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.
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Mots-clés : Groupes de Lie, mécanique des milieux continus, groupes de symétrie, intégrateurs géométriques, modélisation de la turbulence
Géry de Saxcé 1; Dina Razafindralandy 2
@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}, }
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/
[1] Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences, 75, Springer New York, 2012
[2] Lie groups, Graduate texts in mathematics, 225, Springer, 2013 | DOI | Zbl
[3] Theorie der Transformationsgruppen I, Math. Ann., Volume 16 (1880), pp. 441-528 | DOI | MR
[4] Sophus Lie’s 1880 Transformation Group Paper. Vol. I. Sophus Lie’s 1880 transformation group paper, Math. Sci. Press., 1975 | Zbl
[5] Theory of Transformation Groups I: General Properties of Continuous Transformation Groups. A Contemporary Approach and Translation, Springer, 2015 | Zbl
[6] Mémoire sur les conditions de résolubilité des équations par radicaux, Journal de mathématiques pures et appliquées, Volume 1846 (1830), pp. 417-433
[7] Galois Theories, Cambridge studies in advanced mathematics, 72, Cambridge University Press, 2001 | DOI | Zbl
[8] The Mathematical Writings of Évariste Galois, Heritage of European Mathematics, European Mathematical Society, 2011 | DOI | Zbl
[9] Transformation Groups in Differential Geometry, Springer, 1972 | DOI | Zbl
[10] Introduction to symplectic geometry, Springer, 2019 | DOI | Zbl
[11] Global Aspects of Classical Integrable Systems, Birkhäuser, 1997 | DOI | Zbl
[12] Structure of Dynamical Systems. A Symplectic View of Physics, Birkhäuser, 1997 | Zbl
[13] Foundation of Mechanics, Addison-Wesley Publishing Group, 1987
[14] Symplectic techniques in physics, Cambridge University Press, 1984 | Zbl
[15] An invitation to multisymplectic geometry, J. Geom. Phys., Volume 142 (2019), pp. 9-36 | DOI | MR | Zbl
[16] World invariant kinematics, Arch. Ration. Mech. Anal., Volume 1 (1958), pp. 181-211 | DOI | MR | Zbl
[17] Lectures on the foundations of continuum mechanics and thermodynamics, Arch. Ration. Mech. Anal., Volume 52 (1973), pp. 62-92 | DOI | MR | Zbl
[18] 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] Galilean Mechanics and Thermodynamics of Continua, Mechanical Engineering and Solid Mechanics Series, Wiley-ISTE, 2016 | DOI
[20] Géométrie et relativité, Collection Enseignement des Sciences, 6, Hermann, 1964 | Zbl
[21] The Thermodynamics of Irreversible Processes, III. Relativistic Theory of the Simple Fluid, Phys. Rev., II. Ser., Volume 58 (1940), pp. 919-924 | Zbl
[22] Fluid Mechanics, Pergamon Press, 1959
[23] Thermodynamique et géométrie, Differential Geometrical Methods in Mathematical Physics II (Lecture Notes in Mathematics), Volume 676, Springer, 1976, pp. 369-397 | DOI | Zbl
[24] Thermodynamique relativiste des fluides, Rend. Semin. Mat., Torino, Volume 35 (1978), pp. 21-34 | Zbl
[25] Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste, Ph. D. Thesis, University of Poitiers, Poitiers, France (1978)
[26] Relativistic thermodynamics of continua, Int. J. Eng. Sci., Volume 19 (1981), pp. 589-601 | DOI | MR | Zbl
[27] Canonical frame-indifferent transport operators with the four-dimensional formalism of differential geometry, Comput. Mater. Sci., Volume 77 (2013), pp. 120-130 | DOI
[28] 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] 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] On unitary representation of continuous groups, Ann. Math., Volume 59 (1954), pp. 1-46 | DOI | MR | Zbl
[31] AV-differential geometry: Poisson and Jacobi structures, J. Geom. Phys., Volume 52 (2004), pp. 398-446 | DOI | MR | Zbl
[32] 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] 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] On the polynomial invariants of elasticity tensors, J. Elasticity, Volume 34 (1994) no. 2, pp. 97-110 | DOI | MR | Zbl
[35] Symmetry classes for elasticity tensors, J. Elasticity, Volume 43 (1996) no. 2, p. 81-–108 | DOI | MR | Zbl
[36] Symmetry classes for even-order tensors, Math. Mech. Complex Syst., Volume 1 (2013) no. 2, pp. 177-210 | DOI | Zbl
[37] 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] A unified approach to invariants of plane elasticity tensors, Meccanica, Volume 49 (2014) no. 9, pp. 2001-2012 | DOI | MR | Zbl
[39] 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] Anisotropy of plane complex elastic bodies, Int. J. Solids Struct., Volume 47 (2010) no. 9, pp. 1154-1166 | DOI | Zbl
[41] Stiffness design of laminates using the polar method, Int. J. Solids Struct., Volume 38 (2001) no. 50-51, pp. 9281-9294 | DOI | Zbl
[42] Distance to plane elasticity orthotropy by Euler–Lagrange method, C. R. Mécanique, Volume 350 (2022), pp. 413-430 | DOI
[43] Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law, Symmetry, Volume 12 (2020) no. 4, pp. 674-702 | DOI
[44] A damage model based on Kelvin eigentensors and Curie principle, Mech. Mater., Volume 44 (2012), pp. 23-24 | DOI
[45] 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] Applications of Lie groups to differential equations, Graduate Texts in Mathematics, 107, Springer, 1986 | DOI | Zbl
[47] 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] Group Analysis of the Boltzmann and Vlasov Equations, Theor. Math. Phys., Volume 203 (2020) no. 3, pp. 794-823 | DOI | MR | Zbl
[49] 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] The Symmetry Approach to Classification of Integrable Equations, What Is Integrability? (Vladimir E. Zakharov, ed.), Springer, 1991, pp. 115-184 | DOI | Zbl
[51] A complete Lie symmetry classification of a class of -dimensional reaction-diffusion-convection equations, Commun. Nonlinear Sci. Numer. Simul., Volume 92 (2021), 105466 | DOI | MR | Zbl
[52] CRC handbook of Lie group analysis of differential equations. Volume 1: Symmetries, exact solutions and conservation laws, CRC Press, 1994
[53] Lie symmetries and group classification of a class of time fractional evolution systems, J. Math. Phys., Volume 56 (2015), 123504 | DOI | MR | Zbl
[54] Lie Symmetry Analysis of Fractional Differential Equations, CRC Press, 2020 | DOI | Zbl
[55] Lie Symmetries of Differential Equations: Classical Results and Recent Contributions, Symmetry, Volume 2 (2010) no. 2, pp. 658-706 | DOI | MR | Zbl
[56] 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] 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] On symmetries and conservation laws of a Gardner equation involving arbitrary functions, Appl. Math. Comput., Volume 290 (2016), pp. 125-134 | MR | Zbl
[59] Invariant properties of Timoshenko beam equations, Int. J. Eng. Sci., Volume 33 (1995) no. 14, pp. 2103-2114 | DOI | MR | Zbl
[60] Cosmology through Noether Symmetry Approach, Symmetry, Volume 10 (2018) no. 12, 719 | DOI | Zbl
[61] Noether symmetries and stability of ideal gas solution in Galileon Cosmology, Phys. Rev. D, Volume 95 (2017) no. 6, 064031 | DOI | MR
[62] 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] Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin–Voigt Viscoelasticity Equation, Symmetry, Volume 11 (2019) no. 7, 840 | Zbl
[64] 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] 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] Lie Symmetry Classification of the Generalized Nonlinear Beam Equation, Symmetry, Volume 9 (2017) no. 7, 115 | DOI | MR | Zbl
[67] Euler–Poincaré Dynamics of Perfect Complex Fluids, Geometry, Mechanics, and Dynamics, Springer, New York, 2002, pp. 169-180 | DOI | Zbl
[68] Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, 12, Wiley-Interscience, 2009 | DOI | Zbl
[69] 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] 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] Vorticies and invariant surfaces generated by symmetries for the 3D Navier–Stokes equation, Physica A, Volume 286 (2000), pp. 79-108 | DOI | Zbl
[72] New similarity reductions and exact solutions for helically symmetric viscous flows, Phys. Fluids, Volume 32 (2020) no. 5, 053604 | DOI
[73] Invariante Variationsprobleme, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1918), pp. 235-257 | Zbl
[74] 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] 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] Energy-momentum tensors in nonsimple elastic dielectrics, J. Elasticity, Volume 42 (1996), pp. 275-281 | DOI | MR | Zbl
[77] 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] 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] On material equations in second gradient electroelasticity, J. Elasticity, Volume 67 (2002) no. 3, pp. 205-227 | DOI | MR | Zbl
[80] On conservation integrals in micropolar elasticity, Philos. Mag., Volume 83 (2003) no. 11, pp. 1365-1377 | DOI
[81] 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] Conservation laws in elasticity. I: General results, Arch. Ration. Mech. Anal., Volume 85 (1984) no. 2, pp. 111-129 | DOI | MR | Zbl
[83] Conservation and Balance Laws in Linear Elasticity of Grade Three, J. Elasticity, Volume 94 (2009) no. 1, pp. 69-85 | DOI | MR | Zbl
[84] 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] Mechanics in Material Space with Applications to Defect and Fracture Mechanics, Springer, 2000 | Zbl
[86] Material Inhomogeneities in Elasticity, Applied Mathematics and Mathematical Computation, 3, Chapman & Hall, 1993 | DOI | Zbl
[87] 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] Symmetries and Related Physical Balances for Discontinuous Flow Phenomena within the Framework of Lagrange Formalism, Symmetry, Volume 13 (2021) no. 9, 1662 | DOI
[89] Conservation laws of the two-dimensional gas dynamics equations, Int. J. Non-Linear Mech., Volume 112 (2019), pp. 126-132 | DOI
[90] 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] 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] Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws, Lecture Notes in Physics, 946, Springer, 2018 | Zbl
[93] 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] 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] Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics, Symmetry, Volume 2 (2010) no. 2, pp. 970-998 | DOI | Zbl
[96] Extending Noether’s theorem by quantifying the asymmetry of quantum states, Nat. Commun., Volume 5 (2014) no. 1, p. 3821 | DOI
[97] Symmetries and Conservation Laws in Particle Physics: An Introduction to Group Theory for Particle Physicists, Imperial College Press, 2010 | DOI | Zbl
[98] 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] 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] Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer, 2006 | Zbl
[101] A review of some geometric integrators, Adv. Model. Simul. Eng. Sci., Volume 5 (2018) no. 1, p. 16 | DOI
[102] Classical Field Theory, Dover Books on Physics, Dover Publications, 2008
[103] 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] Symmetry in Turbulence Simulation, Numerical Simulation Research Progress, Nova Science Publishers, 2009, pp. 161-207
[105] Application of equivalence transformations to inertial subrange of turbulence, Lie Groups Appl., Volume 1 (1994) no. 1, pp. 232-240 | MR | Zbl
[106] A unified Approach for Symmetries in Plane Parallel Turbulent Shear Flows, J. Fluid Mech., Volume 427 (2001), pp. 299-328 | DOI | Zbl
[107] Scaling Laws of Turbulent Diffusion – An Experimental Validation, 14 Int. Symp. on Applications of Laser Techniques to Fluid Mechanichs (2008)
[108] 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] 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] 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] New symmetry-induced scaling laws of passive scalar transport in turbulent plane jets, J. Fluid Mech., Volume 919 (2021), A5 | DOI | MR | Zbl
[112] Velocity distributions in plane turbulent channel flows, J. Fluid Mech., Volume 100 (1980) no. 1, pp. 1-29 | DOI
[113] 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] 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] New Perspectives in Turbulence: Scaling Laws, Asymptotics, and Intermittency, SIAM Rev., Volume 40 (1998) no. 2, pp. 265-291 | DOI | MR | Zbl
[116] Invariant modeling in large-eddy simulation of turbulence, Annual Research Briefs 1997, Stanford University, 1997, pp. 3-22
[117] Consequences of Symmetries on the Analysis and Construction of Turbulence Models, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 2 (2006), 052 | MR | Zbl
[118] A scale invariance criterion for LES parametrizations, Meteorologische Zeitschrift, Volume 24 (2015) no. 1, pp. 3-13 | DOI
[119] Symmetries and turbulence modeling, Physics of Fluids, Volume 32 (2020) no. 2, 025108 | DOI
[120] Subgrid models preserving the symmetry group of Navier–Stokes equations, C. R. Mécanique, Volume 333 (2005), pp. 481-486 | DOI | Zbl
[121] Consequences of symmetries on the analysis and construction of turbulence models, International Conference on Symmetry in Non-linear Mathematical Physics (2005)
[122] 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] 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] Invariant subgrid modelling in large-eddy simulation of heat convection turbulence, Theor. Comput. Fluid Dyn. (2007) | DOI | Zbl
[125] 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] The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling, Symmetry, Volume 2 (2010), pp. 848-867 | DOI | MR | Zbl
[127] Symbolic Computation and Differential Equations: Lie Symmetries, J. Symb. Comput., Volume 29 (2000) no. 2, pp. 95-116 | MR | Zbl
[128] A Software Package for Lie Algebraic Computations, SIAM Rev., Volume 47 (2005) no. 4, pp. 722-745 | DOI | MR | Zbl
[129] 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] Group Analysis of Differential Equations, Academic Press Inc., 1982 | Zbl
[132] Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168, Springer, 2010 | DOI | Zbl
[133] 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] Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., Volume 48 (2007) no. 3, 033504 | DOI | MR | Zbl
[135] 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] 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] 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] 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] 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] Lagrangian approach to evolution equations: symmetries and conservation laws, Nonlinear Dyn., Volume 36 (2004) no. 1, pp. 29-40 | DOI | MR | Zbl
[141] A new conservation theorem, J. Math. Anal. Appl., Volume 333 (2007), pp. 311-328 | DOI | MR | Zbl
[142] 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] 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] Large eddy simulation for incompressible flows. An introduction, Scientific Computation, Springer, 2006 | Zbl
[145] 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] Lie Groups and Mechanics: An Introduction, J. Nonlinear. Math. Phys., Volume 11 (2004) no. 4, pp. 480-498 | DOI | MR | Zbl
[147] Quantum Theory, Groups and Representations: An Introduction, Springer, 2017 | DOI | Zbl
[148] Lie-group methods, Acta Numerica, Volume 9 (2000), pp. 215–-365 | DOI | Zbl
[149] 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] Lie Group Spectral Variational Integrators, Found. Comput. Math., Volume 17 (2017) no. 1, pp. 199-257 | DOI | MR | Zbl
[151] Topics in structure-preserving discretization, Acta Numer., Volume 20 (2011), pp. 1-119 | DOI | MR | Zbl
[152] 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] Variable stepsize commutator free Lie group integrators, Numer Algor., Volume 82 (2019), pp. 1359-1376 | DOI | Zbl
[154] Lie group integrators for mechanical systems, Int. J. Comput. Math., Volume 99 (2022) no. 1, pp. 58-88 | DOI | MR | Zbl
[155] Geometry-preserving Lie Group Integrators For Differential Equations On The Manifold Of Symmetric Positive Definite Matrices (2022) (preprint, arXiv:2210.08842) | DOI
[156] 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] Applications of Lie Groups to Difference Equations, Differential and Integral Equations and Their Applications, 8, Chapman & Hall, 2010 | DOI | Zbl
[158] Invariant meshless discretization schemes, J. Phys. A. Math. Theor., Volume 46 (2013) no. 6, 062001 | MR | Zbl
[159] 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] The symmetry-preserving difference schemes and exact solutions of some high-dimensional differential equations, Appl. Math. Lett., Volume 112 (2021), 106813 | MR | Zbl
[161] Lie group stability of finite difference schemes (2006) (preprint, arXiv:math/0608757v1) | DOI
[162] Comparison of some Lie-symmetry-based integrators, J. Comput. Phys., Volume 230 (2011) no. 5, pp. 2174-2188 | DOI | MR | Zbl
[163] Lie groups and numerical solutions of differential equations, Acta Polytechnica, Volume 53 (2013) no. 5, pp. 438–-443 | DOI
[164] Invariantization of Numerical Schemes Using Moving Frames, BIT, Volume 47 (2007) no. 3, pp. 525-546 | DOI | MR | Zbl
[165] A new construction for invariant numerical schemes using moving frames, C. R. Mécanique, Volume 338 (2010) no. 2, pp. 97-101 | DOI | Zbl
[166] 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] Lie symmetries and the integration of difference equations, Phys. Lett., A, Volume 184 (1993) no. 1, pp. 64-70 | DOI | MR | Zbl
[168] Lie group formalism for difference equations, J. Phys. A, Math. Gen., Volume 30 (1997) no. 2, pp. 633-649 | DOI | MR | Zbl
[169] 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] Lie group computation of finite difference schemes (2006) (preprint, arXiv:math/0611895) | DOI
[171] Lie group study of finite difference schemes, Discrete and Continuous Dynamical Systems (2007) no. Suppl., pp. 495-505 | MR | Zbl
[172] 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] A continuous/discrete fractional Noether’s theorem, Commun. Nonlinear Sci. Numer. Simul., Volume 18 (2013) no. 4, pp. 878-887 | DOI | MR | Zbl
[174] Discrete Field Theory: Symmetries and Conservation Laws, Math. Phys. Anal. Geom., Volume 26 (2023) no. 19, 19 | MR | Zbl
[175] Noether’s theorem on time scales, J. Math. Anal. Appl., Volume 342 (2008) no. 2, pp. 1220-1226 | DOI | MR | Zbl
[176] Noether’s-type theorems on time scales, J. Math. Phys., Volume 61 (2020) no. 11, 113502 | MR | Zbl
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