Abstract
A brief historical overview is given which discusses the development of classical stability concepts, starting in the seventeenth century and finally leading to the concept of Lyapunov stability at the beginning of the twentieth century. The aim of the paper is to find out how various scientists thought about stability and to which extent their work is related to the stability concepts bearing their names, i.e. Lagrange, Poisson and Lyapunov stability. To this end, excerpts of original texts are discussed in detail. Furthermore, the relationship between the various works is addressed.
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Barrow-Green, J.: Poincaré and the Three-Body Problem. History of Mathematics, vol. 11. American Mathematical Society, Providence (1997)
Bennet, S.: A History of Control Engineering 1800–1930. Peter Peregrinus, Stevenage (1979)
Bernoulli, D.: Commentationes de statu aequilibrii corporum humido insidentium. Comment. Acad. Scient. Imp. Petrop. X, 147–163 (1738) (1747)
Bissel, C.C.: Stodola, Hurwitz and the genesis of the stability criterion. Int. J. Control 50(6), 2313–2332 (1989)
Bissel, C.C.: Russian and soviet contributions to the development of control engineering: a celebration of the Lyapunov centenary. Trans. Inst. Meas. Control 14(4), 170–178 (1992)
Bouguer, P.: Traité du Navire, de sa Construction et de ses Mouvemens. Jombert, Paris (1746)
Crone, E., Dijksterhuis, E.J., Forbes, R.J., Minnaert, M.G.J., Pannekoek, A.: The principal works of Simon Stevin. Swets & Zeitlinger, Amsterdam (1955)
Darrigol, O.: Stability and instability in nineteenth-century fluid mechanics. Rev. Hist. Math. 8, 5–65 (2002)
Dijksterhuis, E.J.: De Mechanisering van het Wereldbeeld. Meulenhoff, Amsterdam (2000) (1950)
Euler, L.: Methodus inveniendi lineas curvas maximi minive proprietate gaudentes. Bousquet, Lausanne & Geneva (1744)
Euler, L.: Scientia navalis seu tractatus de construendis ac dirigendis, volumes 1 and 2. Academiae Scientarum, St. Petersburg (1749)
Fraser, C.G.: Mathematical technique and physical conception in Euler’s investigation of the elastica. Centaurus 34(3), 211–246 (1991)
Fuller, A.T.: Lyapunov centenary issue. Int. J. Control 55(3), 521–527 (1992)
Fuss, P.H.: Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle, Tome II. The Sources of Science, vol. 35. Johnson reprint corporation, New York (1968), originally printed in St. Pétersbourg 1843
Gray, J.J.: Poincaré, topological dynamics, and the stability of the solar system. In: Harman, P.M., Shapiro, A.E. (eds.) The Investigation of Difficult Things: Essays on Newton and the History of the Exact Sciences in Honour of D.T. Whiteside, pp. 503–524. Cambridge University Press, Cambridge (1992)
Harman, P.M.: The Scientific Letters and Papers of James Clerk Maxwell, vols. 1–3. Cambridge University Press, Cambridge (1990) (1995) (2002)
Huygens, C.: De iis quae liquido supernatant libri tres, 1650. In: Œuvres Complètes de Christiaan Huygens, Tome 11. Martinus Nijhoff, Dordrecht (1908). The Hague, 1650
Jacobi, C.G.J.: Vorlesungen über Dynamik (edited by A. Clebsch, reprinted by Chelsea, New York, 1969), Berlin (1866)
Jury, E.I.: Remembering four stability theory pioneers of the nineteenth century. IEEE Trans. Automat. Contr. 43(10), 821–823 (1996)
Lagrange, J.L.: Sur le mouvement des nœuds des orbites planétaires. Mémoires de l’Académie Royale des Sciences et Belles-Lettres, Année 1774, 276–307 (1776)
Lagrange, J.L.: Méchanique Analytique. La Veuve Desaint, Paris (1788)
Laplace, P.S.: Sur le principe de la gravitation universelle et sur les inégalités séculaires des planètes qui en dépendent. In: Œuvres Complètes de Laplace, Tome VIII, pp. 201–275. Gauthier-Villars, Paris (1773) (1841)
Laplace, P.S.: Sur l’équation séculaire de la lune. Mémoires de l’Académie Royale des Sciences de Paris, Année 1786, 243–271 (1788)
Laplace, P.S.: Traité de mécanique céleste, Tome IV. Coucier, Paris (1825)
Laplace, P.S.: Exposition du système du monde. Bachelier, Paris (1836)
Lützen, J.: Joseph Liouville’s work on the figures of equilibrium of a rotating mass of fluid. Arch. History Exact Sci. 30(2), 113–166 (1984)
Lejeune-Dirichlet, J.P.G.: Über die Stabilität des Gleichgewichts. CRELLE, J. Reine Angew. Math. 32, 85–88 (1846)
Leonov, G.A., Burkin, I.M., Shepeljavyi, A.I.: Frequency Methods in Oscillation Theory. Mathematics and its Applications, vol. 357. Kluwer Academic, Dordrecht (1996)
Liapounoff, A.: Problème général de la stabilité du mouvement. Annales de la faculté des sciences de Toulouse, 2e série 9, 203–474 (1907)
Linsebarth, H.: Abhandlungen über das Gleichgewicht und die Schwingungen der ebenen elastischen Kurven. Ostwald’s Klassiker der Exakten Wissenschaften, vol. 175. Wilhelm Engelmann, Leipzig (1910)
Liouville, J.: Sur la stabilité de l’équilibre des mers. C. R. Acad. Sci. Paris 15, 903–907 (1842)
Loriá, A., Panteley, E.: Stability, told by its developers. In: Loriá, A., Lamnabhi-Lagarrigue, F., Panteley, E. (eds.) Advanced Topics in Control Systems Theory. Lecture Notes in Control and Information Sciences, vol. 328, pp. 199–258. Springer, London (2006) Chapter 6
Lyapunov, A.M.: Problème général de la stabilité du mouvement. Annals of Mathematics Studies, vol. 17. Princeton University Press, Princeton (1947)
Lyapunov, A.M.: The general problem of the stability of motion. In: Fuller, A.T. (ed.) The General Problem of the Stability of Motion. Taylor & Francis, London (1992)
Lyapunov, A.M. (Ляпунов, А.М.): A general task about the stability of motion (Общая задача об устойчивости движения). Kharkov Mathematical Society, University of Kharkov (1892)
Lyapunov, A.M. (Ляпунов, А.М.): A general task about the stability of motion (Общая задача об устойчивости движения). Государственное Издательство Технико-Теоретической Литературы, Moscow (1950)
Maxwell, J.C.: On governors. Proc. R. Soc. Lond. 16, 270–283 (1868)
Nowacki, H.: Leonhard Euler and the theory of ships. Max Planck Society—eDocument Server, http://edoc.mpg.de/ID:314833.0 (2007)
Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, Tome I, II, III. Gautier-Villars, Paris (1892) (1893) (1899)
Poisson, S.D.: Mémoire sur les inégalités séculaires des moyens mouvements des planètes. Journal de l’École Polytechnique XV(1), 1–56 (1808)
Routh, E.J.: A Treatise on the Stability of a Given State of Motion. Macmillan, London (1877)
Stevin, S.: Wisconstighe Ghedachtenissen. Bouwensz, Leiden (1608). Digital Library of the Royal Netherlands Academy of Arts and Sciences (KNAW)
Stodola, A.: Über die Regulierung von Turbinen. Schweizerische Bauzeitung 22, 113–117 (1893)
Stodola, A.: Über die Regulierung von Turbinen. Schweizerische Bauzeitung 23, 108–112 (1894)
Suzuki, J.: A history of the stability problem in celestial mechanics, from Newton to Laplace (1687–1787). PhD thesis, Boston University (1996)
Torricelli, E.: Opera Geometrica. Florentiæ typis Amatoris Masse & Laurentij de Landis, Florence (1644). Max Planck Institute for the History of Science (digital library)
Vyshnegradsky, I.A. (Вышнеградский, И.А.): On controllers of direct action (О регуляторах прямого действия). Изв. Санкт-Петербургского технологического института, pp. 21–62 (1877)
Zhukovskii, N.E. (Жуковский, Н.Е.): On the stability of motion (О прочности движения). Ученые записки Московского университета, отдел физико-математический 4, 10–21 (1882)
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Leine, R.I. The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability. Nonlinear Dyn 59, 173–182 (2010). https://doi.org/10.1007/s11071-009-9530-z
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DOI: https://doi.org/10.1007/s11071-009-9530-z