Stability of Potential Systems to General Positional Perturbations

Abstract

This paper shows that a potential system with one multiple eigenvalue of multiplicity two or more can assuredly be made unstable by infinitesimal positional perturbations. The explicit nature of these pertubatory forces is provided. It is shown that the matrices that describe these perturbatory forces are not required to commute with the potential matrix. The general positional perturbatory forces that bring about instability in the perturbed potential system are shown to include, as special cases, circulatory forces that do and that do not commute with the potential matrix. The condition on commutativity of matrices is quite stringent, and its removal has a significant effect on generalizing the conditions that lead to instability. The paper expands the generalized Merkin instability result to include more general, noncirculatory positional perturbations, and it eliminates restrictions imposed on the perturbations by commutation requirements. The structure of infinitesimal as well as finite perturbatory matrices that guarantee instability is obtained. Practical implications of the mathematical results to natural and engineered systems are given. It is pointed out that potential systems with two “nearly” equal frequencies of vibration are, in general, susceptible to instability, created by “small” perturbatory forces, where the terms nearly and small are quantified.