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Derivation of classical mechanics in an energetic framework via conservation and relativity. (English) Zbl 1452.70015
Summary: The notions of conservation and relativity lie at the heart of classical mechanics, and were critical to its early development. However, in Newton’s theory of mechanics, these symmetry principles were eclipsed by domain-specific laws. In view of the importance of symmetry principles in elucidating the structure of physical theories, it is natural to ask to what extent conservation and relativity determine the structure of mechanics. In this paper, we address this question by deriving classical mechanics – both nonrelativistic and relativistic – using relativity and conservation as the primary guiding principles. The derivation proceeds in three distinct steps. First, conservation and relativity are used to derive the asymptotically conserved quantities of motion. Second, in order that energy and momentum be continuously conserved, the mechanical system is embedded in a larger energetic framework containing a massless component that is capable of bearing energy (as well as momentum in the relativistic case). Imposition of conservation and relativity then results, in the nonrelativistic case, in the conservation of mass and in the frame-invariance of massless energy; and, in the relativistic case, in the rules for transforming massless energy and momentum between frames. Third, a force framework for handling continuously interacting particles is established, wherein Newton’s second law is derived on the basis of relativity and a staccato model of motion-change. Finally, in light of the derivation, we elucidate the structure of mechanics by classifying the principles and assumptions that have been employed according to their explanatory role, distinguishing between symmetry principles and other types of principles (such as compositional principles) that are needed to build up the theoretical edifice.
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
70-03 History of mechanics of particles and systems
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[1] Wigner, EP, The unreasonable effectiveness of mathematics in the natural sciences, Commun. Pure Appl. Math., 13, 1 (1960) · Zbl 0102.00703
[2] Darrigol, O., Deducing Newton’s second law from relativity principles: a forgotten history, Arch. Hist. Exact Sci. (2019) · Zbl 1435.01012
[3] Schütz, J.R.: Prinzip der absoluten erhaltung der energie, Nachr. v. d. Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Kl. 2, 110 (1897) · JFM 28.0637.02
[4] Laue, M., Zur Dynamik der Relativitätstheorie, Ann. Phys., 35, 524 (1911) · JFM 42.0725.02
[5] Giulini, D., Laue’s theorem revisited: energy-momentum tensors, symmetries, and the habitat of globally conserved quantities, Int. J. Geom. Methods Modern Phys., 15, 1850182 (2018) · Zbl 1431.53092
[6] Ohanian, HC, Did Einstein prove \(E = mc^2\)?, Stud. Hist. Phil. Sci., 40, 167 (2009) · Zbl 1228.83008
[7] Iltis, CM, The controversy over living force: Leibniz to d’Alembert (1967), Madison: University of Wisconsin, Madison
[8] Iltis, CM, D’Alembert and the vis viva controversy, Stud. Hist. Philos. Sci., 1, 135 (1970) · Zbl 0216.28502
[9] Iltis, CM, Leibniz and the vis viva controversy, Isis, 62, 21 (1971) · Zbl 0228.01011
[10] Simon, H., Axioms of Newtonian mechanics, Philos. Mag., 38, 888 (1947) · Zbl 0030.32307
[11] Desloge, EA, Classical Mechanics (1982), New York: Wiley, New York · Zbl 0564.70001
[12] Dugas, R., History of Mechanics (2011), Mineola: Dover Publications, Mineola
[13] Einstein, A., Elementary derivation of the equivalence of mass and energy, Bull. Am. Math. Soc., 41, 223 (1935) · Zbl 0011.28108
[14] Einstein, A., Does the inertia of a body depend upon its energy content?, Ann. Phys., 18, 639 (1905)
[15] Ehlers, J.; Rindler, W.; Penrose, R., Energy conservation as the basis of relativistic mechanics II, Am. J. Phys., 33, 995 (1965)
[16] Lalan, MV, Sur une définition axiomatique de l’impulsion et de l’énergie, C. R. Acad. Sci., 198, 1211 (1934) · JFM 60.1423.04
[17] Einstein, A.: On the electrodynamics of moving bodies. Ann. Phys. 17 (1905) · JFM 36.0920.02
[18] Planck, M.: Das Prinzip der Relativität und die Grundgleichungen der Mechanik, Verhandlungen Deutsche Physikalische Gesellschaft 8, 136 (1906), English translation available at https://en.wikisource.org/wiki/Translation:The_Principle_of_Relativity_and_the_Fundamental_Equations_of_Mechanics. Accessed 22 Nov 2019 · JFM 37.0721.04
[19] Tolman, RC, Non-Newtonian mechanics, the mass of a moving body, Philos. Mag., 23, 375 (1912) · JFM 43.0785.03
[20] Graney, CM, Mass, speed, direction: John Buridan’s 14th century concept of momentum, Phys. Teacher, 51, 411 (2013)
[21] Clagett, M., The Science of Mechanics in the Middle Ages (1961), Madison: University of Wisconsin Press, Madison · Zbl 0093.00301
[22] Grant, E., A Source Book in Medieval Science (1974), Cambridge: Harvard University Press, Cambridge
[23] Mach, E., The Science of Mechanics: A Critical and Historical Account of Its Development (1919), Chicago: Open Court Pub. Co., Chicago · JFM 33.0718.13
[24] Desloge, EA, Conservation laws for classical and relativistic collisions, I, Int. J. Theor. Phys., 15, 349 (1976)
[25] Desloge, EA, Conservation laws for classical and relativistic collisions, II, Int. J. Theor. Phys., 15, 357 (1976)
[26] Sahoo, PK; Kannappan, P., Introduction to Functional Equations (2011), Boca Raton: CRC Press, Boca Raton · Zbl 1223.39012
[27] Arzeliès, H., Relativistic Point Dynamics (1971), Oxford: Pergamon Press, Oxford
[28] Sonego, S.; Pin, M., Deriving relativistic momentum and energy, Eur. J. Phys., 26, 33 (2005) · Zbl 1065.83502
[29] Goyal, P.: On the explanatory role of physical principles (in preparation)
[30] Goyal, P., Information-geometric reconstruction of quantum theory, Phys. Rev. A, 78, 052120 (2008)
[31] Goyal, P.: Origin of the correspondence rules of quantum theory (2010). arXiv:0910.2444
[32] Aczél, J., Lectures on Functional Equations and their Application (1966), New York: Academic Press, New York · Zbl 0139.09301
[33] Aczél, J.; Dhombres, J., Functional equations in several variables (1989), Cambridge: Cambridge University Press, Cambridge · Zbl 0685.39006
[34] Descartes, R., Principles of Philosophy (1982), Dordrecht: Kluwer, Dordrecht
[35] Westfall, RS, Force in Newton’s Physics: The Science of Dynamics in the Seventeenth Century (1971), Cambridge: Science History Publications, Cambridge · Zbl 0242.01009
[36] Barbour, JB, The Discovery of Dynamics (2001), Oxford: Oxford University Press, Oxford
[37] Leibniz, G.W.: A brief demonstration of a notable error of Descartes and others concerning a natural law, Acta eruditorum, 161 (1686), English translation, with supplements, in Philosophical papers and letters, 2nd edition, ed. L. E. Loemker, pp. 296-302
[38] Scott, WL, The Conflict Between Atomism and Conservation Theory 1644 to 1960 (1970), London: Macdonald, London
[39] Wren, C., Lex naturae de collisione corporum, Philos. Trans. R. Soc., 3, 867 (1669)
[40] Huygens, C., Regles du mouvement dans la recontre des corps, Philos. Trans. R. Soc., 4, 925 (1669)
[41] Huygens, C.: Regles du mouvement dans la recontre des corps. J. Scavans 19 (1669)
[42] Wallis, J., A summary account of the general laws of motion, Philos. Trans. R. Soc., 3, 864 (1669)
[43] Kuhn, TS; Kuhn, TS, Energy conservation as an example of simultaneous discovery, The Essential Tension (1977), Chicago: University of Chicago Press, Chicago
[44] Wolff, J., Are conservation laws metaphysically necessary?, Philos. Sci., 8, 898 (2013)
[45] Beers, BL, Geometric nature of Lagrange’s equations, Am. J. Phys., 40, 1636 (1972)
[46] Landau, LD; Lifshitz, EM, Mechanics (2000), Oxford: Butterworth Heinemann, Oxford · Zbl 0081.22207
[47] Goyal, P.; Knuth, KH; Skilling, J., Origin of complex quantum amplitudes and Feynman’s rules, Phys. Rev. A, 81, 022109 (2010)
[48] Goyal, P., Informational approach to the quantum symmetrization postulate, New J. Phys., 17, 013043 (2015)
[49] Khiari, C.: Newton’s laws of motion revisited: some epistemological and didactic problems. Lat. Am. J. Phys. Educ. 5 (2011)
[50] Papachristou, CJ, Foundations of Newtonian dynamics: an axiomatic approach for the thinking student, Nausivios Chora, 4, 153 (2012)
[51] Herrmann, F., Pohlig, M., Schwarze, H.: The Karlsruhe physics course: mechanics (2019)
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