Quantum Studies: Mathematics and Foundations

, Volume 3, Issue 3, pp 231–236 | Cite as

Small oscillations of the pendulum, Euler’s method, and adequality

  • Vladimir Kanovei
  • Karin U. Katz
  • Mikhail G. Katz
  • Tahl Nowik
Regular Paper


Small oscillations evolved a great deal from Klein to Robinson. We propose a concept of solution of differential equation based on Euler’s method with infinitesimal mesh, with well-posedness based on a relation of adequality following Fermat and Leibniz. The result is that the period of infinitesimal oscillations is independent of their amplitude.


Harmonic motion Infinitesimal Pendulum Small oscillations 


  1. 1.
    Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., McGaffey, T., Schaps, D., Sherry, D., Shnider, S.: Is mathematical history written by the victors? Not. Am. Math. Soc. 60(7), 886–904 (2013). http://www.ams.org/notices/201307/rnoti-p886.pdf. arXiv:1306.5973
  2. 2.
    Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., McGaffey, T., Reeder, P., Schaps, D., Sherry, D., Shnider, S.: Interpreting the infinitesimal mathematics of Leibniz and Euler. J. Gen. Philos. Sci. (2016). doi:10.1007/s10838-016-9334-z
  3. 3.
    Bascelli, T., Bottazzi, E., Herzberg, F., Kanovei, V., Katz, K., Katz, M., Nowik, T., Sherry, D., Shnider, S.: Fermat, Leibniz, Euler, and the gang: the true history of the concepts of limit and shadow. Not. Am. Math. Soc. 61(8), 848–864 (2014)Google Scholar
  4. 4.
    Bascelli, T., Błaszczyk, P., Kanovei, V., Katz, K., Katz, M., Schaps, D., Sherry, D.: Leibniz vs Ishiguro: closing a quarter-century of syncategoremania. HOPOS J. Int. Soc. Hist. Philos. Sci. 6(1) (2016). doi:10.1086/685645. arXiv:1603.07209
  5. 5.
    Borovik, A., Katz, M.: Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus. Found. Sci. 17(3), 245–276 (2012). doi:10.1007/s10699-011-9235-x
  6. 6.
    Grobman, D.: Homeomorphisms of systems of differential equations. Doklady Akademii Nauk SSSR 128, 880–881 (1959)Google Scholar
  7. 7.
    Hartman, P.: A lemma in the theory of structural stability of differential equations. Proc. Am. Math. Soc. 11(4), 610–620 (1960)Google Scholar
  8. 8.
    Katz, K., Katz, M.: Cauchy’s continuum. Perspect. Sci. 19(4), 426–452 (2011). arXiv:1108.4201. http://www.mitpressjournals.org/doi/abs/10.1162/POSC_a_00047
  9. 9.
    Katz, K., Katz, M.: A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. Found. Sci. 17(1), 51–89 (2012). doi:10.1007/s10699-011-9223-1. arXiv:1104.0375
  10. 10.
    Katz, M., Schaps, D., Shnider, S.: Almost equal: the method of adequality from Diophantus to Fermat and beyond. Perspect. Sci. 21(3), 283–324 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Katz, M., Sherry, D.: Leibniz’s infinitesimals: their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis 78(3), 571–625 (2013). doi:10.1007/s10670-012-9370-y. arXiv:1205.0174
  12. 12.
    Keisler, H.J.: Elementary Calculus: An Infinitesimal Approach, 2d edn. Prindle, Weber & Schimidt, Boston (1986). http://www.math.wisc.edu/~keisler/calc.html
  13. 13.
    Klein, F.: Elementary Mathematics from an Advanced Standpoint. Vol. I. Arithmetic, Algebra, Analysis. Translation by E. R. Hedrick and C. A. Noble [Macmillan, New York, 1932] from the third German edition [Springer, Berlin, 1924]. Originally published as Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1908)Google Scholar
  14. 14.
    Lobry, C., Sari, T.: Non-standard analysis and representation of reality. Internat. J. Control 81(3), 517–534 (2008)Google Scholar
  15. 15.
    Nowik, T., Katz, M.: Differential geometry via infinitesimal displacements. J. Logic Anal. 7(5), 1–44 (2015). http://www.logicandanalysis.org/index.php/jla/article/view/237/106. arXiv:1405.0984
  16. 16.
    Pražák, D., Rajagopal, K., Slavík, J.: A non-standard approach to a constrained forced oscillator. Preprint (2016)Google Scholar
  17. 17.
    Robinson, A.: Non-Standard Analysis. North-Holland Publishing, Amsterdam (1966)Google Scholar
  18. 18.
    Stroyan, K.: Advanced Calculus Using Mathematica: NoteBook Edition (2015)Google Scholar
  19. 19.
    Tao, T.: Hilbert’s Fifth Problem and Related Topics. Graduate Studies in Mathematics, vol. 153. American Mathematical Society, Providence (2014)Google Scholar
  20. 20.
    Tao, T., Van Vu, V.: Sum-avoiding sets in groups. arXiv:1603.03068 (2016)

Copyright information

© Chapman University 2016

Authors and Affiliations

  • Vladimir Kanovei
    • 1
    • 2
  • Karin U. Katz
    • 3
  • Mikhail G. Katz
    • 3
  • Tahl Nowik
    • 3
  1. 1.IPPIMoscowRussia
  2. 2.MIITMoscowRussia
  3. 3.Department of MathematicsBar Ilan UniversityRamat GanIsrael

Personalised recommendations