Whittaker’s analytical dynamics: a biography. (English) Zbl 1296.70003
The author states that, due to the development of the symplectic geometry as the perfect language of mechanics, in the 20th century mechanics has undergone such a dramatic change that books written before the 1930s “look hopelessly dated to present day readers”. Therefore, the question arises: what caused the longevity of E. T. Whittaker’s [A treatise on the analytical dynamics of particles and rigid bodies; with an introduction to the problem of three bodies. Cambridge: At the University Press; London: Clay (1904; JFM 35.0682.01)] written already in 1904. To answer this question the author follows the development from E. T. Whittaker’s [“Report on the progress of the solution of the problem of three bodies”, Brit. Ass. Rep., 121–159 (1899; JFM 30.0649.01)] over the influence it had on Dirac’s version of quantum mechanics up to the meaning it has in our days. The paper starts with biographical notes that can help to understand how Whittaker approached this topic. It follows a chronology of the book “Analytical Dynamics” itself [E. T. Whittaker, Analytische Dynamik der Punkte und starren Körper. Berlin: Springer (1924; JFM 50.0528.05)], i.e., its editions and translations. In the next section, the content of the book is discussed, where, in virtue of its great role for modern mechanics, those chapters of the book that concern Hamiltonian mechanics are in the centre of interest. This great role shows in the mentioned meaning for the Dirac’s work and, moreover, in that these chapters “present, in incipient form, the symplectic approach that we now use”. After “a state-of the-art introduction to principles of dynamics as they stood in the first years of the twentieth century” and a section on Hamiltonian systems and contact transformations (building essentially the contents of Part I of Whittaker’s book) it follows, as a second part, a description of applications of the principles to celestial mechanics, in particular to the problem of three bodies. In a fourth section of the paper, the reaction of the mathematical community to “Analytical dynamics” is examined by presenting and commenting reviews of its different editions. The final section is devoted to the book’s style (influenced by Jacobi, Lie and Poincaré) and its impact on the development of mechanics, which was always given, but whose character changed with the passing of time.
Reviewer: Horst-Heino von Borzeszkowski (Berlin)
MSC:
| 70-03 | History of mechanics of particles and systems |
| 01A60 | History of mathematics in the 20th century |
| 81-03 | History of quantum theory |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 81S10 | Geometry and quantization, symplectic methods |
Keywords:
Whittaker; Hamiltonian mechanics; contact transformations; three-body problem; symplectic geometry, quantum mechanicsBiographic References:
Whittaker, Edmund T.References:
| [1] | Ahmed, N., and Q.K. Ghori. 1984. Levi-Civita’s theorem for dynamical equations with constraint multipliers. Archive for Rational Mechanics and Analysis 85: 1–13. · Zbl 0546.70017 |
| [2] | Abraham, R. 1967. Foundations of mechanics: A mathematical exposition of classical mechanics with an introduction to the qualitative theory of dynamical systems and applications to the three-body problem, with the assistance of Jerrold E. Marsden. New York: W.A. Benjamin. · Zbl 0158.42901 |
| [3] | Arnold, V.I. 1989. Mathematical methods of classical mechanics, 2nd ed. Berlin: Springer. |
| [4] | Arnold, V.I., V.V. Kozlov, and A.I. Neishadt. 1997. Mathematical aspects of classical and celestial mechanics, 2nd ed. Berlin: Springer. |
| [5] | Barrow-Green, J. 2002. Whittaker and Watson’s ’Modern Analysis’. European Mathematical Society Newsletter 45: 14–15. |
| [6] | Barrow-Green, J. 1997. Poincaré and the three body problem. American Mathematical Society and London Mathematical Society. Providence: AMS Press. · Zbl 0877.01022 |
| [7] | Bairstow, L.E. 1933. George Hartley Bryan. Obit Not Fell R Soc 1: 139–142. · doi:10.1098/rsbm.1933.0011 |
| [8] | Besant, W.H. 1893. A treatise on dynamics. Cambridge: Deighton Bell. |
| [9] | Birkhoff, G.D. 1920. Review of a treatise of analytical dynamics. Bulletin of the American Mathematical Society 26: 183. · JFM 47.0368.01 · doi:10.1090/S0002-9904-1920-03290-8 |
| [10] | Birkhoff, G.D. 1927. Dynamical systems. Colloquium Publications IX, American Mathematical Society (1927). · JFM 53.0732.01 |
| [11] | Bryan, G.H. 1905. Three Cambridge mathematical works. Nature 71: 601–603. · doi:10.1038/071601a0 |
| [12] | Bryan, G.H. 1918. Analytical dynamics. Nature 100: 363–364. · doi:10.1038/100363a0 |
| [13] | Burnside, W. 1897. Theory of groups of finite order. Cambridge: Cambridge University Press. · JFM 28.0118.03 |
| [14] | Cartan, E. 1922. Leçons sur les Invariant Intégraux. Paris: Hermann. |
| [15] | Cherry, T.M. 1928. Review of Birkhoff’s dynamical systems and the third edition of Whittaker’s analytical dynamics. Mathematical Gazette 195: 198–199. · doi:10.2307/3603797 |
| [16] | Chorlay, R. 2011. ”Local-Global”: the first twenty years. Archive for History of Exact Sciences 65: 1–66. · Zbl 1235.01014 · doi:10.1007/s00407-010-0070-1 |
| [17] | Creese, M.R.S., and T.M. Creese. 2004. Ladies in the laboratory 2. New York: Scarecrow Press. · Zbl 0143.15501 |
| [18] | Crowe, M.J. 1967. A history of vector analysis: The evolution of the idea of a vectorial system. New York: Dover. · Zbl 0165.00303 |
| [19] | Culverwell, The discrimination of maxima and minima values of single integrals with any number of dependent variables and any continuous restrictions of the variables, the limiting values of the variables being supposed given, Proc. London Math. Soc. 23 (1892), 241–265. · JFM 24.0337.01 |
| [20] | Diliberto, S.P. 1958. Review of Carl Ludwig Siegel’s Vorlesungen über Himmelsmechanik. Bulletin of the American Mathematical Society 64: 192–196. · doi:10.1090/S0002-9904-1958-10205-0 |
| [21] | Dirac, P.A.M. 1926. The fundamental equations of quantum mechanics. Proceedings of the Royal Society A 109: 642–653; also in [105, 307–320]. · JFM 51.0729.01 |
| [22] | Dirac, P.A.M. 1926. Quantum mechanics and a preliminary investigation of the Hydrogen atom. Proceedings of the Royal Society A 110: 561–569; also in [105, p. 417–427]. · JFM 52.0974.03 |
| [23] | Dirac, P.A.M. 1947. The principles of quantum mechanics, 3rd ed. Oxford: Oxford University Press. · Zbl 0030.04801 |
| [24] | Fictitious problems in mathematics. Nature (72) (1905), pp. 56, 78, 102, 127 and 175. |
| [25] | GBM 1917. DR. W. H. BESANT, F.R.S. Nature 99: 310–311. |
| [26] | Geiges, H. 2001. A brief history of contact geometry and topology. Expositiones Mathematicae 19: 25–53. · Zbl 1048.53001 |
| [27] | Goldstein, H. 1950. Classical mechanics. Reading: Addison-Wesley. · Zbl 0043.18001 |
| [28] | Goriely, A. 2001. Integrability and nonintegrability of dynamical systems, advanced series in nonlinear dynamics, vol. 19. Singapore: World Scientific. · Zbl 1002.34001 |
| [29] | Gray, C.G., and E.F. Taylor. 2007. When action is not least. American Journal of Physics 75: 434–458. |
| [30] | Hamilton, W.R. 1828. Theory of systems of rays. Transactions of the Royal Irish Academy 15: 69–174. |
| [31] | Hamilton, W.R. 1830. Supplement to an essay on the theory of systems of rays. Transactions of the Royal Irish Academy 16(part I): 1–61. |
| [32] | Hamilton, W.R. 1831. Second supplement to an essay on the theory of systems of rays. Transactions of the Royal Irish Academy 16(part II): 93–125. |
| [33] | Hamilton, W.R. 1837. Third supplement to an essay on the theory of systems of rays. Transactions of the Royal Irish Academy 17: 1–144. |
| [34] | Hawkins, T. 1991. Jacobi and the birth of Lie’s theory of groups. Archive for History of Exact Sciences 42: 187–278. · Zbl 0767.01019 · doi:10.1007/BF00375135 |
| [35] | Hermann, R. 1962. Lectures on Hamilton-Jacobi-Lie theory and the calculus of variations, I, Notes. Berkeley: University of California, Berkeley. |
| [36] | Hill, G.W. 1886. On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and the moon. Cambridge, MA: Press of John Wilson & Son; also Acta 8, 1–36 and in (Hill 1905–1907, Volume 1, pp. 243–270). · JFM 18.1106.01 |
| [37] | Hill, G.W. 1905–1907. The collected mathematical works of George William Hill. The Carnegie Institution of Washington. |
| [38] | Horneffer, K. 1988. Review of the 1988 reprint of a treatise on the analytical dynamics of particles and rigid bodies, Zentralblatt Zbl 0665.70002. |
| [39] | Hunsaker, J.C. 1916. Dynamical stability of aeroplanes. Proceedings of the National Academy of Sciences of the United States of America 2: 278–283. · doi:10.1073/pnas.2.5.278 |
| [40] | Hunsaker, J., and S. MacLane. 1973. Edwin Bidwell Wilson (1879–1964). Biographical Memoir of the National Academy of Sciences. |
| [41] | Jacobi, C.G.J. 1837. Zur Theorie der Variationsrechnung und der Differentialgleichungen. Journal für die angewandte Mathematik 17: 68–82; also in (Jacobi 1881, Volume IV, pp. 39–55). · ERAM 017.0558cj |
| [42] | Jacobi, C.G.J. 1838. Sur le calcul des variations et sur la Thórie des équations différentielles. Journal de Mathématiques Pures et Appliquées, Série 1(3): 44–59. |
| [43] | Jacobi, C.G.J. 2009. Vorlesungen über analytische Mechanik, second edition, edited by A. Clebsch, Georg Reimer (1866); translated as Jacobi’s Lectures on Dynamics, Hindustan Book Agency. |
| [44] | Jacobi, C.G.J. 1881. Gesammelte Werke, edited by C. W. Borchardt, E. Lottner, K. T. W. Weierstrass, 7 volumes. · JFM 13.0020.02 |
| [45] | Jordan, C. 1870. Traité des substituitions. Paris: Gauthier-Villars. |
| [46] | Jost, R. 1964. Poisson brackets: (an unpedagogical lecture). Review of Modern Physics 36: 572–579. · doi:10.1103/RevModPhys.36.572 |
| [47] | Jourdain, E.P. 1917. Review of the first edition of ”A Treatise on the Analytical Dynamics of Particles and Rigid Bodies”. Mathematical Gazette 131: 145–146. · JFM 46.0035.16 · doi:10.2307/3603175 |
| [48] | Jourdain, E.P. 1917–1918. Review of the first edition of ”A Treatise on the Analytical Dynamics of Particles and Rigid Bodies”. Science Progress in the Twentieth Century 12: 345. |
| [49] | Julliard-Tosel, E. 2000. Bruns’ theorem: The proof and some generalizations. Celestial Mechanics and Dynamical Astronomy 76: 241–281. · Zbl 0993.70012 · doi:10.1023/A:1008346516349 |
| [50] | Kline, M. 1990. Mathematical thought from ancient to modern times, vol. 2. Oxford: Oxford University Press. · Zbl 0784.01048 |
| [51] | Kosmann-Schwarzbach, Y. 2013, La géométrie de Poisson, création du XX $${}ê$$ e siécle. In Siméon-Denis Poisson: Les mathématiques au service de la science, edited by Yvette Kosmann-Schwarzbach, Les Éditions de l’École polytechnique, pp. 129–172. |
| [52] | Kragh, H. 1990. Dirac: A scientific biography. Cambridge: Cambridge University Press. |
| [53] | Lagrange, J.-L. 1760/1761. Application de la méthode exposée dans la mémoire précédent a la solution de différents problèmes de dynamique, Miscellanea Taurinensia 2: 196–268; also in [59, t. I (1867), 365–468]. |
| [54] | Lagrange, J.-L. 1808. Mémoire sur las théorie des variations des éléments des planètes et en particulier des grandes axes de leurs orbites, Mém. de l’Institute de France, also in [59, t. VI, 711–768]. |
| [55] | Lagrange, J.-L. 1810. Second mémoire sur las théorie de la variation des arbitraires dans le problèmees de mécanique, dans lequel on simplifie l’application des formules générales a ces problèmes, Mém. de l’Institute de France, also in (Lagrange 1867–1892, t. VI, 809–816). |
| [56] | Lagrange, J.-L. Mécanique Analytique, quatrième édition, publié par Gaston Darboux, Gauthier-Villars t. I (1888), t. II (1889). |
| [57] | Lagrange, J.-L. 1867–1892. Oeuvres de Lagrange, Gauthier-Villars. |
| [58] | Lampe, E. 1918. Review of the first edition of ”A Treatise on the Analytical Dynamics of Particles and Rigid Bodies”, Jahrbuch über die Fortschritte der Mathematik, 35.0682.01. |
| [59] | Landau, L.D., and E.M. Lifschitz. 1976. Mechanics, course o theoretical physics, vol. 1, 3rd ed. Amsterdam: Elsevier. |
| [60] | Levi-Civita, T., and U. Amaldi. 1927. Lezioni di Meccanica Razionale, vol. II, part 2. Bologna: Nicola Zanichelli. · JFM 49.0570.01 |
| [61] | Lie, S. 1872. Zur Theorie partieller Differentialgleichungen, Göttinger Nachrichten, 480. |
| [62] | Liebmann, H. 1914. Berührungstransformation in Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen III.D.7: 441–502. |
| [63] | Liouville, J. 1855. Note sur les équations de la Dynamique. Journal des Mathématiques Pures et Appliquées XX: 137–138. |
| [64] | Love, A.E.H. 1929. George Hartley Bryan. Journal of London Mathematical Society Series 1(4): 238–240. · JFM 55.0021.10 · doi:10.1112/jlms/s1-4.3.238 |
| [65] | Lützen, J. 1990. Joseph Liouville 1809–1882: master of pure and applied mathematics, Studies in the History of Mathematics and Physical Sciences, 15. Berlin: Springer. · Zbl 0701.01015 |
| [66] | Mackey, G.W. 1963. Mathematical foundations of quantum mechanics. New York: W.A. Benjamin. · Zbl 0114.44002 |
| [67] | Marsden, J. 1993. Steve Smale and geometric mechanics in Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, 499–516. · Zbl 0812.70012 |
| [68] | McCrea, W.H. 1957. Edmund Taylor Whittaker. Journal of the London Mathematical Society 32: 234–256. · Zbl 0215.31703 · doi:10.1112/jlms/s1-32.2.234 |
| [69] | Marle, C.-M. 2009. The inception of symplectic geometry: the works of Lagrange and Poisson during the years 1808–1810. Letters in Mathematical Physics 90: 3–21. · Zbl 1189.53003 · doi:10.1007/s11005-009-0347-y |
| [70] | Martin, D. 1958. Sir Edmund Taylor Whittaker, F.R.S. Proceedings of the Edinburgh Mathematical Society (2) 11: 1–10. · Zbl 0083.24510 · doi:10.1017/S0013091500014334 |
| [71] | Mathieu, É. 1874. Mémoire sur les équations différentielles canoniques de la mécanique. Journal des Mathématiques Pures et Appliquées 19: 265–306. |
| [72] | Maxwell, J.C. 1954. A treatise on electricity and magnetism, vol. 1. New York: Dover. · Zbl 0056.20612 |
| [73] | Philip Edward Bertrand Jourdain. Proc London Math Soc 19: 59–60 (1921). |
| [74] | Poincaré, H. Mémoire sur les fonctions zétafuchsiennes. Acta Mathematica IV (1884), 211–278; also in (Poincaré 1916–1954, Volume 2, pp. 402–462). |
| [75] | Poincaré, H. 1890. Sur le problème des trois corps et les équations de la dynamique. Acta 13 1–270 also in (Poincaré 1916–1954, Volume VII, 262–479). |
| [76] | Poincaré, H. 1899. Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, t. I (1892), t. II (1893), t. III (1899). |
| [77] | Poincaré, H. 1910. Leçons de mécanique céleste : professées à la Sorbonne, Gauthier-Villars t. I (1905), t. II partie 1 (1907), t. II partie 2 (1909), t. III (1910). |
| [78] | Poincaré, H. 1916–1954. Oeuvres de Henri Poincaré, 12 volumes, Gauthier-Villars. |
| [79] | Poisson, S.-D. 1809. Sur la variation des constants arbitraires dans le question de Mécanique. Journal de l’Ecole Polytechnique, 15éme cahier VIII: 266–344. |
| [80] | Prange, G. 1935. Die Allgemeinen Integrationsmethoden der Analytischen Mechanik, Encyklopädie der Mathematischen Wissenschaften IV.12.U13, 505–804. |
| [81] | Pulte, H. 2005. Joseph Louis Lagrange, Méchanique Analitique, First Edition (1788), in Landmark Writings in Western Mathematics 1640–1940, edited by I. Grattan-Guiness, Elsevier. |
| [82] | Reeb, G. 1952. Sur certaines propriétés topologiques des trajectoires des systémes dynamiques, Acad. Roy. Belgique, Cl. Sci., Mém., Coll. 8 $$\^\(\backslash\)circ $$ 27(9), 64 S. · Zbl 0048.32903 |
| [83] | Routh, E.J. 1860. An elementary treatise on the dynamics of a system of rigid bodies: With numerous examples. London: MacMillan. |
| [84] | Routh, E.J. 1882. The elementary part of a treatise on the dynamics of a system of rigid bodies, being part I of a treatise on the whole subject. London: MacMillan. |
| [85] | Routh, E.J. 1884. The advanced part of a treatise on the dynamics of a system of rigid bodies, being part II of a treatise on the whole subject. London: MacMillan. |
| [86] | Routh, E.J. 1891. A treatise on analytic statics. Cambridge: Cambridge University Press. · JFM 23.0919.02 |
| [87] | Routh, E.J. 1898. A treatise on dynamics of a particle with numerous examples. Cambridge: Cambridge University Press. · JFM 29.0581.01 |
| [88] | Siegel, C.L. 1943. Symplectic geometry. American Journal of Mathematics 65: 1–86. · Zbl 0138.31401 · doi:10.2307/2371774 |
| [89] | Siegel, C.L. 1956. Vorlesungen über Himmelsmechanik. Berlin: Springer. · Zbl 0070.45403 |
| [90] | Siegel, C.L., and J. Moser. 1971. Lectures on celestial mechanics, translation by Charles I. Kalme. Berlin: Springer. · Zbl 0312.70017 |
| [91] | Smale, S. 1967. Differentiable dynamical systems. Bulletin of the American Mathematical Society 73: 747–817. · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1 |
| [92] | Sneddon, I.N. 1980. Review of ”Mathematical methods of classical mechanics” and ”A course of mathematical physics, vol. 1: Classical dynamical systems”. Bulletin of the American Mathematical Society (N.S.) 2: 346–352. · doi:10.1090/S0273-0979-1980-14755-2 |
| [93] | Souriau, J.M. 1970. Structure des systèmes dynamiques. Paris: Dunod. · Zbl 0186.58001 |
| [94] | Sommerfeld, A. 1921. Atombau und Spektrallinien, 2nd ed. Braunschweig: Friedr. Vieweg & Sohn. · JFM 55.1173.02 |
| [95] | Sternberg, S. 1964. Lectures on differential geometry. Englewood Cliffs: Prentice-Hall. · Zbl 0129.13102 |
| [96] | Sternberg, S. 1980. Review of Ralph Abraham and Jerrold E. Marsden, ”Foundations of mechanics”. Bulletin of the American Mathematical Society (N.S.) 2: 378–387. · doi:10.1090/S0273-0979-1980-14771-0 |
| [97] | Temple, G. 1956. Edmund Taylor Whittaker. Biographical memoirs of fellows of the royal society 2: 299–325. |
| [98] | Thirring, W. 1978. A course of mathematical physics, vol. 1, translated by Evans M. Harrel. Berlin: Springer. · Zbl 0387.70001 |
| [99] | Todhunter, I. 1962. A history of the calculus of variations during the nineteenth century. Berlin: Chelsea. · Zbl 0124.24402 |
| [100] | Truesdell, C. 1984. An idiot’s fugitive essays on science. Berlin: Springer. · Zbl 0599.01013 |
| [101] | van Kampen, E.R., and A. Wintner. 1936. On the canonical transformations of Hamiltonian systems. American Journal of Mathematics 58: 851–863. · Zbl 0015.18105 · doi:10.2307/2371255 |
| [102] | van der Waerden, B.L. 1967. Sources of quantum mechanics. New York: Dover. · Zbl 0151.43604 |
| [103] | Verhulst, F. 2012. Henri Poincaré: impatient genius. Berlin: Springer. · Zbl 1272.01003 |
| [104] | Warwick, A. 2003. Masters of theory: Cambridge and the rise of mathematical theory, Chicago. · Zbl 1127.01001 |
| [105] | Watson, G.N., and E.T. Whittaker. 1963. A course of modern analysis, 4th ed. Cambridge: Cambridge University Press. · Zbl 0108.26903 |
| [106] | Weyl, H. 1939. The classical groups. Englewood Cliffs: Princeton. · Zbl 0020.20601 |
| [107] | Whittaker, E.T. 1900. Report on the progress of the solution of the problem of three bodies, in Report of the Sixty-Ninth Meeting of the British Association for the Advancement of Science held at Dover in September 1899, John Murray. |
| [108] | Whittaker, E.T. 1902. On the solution of dynamical problems in terms of trigonometric series. Proceedings of the London Mathematical Society 34: 206–221. · JFM 33.0746.01 |
| [109] | Whittaker, E.T. 1940. The Hamiltonian revival. The Mathematical Gazette 24(260): 153–158. · doi:10.2307/3605704 |
| [110] | Whittaker, E.T. 1904. A treatise on the analytical dynamics of particles and rigid bodies, 1st ed. Cambridge: Cambridge University Press. · JFM 35.0682.01 |
| [111] | Whittaker, E.T. 1910. A history of the theories of aether and electricity from the age of Descartes to the close of the nineteenth century. London: Longmans. · JFM 41.0072.01 |
| [112] | Whittaker, E.T. 1917. A treatise on the analytical dynamics of particles and rigid bodies, 2nd edn. Cambridge: Cambridge University Press. Reprinted by Pranava Books (2008). |
| [113] | Whittaker, E.T. 1999. A treatise on the analytical dynamics of particles and rigid bodies, reprint of the fourth edition with a forward by W. H. McCrea, Cambridge University Press, Cambridge. |
| [114] | Whittaker, E.T. 1916–1917. On the Adelphic integral of the differential equations of dynamics. Proceedings of the Royal Society of Edinburgh 37: 95–116. |
| [115] | Whittaker, E.T. 1951. A history of the theories of aether and electricity: The classical theories. London: Nelson. · Zbl 0043.24503 |
| [116] | Whittaker, E.T. 1953. A history of the theories of aether and electricity: The modern theories 1900–1926. London: Nelson. · Zbl 0052.40802 |
| [117] | Wilson, E.B. 1901. Vector analysis: A text-book for the use of students of Mathematics and Physics founded upon the lectures of J. Willard Gibbs. New York: Charles Scribner’s Sons. |
| [118] | Wilson, E.B. 1906. Review of the first edition of ”A Treatise on the Analytical Dynamics of Particles and Rigid Bodies”. Bulletin of the American Mathematical Society 12: 451–458. · doi:10.1090/S0002-9904-1906-01372-6 |
| [119] | Wintner, A. 1934. On the linear conservative dynamical systems. Annali di Matematica Seria IV 13: 14–112. · JFM 60.1338.03 |
| [120] | Wintner, A. 1941. The analytical foundations of celestial mechanics. Englewood Cliffs: Princeton University Press. · Zbl 0026.02302 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
