pi_{0}} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}}} $, $ {\beta = \mathrm{d}{x_{1}} + \mathrm{d}{x_{2}}} $
5 $ {\Pi_{0}} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}}} $ 10 $ {\Pi_{0}} $, $ {\lambda = \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{2}}} $
Function Input Function Input
1 $ {\Pi_{0}} $ 6 $ {\Pi_{0}} $, $ {W = x_{2} \frac{\partial}{\partial x_{1}} - x_{1} \frac{\partial}{\partial x_{2}}} $
2 $ {\Pi_{0}} $ 7 $ {\Pi_{0}} $, $ {f=1} $
3 $ {\Pi_{0}} $, $ {h = x_{1}^{2} + x_{2}^{2}} $ 8 $ {\Pi_{0}} $, $ {f=1} $
4 $ {\Pi_{0}} $, $ {f = x_{1}^{2} + x_{2}^{2}} $, $ {g=x_{1} + x_{2}} $ 9 $ {\Pi_{0}} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}}} $, $ {\beta = \mathrm{d}{x_{1}} + \mathrm{d}{x_{2}}} $
5 $ {\Pi_{0}} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}}} $ 10 $ {\Pi_{0}} $, $ {\lambda = \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{2}}} $
Table 3.3.  Summary of the time performance of $\mathtt{NumPoissonGeometry}$ functions, in dimension two
Function Points in mesh/Processing time (in seconds)
$ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
1. $\mathtt{num\_bivector\_field}$ 0.004 $ \pm $ 0.689 0.038 $ \pm $ 0.009 0.356 $ \pm $ 0.002 3.545 $ \pm $ 0.026 35.711 $ \pm $ 0.164
2. $\mathtt{num\_bivector\_to\_matrix}$ 0.006 $ \pm $ 3.633 0.046 $ \pm $ 0.001 0.438 $ \pm $ 0.004 4.442 $ \pm $ 0.037 45.155 $ \pm $ 1.466
3. $\mathtt{num\_hamiltonian\_vf}$ 0.014 $ \pm $ 0.001 0.112 $ \pm $ 0.006 1.096 $ \pm $ 0.021 10.867 $ \pm $ 0.044 108.460 $ \pm $ 0.726
4. $\mathtt{num\_poisson\_bracket}$ 0.021 $ \pm $ 0.006 0.169 $ \pm $ 0.001 1.652 $ \pm $ 0.008 16.721 $ \pm $ 0.049 168.110 $ \pm $ 1.637
5. $\mathtt{num\_sharp\_morphism}$ 0.014 $ \pm $ 0.658 0.111 $ \pm $ 0.001 1.068 $ \pm $ 0.007 10.725 $ \pm $ 0.142 107.275 $ \pm $ 0.667
6. $\mathtt{num\_coboundary\_operator}$ 0.001 $ \pm $ 0.087 0.008 $ \pm $ 0.001 0.084 $ \pm $ 0.006 0.848 $ \pm $ 0.011 8.638 $ \pm $ 0.045
7. $\mathtt{num\_modular\_vf}$ 0.004 $ \pm $ 0.754 0.030 $ \pm $ 0.009 0.280 $ \pm $ 0.001 2.805 $ \pm $ 0.016 28.057 $ \pm $ 0.107
8. $\mathtt{num\_curl\_operator}$ 0.022 $ \pm $ 0.009 0.196 $ \pm $ 0.024 1.923 $ \pm $ 0.004 18.487 $ \pm $ 0.136 182.774 $ \pm $ 1.260
9. $\mathtt{num\_one\_forms\_bracket}$ 0.058 $ \pm $ 0.006 0.420 $ \pm $ 0.007 4.278 $ \pm $ 0.027 43.257 $ \pm $ 0.071 434.450 $ \pm $ 0.589
10. $\mathtt{num\_gauge\_transformation}$ 0.051 $ \pm $ 0.001 0.446 $ \pm $ 0.010 4.380 $ \pm $ 0.016 43.606 $ \pm $ 0.212 434.704 $ \pm $ 1.234
Function Points in mesh/Processing time (in seconds)
$ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
1. $\mathtt{num\_bivector\_field}$ 0.004 $ \pm $ 0.689 0.038 $ \pm $ 0.009 0.356 $ \pm $ 0.002 3.545 $ \pm $ 0.026 35.711 $ \pm $ 0.164
2. $\mathtt{num\_bivector\_to\_matrix}$ 0.006 $ \pm $ 3.633 0.046 $ \pm $ 0.001 0.438 $ \pm $ 0.004 4.442 $ \pm $ 0.037 45.155 $ \pm $ 1.466
3. $\mathtt{num\_hamiltonian\_vf}$ 0.014 $ \pm $ 0.001 0.112 $ \pm $ 0.006 1.096 $ \pm $ 0.021 10.867 $ \pm $ 0.044 108.460 $ \pm $ 0.726
4. $\mathtt{num\_poisson\_bracket}$ 0.021 $ \pm $ 0.006 0.169 $ \pm $ 0.001 1.652 $ \pm $ 0.008 16.721 $ \pm $ 0.049 168.110 $ \pm $ 1.637
5. $\mathtt{num\_sharp\_morphism}$ 0.014 $ \pm $ 0.658 0.111 $ \pm $ 0.001 1.068 $ \pm $ 0.007 10.725 $ \pm $ 0.142 107.275 $ \pm $ 0.667
6. $\mathtt{num\_coboundary\_operator}$ 0.001 $ \pm $ 0.087 0.008 $ \pm $ 0.001 0.084 $ \pm $ 0.006 0.848 $ \pm $ 0.011 8.638 $ \pm $ 0.045
7. $\mathtt{num\_modular\_vf}$ 0.004 $ \pm $ 0.754 0.030 $ \pm $ 0.009 0.280 $ \pm $ 0.001 2.805 $ \pm $ 0.016 28.057 $ \pm $ 0.107
8. $\mathtt{num\_curl\_operator}$ 0.022 $ \pm $ 0.009 0.196 $ \pm $ 0.024 1.923 $ \pm $ 0.004 18.487 $ \pm $ 0.136 182.774 $ \pm $ 1.260
9. $\mathtt{num\_one\_forms\_bracket}$ 0.058 $ \pm $ 0.006 0.420 $ \pm $ 0.007 4.278 $ \pm $ 0.027 43.257 $ \pm $ 0.071 434.450 $ \pm $ 0.589
10. $\mathtt{num\_gauge\_transformation}$ 0.051 $ \pm $ 0.001 0.446 $ \pm $ 0.010 4.380 $ \pm $ 0.016 43.606 $ \pm $ 0.212 434.704 $ \pm $ 1.234
Table 3.4.  Input data used for the time performance tests of functions 1–11 in Table 3.5
Function Input
1 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $
2 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $
3 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {h = x_{1}^{2} + x_{2}^{2} - x_{3}^{2}} $
4 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f = x_{1}^{2} + x_{2}^{2} - x_{3}^{2}} $, $ {g=x_{1} + x_{2} + x_{3}} $
5 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}} - x_{3} \mathrm{d}{x_{3}}} $
6 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {W = e^{{-1}/{(x_1^2 + x_2^2 - x_3^2)^2}} [ {x_1x_{3}}/(x_1^2 + x_2^2)\frac{\partial}{\partial{x_{1}}} + {x_2x_{3}}/(x_1^2 + x_2^2)\frac{\partial}{\partial{x_{2}}} + \frac{\partial}{\partial{x_{3}}} ]} $
7 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f=1} $
8 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f=1} $
9 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}} - x_{3} \mathrm{d}{x_{3}}} $, $ {\beta = \mathrm{d}{x_{1}} + \mathrm{d}{x_{2}} + \mathrm{d}{x_{3}}} $
10 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\lambda = (x_{2}-x_{1}) \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{2}} + (x_{3}-x_{1}) \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{3}} + (x_{2}-x_{3}) \mathrm{d}{x_{2}} \wedge \mathrm{d}{x_{3}}} $
11 $ -\Pi_{\mathfrak{sl}(2)} $
Function Input
1 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $
2 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $
3 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {h = x_{1}^{2} + x_{2}^{2} - x_{3}^{2}} $
4 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f = x_{1}^{2} + x_{2}^{2} - x_{3}^{2}} $, $ {g=x_{1} + x_{2} + x_{3}} $
5 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}} - x_{3} \mathrm{d}{x_{3}}} $
6 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {W = e^{{-1}/{(x_1^2 + x_2^2 - x_3^2)^2}} [ {x_1x_{3}}/(x_1^2 + x_2^2)\frac{\partial}{\partial{x_{1}}} + {x_2x_{3}}/(x_1^2 + x_2^2)\frac{\partial}{\partial{x_{2}}} + \frac{\partial}{\partial{x_{3}}} ]} $
7 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f=1} $
8 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f=1} $
9 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}} - x_{3} \mathrm{d}{x_{3}}} $, $ {\beta = \mathrm{d}{x_{1}} + \mathrm{d}{x_{2}} + \mathrm{d}{x_{3}}} $
10 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\lambda = (x_{2}-x_{1}) \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{2}} + (x_{3}-x_{1}) \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{3}} + (x_{2}-x_{3}) \mathrm{d}{x_{2}} \wedge \mathrm{d}{x_{3}}} $
11 $ -\Pi_{\mathfrak{sl}(2)} $
Table 3.5.  Summary of the time performance of $\mathtt{NumPoissonGeometry}$ functions in dimension 3
Function Points in mesh/Processing time (in seconds)
$ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
1. $\mathtt{num\_bivector\_field}$ 0.009 $ \pm $ 0.009 0.051 $ \pm $ 0.004 0.496 $ \pm $ 0.002 4.984 $ \pm $ 0.023 49.565 $ \pm $ 0.222
2. $\mathtt{num\_bivector\_to\_matrix}$ 0.008 $ \pm $ 3.164 0.057 $ \pm $ 0.002 0.553 $ \pm $ 0.019 5.442 $ \pm $ 0.023 55.249 $ \pm $ 1.690
3. $\mathtt{num\_hamiltonian\_vf}$ 0.017 $ \pm $ 0.002 0.129 $ \pm $ 0.001 1.263 $ \pm $ 0.022 12.518 $ \pm $ 0.064 126.091 $ \pm $ 0.583
4. $\mathtt{num\_poisson\_bracket}$ 0.036 $ \pm $ 0.001 0.299 $ \pm $ 0.010 2.936 $ \pm $ 0.067 29.600 $ \pm $ 0.933 292.625 $ \pm $ 6.094
5. $\mathtt{num\_sharp\_morphism}$ 0.017 $ \pm $ 0.006 0.128 $ \pm $ 0.005 1.252 $ \pm $ 0.005 12.384 $ \pm $ 0.038 124.851 $ \pm $ 1.809
6. $\mathtt{num\_coboundary\_operator}$ 1.589 $ \pm $ 0.016 1.705 $ \pm $ 0.029 2.815 $ \pm $ 0.032 12.972 $ \pm $ 0.166 111.034 $ \pm $ 1.365
7. $\mathtt{num\_modular\_vf}$ 0.050 $ \pm $ 0.001 0.103 $ \pm $ 0.004 0.645 $ \pm $ 0.006 6.025 $ \pm $ 0.013 59.652 $ \pm $ 0.146
8. $\mathtt{num\_curl\_operator}$ 0.019 $ \pm $ 0.010 0.129 $ \pm $ 0.027 1.199 $ \pm $ 0.032 10.911 $ \pm $ 0.181 105.841 $ \pm $ 1.230
9. $\mathtt{num\_one\_forms\_bracket}$ 0.093 $ \pm $ 0.001 0.738 $ \pm $ 0.007 7.285 $ \pm $ 0.159 72.802 $ \pm $ 1.474 724.514 $ \pm $ 13.594
10. $\mathtt{num\_gauge\_transformation}$ 0.051 $ \pm $ 0.001 0.445 $ \pm $ 0.010 4.395 $ \pm $ 0.013 43.794 $ \pm $ 0.173 437.326 $ \pm $ 0.824
11. $\mathtt{num\_linear\_normal\_form\_R3}$ 0.016 $ \pm $ 0.438 0.061 $ \pm $ 0.002 0.504 $ \pm $ 0.012 4.903 $ \pm $ 0.017 48.786 $ \pm $ 0.219
Function Points in mesh/Processing time (in seconds)
$ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
1. $\mathtt{num\_bivector\_field}$ 0.009 $ \pm $ 0.009 0.051 $ \pm $ 0.004 0.496 $ \pm $ 0.002 4.984 $ \pm $ 0.023 49.565 $ \pm $ 0.222
2. $\mathtt{num\_bivector\_to\_matrix}$ 0.008 $ \pm $ 3.164 0.057 $ \pm $ 0.002 0.553 $ \pm $ 0.019 5.442 $ \pm $ 0.023 55.249 $ \pm $ 1.690
3. $\mathtt{num\_hamiltonian\_vf}$ 0.017 $ \pm $ 0.002 0.129 $ \pm $ 0.001 1.263 $ \pm $ 0.022 12.518 $ \pm $ 0.064 126.091 $ \pm $ 0.583
4. $\mathtt{num\_poisson\_bracket}$ 0.036 $ \pm $ 0.001 0.299 $ \pm $ 0.010 2.936 $ \pm $ 0.067 29.600 $ \pm $ 0.933 292.625 $ \pm $ 6.094
5. $\mathtt{num\_sharp\_morphism}$ 0.017 $ \pm $ 0.006 0.128 $ \pm $ 0.005 1.252 $ \pm $ 0.005 12.384 $ \pm $ 0.038 124.851 $ \pm $ 1.809
6. $\mathtt{num\_coboundary\_operator}$ 1.589 $ \pm $ 0.016 1.705 $ \pm $ 0.029 2.815 $ \pm $ 0.032 12.972 $ \pm $ 0.166 111.034 $ \pm $ 1.365
7. $\mathtt{num\_modular\_vf}$ 0.050 $ \pm $ 0.001 0.103 $ \pm $ 0.004 0.645 $ \pm $ 0.006 6.025 $ \pm $ 0.013 59.652 $ \pm $ 0.146
8. $\mathtt{num\_curl\_operator}$ 0.019 $ \pm $ 0.010 0.129 $ \pm $ 0.027 1.199 $ \pm $ 0.032 10.911 $ \pm $ 0.181 105.841 $ \pm $ 1.230
9. $\mathtt{num\_one\_forms\_bracket}$ 0.093 $ \pm $ 0.001 0.738 $ \pm $ 0.007 7.285 $ \pm $ 0.159 72.802 $ \pm $ 1.474 724.514 $ \pm $ 13.594
10. $\mathtt{num\_gauge\_transformation}$ 0.051 $ \pm $ 0.001 0.445 $ \pm $ 0.010 4.395 $ \pm $ 0.013 43.794 $ \pm $ 0.173 437.326 $ \pm $ 0.824
11. $\mathtt{num\_linear\_normal\_form\_R3}$ 0.016 $ \pm $ 0.438 0.061 $ \pm $ 0.002 0.504 $ \pm $ 0.012 4.903 $ \pm $ 0.017 48.786 $ \pm $ 0.219
Table 3.6.  Mean time in seconds (with standard deviation) it takes to evaluate the $\mathtt{num\_flaschka\_ratiu\_bivector}$ method on a irregular mesh on $ \mathbf{R}^{{4}} $ with $ 10^{\kappa} $ points, computed by taking twenty-five samples, for $ {\kappa = 3, \ldots, 7} $
Function Points in mesh/Processing time (in seconds)
$ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
$\mathtt{num\_flaschka\_ratiu\_bivector}$ 0.0158 $ \pm $ 0.105 0.057 $ \pm $ 0.003 0.505 $ \pm $ 0.003 4.993 $ \pm $ 0.029 49.563 $ \pm $ 0.207
Function Points in mesh/Processing time (in seconds)
$ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
$\mathtt{num\_flaschka\_ratiu\_bivector}$ 0.0158 $ \pm $ 0.105 0.057 $ \pm $ 0.003 0.505 $ \pm $ 0.003 4.993 $ \pm $ 0.029 49.563 $ \pm $ 0.207
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