Instability conditions for circulatory and gyroscopic conservative systems

TL;DR

This paper introduces a method to determine instability conditions for circulatory and gyroscopic conservative systems, including applications to charged particles in electromagnetic fields, advancing understanding of their stability properties.

Contribution

It presents a novel method based on Gramians of vectors derived from characteristic polynomial roots to generate sufficient instability conditions for these systems.

Findings

New instability criteria for general circulatory systems

Application of the method to charged particle motion in electromagnetic fields

Enhanced understanding of stability conditions in gyroscopic systems

Abstract

We give a method which generates sufficient conditions for instability of equilibria for circulatory and gyroscopic conservative systems. The method is based on the Gramians of a set of vectors whose coordinates are powers of the roots of the characteristic polynomial for the studied systems. New instability results are obtained for general circulatory and gyroscopic conservative systems. We also apply this method for studying the instability of motion for a charged particle in a stationary electromagnetic field.