Lie groups and continuum mechanics: where do we stand today?

Géry de Saxcé; Dina Razafindralandy

doi:10.5802/crmeca.242

Lie groups and continuum mechanics: where do we stand today?

Géry de Saxcé ¹ ; Dina Razafindralandy ²

¹ Univ. Lille, CNRS, Centrale Lille, UMR 9013 - LaMcube - Laboratoire de mécanique multiphysique multiéchelle, F59000, Lille, France
² Laboratoire des Sciences de l’Ingénieur pour l’Environnement (LaSIE) - UMR CNRS 7356, Pôle Sciences et Technologie, Avenue Michel Crépeau, F17042 La Rochelle Cedex 1, France

Comptes Rendus. Mécanique, The French “Année de la Mécanique”: some views on recent advances in solid and fluid mechanics, Volume 351 (2023) no. S3, pp. 135-159.

Abstract
Résumé

The geometric methods have experienced a fast growth in the past few decades. In this survey, we discuss the use of Lie groups in continuum mechanics. We address both the theoretical and numerical aspects. We explore the classical symmetry groups of the mechanics, the covariant form of the equations and the symmetry group of constitutive laws. We consider the Lie symmetry group of the equations of a mechanical problem and investigate how to take advantage of them in developping analytical models (self-similar solutions, conservation laws, turbulence, ...) of the physical phenomena encoded in these equations. Lastly, we present a method of constructing robust numerical integrators from the knowledge of the Lie symmetry group of the equations.

Les méthodes géométriques ont connu une croissance rapide au cours des dernières décennies. Dans cette étude, nous discutons de l’utilisation des groupes de Lie en mécanique des milieux continus. Nous abordons à la fois les aspects théoriques et numériques. Nous explorons les groupes de symétrie classiques de la mécanique, la forme covariante des équations et le groupe de symétrie des lois constitutives. Nous considérons le groupe de symétrie de Lie des équations d’un problème de mécanique et montrons comment en tirer profit dans le développement de modèles analytiques (solutions auto-similaires, lois de conservation, turbulence, etc.) des phénomènes physiques encodés dans ces équations. Enfin, nous présentons une méthode de construction d’intégrateurs numériques robustes à partir de la connaissance du groupe de symétrie de Lie des équations.

Received: 2023-12-04
Revised: 2024-02-12
Accepted: 2024-02-13
Online First: 2024-05-06
Published online: 2024-08-26

DOI: 10.5802/crmeca.242

Keywords: Lie groups, continuum mechanics, symmetry groups, geometric integrators, turbulence modelling
Mots-clés : Groupes de Lie, mécanique des milieux continus, groupes de symétrie, intégrateurs géométriques, modélisation de la turbulence

Author's affiliations:

Géry de Saxcé ¹; Dina Razafindralandy ²

¹ Univ. Lille, CNRS, Centrale Lille, UMR 9013 - LaMcube - Laboratoire de mécanique multiphysique multiéchelle, F59000, Lille, France
² Laboratoire des Sciences de l’Ingénieur pour l’Environnement (LaSIE) - UMR CNRS 7356, Pôle Sciences et Technologie, Avenue Michel Crépeau, F17042 La Rochelle Cedex 1, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMECA_2023__351_S3_135_0,
     author = {G\'ery de Saxc\'e and Dina Razafindralandy},
     title = {Lie groups and continuum mechanics: where do we stand today?},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {135--159},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {351},
     number = {S3},
     year = {2023},
     doi = {10.5802/crmeca.242},
     language = {en},
}

TY  - JOUR
AU  - Géry de Saxcé
AU  - Dina Razafindralandy
TI  - Lie groups and continuum mechanics: where do we stand today?
JO  - Comptes Rendus. Mécanique
PY  - 2023
SP  - 135
EP  - 159
VL  - 351
IS  - S3
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.242
LA  - en
ID  - CRMECA_2023__351_S3_135_0
ER  -

%0 Journal Article
%A Géry de Saxcé
%A Dina Razafindralandy
%T Lie groups and continuum mechanics: where do we stand today?
%J Comptes Rendus. Mécanique
%D 2023
%P 135-159
%V 351
%N S3
%I Académie des sciences, Paris
%R 10.5802/crmeca.242
%G en
%F CRMECA_2023__351_S3_135_0

Géry de Saxcé; Dina Razafindralandy. Lie groups and continuum mechanics: where do we stand today?. Comptes Rendus. Mécanique, The French “Année de la Mécanique”: some views on recent advances in solid and fluid mechanics, Volume 351 (2023) no. S3, pp. 135-159. doi : 10.5802/crmeca.242. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.242/

References
Cited by

[1] R. Abraham; J. E. Marsden; R. Ratiu Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences, 75, Springer New York, 2012

[2] Daniel Bump Lie groups, Graduate texts in mathematics, 225, Springer, 2013 | DOI | Zbl

[3] Sophus Lie Theorie der Transformationsgruppen I, Math. Ann., Volume 16 (1880), pp. 441-528 | DOI | MR

[4] S. Lie; R. Hermann Sophus Lie’s 1880 Transformation Group Paper. Vol. I. Sophus Lie’s 1880 transformation group paper, Math. Sci. Press., 1975 | Zbl

[5] Sophus Lie; J. Merker Theory of Transformation Groups I: General Properties of Continuous Transformation Groups. A Contemporary Approach and Translation, Springer, 2015 | Zbl

[6] Evariste Galois Mémoire sur les conditions de résolubilité des équations par radicaux, Journal de mathématiques pures et appliquées, Volume 1846 (1830), pp. 417-433

[7] F. Borceux; G. Janelidze Galois Theories, Cambridge studies in advanced mathematics, 72, Cambridge University Press, 2001 | DOI | Zbl

[8] P. Neumann The Mathematical Writings of Évariste Galois, Heritage of European Mathematics, European Mathematical Society, 2011 | DOI | Zbl

[9] S. Kobayashi Transformation Groups in Differential Geometry, Springer, 1972 | DOI | Zbl

[10] J. L. Koszul; Z. Yi Ming Introduction to symplectic geometry, Springer, 2019 | DOI | Zbl

[11] R. H. Cushman; L. M. Bates Global Aspects of Classical Integrable Systems, Birkhäuser, 1997 | DOI | Zbl

[12] J.-M. Souriau Structure of Dynamical Systems. A Symplectic View of Physics, Birkhäuser, 1997 | Zbl

[13] R. Abraham; J. Marsden Foundation of Mechanics, Addison-Wesley Publishing Group, 1987

[14] V. Guillemin; S. Sternberg Symplectic techniques in physics, Cambridge University Press, 1984 | Zbl

[15] L. Ryvkin; T. Wurzbacher An invitation to multisymplectic geometry, J. Geom. Phys., Volume 142 (2019), pp. 9-36 | DOI | MR | Zbl

[16] R. Toupin World invariant kinematics, Arch. Ration. Mech. Anal., Volume 1 (1958), pp. 181-211 | DOI | MR | Zbl

[17] W. Noll Lectures on the foundations of continuum mechanics and thermodynamics, Arch. Ration. Mech. Anal., Volume 52 (1973), pp. 62-92 | DOI | MR | Zbl

[18] H. P. Künzle Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 4 (1972), pp. 337-362

[19] G. de Saxcé; C. Vallée Galilean Mechanics and Thermodynamics of Continua, Mechanical Engineering and Solid Mechanics Series, Wiley-ISTE, 2016 | DOI

[20] J.-M. Souriau Géométrie et relativité, Collection Enseignement des Sciences, 6, Hermann, 1964 | Zbl

[21] C. Eckart The Thermodynamics of Irreversible Processes, III. Relativistic Theory of the Simple Fluid, Phys. Rev., II. Ser., Volume 58 (1940), pp. 919-924 | Zbl

[22] L. Landau; E. Lifchitz Fluid Mechanics, Pergamon Press, 1959

[23] J.-M. Souriau Thermodynamique et géométrie, Differential Geometrical Methods in Mathematical Physics II (Lecture Notes in Mathematics), Volume 676, Springer, 1976, pp. 369-397 | DOI | Zbl

[24] J.-M. Souriau Thermodynamique relativiste des fluides, Rend. Semin. Mat., Torino, Volume 35 (1978), pp. 21-34 | Zbl

[25] C. Vallée Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste, Ph. D. Thesis, University of Poitiers, Poitiers, France (1978)

[26] C. Vallée Relativistic thermodynamics of continua, Int. J. Eng. Sci., Volume 19 (1981), pp. 589-601 | DOI | MR | Zbl

[27] E. Rouhaud; B. Panicaud; R. Kerner Canonical frame-indifferent transport operators with the four-dimensional formalism of differential geometry, Comput. Mater. Sci., Volume 77 (2013), pp. 120-130 | DOI

[28] B. Panicaud; E. Rouhaud A frame-indifferent model for a thermo-elastic material beyond the three-dimensional Eulerian and Lagrangian descriptions, Contin. Mech. Thermodyn., Volume 26 (2014), pp. 79-93 | DOI | MR | Zbl

[29] G. Altmeyer; B. Panicaud; E. Rouhaud; W. Mingchuan; A. Roos; R. Kerner Viscoelasticity behavior for finite deformations, using a consistent hypoelastic model based on Rivlin materials, Contin. Mech. Thermodyn., Volume 28 (2016), pp. 1741-1758 | DOI | MR | Zbl

[30] V. Bargmann On unitary representation of continuous groups, Ann. Math., Volume 59 (1954), pp. 1-46 | DOI | MR | Zbl

[31] K. Grabowska; J. Grabowski; P. Urbanski AV-differential geometry: Poisson and Jacobi structures, J. Geom. Phys., Volume 52 (2004), pp. 398-446 | DOI | MR | Zbl

[32] É. Cartan Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Ann. Sci. Éc. Norm. Supér., Volume 40 (1923), pp. 325-412 | DOI | Numdam | Zbl

[33] É. Cartan Sur les variétés à connexion affine et la théorie de la relativité généralisée. I. (suite), Ann. Sci. Éc. Norm. Supér., Volume 41 (1924), pp. 1-25 | DOI | Zbl

[34] J. P. Boehler; A. Kirillov; E. T. Onat On the polynomial invariants of elasticity tensors, J. Elasticity, Volume 34 (1994) no. 2, pp. 97-110 | DOI | MR | Zbl

[35] S. Forte; M. Vianello Symmetry classes for elasticity tensors, J. Elasticity, Volume 43 (1996) no. 2, p. 81-–108 | DOI | MR | Zbl

[36] M. Olive; N. Auffray Symmetry classes for even-order tensors, Math. Mech. Complex Syst., Volume 1 (2013) no. 2, pp. 177-210 | DOI | Zbl

[37] G. de Saxcé; C. Vallée Invariant measures of the lack of symmetry with respect to the symmetry groups of 2D elasticity tensors, J. Elasticity, Volume 111 (2013) no. 1, pp. 21-39 | DOI | MR | Zbl

[38] S. Forte; M. Vianello A unified approach to invariants of plane elasticity tensors, Meccanica, Volume 49 (2014) no. 9, pp. 2001-2012 | DOI | MR | Zbl

[39] G. Verchery Les invariants des tenseurs d’ordre 4 du type de l’élasticité, Proceedings of Euromech 115, Villard-de-Lance (1979) (Colloques Internationaux du CNRS), Volume 295, Edition du CNRS (1982), pp. 93-104 | Zbl

[40] P. Vannucci; G. Verchery Anisotropy of plane complex elastic bodies, Int. J. Solids Struct., Volume 47 (2010) no. 9, pp. 1154-1166 | DOI | Zbl

[41] P. Vannucci; G. Verchery Stiffness design of laminates using the polar method, Int. J. Solids Struct., Volume 38 (2001) no. 50-51, pp. 9281-9294 | DOI | Zbl

[42] A. Antonelli; B. Desmorat; B. Kolev; R. Desmorat Distance to plane elasticity orthotropy by Euler–Lagrange method, C. R. Mécanique, Volume 350 (2022), pp. 413-430 | DOI

[43] H. Abdoul-Anziz; N. Auffray; B. Desmorat Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law, Symmetry, Volume 12 (2020) no. 4, pp. 674-702 | DOI

[44] M. François A damage model based on Kelvin eigentensors and Curie principle, Mech. Mater., Volume 44 (2012), pp. 23-24 | DOI

[45] P. Curie Sur la symétrie dans les phénomènes physiques, symétrie d’un champélectrique et d’un champ magnétique, J. Phys. Theor. Appl., Volume 3 (1894) no. 1, pp. 393-415 | DOI | Zbl

[46] P. Olver Applications of Lie groups to differential equations, Graduate Texts in Mathematics, 107, Springer, 1986 | DOI | Zbl

[47] S. V. Meleshko Complete group classification of the two-Dimensional shallow water equations with constant coriolis parameter in Lagrangian coordinates, Commun. Nonlinear Sci. Numer. Simul., Volume 89 (2020), 105293 | DOI | MR | Zbl

[48] K. S. Platonova; A. V. Borovskikh Group Analysis of the Boltzmann and Vlasov Equations, Theor. Math. Phys., Volume 203 (2020) no. 3, pp. 794-823 | DOI | MR | Zbl

[49] T. Özer Symmetry group classification for two-dimensional elastodynamics problems in nonlocal elasticity, Int. J. Eng. Sci., Volume 41 (2003) no. 18, pp. 2193-2211 | DOI | MR

[50] A. V. Mikhailov; A. B. Shabat; V. V. Sokolov The Symmetry Approach to Classification of Integrable Equations, What Is Integrability? (Vladimir E. Zakharov, ed.), Springer, 1991, pp. 115-184 | DOI | Zbl

[51] R. Cherniha; M. Serov; Y. Prystavka A complete Lie symmetry classification of a class of $(1 + 2)$ -dimensional reaction-diffusion-convection equations, Commun. Nonlinear Sci. Numer. Simul., Volume 92 (2021), 105466 | DOI | MR | Zbl

[52] N. H. Ibragimov CRC handbook of Lie group analysis of differential equations. Volume 1: Symmetries, exact solutions and conservation laws, CRC Press, 1994

[53] Q. Huang; S. Shen Lie symmetries and group classification of a class of time fractional evolution systems, J. Math. Phys., Volume 56 (2015), 123504 | DOI | MR | Zbl

[54] M. S. Hashemi; D. Baleanu Lie Symmetry Analysis of Fractional Differential Equations, CRC Press, 2020 | DOI | Zbl

[55] F. Oliveri Lie Symmetries of Differential Equations: Classical Results and Recent Contributions, Symmetry, Volume 2 (2010) no. 2, pp. 658-706 | DOI | MR | Zbl

[56] M. S. Bruzón; M. L. Gandarias; R. de la Rosa Conservation Laws of a Family of Reaction-Diffusion-Convection Equations, Localized Excitations in Nonlinear Complex Systems: Current State of the Art and Future Perspectives (R. Carretero-González; J. Cuevas-Maraver; D. Frantzeskakis; N. Karachalios; P. Kevrekidis; F. Palmero-Acebedo, eds.) (Nonlinear Systems and Complexity), Volume 7, Springer, 2014, pp. 403-417 | DOI | Zbl

[57] M. L. Gandarias; C. Khalique Symmetries, solutions and conservation laws of a class of nonlinear dispersive wave equations, Commun. Nonlinear Sci. Numer. Simul., Volume 32 (2016), pp. 114-121 | DOI | MR | Zbl

[58] R. de la Rosa; M. L. Gandarias; M. Bruzon On symmetries and conservation laws of a Gardner equation involving arbitrary functions, Appl. Math. Comput., Volume 290 (2016), pp. 125-134 | MR | Zbl

[59] P. Djondjorov Invariant properties of Timoshenko beam equations, Int. J. Eng. Sci., Volume 33 (1995) no. 14, pp. 2103-2114 | DOI | MR | Zbl

[60] U. Camci $F (R, G)$ Cosmology through Noether Symmetry Approach, Symmetry, Volume 10 (2018) no. 12, 719 | DOI | Zbl

[61] N. Dimakis; A. Giacomini; N. Jamal; G. Leon; A. Paliathanasis Noether symmetries and stability of ideal gas solution in Galileon Cosmology, Phys. Rev. D, Volume 95 (2017) no. 6, 064031 | DOI | MR

[62] V. Vassilev; P. Djondjorov; I. Mladenov Lie Group Analysis of the Willmore and Membrane Shape Equations, Lecture Notes in Applied and Computational Mechanics, 73, Springer (2014), pp. 365-376 | Zbl

[63] Almudena P. Marquez; M. S. Bruzón Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin–Voigt Viscoelasticity Equation, Symmetry, Volume 11 (2019) no. 7, 840 | Zbl

[64] S Senashov; O. Gomonova; I. Savostyanova; O. Cherepanova New Classes of Solutions of Dynamical Problems of Plasticity, J. Sib. Fed. Univ., Math. Phys., Volume 13 (2020) no. 6, pp. 792-796 | DOI | MR | Zbl

[65] A. Halder; A. Paliathanasis; P. Leach Similarity solutions and conservation laws for the Beam Equations: a complete study, Acta Polytech. CTU Proc., Volume 60 (2020) no. 2, pp. 98-110 | DOI

[66] D. Huang; X. Li; S. Yu Lie Symmetry Classification of the Generalized Nonlinear Beam Equation, Symmetry, Volume 9 (2017) no. 7, 115 | DOI | MR | Zbl

[67] Darryl D. Holm Euler–Poincaré Dynamics of Perfect Complex Fluids, Geometry, Mechanics, and Dynamics, Springer, New York, 2002, pp. 169-180 | DOI | Zbl

[68] D. Holm; T. Schmah; C. Stoica Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, 12, Wiley-Interscience, 2009 | DOI | Zbl

[69] W. Fushchych; R. Popowych Symmetry reduction and exact solutions of the Navier–Stokes equations. I, J. Nonlinear. Math. Phys., Volume 1 (1994) no. 1, pp. 75-113 | DOI | MR | Zbl

[70] W. Fushchych; R. Popowych Symmetry reduction and exact solutions of the Navier–Stokes equations. II, J. Nonlinear. Math. Phys., Volume 1 (1994) no. 2, pp. 158-188 | DOI | MR | Zbl

[71] V. Grassi; R. A. Leo; G. Soliani; P. Tempesta Vorticies and invariant surfaces generated by symmetries for the 3D Navier–Stokes equation, Physica A, Volume 286 (2000), pp. 79-108 | DOI | Zbl

[72] D. Dierkes; A. Cheviakov; M. Oberlack New similarity reductions and exact solutions for helically symmetric viscous flows, Phys. Fluids, Volume 32 (2020) no. 5, 053604 | DOI

[73] E. Nœther Invariante Variationsprobleme, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1918), pp. 235-257 | Zbl

[74] E. Nœther; A. Tavel Invariant variation problems, Transp. Theory Stat. Phys., Volume 1 (1971) no. 3, pp. 183-207 (english traduction of the Nœther’s original paper [noether18] in 1918.) | DOI | MR | Zbl

[75] Y. Kosmann-Schwarzbach The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, 2011 | DOI | Zbl

[76] Y.-N. Huang; R. C. Batra Energy-momentum tensors in nonsimple elastic dielectrics, J. Elasticity, Volume 42 (1996), pp. 275-281 | DOI | MR | Zbl

[77] P. Yu; H. Wang; J. Chen; S. Shengping Conservation laws and path-independent integrals in mechanical-diffusion-electrochemical reaction coupling system, J. Mech. Phys. Solids, Volume 104 (2017), pp. 57-70 | DOI | MR | Zbl

[78] S. El Kabir; F. Dubois; R. Moutou Pitti; N. Recho; Y. Lapusta A new analytical generalization of the J and G-theta integrals for planar cracks in a three-dimensional medium, Theor. Appl. Fract. Mech., Volume 94 (2018), pp. 101-109 | DOI

[79] V. K. Kalpakides; E. K. Agiasofitou On material equations in second gradient electroelasticity, J. Elasticity, Volume 67 (2002) no. 3, pp. 205-227 | DOI | MR | Zbl

[80] V. Lubardpa; X. Markenscoff On conservation integrals in micropolar elasticity, Philos. Mag., Volume 83 (2003) no. 11, pp. 1365-1377 | DOI

[81] M. Lazar; H. Kirchner The Eshelby stress tensor, angular momentum tensor and scaling flux in micropolar elasticity, Int. J. Solids Struct., Volume 44 (2007) no. 14-15, pp. 4613-4620 | DOI | Zbl

[82] P. Olver Conservation laws in elasticity. I: General results, Arch. Ration. Mech. Anal., Volume 85 (1984) no. 2, pp. 111-129 | DOI | MR | Zbl

[83] E. Agiasofitou; M. Lazar Conservation and Balance Laws in Linear Elasticity of Grade Three, J. Elasticity, Volume 94 (2009) no. 1, pp. 69-85 | DOI | MR | Zbl

[84] S. Li; A. Gupta; X. Markenscoff Conservation Laws of Linear Elasticity in Stress Formulations, Proc. R. Soc. A: Math. Phys. Eng. Sci., Volume 461 (2005) no. 2053, pp. 99-116 | DOI | MR | Zbl

[85] R. Kienzler; G. Herrmann Mechanics in Material Space with Applications to Defect and Fracture Mechanics, Springer, 2000 | Zbl

[86] G. Maugin Material Inhomogeneities in Elasticity, Applied Mathematics and Mathematical Computation, 3, Chapman & Hall, 1993 | DOI | Zbl

[87] R. Shankar Symmetries and conservation laws of the Euler equations in Lagrangian coordinates, J. Math. Anal. Appl., Volume 447 (2017) no. 2, pp. 867-881 | DOI | MR | Zbl

[88] Marcel Mellmann; Markus Scholle Symmetries and Related Physical Balances for Discontinuous Flow Phenomena within the Framework of Lagrange Formalism, Symmetry, Volume 13 (2021) no. 9, 1662 | DOI

[89] E. Kaptsov; S. S. Meleshko Conservation laws of the two-dimensional gas dynamics equations, Int. J. Non-Linear Mech., Volume 112 (2019), pp. 126-132 | DOI

[90] W. Nakpim; S. V. Meleshko Conservation laws of the one-dimensional equations of relativistic gas dynamics in Lagrangian coordinates, Int. J. Non-Linear Mech., Volume 124 (2020), 103496 | DOI

[91] A. Yahalom; H. Qin Noether currents for Eulerian variational principles in non-barotropic magnetohydrodynamics and topological conservations laws, J. Fluid Mech., Volume 908 (2021), A4 | DOI | MR | Zbl

[92] G. Webb Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws, Lecture Notes in Physics, 946, Springer, 2018 | Zbl

[93] O. Kelbin; A. Cheviakov; M. Oberlack New conservation laws of helically symmetric, plane and rotationally symmetric viscous and inviscid flows, J. Fluid Mech., Volume 721 (2013), pp. 340-366 | DOI | MR | Zbl

[94] C. Cotter; D. Holm On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities, Found. Comput. Math., Volume 13 (2013), pp. 457-477 | DOI | Zbl

[95] L. Fatibene; M. Francaviglia; S. Mercadante Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics, Symmetry, Volume 2 (2010) no. 2, pp. 970-998 | DOI | Zbl

[96] I. Marvian; R. Spekkens Extending Noether’s theorem by quantifying the asymmetry of quantum states, Nat. Commun., Volume 5 (2014) no. 1, p. 3821 | DOI

[97] S. Haywood Symmetries and Conservation Laws in Particle Physics: An Introduction to Group Theory for Particle Physicists, Imperial College Press, 2010 | DOI | Zbl

[98] H. R. Brown; P. Holland Simple applications of Noether’s first theorem in quantum mechanics and electromagnetism, Am. J. Phys., Volume 72 (2004) no. 1, pp. 34-39 | DOI | MR | Zbl

[99] M. Thieullen; J.-C. Zambrini Probability and quantum symmetries. I. The theorem of Nœther in Schrödinger’s euclidean quantum mechanics, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 67 (1997) no. 3, pp. 297-338 | Numdam | Zbl

[100] W. Hairer; G. Wanner; C. Lubich Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer, 2006 | Zbl

[101] D. Razafindralandy; A. Hamdouni; M. Chhay A review of some geometric integrators, Adv. Model. Simul. Eng. Sci., Volume 5 (2018) no. 1, p. 16 | DOI

[102] D. E. Soper Classical Field Theory, Dover Books on Physics, Dover Publications, 2008

[103] J.-F. Ganghoffer Symmetries in Mechanics: From Field Theories to Master Responses in the Constitutive Modeling of Materials, Similarity and Symmetry Methods. Applications in Elasticity and Mechanics of Materials (J.-F. Ganghoffer; I. Mladenov, eds.), Springer, 2014, pp. 271-351 | DOI | Zbl

[104] D. Razafindralandy; A. Hamdouni; M. Chhay Symmetry in Turbulence Simulation, Numerical Simulation Research Progress, Nova Science Publishers, 2009, pp. 161-207

[105] G. Ünal Application of equivalence transformations to inertial subrange of turbulence, Lie Groups Appl., Volume 1 (1994) no. 1, pp. 232-240 | MR | Zbl

[106] M. Oberlack A unified Approach for Symmetries in Plane Parallel Turbulent Shear Flows, J. Fluid Mech., Volume 427 (2001), pp. 299-328 | DOI | Zbl

[107] M. Kinzel; M. Holzner; B. Lüthi; C. Tropea; W. Kinzelbach; M. Oberlack Scaling Laws of Turbulent Diffusion – An Experimental Validation, 14 $^{t h}$ Int. Symp. on Applications of Laser Techniques to Fluid Mechanichs (2008)

[108] D. Razafindralandy; A. Hamdouni; N. Al Sayed Lie-symmetry group and modeling in non-isothermal fluid mechanics, Phys. A: Stat. Mech. Appl., Volume 391 (2012) no. 20, pp. 4624-4636 | DOI | MR

[109] H. Sadeghi; M. Oberlack; M. Gauding On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation, J. Fluid Mech., Volume 854 (2018), pp. 233-260 | DOI | MR | Zbl

[110] H. Sadeghi; M. Oberlack New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence, J. Fluid Mech., Volume 899 (2020), A10 | DOI | MR | Zbl

[111] H. Sadeghi; M. Oberlack; M. Gauding New symmetry-induced scaling laws of passive scalar transport in turbulent plane jets, J. Fluid Mech., Volume 919 (2021), A5 | DOI | MR | Zbl

[112] M. M. El Telbany; A. J. Reynolds Velocity distributions in plane turbulent channel flows, J. Fluid Mech., Volume 100 (1980) no. 1, pp. 1-29 | DOI

[113] G. I. Barenblatt Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis, J. Fluid Mech., Volume 248 (1993), pp. 513-520 | DOI | MR | Zbl

[114] G. I. Barenblatt; V. M. Prostokishin Scaling laws for fully developed turbulent shear flows. Part 2. Processing of experimental data, J. Fluid Mech., Volume 248 (1993), pp. 521-529 | DOI | Zbl

[115] G. I. Barenblatt; A. J. Chorin New Perspectives in Turbulence: Scaling Laws, Asymptotics, and Intermittency, SIAM Rev., Volume 40 (1998) no. 2, pp. 265-291 | DOI | MR | Zbl

[116] M. Oberlack Invariant modeling in large-eddy simulation of turbulence, Annual Research Briefs 1997, Stanford University, 1997, pp. 3-22

[117] D. Razafindralandy; A. Hamdouni Consequences of Symmetries on the Analysis and Construction of Turbulence Models, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 2 (2006), 052 | MR | Zbl

[118] U. Schaefer-Rolffs; R. Knöpfel; E. Becker A scale invariance criterion for LES parametrizations, Meteorologische Zeitschrift, Volume 24 (2015) no. 1, pp. 3-13 | DOI

[119] D. Klingenberg; M. Oberlack; D. Pluemacher Symmetries and turbulence modeling, Physics of Fluids, Volume 32 (2020) no. 2, 025108 | DOI

[120] D. Razafindralandy; A. Hamdouni Subgrid models preserving the symmetry group of Navier–Stokes equations, C. R. Mécanique, Volume 333 (2005), pp. 481-486 | DOI | Zbl

[121] D. Razafindralandy; A. Hamdouni Consequences of symmetries on the analysis and construction of turbulence models, $6^{t h}$ International Conference on Symmetry in Non-linear Mathematical Physics (2005)

[122] D. Razafindralandy; A. Hamdouni; C. Béghein A class of subgrid-scale models preserving the symmetry group of Navier–Stokes equations, Commun. Nonlinear Sci. Numer. Simul., Volume 12 (2007) no. 3, pp. 243-253 | DOI | Zbl

[123] D. Razafindralandy; A. Hamdouni; M. Oberlack Analysis and development of subgrid turbulence models preserving the symmetry properties of the Navier–Stokes equations, Eur. J. Mech. B Fluids, Volume 26 (2007), pp. 531-550 | DOI | MR | Zbl

[124] D. Razafindralandy; A. Hamdouni Invariant subgrid modelling in large-eddy simulation of heat convection turbulence, Theor. Comput. Fluid Dyn. (2007) | DOI | Zbl

[125] D. Razafindralandy; A. Hamdouni Analysis of subgrid models of heat convection by symmetry group theory, C. R. Mécanique, Volume 335 (2007) no. 4, pp. 225-230 | DOI | Zbl

[126] N. Al Sayed; A. Hamdouni; E. Liberge; D. Razafindralandy The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling, Symmetry, Volume 2 (2010), pp. 848-867 | DOI | MR | Zbl

[127] K. Vu; J. Carminati Symbolic Computation and Differential Equations: Lie Symmetries, J. Symb. Comput., Volume 29 (2000) no. 2, pp. 95-116 | MR | Zbl

[128] Miguel Torres-Torriti; Hannah Michalska A Software Package for Lie Algebraic Computations, SIAM Rev., Volume 47 (2005) no. 4, pp. 722-745 | DOI | MR | Zbl

[129] A. Cheviakov Symbolic Computation of Nonlocal Symmetries and Nonlocal Conservation Laws of Partial Differential Equations Using the GeM Package for Maple, Similarity and Symmetry Methods: Applications in Elasticity and Mechanics of Materials (J.-F. Ganghoffer; I. Mladenov, eds.) (Lecture Notes in Applied and Computational Mechanics), Volume 73, Springer, 2014, pp. 165-184 | DOI | Zbl

[130] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (M. Abramowitz; I. Stegun, eds.), U. S. Government Printing Office, 1964 | DOI | MR | Zbl

[131] L. Ovsiannikov Group Analysis of Differential Equations, Academic Press Inc., 1982 | Zbl

[132] G. W. Bluman; A. Cheviakov; S. Anco Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168, Springer, 2010 | DOI | Zbl

[133] G. Frederico; D. Torres A formulation of Noether’s theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl., Volume 334 (2007) no. 2, pp. 834-846 | DOI | MR | Zbl

[134] J. Cresson Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., Volume 48 (2007) no. 3, 033504 | DOI | MR | Zbl

[135] E. Buhe; G. W. Bluman; C. Alatancang; H. Yulan Some Approaches to the Calculation of Conservation Laws for a Telegraph System and Their Comparisons, Symmetry, Volume 10 (2018) no. 6, 182 | DOI | Zbl

[136] T. Wolf A comparison of four approaches to the calculation of conservation laws, Eur. J. Appl. Math., Volume 13 (2002) no. 2, pp. 129-152 | DOI | MR | Zbl

[137] S. Anco Generalization of Noether’s Theorem in Modern Form to Non-variational Partial Differential Equations, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science (Makarov R. Melnik R.; Belair J, eds.) (Fields Institute Communications), Volume 79, Springer, 2017, pp. 119-182 | DOI | Zbl

[138] S. Anco; G. W. Bluman Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications, Eur. J. Appl. Math., Volume 13 (2002), pp. 545-566 | DOI | Zbl

[139] S. Anco; G. W. Bluman Direct construction method for conservation laws of partial differential equations. Part II: General treatment, Eur. J. Appl. Math., Volume 13 (2002), pp. 567-585 | DOI | Zbl

[140] N. H. Ibragimov; T. Kolsrud Lagrangian approach to evolution equations: symmetries and conservation laws, Nonlinear Dyn., Volume 36 (2004) no. 1, pp. 29-40 | DOI | MR | Zbl

[141] N. H. Ibragimov A new conservation theorem, J. Math. Anal. Appl., Volume 333 (2007), pp. 311-328 | DOI | MR | Zbl

[142] N. H. Ibragimov Construction of conservation laws using symmetries, Similarity and symmetry methods. Applications in elasticity and mechanics of materials (J.-F. Ganghoffer et al., eds.) (Lecture Notes in Applied and Computational Mechanics), Volume 73, Springer, 2014, pp. 61-164 | DOI | Zbl

[143] N. H. Ibragimov Conservation laws and non-invariant solutions of anisotropic wave equations with a source, Nonlinear Anal., Real World Appl., Volume 40 (2018) no. Suppl. C, pp. 82-94 | DOI | MR | Zbl

[144] P. Sagaut Large eddy simulation for incompressible flows. An introduction, Scientific Computation, Springer, 2006 | Zbl

[145] A. Müller; P. Maißer A Lie-Group Formulation of Kinematics and Dynamics of Constrained MBS and Its Application to Analytical Mechanics, Multibody System Dynamics, Volume 9 (2003) no. 4, pp. 311-352 | DOI | MR | Zbl

[146] B. Kolev Lie Groups and Mechanics: An Introduction, J. Nonlinear. Math. Phys., Volume 11 (2004) no. 4, pp. 480-498 | DOI | MR | Zbl

[147] P. Woit Quantum Theory, Groups and Representations: An Introduction, Springer, 2017 | DOI | Zbl

[148] A. Iserles; H. Munthe-Kaas; S. Nørsett; A. Zanna Lie-group methods, Acta Numerica, Volume 9 (2000), pp. 215–-365 | DOI | Zbl

[149] C. J. Budd; A. Iserles; A. Iserles; S.P. Norsett On the solution of linear differential equations in Lie groups, Philos. Trans. R. Soc. A, Volume 357 (1999) no. 1754, pp. 983-1019 | DOI | MR

[150] J. Hall; M. Leok Lie Group Spectral Variational Integrators, Found. Comput. Math., Volume 17 (2017) no. 1, pp. 199-257 | DOI | MR | Zbl

[151] S. Christiansen; H. Munthe-Kaas; B. Owren Topics in structure-preserving discretization, Acta Numer., Volume 20 (2011), pp. 1-119 | DOI | MR | Zbl

[152] E. Celledoni; H. Marthinsen; B. Owren An introduction to Lie group integrators – basics, new developments and applications, J. Comput. Phys., Volume 257 (2014) no. Part B, pp. 1040-1061 | DOI | MR | Zbl

[153] C. Curry; B. Owren Variable stepsize commutator free Lie group integrators, Numer Algor., Volume 82 (2019), pp. 1359-1376 | DOI | Zbl

[154] E. Celledoni; E Çokaj; A. Leone; D. Murari; B. Owren Lie group integrators for mechanical systems, Int. J. Comput. Math., Volume 99 (2022) no. 1, pp. 58-88 | DOI | MR | Zbl

[155] L. Drumetz; A. Reiffers-Masson; Naoufal E. Bekri; F. Vermet Geometry-preserving Lie Group Integrators For Differential Equations On The Manifold Of Symmetric Positive Definite Matrices (2022) (preprint, arXiv:2210.08842) | DOI

[156] V. A. Dorodnitsyn Finite difference models entirely inheriting continuous symmetry of original differential equations, Int. J. Mod. Phys. C, Volume 5 (1994) no. 4, pp. 723-734 | DOI | MR | Zbl

[157] V. A. Dorodnitsyn Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, Chapman & Hall, 2010 | DOI | Zbl

[158] Alexander Bihlo Invariant meshless discretization schemes, J. Phys. A. Math. Theor., Volume 46 (2013) no. 6, 062001 | MR | Zbl

[159] A. Bihlo; J. Jackaman; F. Valiquette On the development of symmetry-preserving finite element schemes for ordinary differential equations, J. Comput. Dyn., Volume 7 (2020) no. 2, pp. 339-368 | DOI | MR | Zbl

[160] Tian-Tian Zhang; Mei-Juan Xu The symmetry-preserving difference schemes and exact solutions of some high-dimensional differential equations, Appl. Math. Lett., Volume 112 (2021), 106813 | MR | Zbl

[161] E. Hoarau; C. David; P. Sagaut; T.-H. Lê Lie group stability of finite difference schemes (2006) (preprint, arXiv:math/0608757v1) | DOI

[162] M. Chhay; E. Hoarau; A. Hamdouni; P. Sagaut Comparison of some Lie-symmetry-based integrators, J. Comput. Phys., Volume 230 (2011) no. 5, pp. 2174-2188 | DOI | MR | Zbl

[163] D. Levi; P. Winternitz Lie groups and numerical solutions of differential equations, Acta Polytechnica, Volume 53 (2013) no. 5, pp. 438–-443 | DOI

[164] P. Kim Invariantization of Numerical Schemes Using Moving Frames, BIT, Volume 47 (2007) no. 3, pp. 525-546 | DOI | MR | Zbl

[165] M. Chhay; A. Hamdouni A new construction for invariant numerical schemes using moving frames, C. R. Mécanique, Volume 338 (2010) no. 2, pp. 97-101 | DOI | Zbl

[166] P. Olver Lectures on Moving Frames, Symmetries and Integrability of Difference Equations (D. Levi; P. Olver; Z. Thomova; P. Winternitz, eds.) (London Mathematical Society Lecture Note Series), Volume 381, Cambridge University Press, 2011, pp. 207-246 | DOI | Zbl

[167] G. R. W. Quispel; R. Sahadevan Lie symmetries and the integration of difference equations, Phys. Lett., A, Volume 184 (1993) no. 1, pp. 64-70 | DOI | MR | Zbl

[168] D. Levi; L. Vinet; P. Winternitz Lie group formalism for difference equations, J. Phys. A, Math. Gen., Volume 30 (1997) no. 2, pp. 633-649 | DOI | MR | Zbl

[169] A. Iserles Numerical analysis in Lie groups, Foundations of Computational Mathematics (Ronald Devore; Arieh Iserles; Endre Süli, eds.) (London Mathematical Society Lecture Note Series), Cambridge University Press, 2001, p. 105-–124 | DOI | Zbl

[170] E. Hoarau; C. David Lie group computation of finite difference schemes (2006) (preprint, arXiv:math/0611895) | DOI

[171] E. Hoarau; P. Sagaut; T. Lê Lie group study of finite difference schemes, Discrete and Continuous Dynamical Systems (2007) no. Suppl., pp. 495-505 | MR | Zbl

[172] P. Hydon; E. Mansfield Extensions of Noether’s Second Theorem: from continuous to discrete systems, Proc. R. Soc. A: Math. Phys. Eng. Sci., Volume 467 (2011) no. 2135, pp. 3206-3221 | DOI | MR | Zbl

[173] L. Bourdin; J. Cresson; I. Greff A continuous/discrete fractional Noether’s theorem, Commun. Nonlinear Sci. Numer. Simul., Volume 18 (2013) no. 4, pp. 878-887 | DOI | MR | Zbl

[174] M. Skopenkov Discrete Field Theory: Symmetries and Conservation Laws, Math. Phys. Anal. Geom., Volume 26 (2023) no. 19, 19 | MR | Zbl

[175] Z. Bartosiewicz; D. Torres Noether’s theorem on time scales, J. Math. Anal. Appl., Volume 342 (2008) no. 2, pp. 1220-1226 | DOI | MR | Zbl

[176] Baptiste Anerot; Jacky Cresson; Khaled Hariz Belgacem; Frederic Pierret Noether’s-type theorems on time scales, J. Math. Phys., Volume 61 (2020) no. 11, 113502 | MR | Zbl

Cited by Sources:

Comments - Policy