Advertisement

Deducing Newton’s second law from relativity principles: A forgotten history

  • Olivier DarrigolEmail author
Article
  • 35 Downloads

Abstract

In French mechanical treatises of the nineteenth century, Newton’s second law of motion was frequently derived from a relativity principle. The origin of this trend is found in ingenious arguments by Huygens and Laplace, with intermediate contributions by Euler and d’Alembert. The derivations initially relied on Galilean relativity and impulsive forces. After Bélanger’s Cours de mécanique of 1847, they employed continuous forces and a stronger relativity with respect to any commonly impressed motion. The name “principle of relative motions” and the very idea of using this principle as a constructive tool were born in this context. The consequences of Poincaré’s and Einstein’s awareness of this approach are analyzed. Lastly, the legitimacy and significance of a relativity-based derivation of Newton’s second law are briefly discussed in a more philosophical vein.

Notes

Acknowledgements

I thank Jürgen Renn for useful advice and comments.

Compliance with ethical standards

Conflict of interest

The author states that there is no conflict of interest.

References

  1. Adam, Charles, and Paul Tannery. 1897. Œuvres de Descartes, vol. 1. Paris: Cerf.Google Scholar
  2. Appell, Paul. 1893. Traité de mécanique rationnelle. Paris: Gauthier-Villars.zbMATHGoogle Scholar
  3. Appell, Paul. 1902. Traité de mécanique rationnelle, 2nd ed. Paris: Gauthier-Villars.zbMATHGoogle Scholar
  4. Barbour, Julian. 2001. The discovery of dynamics a study from a Machian point of view of the discovery and the structure of dynamical theories. Oxford: Oxford University Press.Google Scholar
  5. Beeckman, Isaac. 1939–1953. Journal tenu par Isaac Beeckman de 1604 à 1634, publié avec une introduction et des notes, ed. by Cornelis de Waard. 4 vols. La Haye: Nijhoff.Google Scholar
  6. Bélanger, Jean-Baptiste. 1847. Cours de mécanique. Paris: Carilian Goeury.Google Scholar
  7. Bélanger, Jean-Baptiste. 1848. Lehrbuch der mechanik und ihrer Anwendungen auf das ingenieurwesen. Ludwigsburg: Neubert.Google Scholar
  8. Bernoulli, Daniel. 1726. Examen principiorum mechanicae, et demonstrationes geometricae de compositione et resolutione virum. Commentarii Academiae scientiarum imperialis Petropolitanae 1:126–142.Google Scholar
  9. Bertoloni Meli, Domenico. 1993. The emergence of reference frames and the transformation of mechanics in the Enlightenment. Historical Studies in the Physical Sciences 23: 301–335.CrossRefGoogle Scholar
  10. Blondlot, René. 1901. Exposé des principes de la mécanique. Bibliothèque du Congrès international de philosophie, vol. 3 (Paris: Armand Colin), pp. 445–455.Google Scholar
  11. Boltzmann, Ludwig. 1897–1920. Vorlesungen über die Principe der Mechanik. 3 vols. Leipzig: Barth.Google Scholar
  12. Bonnet, Ossian. 1858. Leçons de mécanique élémentaire. Paris: Mallet-Bachelier.Google Scholar
  13. Bouasse, Henri. 1895. Introduction à l’étude des théories de la mécanique. Paris: Carré.zbMATHGoogle Scholar
  14. Brown, Harvey. 2005. Physical relativity: Space-time structure from a dynamical perspective. Oxford: Oxford University Press.CrossRefzbMATHGoogle Scholar
  15. Buchwald, Jed, and Robert Fox (eds.). 2013. The Oxford handbook in the history of physics. Oxford: Oxford University Press.zbMATHGoogle Scholar
  16. Cajori, Florian (ed.). 1962. Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and his System of the World, 2 vols. Berkeley: University of California Press.Google Scholar
  17. Chatzis, Konstantinos. 1994. Mécanique rationnelle et mécanique des machines. In La formation polytechnicienne, 1794–1994, ed. Bruno Belhoste, Amy Dahan-Dalmenico, and Antoine Picon, 95–108. Paris: Dunod.Google Scholar
  18. Chatzis, Konstantinos. 1995. Un aperçu de la discussion sur les principes de la mécanique rationnelle en france à la fin du siècle dernier. Revue d’histoire des Mathématiques 1: 235–270.MathSciNetzbMATHGoogle Scholar
  19. Coriolis, Gaspard. 1829. Du calcul de l’effet des machines. Paris: Carilian Goeury.Google Scholar
  20. Coriolis, Gaspard. 1844. Traité de la mécanique des corps solides et du calcul de l’effet des machines. Paris: Fain et Thunot.Google Scholar
  21. Daguin, Pierre Adolphe. 1861. Traité élémentaire de physique théorique et expérimentale, 2nd ed. Toulouse: Privat.Google Scholar
  22. d’Alembert, Jean le Rond. 1743. Traité de dynamique, dans lequel les loix de l’équilibre et du mouvement des corps sont réduites au plus petit nombre possible, et démontrées d’une manière nouvelle, et où l’on donne un principe général pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres, d’une manière quelconque. Paris: David.Google Scholar
  23. d’Alembert, Jean le Rond. 1758. Traité de dynamique. 2nd ed. Paris: David.Google Scholar
  24. Damerow, Peter, Gideon Freudenthal, Peter McLaughlin, and Jürgen Renn. 1992. Concept and inference: Descartes and Beeckman on the fall of bodies. In Exploring the limits of preclassical mechanics, eds. P. Damerow, G. Freudenthal, P. McLaughlin, and J. Renn, 8–67. New York: Springer.Google Scholar
  25. Darrigol, Olivier. 1995. Henri Poincaré’s criticism of fin de siècle electrodynamics. Studies in the History and Philosophy of Modern Physics 26: 1–44.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Darrigol, Olivier. 2001. God, waterwheels, and molecules: Saint-Venant’s anticipation of energy conservation. Historical Studies in the Physical and Biological Sciences 31: 285–353.CrossRefGoogle Scholar
  27. Darrigol, Olivier. 2008. The modular structure of physical theories. Synthese 162: 195–223.MathSciNetCrossRefGoogle Scholar
  28. Darrigol, Olivier. 2014. Physics and necessity: Rationalist pursuits from the Cartesian past to the quantum present. Oxford: Oxford University Press.CrossRefGoogle Scholar
  29. De Risi, Vincenzo. 2015. Introduction. In Mathematizing space: The objects of geometry from antiquity to the early modern age, ed. V. de Risi, 1–13. Cham: Birkhäuser.Google Scholar
  30. Deguin, Nicolas. 1853. Cours élémentaire de physique, 8th ed. Paris: Belin.zbMATHGoogle Scholar
  31. Delaunay, Charles. 1856. Traité de mécanique rationnelle. Paris: Masson.Google Scholar
  32. Descartes, René. 1644. Principia philosophiae. Amsterdam: Ludovicus Elzevirius.Google Scholar
  33. Drake, Stillman. 1978. Galileo at work. Chicago: The University of Chicago Press.Google Scholar
  34. Dugas, René. 1950. Histoire de la mécanique. Paris: Dunod. Google Scholar
  35. Duhamel, Jean-Marie Constant. 1845–1846. Cours de mécanique de l’Ecole Polytechnique. 2 vols. Paris: Bachelier.Google Scholar
  36. Einstein, Albert. 1905. Zur Elektrodynamik bewegter Körper. Annalen der Physik 17: 891–921.CrossRefzbMATHGoogle Scholar
  37. Euler, Leonhard. 1736. Mechanica, sive motus scientia analytice exposita. 2 vols. Petersburg: Typographia Academiae Scientiarum.Google Scholar
  38. Finocchiaro, Maurice. 1989. The Galileo affair: A documentary history. Berkeley: University of California Press.Google Scholar
  39. Firode, Alain. 2001. La dynamique de d’Alembert. Paris: Vrin.Google Scholar
  40. Francœur, Louis Benjamin. 1804. Traité de mécanique élémentaire, 3rd ed. Paris: Bachelier.Google Scholar
  41. Francœur, Louis Benjamin. 1807. Traité de mécanique élémentaire, 4th ed. Paris: Bachelier.Google Scholar
  42. Francœur, Louis Benjamin. 1825. Traité de mécanique élémentaire, 5th ed. Paris: Bachelier.Google Scholar
  43. Gabbey, Alan. 1980. Huygens and mechanics. In Studies on Chirstiaan Huygens, ed. H. Bos, M. Rudwick, H. Snelders and R. Visser, 166–199. Lisse: Swets and Zeitlinger.Google Scholar
  44. Galilei, Galileo. 1632. Dialogo sopra i due massimi sistemi del mondo. Florence: Landini. (trans: Drake, S.). Berkeley: University of California Press. 1964.Google Scholar
  45. Galilei, Galileo. 1638. Discorsi e dimostrazioni matematiche intorno a due nuove scienze. Leiden: Elzevir.Google Scholar
  46. Garber, Daniel. 1992. Descartes’ metaphysical physics. Chicago: The University of Chicago Press.Google Scholar
  47. Giere, Ronald. 1988. Explaining science: A cognitive approach. Chicago: The University of Chicago Press.CrossRefGoogle Scholar
  48. Grattan-Guinness, Ivor. 1984. Work of the workers: Advances in engineering mechanics and instruction in France, 1800–1830. Annals of Science 41: 1–33.MathSciNetCrossRefzbMATHGoogle Scholar
  49. Grattan-Guinness, Ivor. 2005. ‘Exposition du système du monde’ and ‘Traité de méchanique céleste’. In Landmark writings in Western mathematics 1640–1940, ed. I. Grattan-Guinness, 242–257. Amsterdam: Elsevier.CrossRefGoogle Scholar
  50. Hankins, Thomas. 1970. Jean d’Alembert: Science and the enlightenment. Oxford: Clarendon Press.zbMATHGoogle Scholar
  51. Heilbron, John. 2010. Galileo. Oxford: Oxford University Press.Google Scholar
  52. Herivel, John. 1965. The background to Newton’s Principia. Oxford: Clarendon.zbMATHGoogle Scholar
  53. Huygens, Christiaan. [1659]. Untitled MS on pendulum motion. In Huygens 1888–1950, vol. 17, pp. 126–137.Google Scholar
  54. Huygens, Christiaan. 1669. Extrait d’une lettre de M. Hugens à l’auteur du journal. Journal des sçavans (18 March 1669), 22–24.Google Scholar
  55. Huygens, Christiaan. 1673. Horologium oscillatorium, sive de motu pendulorum ad horologia aptato demonstrationes geometricae. Paris: Muguet.Google Scholar
  56. Huygens, Christiaan. 1703. De motu corporum ex percussione. In Christiani Hugenii Zelemii, dum viveret, toparchae opuscula postuma, quae continent dioptricam. Commentarios de vitris figurandis. Dissertationem de corona & parheliis. Tractatum de motu. Tractatum de vi centrifuga. Descriptionem automati planetarii (Leiden: Boutesteyn, 1703), 369-400. Translated by Michael Mahoney at http://www.princeton.edu/~hos/Mahoney/texts/huygens/impact/huyimpct.html. Accessed 7 May 2019.
  57. Huygens, Christiaan. 1888–1950. Œuvres complètes publiées par la Société hollandaise des sciences. 22 vols. La Haye: Nijhof.Google Scholar
  58. Jamin, Jules. 1858. Cours de physique de l’École polytechnique. Paris: Mallet-Blanchet.zbMATHGoogle Scholar
  59. Jammer, Max. 1957. Concepts of force: A study in the foundation of dynamics. Cambridge: Harvard University Press.Google Scholar
  60. Jouguet, Émile. 1908. Lectures de mécanique. 2 vols. Paris: Gauthier-Villars.Google Scholar
  61. Kirchhoff, Gustav. 1876. Vorlesungen über mathematische Physik, vol. 1: Vorlesungen über Mechanik. Leipzig: Teubner.Google Scholar
  62. Laplace, Pierre Simon de. 1796. Exposition du système du monde, vol. 1. Paris: Cercle Social.Google Scholar
  63. Laplace, Pierre Simon de. 1799. Traité de mécanique céleste, vol. 1. Paris: Duprat.Google Scholar
  64. Laplace, Pierre Simon de. 1809. The system of the world, (trans: Pond, J.). vol. 1. London: Phillips.Google Scholar
  65. Lecornu, Léon. 1914. Cours de mécanique professé à l’École Polytechnique. Paris: Gauthier-Villars.zbMATHGoogle Scholar
  66. Love, Augustus. 1897. Theoretical mechanics: An introductory treatise on the principles of dynamics with applications and numerous examples. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  67. Mach, Ernst. 1883. Die Mechanik in ihrer Entwicklung. Historisch-kritisch dargestellt. Leipzig: Brockhaus.Google Scholar
  68. Maltese, Giulio. 2000. On the relativity of motion in Leonhard Euler’s science. Archive for History of Exact Sciences 54: 319–348.MathSciNetCrossRefzbMATHGoogle Scholar
  69. Martínez, Alberto. 2009. Kinematics: The lost origins of Einstein’s relativity. Baltimore: Johns Hopkins University Press.zbMATHGoogle Scholar
  70. Mascart, Éleuthère. 1866. 1866 Éléments de mécanique. Paris: Hachette.zbMATHGoogle Scholar
  71. Moatti, Alexandre. 2014. Le mystère Coriolis. Paris: CNRS editions.Google Scholar
  72. Newton, Isaac. [c. 1665].Waste Book, MS Add. 4004, Cambridge University Library, Cambridge, UK. In The Newton Project. http://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00220.
  73. Newton, Isaac. 1687. Philosophiae naturalis principia mathematica. London: Streater & Smith.Google Scholar
  74. Newton, Isaac. 1729. The mathematical principles of natural philosophy. (trans: Motte, A.). 2 vols. London: Motte.Google Scholar
  75. Péclet, Eugène. 1823. Cours de physique. Marseille: Ricard.Google Scholar
  76. Péclet, Eugène. 1838. Traité élémentaire de physique. Paris: Hachette.Google Scholar
  77. Pinaud, Auguste. 1846. Programme d’un cours élémentaire de physique, 4th ed. Paris: Hachette.Google Scholar
  78. Poincaré, Henri. 1900. La théorie de Lorentz et le principe de la réaction. In Recueil de travaux offerts par les auteurs à H.A. Lorentz à l’occasion du 25ème anniversaire de son doctorat le 11 décembre 1900; Archives néerlandaises, 5: 252–278.Google Scholar
  79. Poincaré, Henri. 1901. Sur les principes de la mécanique. Bibliothèque du Congrès international de philosophie, vol. 3 (Paris: Armand Colin), 457–494.Google Scholar
  80. Poincaré, Henri. 1902. La science et l’hypothèse. Paris: Flammarion.zbMATHGoogle Scholar
  81. Poincaré, Henri. 1904. Wissenschaft und Hypothese. Leipzig: Teubner.zbMATHGoogle Scholar
  82. Poisson, Siméon Denis. 1811. Traité de mécanique, vol. 1. Paris: veuve Courcier.Google Scholar
  83. Poissson, Siméon Denis. 1833. Traité de mécanique, vol. 1, 2nd ed.Google Scholar
  84. Privat-Deschanel, Augustin. 1869. Traité élémentaire de physique. Paris: Hachette.Google Scholar
  85. Pulte, Helmut. 1989. Das Prinzip der kleinsten Wirkung und die Kraftkonzeptionen der rationalen Mechanik: Eine Untersuchung zur Grundproblematik bei Leonhard Euler, Pierre Louis Moreau de Maupertuis und Joseph Louis Lagrange. Stuttgart: Steiner.zbMATHGoogle Scholar
  86. Resal, Henry. 1862. Éléments de mécanique. Paris: Mallet-Bachelier.Google Scholar
  87. Saint-Venant, Adhémar Barré de. 1851. Principes de mécanique fondés sur la cinématique. Paris: Bachelier.Google Scholar
  88. Saunders, Simon. 2013. Rethinking Newton’s Principia. Philosophy of Science 80: 22–48.CrossRefGoogle Scholar
  89. Simon, Josep. 2013. Physics textbooks and textbook physics in the nineteenth and twentieth centuries. In Buchwald and Fox 2013, pp. 651–678.Google Scholar
  90. Smeenk, Chris, and Eric Schliesser. 2013. Newton’s Principia. In Buchwald and Fox 2013, pp. 109–165.Google Scholar
  91. Stachel, John, et al. (eds.). 1989. The collected papers of Albert Einstein. Vol. 2, The Swiss years: writings, 1900–1909. Princeton: Princeton University Press.Google Scholar
  92. Stallo, John Bernhard. 1882. The concepts and theories of modern physics. New York: Appleton.zbMATHGoogle Scholar
  93. Stan, Marius. 2016. Huygens on inertial structure and relativity. Philosophy of Science 83: 277–298.CrossRefGoogle Scholar
  94. Stein, Howard. 1977. Some philosophical prehistory of Einstein’s general relativity. In Foundations of space-time theories, ed. John Earman, Clark Glymour, and John Stachel, 3–49. University of Minnesota Press.Google Scholar
  95. Sturm, Charles. 1861. Cours de mécanique de l’École Polytechnique. Paris: Mallet-Bachelier.Google Scholar
  96. Suppe, Frederick. 1974. The structure of scientific theories. Urbana: The University of Illinois Press.Google Scholar
  97. Swerdlow, Noel Mark. 2013. Galileo’s mechanics of natural motion and projectiles. Buchwald and Fox 2013: 25–55.Google Scholar
  98. Tait, Peter Guthrie. 1884. Note on reference frames. Proceedings of the Royal Society of Edinburgh 22: 743–745.CrossRefzbMATHGoogle Scholar
  99. Thomson, James. 1884a. On the law of inertia; the principle of chronometry, and the principle of absolute clinural rest, and of absolute rotation. Proceedings of the Royal Society of Edinburgh 22: 568–578.zbMATHGoogle Scholar
  100. Thomson, James. 1884b. A problem on point-motions for which a reference-frame can so exist as to have the motions of the points relative to it, rectilinear and mutually proportional. Proceedings of the Royal Society of Edinburgh 22: 730–743.CrossRefzbMATHGoogle Scholar
  101. Torretti, Roberto. 1983. Relativity and geometry. Oxford: Pergamon Press.zbMATHGoogle Scholar
  102. van Fraassen, Bas. 1980. The scientific image. Oxford: Clarendon Press.CrossRefGoogle Scholar
  103. Vilain, Christiane. 1996. La mécanique de Christiaan Huygens: La relativité du mouvement au XVII e siècle. Paris: Blanchard.zbMATHGoogle Scholar
  104. Violle, Jules. 1883–1892. Cours de physique. 2 vols. Paris: Masson.Google Scholar
  105. Violle, Jules. 1892–1897. Lehrbuch der Physik. 2 vols. Berlin: Springer.Google Scholar
  106. Volgraff, Johan Adriaan. 1929. Avertissement. In Huygens 1888–1950, vol. 16, pp. 171–178.Google Scholar
  107. Wallis, John. 1668. A summary account given by Dr. John Wallis of the general laws of motion. Philosophical Transactions of the Royal Society of London 3: 864–866.Google Scholar
  108. Westfall, Richard. 1980. Never at rest: A biography of Isaac Newton. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  109. Whewell, William. 1819. An elementary treatise on dynamics. Cambridge: Whittaker.Google Scholar
  110. Whittaker, Edmund. 1904. A treatise on the analytical dynamics of particles and rigid bodies. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  111. Wren, Christoffer. 1668. Lex naturae de collisione corporum. Philosophical Transactions of the Royal Society of London 3: 867–868.Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CNRSUMR SPhereParisFrance

Personalised recommendations