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A.N. Shelaev
The analysis of non-linear systems - the oscillating and rotating mathematical pendulum by means of the sized scaling method

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Abstract

It is shown that even for such non-linear system, as the mathematical (simple) pendulum (MP) [1,2], known more than 350 years, it is possible to receive by means of the method of sized scaling (SS) a row of new nontrivial results. In the SS method along with well-known scale for the oscillation frequency M0 = √g/L (g is the free fall acceleration, L is the MP length) may be introduced, entering from a basic scales M1 = V/L (V is the bob speed in the lower point of a path) and M-1 = g/L, M0 = √M1•M-1, and then as there geometric averages the infinite set of other scales. At the same time, using parameters K and k = 1/K in the elliptic functions and integrals (K2 = V2/4gL is equal to the relation of the kinetic energy of the MP bob in the lower point of a path to the maximum potential energy in the upper point), it is possible to describe the dynamics both of the oscillating and rotating MP in spite of the fact that it is topologically different movements. It is essential also that for the MP it is found a number of exact and interesting ratios which are expressed through the gold ratio constants ф = (-1 + √5)/2 and Ф = (1+ √5)/2. Besides, it is shown that basic scales can form the so-called Kepler’s and meta-triangles found, in particular, in geometry of the Great Pyramid of Cheops.


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A.N. Shelaev, The analysis of non-linear systems - the oscillating and rotating mathematical pendulum by means of the sized scaling method // «Академия Тринитаризма», М., Эл № 77-6567, публ.22421, 21.08.2016

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