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Based on the analysis of establishing dynamic equations by using Newton's mechanics, Lagrange's, and Hamilton's mechanics, a new idea of establishing elastodynamic equations under the framework of energy conservation is proposed. Firstly, Newton’s second law is used to derive wave equations of motion. Secondly, Lagrange's equation, Hamilton's canonical equations, and the corresponding dynamical equations in a continuum medium are derived by using Hamilton’s variational principle. Thirdly, under the framework of energy conservation, Lagrange's equation, Hamilton's canonical equations, and the acoustic dynamic equations of the continuum are established, and the results are proved to be consistent with those derived from classical mechanics. Some fuzzy understandings when using Hamilton's variational principle to establish Lagrange’s equation and Hamilton’s canonical equation, are clarified. A series of dynamical equations established under the framework of energy conservation provides an alternative way to characterize and represent the propagation characteristics of wave motions in various complex media without involving the variational principle of functional extremum. Finally, as an application example, the differential equation of elastodynamics in a viscoelastic medium is given under the framework of energy conservation.
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Keywords:
- dynamical equation /
- Hamilton variational principle /
- Lagrange equation /
- law of conservation of energy
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Google Scholar
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Google Scholar
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图 1 物理模型示意图
Figure 1. Schematic diagram of physical model.
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[1] Gurtin M E 1964 Arch. Rational Mech. Anal. 16 34
Google Scholar
[2] Tiersten H F 1969 Linear Piezoelectric Plate Vibrations (New York: Plenum Press) pp33–35, pp43–46
[3] Achenbach J D 1975 Wave Propagation in Elastic Solids (Netherland: Elsevier) pp51–65
[4] Babich V M, Kiselev A P 2015 Elastic Waves High Frequency Theory (Boca Raton: CRC Press) pp8–10
[5] Shtrikman Z S 1962 J. Mech. Phys. Solids 10 335
Google Scholar
[6] 张海澜 1985 声学学报 10 223
Zhang H L 1985 Acta Acustica 10 223
[7] Shen H C, Li S M 2006 Classical Mechanics (Hefei: University of Science and Technology of China Press) pp233, 240 (in Chinese) [沈惠川, 李书明 2006 经典力学(合肥: 中国科学技术大学出版社) 第233, 240页
[8] EerNisse E P, Holland R 1967 Proceedings of the IEEE p1524
[9] Luan P 2018 J. Phys. Commun. 2 075016
Google Scholar
[10] Civelek C, Bechteler T F 2008 Int. J. Eng. Sci. 46 1218
Google Scholar
[11] Luan P 2020 Crystals 10 863
Google Scholar
[12] Gueorguiev V G, Maeder A 2021 Symmetry 13 522
Google Scholar
[13] Moiseiwitsch B L 2004 Variational Principles (New York: Dover Publications) pp82–83
[14] Cline D 2019 Variational Principles in Classic Mechanics (Rochester: University of Rochester) pp181–184, 443
[15] 唐立民 1991 计算结构力学及其应用 8 343
Google Scholar
Tang L M 1991 Chin. J. Comput. Mech. 8 343
Google Scholar
[16] Landau L D, Lifshitz E M 1976 Mechanics (Oxford: Butterworth-Heinemann) p14, 131
[17] Lanczos C 1986 The Variational Principles of Mechanics (4th Ed.) (New York: Dover) pp120–122
[18] Arnold I V 1997 Mathematical Methods of Classical Mechanics (2nd Ed.) (New York: Springer) pp59–60
[19] Goldstein H, Poole C P, Safko J L 2013 Classical Mechanics (3rd Ed.) (Essex: Pearson Education Limited) p35
[20] Morita S 2016 World J. Mech. 6 84
Google Scholar
[21] Huang Y C 2003 Mech. Res. Commun. 30 567
Google Scholar
[22] 黄永畅 2005 54 3473
Google Scholar
Huang Y C 2005 Acta. Phys. Sin. 54 3473
Google Scholar
[23] Huang Y C, Lee X G, Shao M X 2006 Mod. Phys. Lett. A 21 1107
Google Scholar
[24] Huang C and Huang Y C 2020 doi: 10.20944/preprints202008.0334.v3
[25] Bondar D I, Cabrera R, Lompay R R, Ivanov M Y, Rabitz H A 2012 Phys. Rev. Lett. 109 190403
Google Scholar
[26] Morse P M, Feshbach H 1953 Methods of Theoretical Physics (York: The Maple Press Company) pp151, 280–304
[27] Kim J, Dargush G F, Ju Y K 2013 Int. J. Solids Struct. 50 3418
Google Scholar
[28] Riewe F 1996 Phys. Rev. E. 53 1890
Google Scholar
[29] 林旭升 2002 汕头大学学报(自然科学版) 17 63
Lin X S 2002 J. Shantou Univ. (Nat. Sci. Ed.) 17 63
[30] 张海澜 2012 理论声学(修订版) (北京: 高等教育出版社) 第12页
Zhang H L 2012 Theoretical Acoustics (Beijing: Higher Education Press) p12 (in Chinese)
[31] Zhou P 2015 arXiv: 1512.04487 [physics. gen-ph]
[32] Lindsay G A 1952 Am. J. Phys. 20 86
Google Scholar
[33] Courant R, Hilbert D 1953 Methods of Mathematical Physics (Vol. 1) (New York: Interscience) pp208–211
[34] Gelfand I M, Fomin S V 1963 Calculus of Variations (Englewood Cliffs: Prentice-Hall) p42, 71
[35] Zia R K P, Redish E F, McKay S R 2009 Am. J. Phys. 77 614
Google Scholar
[36] Ansermet J P, Brechet S 2018 Principles of Thermodynamics (New York: Cambridge University Press) p3
[37] Ruderman M S 2019 Fluid Dynamics and Linear Elasticity-A First Course in Continuum Mechanics (Cham: Springer) pp40, 58, 61–62
[38] Maxwell J C 1867 Phil. Trans. Roy. Soc. London 157 49
[39] Carcione J 2015 Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic Porous and Electromagnetic Media (Netherlands: Elsevier) p66
[40] Wang X M, Dodds K, Zhao H B 2006 Explor. Geophys. 37 160
Google Scholar
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