Damped harmonic oscillator revisited: the fastest route to equilibrium
Abstract
Theoretically, solutions of the damped harmonic oscillator asymptotically approach equilibrium, i.e., the zero energy state, without ever reaching it exactly, and the critically damped solution approaches equilibrium faster than the underdamped or the overdamped solution. Experimentally, the systems described with this model reach equilibrium when the system's energy has dropped below some threshold corresponding to the energy resolution of the measuring apparatus. We show that one can (almost) always find an optimal underdamped solution that will reach this energy threshold sooner than all other underdamped solutions, as well as the critically damped solution, no matter how small this threshold is. We also comment on one exception to this for a particular type of initial conditions, when a specific overdamped solution reaches the equilibrium state sooner than all other solutions. We confirm some of our findings experimentally.
Keywords
Cite
@article{arxiv.2301.08222,
title = {Damped harmonic oscillator revisited: the fastest route to equilibrium},
author = {Karlo Lelas and Nikola Poljak and Dario Jukić},
journal= {arXiv preprint arXiv:2301.08222},
year = {2023}
}
Comments
9 pages of main text + supplementary material. The biggest changes compared to the previous version are in section IIIC, that section has been somewhat shortened and written more clearly. Also, some typos and similar errors in supplementary materials have been corrected. The paper is accepted for publication in American Journal of Physics