Optimal dynamical stabilization

TL;DR

This paper investigates the minimal periodic stiffness needed for stability in a mass-spring system and applies findings to control a magnetic compass, revealing quantum-like rules for stability durations.

Contribution

It introduces a novel analysis of stability conditions in time-varying systems and uncovers quantum-analogous rules governing stability durations.

Findings

Stability depends on small, precisely timed durations within each period.

Discrete set of durations predicted by rules similar to quantum mechanics.

Application to trapping a magnetic compass's upside-down state.

Abstract

Stability is a fundamental concept that refers to a system's ability to return close to its original state after disturbances. The minimal conditions for stability when system parameters vary in time, though common in physics, have been largely overlooked. Here, we study the minimal amount of periodic stiffness a linear mass-spring system requires to remain stable and apply our findings to optimally trap the upside-down state of a compass in a time-varying magnetic field. We show that the ability to return close to its original state only needs to be ensured over small but precisely defined durations within each period for the system to achieve dynamic stability. These precise durations form a discrete set, remarkably predicted by rules analogous to those of quantum mechanics. This unexpected connection opens new avenues for controlling dynamical systems.