Hilbert space representation of binary operations on a power-multiplying oscillator

TL;DR

This paper explores how representing a complex oscillating system in Hilbert space using complex numbers reveals properties of the system, especially at specific angles where oscillation translates into displacement.

Contribution

It introduces a Hilbert space framework for analyzing binary operations on a power-multiplying oscillator using complex number representations.

Findings

Complex number representation captures system properties at specific angles.

Physical quantities are expressed in the real plane at certain oscillation points.

Hilbert space approach provides new insights into oscillator behavior.

Abstract

In this study, the properties of an oscillating system composed of a pendulum connected to a seesaw and placed on a moving platform with a certain slope are analyzed. Using complex numbers to collect the information contained in the system proves to be crucial in order to observe the properties described by both cross and dot products. The representation of physical quantities in complex numbers reveals that for certain angles, precisely those where the oscillation translates into a displacement, the properties of the system are expressed in the real plane.