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The mathematical pendulum from Gauß via Jacobi to Riemann. (English) Zbl 1168.70008

Summary: The goal is to introduce double-periodic elliptic functions on the basis of a simple mechanical system, namely the mathematical pendulum. Thereby it is not geometry that is in the foreground, as in Gauß analysis of the lemniscate curve, but rather the calculation of the specific attributes of elliptic functions with the aid of a periodic integrable system. Not the spatial degree of freedom, but the time variable is continued into the complex plane. This will make it possible for us not only to identify the known real period of the pendulum oscillation, but also to detect a second imaginary period. Only then does the solution of the equation of motion become a Jacobi-type elliptic function. Using the Cauchy integral theorem, which Gauß was already familiar with, as well as the simplest Riemannian surface of the function \(w= \sqrt{(1-z^2)(1-k^2z^2)}\), we want to calculate analytic and topological characteristics of the oscillatory motion of a pendulum. Our intent is to show that elliptic functions could have appeared much earlier than 1796 in the literature. Admittedly, for this the field of complex numbers was necessary, as represented in the Gaußian plane of complex numbers. However, Gauß was unwilling to publish his findings because of his “fear of the cry of the Boeotians”.

MSC:

70K99 Nonlinear dynamics in mechanics
33E05 Elliptic functions and integrals
70-03 History of mechanics of particles and systems
01A55 History of mathematics in the 19th century
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References:

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