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

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.

[1]  Kumbakonam R. Rajagopal,et al.  A non-standard approach to a constrained forced oscillator , 2017, J. Log. Anal..

[2]  Tewfik Sari,et al.  Non-standard analysis and representation of reality , 2008, Int. J. Control.

[3]  T. Tao.  Hilbert's Fifth Problem and Related Topics , 2014 .

[4]  Piotr Blaszczyk,et al.  Is mathematical history written by the victors , 2013, 1306.5973.

[5]  Mikhail G. Katz,et al.  Differential geometry via infinitesimal displacements , 2014, J. Log. Anal..

[6]  Vladimir Kanovei,et al.  Interpreting the Infinitesimal Mathematics of Leibniz and Euler , 2016, 1605.00455.

[7]  Terence Tao,et al.  Sum-avoiding sets in groups , 2016, 1603.03068.

[8]  Mikhail G. Katz,et al.  A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography , 2011, 1104.0375.

[9]  P. Hartman.  A lemma in the theory of structural stability of differential equations , 1960 .

[10]  W. Luxemburg.  Non-Standard Analysis , 1977 .

[11]  Emanuele Bottazzi,et al.  Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow , 2014, 1407.0233.

[12]  Piotr Blaszczyk,et al.  Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania , 2016, HOPOS: The Journal of the International Society for the History of Philosophy of Science.

[13]  Alexandre Borovik,et al.  Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus , 2011, 1108.2885.

[14]  Mikhail G. Katz,et al.  Cauchy's Continuum , 2011, Perspectives on Science.

[15]  Mikhail G. Katz,et al.  Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond , 2012, 1205.0174.

[16]  H. Keisler.  Elementary Calculus: An Infinitesimal Approach , 1976 .

[17]  P. Zsombor-Murray,et al.  Elementary Mathematics from an Advanced Standpoint , 1940, Nature.

[18]  Mikhail G. Katz,et al.  Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond , 2012, Perspectives on Science.