Volume 59, Issue 326, February 2023

On the two different formulations of mechanical energy in driven oscillations

Jeong Ryeol Choi^♦, Ji Nny Song

Department of Nano engineering, Kyonggi University, Gwanggyosan-ro, Yeongtong-gu, Suwon, Gyeonggi-do 16227, Republic of Korea

^♦Corresponding author
Department of Nano engineering, Kyonggi University, Gwanggyosan-ro, Yeongtong-gu, Suwon, Gyeonggi-do 16227, Republic of Korea

ABSTRACT

We introduce two different formulae of mechanical energy of harmonic oscillators driven by an external force in this work. In the first formula, the energy is composed of two terms, i.e., the kinetic energy and quadratic-potential-energy terms. Another formula is that it is just the same as the Hamiltonian itself. While the first kind of energy reflects the mechanical energy remained in the system at a certain instant of time, the second energy depends on the linear potential term as well as the terms quadratic in canonical variables. Time evolution of the two energies is analyzed for two specific cases of which driving force is different from each other. The second kind of energy can be negative, whereas the first kind is always positive. The second formula of energy has generally been used in optimal control theory in relation with shortcuts to adiabaticity.

Keywords: Mechanical energy, Harmonic oscillator, Hamiltonian, External force, Classical solutions

Discovery, 2023, 59, e18d1023

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Published: February 2023