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Lagrangian descriptions of dissipative systems: a review. (English) Zbl 07357428

Summary: In this paper, we review classical and recent results on the Lagrangian description of dissipative systems. After having recalled Rayleigh extension of Lagrangian formalism to equations of motion with dissipative forces, we describe Helmholtz conditions, which represent necessary and sufficient conditions for the existence of a Lagrangian function for a system of differential equations. These conditions are presented in different formalisms, some of them published in the last decades. In particular, we state the necessary and sufficient conditions in terms of multiplier factors, discussing the conditions for the existence of equivalent Lagrangians for the same system of differential equations. Some examples are discussed, to show the application of the techniques described in the theorems stated in this paper.

MSC:

74-XX Mechanics of deformable solids
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[1] Onsager, L. Reciprocal relations in irreversible processes. I. Phys Rev (Ser I) 1931; 37(4): 405. · JFM 57.1168.10
[2] Onsager, L. Reciprocal relations in irreversible processes. II. Phys Rev (Ser I) 1931; 38(12): 2265. · Zbl 0004.18303
[3] Cuomo, M. Forms of the dissipation function for a class of viscoplastic models. Math Mech Complex Syst 2017; 5(3): 217-237. · Zbl 1386.74007
[4] Giorgio, I, Del Vescovo, D. Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators. Math Mech Complex Syst 2019; 7(2): 159-174. · Zbl 1458.70004
[5] D’Annibale, F, Luongo, A. Modeling the linear dynamics of continuous viscoelastic systems on their infinite-dimensional central subspace. Math Mech Complex Syst 2020; 8(2): 127-151. · Zbl 1445.74012
[6] Altenbach, H, Eremeyev, V, Morozov, N. Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale. Int J Eng Sci 2012; 59: 83-89. · Zbl 1423.74805
[7] Eremeyev, V, Pietraszkiewicz, W. Thermomechanics of shells undergoing phase transition. J Mech Phys Solids 2011; 59(7): 1395-1412. · Zbl 1270.74125
[8] Mühlich, U, Abali, BE, dell’Isola, F. Commented translation of Erwin Schrödinger’s paper “On the dynamics of elastically coupled point systems” (Zur Dynamik elastisch gekoppelter Punktsysteme). Math Mech Solids. In press; first published online August 5, 2020. DOI: 10.1177/1081286520942955.
[9] Eugster, S, dell’Isola, F. Exegesis of the introduction and Sect. I from “Fundamentals of the mechanics of continua” by E Hellinger. Z Angew Math Mech 2017; 97(4): 477-506.
[10] Eugster, S, dell’Isola, F. Exegesis of Sect. II and III.A from “Fundamentals of the mechanics of continua” by E Hellinger. Z Angew Math Mech 2018; 98(1): 31-68.
[11] Eugster, S, dell’Isola, F. Exegesis of Sect. III.B from “Fundamentals of the mechanics of continua” by E Hellinger. Z Angew Math Mech 2018; 98(1): 69-105.
[12] dell’Isola, F, Andreaus, U, Placidi, L. At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola. Math Mech Solids 2015; 20(8): 887-928. · Zbl 1330.74006
[13] Razavy, M. Classical and quantum dissipative systems. London: Imperial College Press, 2005. · Zbl 1105.82014
[14] Weiss, U. Quantum dissipative systems. 3rd ed. Singapore: World Scientific, 2008. · Zbl 1166.81005
[15] Landau, L, Lifshitz, E. Mechanics (Course of Theoretical Physics, vol. 1). 3rd ed. Oxford: Pergamon Press, 1976.
[16] Lanczos, C. The variational principles of mechanics. 4th ed. New York, NY: Dover Publications, 1949. · Zbl 0037.39901
[17] Wells, D. Lagrangian dynamics. New York, NY: McGraw-Hill, 1967.
[18] Bateman, H. On dissipative systems and related variational principles. Phys Rev (Ser I) 1931; 38(4): 815. · Zbl 0003.01101
[19] Martínez-Pérez, NE, Ramírez, C. On the Lagrangian description of dissipative systems. J Math Phys 2018; 59(3): 032904. · Zbl 1390.70043
[20] von Helmholtz, H. Über die physikalische Bedeutung des Princips der kleinsten Wirkung. J Reine Angew Math 1886; 100: 137-166. · JFM 18.0941.01
[21] Rayleigh, J. The theory of sound. vols 1 & 2. New York, NY: Dover, 1945.
[22] Goldstein, H. Classical mechanics. Reading, MA: Addison Wesley, 1950. · Zbl 0043.18001
[23] Whittaker, ET. A treatise on the analytical dynamics of particles and rigid bodies. 4th ed. Cambridge: Cambridge University Press, 1937. · JFM 63.1286.03
[24] Rosenberg, R. Analytical dynamics of discrete systems. New York, NY: Plenum Press, 1977. · Zbl 0416.70001
[25] Abraham, R, Marsden, J, Ratiu, T, et al. Foundations of mechanics. 2nd ed. Providence, RI: AMS Chelsea Publishing, 2008.
[26] Mestdag, T, Sarlet, W, Crampin, M. Second-order dynamical systems of Lagrangian type with dissipation. Differ Geom Appl 2011; 29: S156-S163. · Zbl 1222.70017
[27] Scerrato, D, Giorgio, I, Madeo, A, et al. A simple non-linear model for internal friction in modified concrete. Int J Eng Sci 2014; 80: 136-152. · Zbl 1423.74651
[28] Kimball, AL, Lovell, DE. Internal friction in solids. Phys Rev (Ser I) 1927; 30(6): 948.
[29] Mayer, A. Die Existenzbedingungen eines kinetischen Potentiales. Ber Kais Ges Wiss Leipzig (Math Phys Class) 1896; 48: 519-529. · JFM 27.0599.03
[30] Darboux, G. Leçons sur la Theorie Generale des Surface. Vol. III. Lignes géodésiques et courbure géodśique. Paramètres différentiels. Déformation des surfaces. Paris: Gauthier-Villars, 1894. · JFM 25.1159.02
[31] Hirsch, A. Die Existenzbedingungen des verallgemeinerten kinetischen Potentials. Math Ann 1898; 50: 429-441. · JFM 29.0603.01
[32] Boehm, K. Die Existenzbedingungen eines von den ersten und zweiten Differentialquotienten der Coordinaten abhängigen kinetischen Potentials. J Reine Angew Math 1900; 121: 124-140. · JFM 30.0644.01
[33] Königsberger, L. Die Principien der Mechanik. Leipzig: Teubner, 1901.
[34] Douglas, J. Solution of the inverse problem of the calculus of variations. Trans Am Math Soc 1941; 50: 71-128. · JFM 67.1038.01
[35] Lopuszanski, J. The inverse variational problem in classical mechanics. Singapore: World Scientific, 1999. · Zbl 0961.70001
[36] Saunders, D. Thirty years of the inverse problem in the calculus of variations. Rep Math Phys 2010; 66: 43-53. · Zbl 1237.49051
[37] Havas, P. The range of application of the Lagrange formalism, 1. Nuovo Cimento (1955-1965) 1957; 5(S3): 363-388. · Zbl 0077.37202
[38] Nigam, K, Banerjee, K. A brief review of Helmholtz conditions, 2016. arXiv arXiv: 1602.01563.
[39] Engels, E. On the Helmholtz conditions for the existence of a Lagrange formalism. Nuovo Cimento B (1971-1996) 1975; 26(2): 481-492.
[40] Lefschetz, S. Differential equations: Geometric theory. 2nd ed. New York, NY: Interscience Publishers, 1962. · Zbl 0080.06401
[41] Sarlet, W. The Helmholtz conditions revisited: A new approach to the inverse problem of Lagrangian dynamics. J Phys A: Math Gen 1982; 15(5): 1503. · Zbl 0537.70018
[42] Olver, P. Applications of Lie groups to differential equations (Graduate Texts in Mathematics, vol. 107). 2nd ed. New York: Springer, 1986. · Zbl 0588.22001
[43] Gel’fand, I, Dorfman, IY. Hamiltonian operators and algebraic structures related to them. Funct Anal Appl 1979; 13(4): 248-262. · Zbl 0437.58009
[44] Crampin, M. On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics. J Phys A: Math Gen 1981; 14(10): 2567. · Zbl 0475.70022
[45] Crampin, M, Prince, G, Thompson, G. A geometric version of the Helmholtz conditions in time dependent Lagrangian dynamics. J Phys A: Math Gen 1984; 17: 1437-1447. · Zbl 0545.58020
[46] Henneaux, M. On the inverse problem of the calculus of variations. J Phys A: Math Gen 1982; 15(3): L93-L96. · Zbl 0475.70024
[47] Henneaux, M. On the inverse problem of the calculus of variations in field theory. J Phys A: Math Gen 1984; 17(1): 75-85. · Zbl 0557.70019
[48] Krupková, O, Prince, G. Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations. In: Demeter Krupka, DS (ed.) Handbook of global analysis. Amsterdam: Elsevier Science, 2008, 837-904. · Zbl 1236.58027
[49] Sarlet, W. Geometric calculus for second-order differential equations and generalizations of the inverse problem of Lagrangian mechanics. Int J Non Linear Mech 2012; 47(10): 1132-1140.
[50] Kielau, G, Maisser, P. A generalization of the Helmholtz conditions for the existence of a first-order Lagrangian. Z Angew Math Mech 2006; 86(9): 722-735. · Zbl 1105.70010
[51] Jungnickel, U, Kielau, G, Maisser, P, et al. A generalization of the Helmholtz conditions for the existence of a first-order Lagrangian using nonholonomic velocities. Z Angew Math Mech 2009; 89(1): 44-53. · Zbl 1156.70010
[52] Crampin, M, Mestdag, T, Sarlet, W. On the generalized Helmholtz conditions for Lagrangian systems with dissipative forces. Z Angew Math Mech 2010; 90(6): 502-508. · Zbl 1241.70022
[53] Hirsch, A. Über eine charakteristische Eigenschaft der Differentialgleichungen der Variationsrechnung. Math Ann 1897; 49: 49-72. · JFM 28.0322.01
[54] Pardo, F. The Helmholtz conditions in terms of constants of motion in classical mechanics. J Math Phys 1989; 30: 2054-2061. · Zbl 0676.70023
[55] Smirnov, V . A course of higher mathematics. vol. 2. Reading, MA: Addison-Wesley, 1964. · Zbl 0121.25904
[56] Do, T, Prince, G. New progress in the inverse problem in the calculus of variations. Differ Geom Appl 2016; 45: 148-179. · Zbl 1333.37070
[57] Prince, G, King, D. The inverse problem in the calculus of variations: Nonexistence of Lagrangians. In: Cantrijn, F, Langerock, B (eds.) Differential geometric methods in mechanics and field theory: Volume in honour of Willy Sarlet. Gent: Academia Press, 131-140.
[58] Hojman, S, Harleston, H. Equivalent Lagrangians: Multidimensional case. J Math Phys 1981; 22(7): 1414-1419. · Zbl 0522.70024
[59] Currie, D, Saletan, E. \(q\)-Equivalent particle Hamiltonians. I. The classical one-dimensional case. J Math Phys 1966; 7: 967.
[60] Crampin, M, Prince, G. Equivalent Lagrangians and dynamical symmetries: Some comments. Phys Lett A 1985; 108(4): 191-194.
[61] Kara, A. Equivalent Lagrangians and the inverse variational problem with applications. Quaest Math 2004; 27(2): 207-216. · Zbl 1089.34514
[62] Hojman, R, Zanelli, J. An acceleration-dependent Lagrangian proof of the conserved traces’ theorem. Il Nuovo Cimento B (1971-1996) 1986; 94(1): 87-92.
[63] Henneaux, M. Equations of motion, commutation relations and ambiguities in the Lagrangian formalism. Ann Phys 1982; 140(1): 45-64. · Zbl 0501.70020
[64] Denman, H. Solution of the Hamilton-Jacobi equation for certain dissipative classical mechanical systems. J Math Phys 1973; 14(3): 326-329. · Zbl 0258.70015
[65] Hojman, SA. Construction of Lagrangian and Hamiltonian structures starting from one constant of motion. Acta Mech 2015; 226(3): 735-744. · Zbl 1357.70027
[66] Patiño, A, Rago, H. On the constant of motion of dissipative systems. Can J Phys 2011; 80: 1-5.
[67] Musielak, Z, Roy, D, Swift, L. Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients. Chaos, Solitons Fractals 2008; 38(3): 894-902. · Zbl 1146.70336
[68] Musielak, ZE. Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J Phys A: Math Theor 2008; 41(5): 055205. · Zbl 1136.37044
[69] Cieśliński, J, Nikiciuk, T. A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients. J Phys A: Math Theor 2010; 43(17): 175205. · Zbl 1273.70024
[70] Lagrange, J. Méchanique analitique. Paris: La Veuve Desaint, 1788.
[71] Thomson, W, Tait, PG. Treatise on natural philosophy. vol. 1. 2nd ed. Cambridge: Cambridge University Press, 1879.
[72] Krupka, D . The Sonin-Douglas problem. In: Zenkov, DV (ed.) The inverse problem of the calculus of variations: Local and global theory. Paris: Atlantis Press, 2015, 31-73. · Zbl 1337.49063
[73] Sonin, NY. On the definition of maximal and minimal properties. Warsaw Univ Izv 1886; 1-2: 1-68.
[74] Andreaus, U, dell’Isola, F, Porfiri, M. Piezoelectric passive distributed controllers for beam flexural vibrations. J Vib Control 2004; 10(5): 625-659. · Zbl 1078.74026
[75] Alessandroni, S, Andreaus, U, dell’Isola, F, et al. Piezo-electromechanical (PEM) Kirchhoff-Love plates. Eur J Mech A Solids 2004; 23(4): 689-702. · Zbl 1065.74562
[76] Giorgio, I, Culla, A, Del Vescovo, D. Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network. Arch Appl Mech 2009; 79(9): 859. · Zbl 1176.74128
[77] Giorgio, I, Galantucci, L, Della Corte, A, et al. Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: Current and upcoming applications. Int J Appl Electromagnet Mech 2015; 47(4): 1051-1084.
[78] Lossouarn, B, Deü, JF, Aucejo, M, et al. Multimodal vibration damping of a plate by piezoelectric coupling to its analogous electrical network. Smart Mater Struct 2016; 25(11): 115042.
[79] Chróścielewski, J, Schmidt, R, Eremeyev, V. Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches. Continuum Mech Thermodyn 2018; 31(1): 147-188.
[80] Santilli, R. Foundations of theoretical mechanics I: The inverse problem in Newtonian mechanics (Texts and Monographs in Physics). Berlin: Springer-Verlag, 1978.
[81] Jacobi, C. Zur Theorie der Variations-Rechnung und der Differential-Gleichungen. J Reine Angew Math 1837; 17: 68-82.
[82] Nucci, MC, Tamizhmani, KM. Lagrangians for dissipative nonlinear oscillators: The method of Jacobi last multiplier. J Nonlinear Math Phys 2010; 17: 167-178. · Zbl 1206.34013
[83] Greenberger, D. A new approach to the problem of dissipation in quantum mechanics. J Math Phys 1979; 20(5): 771-780.
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