On destabilizing effects of two fundamental non-conservative forces

Abstract

In this work we discuss the instabilities in mechanical systems caused by two fundamentally different non-conservative forces, referred to as dissipative and positional forces, each of which may lead to energy dissipation. One of the objectives of this discussion is to recall and to put into the context of current research some of the important classical results by Thomson–Tait–Chetayev and Merkin, which are under-appreciated nowadays: many new examples, e.g. radiation in Hamiltonian systems, the Levitron, etc., appearing in recent literature can be interpreted with the help of these classical results. Next, in the spirit of the Lagrange–Dirichlet theory, we introduce the geometric picture of the phase space corresponding to the effects of destabilization in finite-dimensional systems. On the physical side, our objective is to demonstrate that both of these types of non-conservative forces appear quite commonly and often simultaneously in physical systems. As an illustration, we consider the Levitron, a system in which the dissipative effects have not heretofore been studied. Finally, using Nikolai’s elastic bar model as a paradigm, we discuss the notion of a secondary dissipation-induced instability, when the above two fundamental non-conservative forces interact. This unified way of thinking should help us to understand the intricate links among various mechanical systems through the most fundamental mechanisms in their behavior