Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds

Version 1 : Received: 21 September 2022 / Approved: 22 September 2022 / Online: 22 September 2022 (03:33:14 CEST)

How to cite: Shvalb, N.; Frenkel, M.; Shoval, S.; Bormashenko, E. Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds. Preprints 2022, 2022090331 (doi: 10.20944/preprints202209.0331.v1). Shvalb, N.; Frenkel, M.; Shoval, S.; Bormashenko, E. Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds. Preprints 2022, 2022090331 (doi: 10.20944/preprints202209.0331.v1).

Abstract

Ramsey theory influences the dynamics of mechanical systems, which may be described as abstract complete graphs. We address a mechanical system which is completely interconnected with the two kinds of ideal Hookean springs. The suggested system mechanically corresponds to the cyclic molecules, in which functional groups are interconnected with two kinds of chemical bonds, represented mechanically with two springs k1 and k2. In this paper, we consider a Cyclic system (molecule) built of six equal masses m and two kinds of springs. We pose the following question: what is the minimal number of masses in the such a system in which three masses are constrained to be connected with spring k1 or three masses to be connected with spring k2? The answer to this question is supplied by the Ramsey theory, and it is formally stated as follows: what is the minimal number R3,3? The result emerging from the Ramsey theory is R3,3=6. Thus, in the aforementioned interconnected mechanical system will be necessarily present the triangles (at least one triangle), built of masses and springs. This prediction constitutes the vibrational spectrum of the system. Thus, the Ramsey Theory supplies the selection rules for the vibrational spectra of the cyclic molecules. Symmetrical system built of six vibrating entities is addressed. The Ramsey approach works for 2D and 3D molecules, which may be described as abstract complete graphs.

Keywords

Ramsey theory; complete graph; vibrational spectrum; eigenfrequency; selection rule; cyclic molecule

Subject

PHYSICAL SCIENCES, General & Theoretical Physics

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