The equations of motion for a simple mechanical system, consisting of a pendulum of length / and mass m coupled to a harmonic oscillator formed by an ideal spring and a body of mass m, are derived and solved, using New-tonian mechanics. Interestingly, when the natural frequencies of the two harmonic oscillators are set to the same value omega(O), the eigenfrequencies of the combined system are found to be omega(O) scaled by the golden ratio root 5 + 1/2 and by its reciprocal root 5 - 1/2 This intriguing relationship between the coupled oscillators and the ratio was uncovered serendipitously.
Hidden golden ratio in two coupled harmonic oscillators
R De Luca
2025
Abstract
The equations of motion for a simple mechanical system, consisting of a pendulum of length / and mass m coupled to a harmonic oscillator formed by an ideal spring and a body of mass m, are derived and solved, using New-tonian mechanics. Interestingly, when the natural frequencies of the two harmonic oscillators are set to the same value omega(O), the eigenfrequencies of the combined system are found to be omega(O) scaled by the golden ratio root 5 + 1/2 and by its reciprocal root 5 - 1/2 This intriguing relationship between the coupled oscillators and the ratio was uncovered serendipitously.File in questo prodotto:
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